Radius of α-Spirallikeness of Order cos(α)/2 for Entire Functions

Abstract

We determine the radius of $\alpha$-spirallikeness of order $\cos \left(\alpha \right)/2$ for entire functions represented as infinite products of their positive zeros. The discussion includes several examples featuring special functions such as Gamma functions, Bessel functions, Struve functions, Wright functions, Ramanujan-type entire functions, and q-Bessel functions. We also consider combinations of classical Bessel functions, including both first-order and second-order derivatives. Additionally, several other special functions that can be incorporated into the established classes are described. We utilize Mathematica 12 software to compute the numerical values of the radius for some functions.

1. Introduction and Problem Formation

We start with the historical development of geometric classes, focusing on the class of spirallike functions. First, we consider the well-known class $\mathcal{A}$ of functions f that are analytic and univalent in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$, and that satisfy the normalization condition $f\left(0\right)=f^{\prime }\left(0\right)-1$. The class ${\mathtt{S}}^{*}\left(\alpha \right)$, commonly known as starlike of order $\alpha$, consists of functions $f\in \mathcal{A}$ that fulfill the following characterization:

$$\begin{array}{c}\hfill Re\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\alpha ,z\in \mathbb{D}0\le \alpha <1.\end{array}$$

(1)

The inequality above is associated with the right half-plane. Rønning [1] defined class ${\mathtt{S}}_p^{*}$, which is related to a parabolic region. A function f belongs to ${\mathtt{S}}_p^{*}$ if it satisfies the following condition:

$$\begin{array}{c}\hfill \mathrm{Re}\left({\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right)>\left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}-1\right|,z\in \mathbb{D}.\end{array}$$

(2)

The class ${\mathtt{S}}_p^{*}$ is further generalised in two different ways. In [2], it is described that $f\in {\mathtt{S}}_p^{*}\left(\alpha \right)$, $\alpha <1$ if

$$\begin{array}{c}\hfill Re\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\left|\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}-1\right|+\alpha ,z\in \mathbb{D}.\end{array}$$

(3)

It is shown that the Equation (3) also defines a parabola with its vertex at $w={\displaystyle \frac{1+\alpha }{2}}$. As the value of $\alpha$ increases, the parabola becomes narrower, and it degenerates when $\alpha =1$. For $\alpha =0$, it corresponds to the previous class ${\mathtt{S}}_p^{*}$, and notably, ${\mathtt{S}}_p^{*}\left(\alpha \right)\subset {\mathtt{S}}^{*}\left(0\right)$ for $-1\le \alpha <1$. However, when $\alpha <-1$, the class includes non-univalent functions.

The second generalization given in [3] stated that $f\in {\mathtt{PS}}^{*}\left(\rho \right)$ when $z{f}^{\prime }\left(z\right)/f\left(z\right)\in {\Omega }_{\rho }$ for $z\in \mathbb{D}$, where

$$\begin{array}{cc}\hfill {\Omega }_{\rho }& =\left\{w=u+iv:v^2<4(1-\rho )(u-\rho )\right\}\hfill \\ \hfill & =\left\{w:|w-1|<1-2\rho +Re\left(w\right)\right\}.\hfill \end{array}$$

In this case, $f\in {\mathtt{PS}}^{*}\left(\rho \right)$ if and only if

$$\begin{array}{c}\hfill Re\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\left|\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}-1\right|+1-2\rho ,z\in \mathbb{D},0\le \rho <1.\end{array}$$

(4)

We refer to [1,2,3,4,5] for more details and related properties of the class, ${\mathtt{S}}_p^{*}\left(\alpha \right)$ and ${\mathtt{PS}}^{*}\left(\rho \right)$.

The concept of spirallikeness is the main topic of this article. The function $f\in \mathcal{A}$ can be called spirallike if

$$\begin{array}{c}\hfill Re\left(e^{-i\alpha }{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right)>0,z\in \mathbb{D},\end{array}$$

(5)

for some $\alpha \in (-\pi /2,\pi /2)$. Spirallike functions are essential for analyzing the rotations and symmetries of analytic functions, as they spiral outward from the origin in a controlled manner.

The function $f\left(z\right)$ is considered convex spirallike if $z{f}^{\prime }\left(z\right)$ is spirallike. A function $f\left(z\right)$ is termed uniformly $\alpha$-spirallike if the image of every circular arc ${\mathsf{\Gamma }}_z$ centred at $\eta$ and lying in the region $\mathbb{D}$ is $\alpha$-spirallike with respect to $f\left(\eta \right)$. The collection of all uniformly $\alpha$-spirallike functions is denoted as $USP\left(\alpha \right)$.

Similarly, a function $f\left(z\right)$ is classified as a uniformly convex $\alpha$-spiral if the image of every circular arc ${\mathsf{\Gamma }}_z$ with the centre $\eta$ lying in $\mathbb{D}$ is convex $\alpha$-spirallike. The class of all uniformly convex $\alpha$-spiral functions is referred to as $UCSP\left(\alpha \right)$. These concepts are studied in [6], and the following analytic characterization for functions f in $USP\left(\alpha \right)$ and $UCSP\left(\alpha \right)$ are established:

$$\begin{array}{cc}\hfill f\in USP\left(\alpha \right)& \iff Re\left\{e^{-i\alpha }{\displaystyle \frac{(z-\eta )f^{\prime }\left(z\right)}{f\left(z\right)-f\left(\eta \right)}}\right\}\ge 0,z\ne \eta ,z,\eta \in \mathbb{D}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill f\in UCSP\left(\alpha \right)& \iff Re\left\{e^{-i\alpha }\left(1+{\displaystyle \frac{(z-\eta )f^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right)\right\}\ge 0,z\ne \eta ,z,\eta \in \mathbb{D},\left|\alpha \right|<{\displaystyle \frac{\pi }{2}}.\hfill \end{array}$$

(7)

In addition to the analytic characterization mentioned above, another one-variable characterization for these classes is provided in [6] as follows:

$$\begin{array}{c}\hfill f\in UCSP\left(\alpha \right)\iff Re\left\{e^{-i\alpha }\left(1+z{\displaystyle \frac{f^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right)\right\}\ge \left|z{\displaystyle \frac{f^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right|,z\in \mathbb{D}.\end{array}$$

(8)

The class of functions $F\left(z\right)=z{f}^{\prime }\left(z\right)$, where $f\left(z\right)\in UCSP\left(\alpha \right)$, is a subclass of the spirallike functions, and we denote this by $S{P}_p\left(\alpha \right)$. In fact, the function $f\left(z\right)\in A$ is in $S{P}_p\left(\alpha \right)$ if and only if

$$\begin{array}{c}\hfill Re\left\{e^{-i\alpha }z{\displaystyle \frac{f^{\prime }\left(z\right)}{f\left(z\right)}}\right\}\ge \left|z{\displaystyle \frac{f^{\prime }\left(z\right)}{f\left(z\right)}}-1\right|,z\in \mathbb{D}.\end{array}$$

(9)

This condition is equivalent to

$$\begin{array}{c}\hfill Re\left\{{\displaystyle \frac{e^{-i\alpha }z\frac{f^{\prime }\left(z\right)}{f\left(z\right)}+isin\alpha }{\cos \alpha }}\right\}\ge \left|{\displaystyle \frac{e^{i\alpha }z\frac{f^{\prime }\left(z\right)}{f\left(z\right)}+isin\alpha }{\cos \alpha }}-1\right|,\left|\alpha \right|<{\displaystyle \frac{\pi }{2}}.\end{array}$$

(10)

It is also explained in [6] that the condition (9) is equivalent to state that $z{f}^{\prime }\left(z\right)f\left(z\right)$ lies in the parabolic region

$${\Omega }_{\alpha }=\{w:Re\left(e^{-i\alpha }w\right)>|w-1|\}.$$

For $w\in {\Omega }_{\alpha }$, the functions in the class $S{P}_p\left(\alpha \right)$ are $\alpha$-spirallike of order $\cos \alpha /2$ due to the fact $Re\left(e^{-i\alpha }w\right)\ge \cos \alpha /2$. For $\alpha =0$, the classes, $UCSP\left(\alpha \right)$ and $S{P}_p\left(\alpha \right)$ respectively reduce to the classes $UCV$ and $SP$ introduced and studied by Ronning [1,2].

In line with the work in [2], Selvaraj and Geeth [7] generalized the class $S{P}_p\left(\alpha \right)$ as $S{P}_p(\alpha ,\sigma )$.

Definition 1

([7]). A function $f\left(z\right)$ is in $S{P}_p(\alpha ,\sigma )$ if $f\left(z\right)$ satisfies the analytic characterization

$$\begin{array}{c}\hfill Re\left\{e^{-i\alpha }z{\displaystyle \frac{f^{\prime }\left(z\right)}{f\left(z\right)}}\right\}\ge \left|z{\displaystyle \frac{f^{\prime }\left(z\right)}{f\left(z\right)}}-1\right|+\sigma ,z\in \mathbb{D}.\end{array}$$

(11)

The class $S{P}_p(\alpha ,\sigma )$, as outlined in Definition 1, serves as the cornerstone of this study. To build upon this foundation, we will now explore the radius problem related to spirallike functions. Specifically, we define the $S{P}_p(\alpha ,\sigma )$-radius of a function $\mathtt{f}$ as a real number

$$\begin{array}{c}\hfill {\mathtt{r}}_{\alpha ,\sigma }^{SP}\left(f\right)=\underset{z\in {\mathbb{D}}_r}{sup}\left\{r>0:Re\left\{e^{-i\alpha }{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right\}>\left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}-1\right|+\sigma \right\}.\end{array}$$

(12)

An entire function is a function that is holomorphic everywhere in the complex plane $\mathbb{C}$. These functions are fundamental in complex analysis and find applications in differential equations, number theory, and mathematical physics [8,9]. Every entire function $f\left(z\right)$ has a globally convergent Taylor series expansion as follows:

$$f\left(z\right)=\sum _{n=0}^{\infty }{a}_n{z}^n,a_n={\displaystyle \frac{f^{\left(n\right)}\left(0\right)}{n!}}.$$

(13)

Since the radius of convergence is infinite, entire functions are completely characterized by their Taylor series [10].

The growth of an entire function is classified by its order $\rho$ defined as

$$\rho =\underset{r\to \infty }{lim\; sup}{\displaystyle \frac{loglogM\left(r\right)}{logr}},$$

(14)

where $M\left(r\right)={max}_{\left|z\right|=r}\left|f\left(z\right)\right|$ [11]. For more precise classification, the type $\sigma$ is defined as follows when $\rho =1$:

$$\sigma =\underset{r\to \infty }{lim\; sup}{\displaystyle \frac{logM\left(r\right)}{r}}.$$

(15)

The Weierstrass factorization theorem states that an entire function with prescribed zeros $\left\{a_n\right\}$ can be expressed as an infinite product as follows:

$$f\left(z\right)=e^{g\left(z\right)}\prod _{n=1}^{\infty }{E}_n\left(z\right),$$

(16)

where

• $g\left(z\right)$ is an entire function;
• $E_n\left(z\right)$ are elementary factors that ensure convergence, as follows:

  $$E_n\left(z\right)=\left(1-{\displaystyle \frac{z}{a_n}}\right)e^{z/a_n}.$$

  (17)

A well-known example is the representation of the sine function [10] in terms of its product form as follows:

$$\begin{array}{c}\hfill sin\left(\pi z\right)=\pi z\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{n^2}}\right).\end{array}$$

(18)

The Laguerre–Pólya class consists of entire functions that are limits of polynomials with only real zeros as follows:

$$f\left(z\right)=c{z}^m{e}^{\alpha z+\beta z^2}\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z}{a_n}}\right)e^{z/a_n}.$$

(19)

These functions play a role in stability analysis and the Riemann hypothesis [8,9].

The infinite product representation of entire functions plays a significant role in the study of various subclasses of univalent functions; for example, see [12,13] and the references therein. In this study, we examine three subclasses of $\mathcal{A}$, each defined by a parameter $\nu \in \mathbb{R}$. These subclasses are categorised based on their infinite factorization involving zeros and are denoted as ${\mathbb{H}}_1$, ${\mathbb{H}}_2$, and ${\mathbb{H}}_3$. The analytical representation of these classes is as follows:

$$\begin{array}{cc}\hfill {\mathbb{H}}_1& :=\left\{f_{\nu }\in \mathcal{A}|f_{\nu }\left(z\right):=z\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z}{a_n^2\left(\nu \right)}}\right),\nu \in \mathbb{R}\right\},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\mathbb{H}}_2& :=\left\{{\mathtt{g}}_{\nu }\in \mathcal{A}|{\mathtt{g}}_{\nu }\left(z\right):=z\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{b_n^2\left(\nu \right)}}\right),\nu \in \mathbb{R}\right\},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\mathbb{H}}_3& :=\left\{{\mathtt{h}}_{\nu }\in \mathcal{A}|{\mathtt{h}}_{\nu }\left(z\right):=z\prod _{n=1}^{\infty }{\left(1-{\displaystyle \frac{z^2}{d_n^2\left(\nu \right)}}\right)}^{\mu \left(\nu \right)},\nu \in \mathbb{R}\setminus \left\{0\right\}\right\}.\hfill \end{array}$$

(22)

In this context, $\mu$ is a positive real function of $\nu$. The sequences $\left\{a_n\left(\nu \right)\right\}$, $\left\{b_n\left(\nu \right)\right\}$, and $\left\{d_n\left(\nu \right)\right\}$, respectively, represent the n-th zeros of the functions $f_{\nu }$, ${\mathtt{g}}_{\nu }$, and ${\mathtt{h}}_{\nu }$. The nature of these zeros varies with respect to $\nu$, such that each of the infinite products mentioned converges uniformly on every compact subset of $\mathbb{C}$. These subclasses are introduced and studied in [13] to determine the radius of k-parabolic starlikeness. Several examples provided to demonstrate that the classes ${\mathbb{H}}_1$, ${\mathbb{H}}_2$, and ${\mathbb{H}}_3$ are indeed non-empty. In this study we are going to answer the following problem.

Problem 1.

Find the $S{P}_p(\alpha ,\sigma )$-radius ${\mathtt{r}}_{\alpha ,\sigma }^{SP}\left(f\right)$ of functions for $f\in {\mathbb{H}}_i$, $i=1,2,3$.

The following result proved in [14] is useful for this study.

Lemma 1

([14]). If $\left|z\right|\le r<a<b$, and $\lambda \in [0,1]$, then

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{z}{b-z}}-\lambda {\displaystyle \frac{z}{a-z}}\right|\le {\displaystyle \frac{r}{b-r}}-\lambda {\displaystyle \frac{r}{a-r}}.\end{array}$$

(23)

Consequently, it follows that

$$\begin{array}{c}\hfill Re\left({\displaystyle \frac{z}{b-z}}-\lambda {\displaystyle \frac{z}{a-z}}\right)\le {\displaystyle \frac{r}{b-r}}-\lambda {\displaystyle \frac{r}{a-r}}\end{array}$$

(24)

and

$$\begin{array}{c}\hfill Re\left({\displaystyle \frac{z}{b-z}}\right)\le \left|{\displaystyle \frac{z}{b-z}}\right|\le {\displaystyle \frac{r}{b-r}}.\end{array}$$

(25)

In our study, we need the digamma function. The digamma function $\psi \left(z\right)$ is defined as the logarithmic derivative of the gamma function $\mathsf{\Gamma }\left(z\right)$ as follows:

$$\psi \left(z\right)={\displaystyle \frac{d}{dz}}ln\left(\mathsf{\Gamma }\left(z\right)\right)={\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(z\right)}{\mathsf{\Gamma }\left(z\right)}},$$

(26)

where $\mathsf{\Gamma }\left(z\right)$ is the well-known gamma function.

The digamma function $\psi \left(z\right)$ can be represented as follows:

• Integral form:

  $$\psi \left(z\right)={\int }_0^{\infty }\left({\displaystyle \frac{e^{-t}}{t}}-{\displaystyle \frac{e^{-zt}}{1-e^{-t}}}\right)dt,x>0.$$

  (27)
• Series expansion (for $z>0$):

  $$\psi \left(z\right)=-\gamma +\sum _{n=1}^{\infty }\left({\displaystyle \frac{1}{n}}-{\displaystyle \frac{1}{n+z-1}}\right),$$

  (28)

  where $\gamma$ is the Euler–Mascheroni constant.

For positive integers n, the digamma function is directly related to the harmonic number as follows:

$$\psi \left(n\right)=-\gamma +\sum _{k=1}^{n-1}{\displaystyle \frac{1}{k}}.$$

(29)

For a detailed overview of digamma functions, see [15].

The rest of the paper is organized as follows: In Section 2, we determine the $S{P}_p(\alpha ,\sigma )$-radius for the functions ${\mathtt{f}}_{\nu }\left(z\right)$, ${\mathtt{g}}_{\nu }\left(z\right)$, and ${\mathtt{h}}_{\nu }\left(z\right)$. In Section 3, we present several examples of special functions that belong to the classes ${\mathbb{H}}_1$, ${\mathbb{H}}_2$, and ${\mathbb{H}}_3$. We calculate the $S{P}_p(\alpha ,\sigma )$-radius for various well-known special functions, using the results from Section 2, and we also compute the numerical values with fixed parameters.

2. Main Results

Theorem 1.

Let $\left|\alpha \right|<\pi /2$ and $\sigma \in [0,1)$ such that $\cos \alpha >\sigma$. If for some $\nu \in \mathbb{R}$, the functions $f_{\nu }$, ${\mathtt{g}}_{\nu }$, and ${\mathtt{h}}_{\nu }$ have real positive zeros, then the following statements are true:

1. 

The $S{P}_p(\alpha ,\sigma )$-radius of $f_{\nu }\in {\mathbb{H}}_1$ is ${\mathtt{R}}_f$, where ${\mathtt{R}}_f$ is the unique root of the equation $2r{f}_{\nu }^{\prime }\left(r\right)+(\cos \left(\alpha \right)-2-\sigma )f_{\nu }\left(r\right)=0$ in $(0,a_1^2\left(\nu \right))$;

2. 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathtt{g}}_{\nu }\in {\mathbb{H}}_2$ is ${\mathtt{R}}_{\mathtt{g}}$, where ${\mathtt{R}}_g$ is the unique root of the equation $2r{\mathtt{g}}_{\nu }^{\prime }\left(r\right)+(\cos \left(\alpha \right)-2-\sigma ){\mathtt{g}}_{\nu }\left(r\right)=0$ in $(0,b_1\left(\nu \right))$;

3. 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathtt{h}}_{\nu }\in {\mathbb{H}}_3$ is ${\mathtt{R}}_{\mathtt{h}}$, where ${\mathtt{R}}_h$ is the unique root of the equation $2r{\mathtt{h}}_{\nu }^{\prime }\left(r\right)+(\cos \left(\alpha \right)-2-\sigma ){\mathtt{h}}_{\nu }\left(r\right)=0$ in $(0,d_1\left(\nu \right))$.

Proof. 

To prove the first part, let $f_{\nu }\in {\mathbb{H}}_1$ such that it $f_{\nu }$ has real positive zeros. The logarithmic differentiation of $f_{\nu }$ yields

$$\begin{array}{c}\hfill {\displaystyle \frac{f_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}={\displaystyle \frac{1}{z}}-\sum _{n=1}^{\infty }{\displaystyle \frac{1}{a_n^2\left(\nu \right)-z}}.\end{array}$$

Hence,

$$\begin{array}{c}\hfill {\displaystyle \frac{z{f}_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}=1-\sum _{n=1}^{\infty }{\displaystyle \frac{z}{a_n^2\left(\nu \right)-z}}.\end{array}$$

(30)

Now, we have

$$\begin{array}{cc}\hfill \mathrm{Re}\left(e^{-i\alpha }{\displaystyle \frac{z{f}_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}\right)& =\mathrm{Re}\left(e^{-i\alpha }\right)-\mathrm{Re}\left(e^{-i\alpha }\sum _{n=1}^{\infty }{\displaystyle \frac{z}{a_n^2\left(\nu \right)-z}}\right)\hfill \\ \hfill & \ge \cos \alpha -\left|e^{-i\alpha }\sum _{n=1}^{\infty }{\displaystyle \frac{z}{a_n^2\left(\nu \right)-z}}\right|\hfill \\ \hfill & \ge \cos \alpha -\left|\sum _{n=1}^{\infty }{\displaystyle \frac{z}{a_n^2\left(\nu \right)-z}}\right|.\hfill \end{array}$$

(31)

If $\left|z\right|\le r<a_1^2\left(\nu \right)$, where $a_1^2\left(\nu \right)$ is the smallest zero of $f_{\nu }$, then, using (25), we obtain for all $n\ge 1$

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{z}{a_n^2\left(\nu \right)-z}}\right|\le {\displaystyle \frac{r}{a_n^2\left(\nu \right)-r}}.\end{array}$$

(32)

Hence, using (31) and (32), we deduce

$$\begin{array}{cc}\hfill \mathrm{Re}\left(e^{-i\alpha }{\displaystyle \frac{z{f}_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}\right)& \ge \cos \alpha -\sum _{n=1}^{\infty }{\displaystyle \frac{r}{a_n^2\left(\nu \right)-r}}\hfill \\ \hfill & =\cos \alpha +r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}-1.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill Re\left(e^{-i\alpha }{\displaystyle \frac{z{f}_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}\right)\ge \cos \alpha +r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}-1.\end{array}$$

(33)

On the other hand,

$$\begin{array}{cc}\hfill \left|z{\displaystyle \frac{f_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}-1\right|& =\left|\sum _{n=1}^{\infty }{\displaystyle \frac{z}{a_n^2\left(\nu \right)-z}}\right|\le \sum _{n=1}^{\infty }\left|{\displaystyle \frac{z}{a_n^2\left(\nu \right)-z}}\right|\hfill \\ \hfill & \le \sum _{n=1}^{\infty }{\displaystyle \frac{r}{a_n^2\left(\nu \right)-r}}=1-r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}.\hfill \end{array}$$

(34)

So, for $\left|\alpha \right|<{\displaystyle \frac{\pi }{2}}$ and $0\le \sigma <1$, relations (33) and (34), imply

$$\begin{array}{cc}\hfill Re\left(e^{-i\alpha }{\displaystyle \frac{z{f}_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}\right)-\left|z{\displaystyle \frac{f_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}-1\right|-\sigma & \ge \cos \alpha +r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}-1-1+r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}-\sigma \hfill \\ & =2r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma ).\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill \mathrm{Re}\left(e^{-i\alpha }{\displaystyle \frac{z{f}_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}\right)-\left|z{\displaystyle \frac{f_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}-1\right|-\sigma \ge 2r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma ).\end{array}$$

(35)

Using the minimum principle for harmonic functions along with (34) and (35), we derive

$$\begin{array}{cc}\hfill Re\left(e^{-i\alpha }{\displaystyle \frac{z{f}_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}\right)-\left|z{\displaystyle \frac{f_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}-1\right|-\sigma & \ge Re\left(e^{-i\alpha }{\displaystyle \frac{z{f}_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}\right)-1+r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}-\sigma \hfill \\ & \ge \underset{\left|z\right|=r}{min}\left[Re\left(e^{-i\alpha }{\displaystyle \frac{z{f}_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}\right)-1+r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}-\sigma \right]\hfill \\ & \ge 2r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma ).\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \underset{z\in {\mathbb{D}}_r}{inf}\left\{\mathrm{Re}\left(e^{-i\alpha }{\displaystyle \frac{z{f}_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}\right)-\left|z{\displaystyle \frac{f_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}-1\right|-\sigma \right\}=2r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma ).\end{array}$$

(36)

Note that, the function

$$\begin{array}{c}\hfill {\varphi }_{f_{\nu }}:(0,a_n^2\left(\nu \right))⟶\mathbb{R},{\varphi }_{f_{\nu }}\left(r\right)=2r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma ).\end{array}$$

is strictly decreasing, as

$$\begin{array}{c}\hfill {\varphi }_{f_{\nu }}\left(r\right)=2-2\sum _{n=1}^{\infty }{\displaystyle \frac{r}{a_n^2\left(\nu \right)-r}}+(\cos \alpha -2-\sigma )\end{array}$$

implies that

$$\begin{array}{c}\hfill {\varphi }_{f_{\nu }}^{\prime }\left(r\right)=-2\sum _{n=1}^{\infty }{\displaystyle \frac{a_n^2\left(\nu \right)}{{(a_n^2\left(\nu \right)-r)}^2}}<0.\end{array}$$

Furthermore, if $\alpha \ne 0$ and $\cos \alpha >\sigma$, it follows that

$$\begin{array}{cc}\hfill \underset{r\to 0^{+}}{lim}{\varphi }_{f_{\nu }}\left(r\right)& =\underset{r\to 0^{+}}{lim}\left[2r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma )\right]\hfill \\ & =\underset{r\to 0^{+}}{lim}\left[2-2\sum _{n=1}^{\infty }{\displaystyle \frac{r}{a_n^2\left(\nu \right)-r}}+\cos \alpha -2-\sigma \right]\hfill \\ & =2+\cos \alpha -2-\sigma =\cos \alpha -\sigma >0.\hfill \end{array}$$

On the other hand,

$$\begin{array}{cc}\hfill \underset{r\to {a_1^2\left(\nu \right)}^{-}}{lim}{\varphi }_{f_{\nu }}\left(r\right)& =\underset{r\to {a_1^2\left(\nu \right)}^{-}}{lim}\left[2r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma )\right]\hfill \\ & =\underset{r\to {a_1^2\left(\nu \right)}^{-}}{lim}\left[2-2\sum _{n=1}^{\infty }{\displaystyle \frac{r}{a_n^2\left(\nu \right)-r}}+\cos \alpha -2-\sigma \right]=-\infty .\hfill \end{array}$$

Thus, the equation ${\varphi }_{f_{\nu }}\left(r\right)=0$ has a unique root ${\mathtt{R}}_f$ in the interval $(0,a_n^2\left(\nu \right))$. The same result follows when $\alpha =0$ as $0<\sigma <1$. Therefore,

$$\begin{array}{c}\hfill Re\left(e^{-i\alpha }{\displaystyle \frac{z{f}_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}\right)\ge \left|z{\displaystyle \frac{f_{\nu }^{\prime }\left(z\right)}{f_{\nu }\left(z\right)}}-1\right|+\sigma \forall z\in {\mathbb{D}}_{{\mathtt{R}}_f},\end{array}$$

and ${\mathtt{R}}_f$ is the biggest real number that satisfies this relation.

The proofs of parts 2 and 3 follow the same technique. The logarathmic differentiation of functions $g_{\nu }\in {\mathbb{H}}_2$ and ${\mathtt{h}}_{\nu }\in {\mathbb{H}}_3$ yields

$$\begin{array}{c}\hfill z{\displaystyle \frac{g_{\nu }^{\prime }\left(z\right)}{g_{\nu }\left(z\right)}}=1-\sum _{n=1}^{\infty }{\displaystyle \frac{2z^2}{b_n^2\left(\nu \right)-z^2}},\end{array}$$

(37)

and

$$\begin{array}{c}\hfill z{\displaystyle \frac{{\mathtt{h}}_{\nu }^{\prime }\left(z\right)}{{\mathtt{h}}_{\nu }\left(z\right)}}=1-\mu \left(\nu \right)\sum _{n=1}^{\infty }{\displaystyle \frac{2z^2}{d_{\nu }^2\left(z\right)-z^2}},\end{array}$$

(38)

respectively. A similar argument to part (1) yields

$$\begin{array}{c}\hfill \underset{z\in {\mathbb{D}}_r}{inf}\left\{Re\left(e^{-i\alpha }z{\displaystyle \frac{g_{\nu }^{\prime }\left(z\right)}{g_{\nu }\left(z\right)}}\right)-\left|z{\displaystyle \frac{g_{\nu }^{\prime }\left(z\right)}{g_{\nu }\left(z\right)}}-1\right|-\sigma \right\}\ge 2r{\displaystyle \frac{g_{\nu }^{\prime }\left(r\right)}{g_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma )\forall 0<r<b_1\left(\nu \right),\end{array}$$

(39)

and

$$\begin{array}{c}\hfill \underset{z\in {\mathbb{D}}_r}{inf}\left\{Re\left(e^{-i\alpha }z{\displaystyle \frac{{\mathtt{h}}_{\nu }^{\prime }\left(z\right)}{{\mathtt{h}}_{\nu }\left(z\right)}}\right)-\left|z{\displaystyle \frac{{\mathtt{h}}_{\nu }^{\prime }\left(z\right)}{{\mathtt{h}}_{\nu }\left(z\right)}}-1\right|-\sigma \right\}\ge 2r{\displaystyle \frac{{\mathtt{h}}_{\nu }^{\prime }\left(r\right)}{{\mathtt{h}}_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma )\forall 0<r<d_1\left(\nu \right)\end{array}$$

(40)

respectively, in addition to the mappings

$$\begin{array}{c}\hfill {\varphi }_{g_{\nu }}:(0,b_1\left(\nu \right))⟶\mathbb{R},{\varphi }_{g_{\nu }}\left(r\right)=2r{\displaystyle \frac{g_{\nu }^{\prime }\left(r\right)}{g_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma )\end{array}$$

and

$$\begin{array}{c}\hfill {\varphi }_{{\mathtt{h}}_{\nu }}:(0,d_1\left(\nu \right))⟶\mathbb{R},{\varphi }_{{\mathtt{h}}_{\nu }}\left(r\right)=2r{\displaystyle \frac{{\mathtt{h}}_{\nu }^{\prime }\left(r\right)}{{\mathtt{h}}_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma ).\end{array}$$

Both are strictly decreasing, and when $\cos \alpha >\sigma$, they satisfy

$$\begin{array}{c}\hfill \underset{r\to 0^{+}}{lim}{\varphi }_{g_{\nu }}\left(r\right)=\cos \alpha -\sigma >0,\underset{r\to b_1{\left(\nu \right)}^{-}}{lim}{\varphi }_{g_{\nu }}\left(r\right)=-\infty \\ \hfill \underset{r\to 0^{+}}{lim}{\varphi }_{{\mathtt{h}}_{\nu }}\left(r\right)=\cos \alpha -\sigma >0,\underset{r\to d_1{\left(\nu \right)}^{-}}{lim}{\varphi }_{{\mathtt{h}}_{\nu }}\left(r\right)=-\infty .\end{array}$$

Thus, the equations ${\varphi }_{g_{\nu }}\left(r\right)=0$ and ${\varphi }_{{\mathtt{h}}_{\nu }}\left(r\right)=0$ have unique solutions $R_g$ and $R_{\mathtt{h}}$ in the intervals $(0,b_1\left(\nu \right))$ and $(0,d_1\left(\nu \right))$, respectively. Consequently, the desired results follow directly. □

Remark 1.

Note that when $\sigma \ge \cos \alpha$, the function $f_{\nu }\left(z\right)$ no longer satisfies the spirallikeness condition. To see this, observe that for $z=\left|z\right|=r<a_1\left(\nu \right)$

$$\begin{array}{c}\hfill Re\left(e^{-i\alpha }r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}\right)-\left|r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}-1\right|-\sigma \le (\sigma +1)r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}-(\sigma +1)=(\sigma +1)(r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}-1)<0.\end{array}$$

The last inequality follows from (34). Similarly, the functions $g_{\nu }$ and ${\mathtt{h}}_{\nu }$ also fail to meet this condition under the same circumstances.

3. Application to Special Functions

In this section, we will provide several examples of special functions that are members of the classes ${\mathbb{H}}_1$, ${\mathbb{H}}_2$, and ${\mathbb{H}}_3$, and thus satisfy the results proved in the earlier sections.

3.1. Functions Involving sin Functions

For $\nu \ne 0$, consider the function

$$\begin{array}{cc}\hfill {\mathcal{E}}_{\nu }\left(z\right):& =\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z}{n^2{\pi }^2-{\nu }^2}}\right)={\displaystyle \frac{\nu csc\left(\nu \right)sin\left(\sqrt{{\nu }^2+z}\right)}{\sqrt{{\nu }^2+z}}}.\hfill \end{array}$$

(41)

Example 1.

The normalised function $f_{\nu }\left(z\right)=z{\mathcal{E}}_{\nu }\left(z\right)\in {\mathbb{H}}_1$ with zeros $a_n\left(\nu \right)=\sqrt{n^2{\pi }^2-{\nu }^2}$, $n=1,2,3,\dots$, and $\nu \in (-\pi ,\pi )$.

A calculation yields that

$$f_{\nu }^{\prime }\left(r\right)={\displaystyle \frac{\nu csc\left(\nu \right)\left(\left(2{\nu }^2+r\right)sin\left(\sqrt{{\nu }^2+r}\right)+r\sqrt{{\nu }^2+r}\cos \left(\sqrt{{\nu }^2+r}\right)\right)}{2{\left({\nu }^2+r\right)}^{3/2}}}.$$

Further computation gives us

$$\begin{array}{cc}\hfill & 2r{\displaystyle \frac{f_{\nu }^{\prime }\left(r\right)}{f_{\nu }\left(r\right)}}+(\cos \left(\alpha \right)-2-\sigma )=0\hfill \\ \hfill \Rightarrow & \cos \left(\alpha \right)-{\displaystyle \frac{r}{{\nu }^2+r}}+{\displaystyle \frac{rcot\left(\sqrt{{\nu }^2+r}\right)}{\sqrt{{\nu }^2+r}}}-\sigma =0.\hfill \end{array}$$

From Theorem 1, it follows that the $S{P}_p(\alpha ,\sigma )$-radius of $f_{\nu }\left(z\right)=z{\mathcal{E}}_{\nu }\left(z\right)$ is ${\mathtt{R}}_f={\mathtt{R}}_f$, where ${\mathtt{R}}_f$ is the unique solution of

$$\begin{array}{c}\hfill \cos \left(\alpha \right)-{\displaystyle \frac{r}{{\nu }^2+r}}+{\displaystyle \frac{rcot\left(\sqrt{{\nu }^2+r}\right)}{\sqrt{{\nu }^2+r}}}-\sigma =0,\end{array}$$

(42)

in $(0,{\pi }^2-{\nu }^2)$. To validate the existence of a unique solution for (42), we tabulate below a few cases considering fixed $\sigma ,\nu$, and $\alpha$. Due to the symmetric nature of the cosine function, it is sufficient to consider $\alpha$ in the interval $[0,\pi /2)$. For a specific choice of $\nu$ and $\alpha \in [0,\pi /2)$, Table 1 presents the $S{P}_p(\alpha ,\sigma )$-radius of $f_{\nu }\left(z\right)=z{\mathcal{E}}_{\nu }\left(z\right)$ when for $\sigma =0$

Table 1. Solution of (42) when $\sigma =0$.

	$\mathit{\alpha }=\frac{\mathit{\pi }}{2.1}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{3}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{4}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{6}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{12}$	$\mathit{\alpha }=0$
$\nu =1/2$	0.217114	1.3339	1.81095	2.15037	2.35265	2.41977
$\nu =1$	0.205517	1.25799	1.70511	2.02229	2.21095	2.27349
$\nu =3/2$	0.184955	1.12422	1.51913	1.79777	1.96291	2.01756
$\nu =2$	0.153053	0.919124	1.23563	1.45698	1.58742	1.63046
$\nu =5/2$	0.104929	0.616967	0.822393	0.964072	1.04682	1.07401
$\nu =3$	0.0295322	0.166881	0.219232	0.254557	0.274919	0.281569

Now when $\sigma =1/2$, $\cos \left(\alpha \right)>1/2$ holds when $0\le \alpha <\pi /3$. We present the $S{P}_p(\alpha ,1/2)$ radius of $f_{\nu }\left(z\right)$ in Table 2.

Table 2. Solution of (42) when $\sigma =1/2$.

	$\mathit{\alpha }=\frac{\mathit{\pi }}{3.1}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{3.5}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{4}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{6}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{12}$	$\mathit{\alpha }=0$
$\nu =1/2$	0.084933	0.355255	0.58584	1.00288	1.25143	1.3339
$\nu =1$	0.0804303	0.33613	0.553885	0.946871	1.18055	1.25799
$\nu =3/2$	0.0724421	0.302243	0.497332	0.847966	1.05557	1.12422
$\nu =2$	0.0600326	0.249737	0.409909	0.695755	0.863767	0.919124
$\nu =5/2$	0.0412637	0.170754	0.279026	0.469885	0.580681	0.616967
$\nu =3$	0.011675	0.0478007	0.077436	0.128492	0.157484	0.166881

3.2. Functions Involving Gamma Function

Consider the function

$$\begin{array}{c}\hfill g_{\nu }\left(z\right)=z\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{{(n+\nu )}^2}}\right)={\displaystyle \frac{z{\left(\mathsf{\Gamma }(1+\nu )\right)}^2}{\mathsf{\Gamma }(1-z+\nu )\mathsf{\Gamma }(1+z+\nu )}},\nu >-1.\end{array}$$

(43)

Example 2.

The function $g_{\nu }\in {\mathbb{H}}_2$ with zeros $b_n\left(\nu \right)=n+\nu$.

Notably, this example covers a wide range of functions involving sine or cosine. Specifically, when $\nu =(k+1)/2,k\in \mathbb{N}$, the resulting function $g_{\nu }$ involves cosine, whereas for $\nu =k,k\in \mathbb{N}$, the function $g_{\nu }$ involves sine.

Next, by computing the derivative $g_{\nu }$ for $r>0$, we obtain

$$\begin{array}{c}\hfill g_{\nu }^{\prime }\left(r\right)={\displaystyle \frac{{\left(\mathsf{\Gamma }(1+\nu )\right)}^2(1+r\psi (1-r+\nu )-r\psi (1+r+\nu ))}{\mathsf{\Gamma }(1-r+\nu )\mathsf{\Gamma }(1+r+\nu )}}.\end{array}$$

Here, $\psi \left(z\right)$ is the digamma function defined in (26). Further calculations yield

$$\begin{array}{c}\hfill {\varphi }_{g_{\nu }}\left(r\right)=\cos \alpha +2r\psi (1-r+\nu )-2r\psi (1+r+\nu )-\sigma .\end{array}$$

Since the function $g_{\nu }$ belongs to the class ${\mathbb{H}}_2$, Theorem 1 allows us to determine the radius explicitly as the unique root of the equation

$$\begin{array}{c}\hfill {\varphi }_{g_{\nu }}\left(r\right)=\cos \alpha +2r\psi (1-r+\nu )-2r\psi (1+r+\nu )-\sigma =0.\end{array}$$

(44)

Table 3 and Table 4 display the computed radius for the cases $\sigma =0$ and $\sigma =1/2$ within the interval $(0,b_1\left(\nu \right))$, provided $\cos \alpha >\sigma$ when $\alpha \in [0,\pi /2)$.

Table 3. Solution of (44) when $\sigma =0$.

	$\mathit{\alpha }=\frac{\mathit{\pi }}{2.1}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{3}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{4}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{6}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{12}$	$\mathit{\alpha }=0$
$\nu =1/2$	0.141016	0.359572	0.424628	0.467413	0.491974	0.5
$\nu =1$	0.169886	0.434816	0.514422	0.567044	0.597361	0.607287
$\nu =3/2$	0.194909	0.5	0.592198	0.653334	0.688635	0.700206
$\nu =2$	0.21724	0.558124	0.661526	0.730233	0.769963	0.782996
$\nu =5/2$	0.237569	0.610997	0.724569	0.800142	0.843887	3.82446
$\nu =3$	0.256341	0.659788	0.782727	0.86462	0.912059	0.927635

Table 4. Solution of (44) when $\sigma ={\displaystyle \frac{1}{2}}$.

	$\mathit{\alpha }=\frac{\mathit{\pi }}{3}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{4}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{6}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{12}$	$\mathit{\alpha }=0$
$\nu =1/2$	$5.98052\times {10}^{-9}$	0.233713	0.309042	0.347502	0.359572
$\nu =1$	$5.97063\times {10}^{-9}$	0.281891	0.373271	0.420095	0.434816
$\nu =3/2$	$5.96652\times {10}^{-9}$	0.323641	0.42892	0.482984	0.5
$\nu =2$	$5.96446\times {10}^{-9}$	0.360889	0.478554	0.539065	0.558124
$\nu =5/2$	$5.96329\times {10}^{-9}$	0.39479	0.523715	0.590082	0.610997
$\nu =3$	$5.96256\times {10}^{-9}$	0.426087	0.565397	0.637163	0.659788

For our next example involving the gamma function, consider the function

$$\begin{array}{c}\hfill {\mathtt{h}}_{\nu }\left(z\right)=z\prod _{n=1}^{\infty }{\left(1-{\displaystyle \frac{z^2}{{(n\pi +\nu )}^2}}\right)}^{{\displaystyle \frac{1}{\nu }}}=z{\left({\displaystyle \frac{{\left(\mathsf{\Gamma }\left(\frac{\pi +\nu }{\pi }\right)\right)}^2}{\mathsf{\Gamma }\left(\frac{\pi -z+\nu }{\pi }\right)\mathsf{\Gamma }\left(\frac{\pi +z+\nu }{\pi }\right)}}\right)}^{{\displaystyle \frac{1}{\nu }}},\nu >0.\end{array}$$

(45)

Example 3.

The function ${\mathtt{h}}_{\nu }$ in the class ${\mathbb{H}}_3$ with positive zeros $d_n\left(\nu \right)=n\pi +\nu$.

Using the logarithmic differentiation of (45) along with further calculations, we obtain

$$\begin{array}{c}\hfill {\varphi }_{{\mathtt{h}}_{\nu }}\left(r\right)=2r{\displaystyle \frac{{\mathtt{h}}_{\nu }^{\prime }\left(r\right)}{{\mathtt{h}}_{\nu }\left(r\right)}}+(\cos \alpha -2-\sigma )=\cos \alpha +{\displaystyle \frac{2r}{\pi \nu }}\left[\psi \left({\displaystyle \frac{\pi -r+\nu }{\pi }}\right)-\psi \left({\displaystyle \frac{\pi +r+\nu }{\pi }}\right)\right]-\sigma ,\end{array}$$

where $\psi$ is the digamma function defined in (26). By applying Theorem 1, the equation

$$\begin{array}{c}\hfill {\varphi }_{{\mathtt{h}}_{\nu }}\left(r\right)=0\Rightarrow \cos \alpha +{\displaystyle \frac{2r}{\pi \nu }}\left[\psi \left({\displaystyle \frac{\pi -r+\nu }{\pi }}\right)-\psi \left({\displaystyle \frac{\pi +r+\nu }{\pi }}\right)\right]-\sigma =0\end{array}$$

(46)

has a unique root in the interval $(0,\pi +\nu )$, provided that $\cos \alpha >\sigma$. This result is demonstrated in Table 5, Table 6, and Table 7, respectively, for $\sigma =0,0.5,0.9$.

Table 5. Solution of (46) when $\sigma =0$.

	$\mathit{\alpha }=\frac{\mathit{\pi }}{2.1}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{3}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{4}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{6}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{12}$	$\mathit{\alpha }=0$
$\nu =1/2$	0.262801	0.673514	0.79735	0.879373	0.926697	0.942204
$\nu =1$	0.405913	1.03296	1.21868	1.3405	1.4103	1.43308
$\nu =3/2$	0.536429	1.35759	1.59734	1.75336	1.84223	1.87116
$\nu =2$	0.662165	1.66845	1.95885	2.14657	2.25299	2.28755
$\nu =5/2$	0.7856	1.97244	2.31164	2.52969	2.65281	2.69269
$\nu =3$	0.907782	2.27255	2.65945	2.90697	3.04624	3.09127

Table 6. Solution of (46) when $\sigma ={\displaystyle \frac{1}{2}}$.

	$\mathit{\alpha }=\frac{\mathit{\pi }}{3}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{4}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{6}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{12}$	$\mathit{\alpha }=0$
$\nu =1/2$	$5.9642\times {10}^{-9}$	0.43624	0.577939	0.650641	0.673514
$\nu =1$	$5.96321\times {10}^{-9}$	0.672322	0.888357	0.998448	1.03296
$\nu =3/2$	$5.96256\times {10}^{-9}$	0.886981	1.16958	1.31281	1.35759
$\nu =2$	$5.96211\times {10}^{-9}$	1.0934	1.4394	1.61399	1.66845
$\nu =5/2$	$5.96179\times {10}^{-9}$	1.29581	1.70359	1.90862	1.97244
$\nu =3$	$5.96155\times {10}^{-9}$	1.496	1.96463	2.19955	2.27255

Table 7. Solution of (46) when $\sigma =0.9$.

	$\mathit{\alpha }=\frac{\mathit{\pi }}{3}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{4}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{6}$	$\mathit{\alpha }=\frac{\mathit{\pi }}{12}$	$\mathit{\alpha }=0$
$\nu =1/2$	$-4.71723\times {10}^{-7}$	$-4.71737\times {10}^{-7}$	$5.10716\times {10}^{-9}$	0.246882	0.303837
$\nu =1$	$-4.28641\times {10}^{-6}$	$-4.71737\times {10}^{-7}$	$-3.52521\times {10}^{-7}$	0.381382	0.469099
$\nu =3/2$	$7.15794\times {10}^{-6}$	$-4.71737\times {10}^{-7}$	$-4.71737\times {10}^{-7}$	0.504067	0.61973
$\nu =2$	$7.15794\times {10}^{-6}$	$-4.28647\times {10}^{-6}$	$-4.71737{\times }^{-7}$	0.622273	0.764793
$\nu =5/2$	−10.0437	$2.70565\times {10}^{-6}$	$-4.71737\times {10}^{-7}$	0.738325	0.907171
$\nu =3$	0.0000222881	$7.15794\times {10}^{-6}$	$-4.71737\times {10}^{-7}$	0.853205	1.04808

3.3. Functions Associated with the Normalized Bessel Functions

The positive zeros $j_n\left(\nu \right)$ of the well-known classical Bessel function $J_{\nu }$ follow the increasing order $j_1\left(\nu \right)<j_2\left(\nu \right)<\dots$ for $\nu \ge 0$. Furthermore, the Bessel function can also be represented by

$$\begin{array}{c}\hfill J_{\nu }\left(z\right)={\displaystyle \frac{z^{\nu }}{2^{\nu }\mathsf{\Gamma }(\nu +1)}}\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{j_n^2\left(\nu \right)}}\right).\end{array}$$

(47)

For more details about the classical Bessel functions, see [16].

A logarithmic differentiation of (47) yields

$$\begin{array}{c}\hfill {\displaystyle \frac{z{J}_{\nu }^{\prime }\left(z\right)}{J_{\nu }\left(z\right)}}=\nu -\sum _{n=1}^{\infty }{\displaystyle \frac{2z^2}{j_n^2\left(\nu \right)-z^2}}\end{array}$$

(48)

Then, the following three normalization can be obtained from (47):

$$\begin{array}{cc}\hfill 2^{\nu }\mathsf{\Gamma }(\nu +1)z^{1-{\displaystyle \frac{\nu }{2}}}{J}_{\nu }\left(\sqrt{z}\right)& =z\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z}{j_n^2\left(\nu \right)}}\right);\hfill \\ \hfill 2^{\nu }\mathsf{\Gamma }(\nu +1)z^{1-\nu }{J}_{\nu }\left(z\right)& =z\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{j_n^2\left(\nu \right)}}\right);\hfill \\ \hfill {\left(2^{\nu }\mathsf{\Gamma }(\nu +1)J_{\nu }\left(z\right)\right)}^{{\displaystyle \frac{1}{\nu }}}& =z\prod _{n=1}^{\infty }{\left(1-{\displaystyle \frac{z^2}{j_n^2\left(\nu \right)}}\right)}^{{\displaystyle \frac{1}{\nu }}},\hfill \end{array}$$

which follows the below example for ${\mathbb{H}}_1$, ${\mathbb{H}}_2$, and ${\mathbb{H}}_3$.

Example 4

(The normalized Bessel function). Denote $j_n\left(\nu \right)$ as the n-th zero of the Bessel functions $J_{\nu }\left(z\right)$. Then,

(i) 

${\mathcal{B}}_1(\nu ,z):=2^{\nu }\mathsf{\Gamma }(\nu +1)z^{1-{\displaystyle \frac{\nu }{2}}}{J}_{\nu }\left(\sqrt{z}\right)\in {\mathbb{H}}_1,$ with $a_n\left(\nu \right)=j_n\left(\nu \right)$;

(ii) 

${\mathcal{B}}_2(\nu ,z)=2^{\nu }\mathsf{\Gamma }(\nu +1)z^{1-\nu }{J}_{\nu }\left(z\right)\in {\mathbb{H}}_2$ with $b_n\left(\nu \right)=j_n\left(\nu \right)$;

(iii) 

${\mathcal{B}}_3(\nu ,z)={\left(2^{\nu }\mathsf{\Gamma }(\nu +1)J_{\nu }\left(z\right)\right)}^{{\displaystyle \frac{1}{ \nu }}}=z{\prod }_{n=1}^{\infty }{\left(1-{\displaystyle \frac{z^2}{j_n^2\left(\nu \right)}}\right)}^{{\displaystyle \frac{1}{\nu }}}\in {\mathbb{H}}_3$ with $d_n\left(\nu \right)=j_n\left(\nu \right)$ and $\mu \left(\nu \right):=1/\nu$.

The logarithmic differentiation of ${\mathcal{B}}_i(\nu ,z)$, $i=1,2,3$ yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{z{\mathcal{B}}_1^{\prime }(\nu ,z)}{{\mathcal{B}}_1(\nu ,z)}}& =1-{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{\sqrt{z}{J}_{\nu }^{\prime }\left(\sqrt{z}\right)}{2J_{\nu }\left(\sqrt{z}\right)}}\hfill \\ \hfill {\displaystyle \frac{z{\mathcal{B}}_2^{\prime }(\nu ,z)}{{\mathcal{B}}_2(\nu ,z)}}& =1-\nu +{\displaystyle \frac{z{J}_{\nu }^{\prime }\left(z\right)}{J_{\nu }\left(z\right)}}\hfill \\ \hfill {\displaystyle \frac{z{\mathcal{B}}_3^{\prime }(\nu ,z)}{{\mathcal{B}}_3(\nu ,z)}}& ={\displaystyle \frac{z{J}_{\nu }^{\prime }\left(z\right)}{\nu J_{\nu }\left(z\right)}}\hfill \end{array}$$

Applying the above three relations to Theorem 1 yields the following result.

Theorem 2.

Suppose that for some $\left|\alpha \right|<\pi /2$ and $\sigma \in [0,1)$, the inequality $\cos \alpha >\sigma$ holds. Then,

(i) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{B}}_1(\nu ,z)$ is the smallest positive root of the equation

$$\sqrt{r}{J}_{\nu }^{\prime }\left(\sqrt{r}\right)+(\cos \left(\alpha \right)-\nu -\sigma )J_{\nu }\left(\sqrt{r}\right)=0,$$

in $(0,j_1^2\left(\nu \right))$.

(ii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{B}}_2(\nu ,z)$ is the smallest positive root of the equation

$$2r{J}_{\nu }^{\prime }\left(r\right)+(\cos \left(\alpha \right)-2\nu -\sigma )J_{\nu }\left(r\right)=0,$$

in $(0,j_1\left(\nu \right))$.

(iii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{B}}_3(\nu ,z)$ is the smallest positive root of the equation

$$2r{J}_{\nu }^{\prime }\left(r\right)+\nu (\cos \left(\alpha \right)-2-\sigma )J_{\nu }\left(r\right)=0$$

in $(0,j_1\left(\nu \right))$.

Next, from Theorem 2, we are going to present a few numerical examples of the $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{B}}_i(\nu ,z)$, $i=1,2,3$ for fixed $\alpha ,\sigma$, and $\nu$.

We note here that for $\nu =1/2$,

$$\begin{array}{c}\hfill {\mathcal{B}}_1(1/2,z)=\sqrt{z}sin\left(\sqrt{z}\right),{\mathcal{B}}_2(1/2,z)=sin\left(z\right),\end{array}$$

and for $\nu =-1/2$, we have

$${\mathcal{B}}_3(-1/2,z)=z{sec}^2\left(z\right).$$

This implies the following:

Corollary 1.

Suppose that for some $\left|\alpha \right|<\pi /2$ and $\sigma \in [0,1)$, the inequality $\cos \alpha >\sigma$ holds. Then, the $S{P}_p(\alpha ,\sigma )$-radius of $\sqrt{z}sin\left(\sqrt{z}\right)$ is the smallest positive root of the equation

$$sin\left(\sqrt{r}\right)\left(4\sqrt{r}\cos \left(\alpha \right)-2\sqrt{r}(2\sigma +1)-1\right)+2\sqrt{r}\cos \left(\sqrt{r}\right)=0,$$

in $(0,{\pi }^2)$.

We further investigate the numerical values of the radius for specific values of $\alpha$ and $\sigma$. We discuss four cases, namely by taking $\alpha =0,\pi /6,\pi /4,\pi /3,\pi /2.1$ and the corresponding $\sigma \in [0,\cos (\alpha \left)\right)$. The outcome is presented in Table 8.

Table 8. The $S{P}_p(\alpha ,\sigma )$-radius of $\sqrt{z}sin\left(\sqrt{z}\right)$.

$\mathit{\alpha }=0\Rightarrow \sigma <\cos \left(\alpha \right)=1$
$\sigma$	0	$1/4$	$1/2$	$3/4$	$0.9$
$S{P}_p(0,\sigma )$	$4.97574$	$3.19563$	$1.35853$	$0.463639$	$0.263434$
$\alpha =\pi /6\Rightarrow \sigma <\cos (\pi /6)=0.866025$
$\sigma$	0	$1/4$	$1/2$	$3/4$	$0.8$
$S{P}_p(\pi /6,\sigma )$	$4.08932$	$2.13474$	$0.758539$	$0.278617$	$0.234724$
$\alpha =\pi /4\Rightarrow \sigma <\cos (\pi /4)=0.707107$
$\sigma$	0	$1/4$	$1/2$	$0.6$	$0.7$
$S{P}_p(\pi /4,\sigma )$	$2.8504$	$1.13074$	$0.390812$	$0.270028$	$0.194197$
$\alpha =\pi /3\Rightarrow \sigma <\cos (\pi /3)=0.5$
$\sigma$	0	$0.1$	$0.2$	$0.3$	$0.49$
$S{P}_p(\pi /3,\sigma )$	$1.35853$	$0.880536$	$0.570593$	$0.380166$	$0.195953$
$\alpha =\pi /2.1\Rightarrow \sigma <\cos (\pi /(2.1\left)\right)=0.0747301$
$\sigma$	0	$0.01$	$0.02$	$0.03$	$0.07$
$S{P}_p(\pi /2.1,\sigma )$	$0.241662$	$0.233714$	$0.226116$	$0.21885$	$0.19277$

The next corollary is obtained by taking $\nu =1/2$ on ${\mathcal{B}}_2(\nu ,z)$ in Theorem 2.

Corollary 2.

Suppose that for some $\left|\alpha \right|<\pi /2$ and $\sigma \in [0,1)$, the inequality $\cos \alpha >\sigma$ holds. Then, the $S{P}_p(\alpha ,\sigma )$-radius of $sin\left(z\right)$ is the smallest positive root of the equation

$$sin\left(r\right)\left(\cos \right(\alpha )-\sigma -2)+2r\cos \left(r\right)=0,$$

in $(0,\pi )$.

Similar to the special cases of Corollary 1, we will present the $S{P}_p(\alpha ,\sigma )$-radius of $sin\left(z\right)$ for the specific $\alpha =0,\pi /6,\pi /4,\pi /3,\pi /2.1$ and $\sigma$ in Table 9.

Table 9. The $S{P}_p(\alpha ,\sigma )$-radius of $sin\left(z\right)$.

$\alpha =0\Rightarrow \sigma <\cos \left(\alpha \right)=1$
$\sigma$	0	$1/4$	$1/2$	$3/4$	$0.9$
$S{P}_p(0,\sigma )$	$1.16556$	$1.02188$	$0.844731$	$0.60478$	$0.385368$
$\alpha =\pi /6\Rightarrow \sigma <\cos (\pi /6)=0.866025$
$\sigma$	0	$1/4$	$1/2$	$3/4$	$0.8$
$S{P}_p(\pi /6,\sigma )$	$1.09183$	$0.932264$	$0.727573$	$0.414768$	$0.313666$
$\alpha =\pi /4\Rightarrow \sigma <\cos (\pi /4)=0.707107$
$\sigma$	0	$1/4$	$1/2$	$0.6$	$0.7$
$S{P}_p(\pi /4,\sigma )$	$0.994326$	$0.809405$	$0.551636$	$0.398685$	$0.103211$
$\alpha =\pi /3\Rightarrow \sigma <\cos (\pi /3)=0.5$
$\sigma$	0	$0.1$	$0.2$	$0.3$	$0.49$
$S{P}_p(\pi /3,\sigma )$	$0.844731$	$0.759308$	$0.660857$	$0.542281$	$0.122413$
$\alpha =\pi /2.1\Rightarrow \sigma <\cos (\pi /2.1)=0.0747301$
$\sigma$	0	$0.01$	$0.02$	$0.03$	$0.07$
$S{P}_p(\pi /2.1,\sigma )$	$0.333558$	$0.310595$	$0.28574$	$0.258449$	$0.0842127$

The final corollary is derived by setting $\nu =-1/2$ in ${\mathcal{B}}_3(\nu ,z)$ from Theorem 2.

Corollary 3.

Suppose that for some $\left|\alpha \right|<\pi /2$ and $\sigma \in [0,1)$, the inequality $\cos \alpha >\sigma$ holds. Then, the $S{P}_p(\alpha ,\sigma )$-radius of $z{sec}^2\left(z\right)$ is the smallest positive root of the equation

$$\cos \left(r\right)(\sigma -\cos (\alpha \left)\right)-4rsin\left(r\right)=0,$$

in $(0,\pi )$.

As earlier, the special case of Corollary 3 is presented in Table 10.

Table 10. The $S{P}_p(\alpha ,\sigma )$-radius of $z{sec}^2\left(z\right)$.

$\alpha =0\Rightarrow \sigma <\cos \left(\alpha \right)=1$
$\sigma$	0	$1/4$	$1/2$	$3/4$	$0.9$
$S{P}_p(0,\sigma )$	$3.06008$	$3.08081$	$3.10131$	$3.12157$	$3.13361$
$\alpha =\pi /6\Rightarrow \sigma <\cos (\pi /6)=0.866025$
$\sigma$	0	$1/4$	$1/2$	$3/4$	$0.8$
$S{P}_p(\pi /6,\sigma )$	$3.07121$	$3.09182$	$3.1122$	$3.13233$	$3.13633$
$\alpha =\pi /4\Rightarrow \sigma <\cos (\pi /4)=0.707107$
$\sigma$	0	$1/4$	$1/2$	$0.6$	$0.7$
$S{P}_p(\pi /4,\sigma )$	$3.08434$	$3.1048$	$3.12503$	$3.13305$	$3.14103$
$\alpha =\pi /3\Rightarrow \sigma <\cos (\pi /3)=0.5$
$\sigma$	0	$0.1$	$0.2$	$0.3$	$0.49$
$S{P}_p(\pi /3,\sigma )$	$3.10131$	$3.10944$	$3.11754$	$3.1256$	$3.1408$
$\alpha =\pi /2.1\Rightarrow \sigma <\cos (\pi /(2.1\left)\right)=0.0747301$
$\sigma$	0	$0.01$	$0.02$	$0.03$	$0.07$
$S{P}_p(\pi /2.1,\sigma )$	$3.13563$	$3.13643$	$3.13723$	$3.13803$	$3.14122$

3.4. Examples Involving Derivatives of Bessel Functions

For the next example, consider the function

$$\begin{array}{c}\hfill N_{\nu }\left(z\right)=a{z}^2{J}_{\nu }^{″}\left(z\right)+bz{J}_{\nu }\left(z\right)+c,a,b,c\in \mathbb{R}.\end{array}$$

(49)

This function was first introduced in [17], where its zeros were analyzed in detail. Furthermore, a recent study [18], investigated the radius of uniformly convex $\gamma$-spirallikeness of $N_{\nu }$. From the relation between Bessel function $J_{\nu }$ and $N_{\nu }$, the latter can be represented as follows

$$\begin{array}{c}\hfill N_{\nu }\left(z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{Q(2n+\nu ){(-1)}^n}{n!\mathsf{\Gamma }(n+\nu +1)}}{\left({\displaystyle \frac{z}{2}}\right)}^{2n+\nu },\end{array}$$

where $Q\left(\nu \right)=a\nu (\nu -1)+b\nu +c,a,b,c\in \mathbb{R}$. When $c=0$ and $a\ne b$ or $c>0$ and $b>a$, the zeros of $N_{\nu }$ are real for $\nu \ge max\{0,{\nu }_0\}$, where ${\nu }_0$ presents the largest real root of $Q\left(\nu \right)$. Moreover, in this case, the function $N_{\nu }$ can be written as follows [18].

$$\begin{array}{c}\hfill N_{\nu }\left(z\right)={\displaystyle \frac{Q\left(\nu \right)z^{\nu }}{2^{\nu }\mathsf{\Gamma }(\nu +1)}}\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{{\lambda }_n^2\left(\nu \right)}}\right),\end{array}$$

(50)

where ${\lambda }_n\left(\nu \right)$ is the nth positive zero of $N_{\nu }$. For further details about the function $N_{\nu }$ and its zeros, refer to [18] and the references therein. In the next example, we present three normalizations of $N_{\nu }\left(z\right)$ to include it in the classes ${\mathbb{H}}_i$, $i=1,2,3$, which were introduced in [18].

Example 5.

For $\nu \ge max\{0,{\nu }_0\}$, where ${\nu }_0$ presents the largest real root of $Q\left(\nu \right)$, define

(i) 

${\mathcal{N}}_1(\nu ,z)={\displaystyle \frac{2^{\nu }\mathsf{\Gamma }(\nu +1)z^{1-\frac{\nu }{2}}}{Q\left(\nu \right)}}{N}_{\nu }\left(\sqrt{z}\right)=z{\prod }_{n=1}^{\infty }\left(1-{\displaystyle \frac{z}{{\lambda }_n^2\left(\nu \right)}}\right)\in {\mathbb{H}}_1$ with $a_n\left(\nu \right)={\lambda }_n\left(\nu \right)$;

(ii) 

${\mathcal{N}}_2(\nu ,z)={\displaystyle \frac{2^{\nu }\mathsf{\Gamma }(\nu +1)z^{1-\nu }}{Q\left(\nu \right)}}{N}_{\nu }\left(z\right)=z{\prod }_{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{{\lambda }_n^2\left(\nu \right)}}\right)\in {\mathbb{H}}_2$ with $b_n\left(\nu \right)={\lambda }_n\left(\nu \right)$;

(iii) 

${\mathcal{N}}_3(\nu ,z)={\left[{\displaystyle \frac{2^{\nu }\mathsf{\Gamma }(\nu +1)}{Q\left(\nu \right)}}{N}_{\nu }\left(z\right)\right]}^{{\displaystyle \frac{1}{\nu }}}=z{\prod }_{n=1}^{\infty }{\left(1-{\displaystyle \frac{z^2}{{\lambda }_n^2\left(\nu \right)}}\right)}^{{\displaystyle \frac{1}{\nu }}}\in {\mathbb{H}}_3$ with $d_n\left(\nu \right)={\lambda }_n\left(\nu \right)$.

A direct application of Theorem 1, combined with the relationships between ${\mathcal{N}}_i$ and $N_{\nu }$ for $i=1,2,3$, leads to the following result:

Theorem 3.

Let $\left|\alpha \right|<\pi /2$ and $\sigma \in [0,1)$ such that $\cos \alpha >\sigma$. For $\nu \ge max\{0,{\nu }_0\}$, where ${\nu }_0$ is the largest real root of $Q\left(\nu \right)$, the following statements are true:

(i) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{N}}_1(\nu ,z)$ is the smallest positive root of the equation

$$\sqrt{r}{N}_{\nu }^{\prime }\left(\sqrt{r}\right)+(\cos \left(\alpha \right)-\nu -\sigma )N_{\nu }\left(\sqrt{r}\right)=0,$$

in $(0,{\lambda }_1^2\left(\nu \right))$;

(ii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{N}}_2(\nu ,z)$ is the smallest positive root of the equation

$$2r{N}_{\nu }^{\prime }\left(r\right)+(\cos \left(\alpha \right)-2\nu -\sigma )N_{\nu }\left(r\right)=0,$$

in $(0,{\lambda }_1\left(\nu \right))$;

(iii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{N}}_3(\nu ,z)$ is the smallest positive root of the equation

$$2r{N}_{\nu }^{\prime }\left(r\right)+\nu (\cos \left(\alpha \right)-2-\sigma )N_{\nu }\left(r\right)=0,$$

in $(0,{\lambda }_1\left(\nu \right))$.

In the following example, we find the $S{P}_p(\alpha ,\sigma )$-radius for the functions ${\mathcal{N}}_i(\nu ,z), i=1,2,3$ when $\nu =1/2$, considering different values of $a,b,c\in \mathbb{R}$ that satisfy either $c=0$ and $a\ne b$ or $c>0$ and $b>a$.

Example 6.

When $\nu =1/2$, the function $N_{1/2}$ can be expressed in terms of sine and cosine functions. Similarly, the associated functions defined here can also be represented in terms of trigonometric functions, in particular,

$$\begin{array}{cc}\hfill N_{\frac{1}{2}}\left(z\right)& ={\displaystyle \frac{4(b-a)z\cos \left(z\right)+(-2b+4c+a(3-4z^2))sin\left(z\right)}{2\sqrt{2\pi }\sqrt{z}}};\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill {\mathcal{N}}_1(\frac{1}{2},z)& ={\displaystyle \frac{4(a-b)z\cos \left(\sqrt{z}\right)+\sqrt{z}(-3a+2b-4c+4az)sin\left(\sqrt{z}\right)}{a-2b+4c}};\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill {\mathcal{N}}_2(\frac{1}{2},z)& ={\displaystyle \frac{4(a-b)z\cos \left(z\right)+(-3a+2b-4c+4a{z}^2)sin\left(z\right)}{a-2b+4c}};\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill {\mathcal{N}}_3(\frac{1}{2},z)& ={\displaystyle \frac{{\left(4(a-b)z\cos \left(z\right)+(-3a+2b-4c+4a{z}^2)sin\left(z\right)\right)}^2}{{(a-2b+4c)}^{2}z}}.\hfill \end{array}$$

(54)

Based on Theorem 3, the $S{P}_p(\alpha ,1/2)$-radius of the functions ${\mathcal{N}}_i(1/2,z),i=1,2,3$ is given by the smallest positive root of the corresponding equations within the interval $(0,{\lambda }_1^2(1/2))$, for ${\mathcal{N}}_1(1/2,z)$ and $(0,{\lambda }_1(1/2))$ for the remaining functions, where, ${\lambda }_1(1/2)$ is the first positive zero of $N_{1/2}\left(z\right)$. Numerical calculations of the values of ${\lambda }_1(1/2)$ are presented in Table 11.

Table 11. The values of ${\lambda }_1(1/2)$ corresponding to some values of $a,b,c$.

	$\mathit{c}=0,\mathit{b}=1$			$\mathit{c}=0,\mathit{a}=1$			$\mathit{a}=0,\mathit{b}=1$
	$\mathit{a}=\mathbf{2}$	$\mathit{a}=\mathbf{3}$	$\mathit{a}=\mathbf{4}$	$\mathit{b}=\mathbf{2}$	$\mathit{b}=\mathbf{3}$	$\mathit{b}=\mathbf{4}$	$\mathit{c}=\mathbf{2}$	$\mathit{c}=\mathbf{3}$	$\mathit{c}=\mathbf{4}$
${\lambda }_1(1/2)$	2.96464	2.89936	2.86553	0.74635	0.857045	0.921336	1.40771	1.62929	1.81561

Table 12, Table 13 and Table 14 represent the $S{P}_p(\alpha ,\sigma )$-radius of the functions ${\mathcal{N}}_1,{\mathcal{N}}_2,$ and ${\mathcal{N}}_3$ in the case $\sigma =0$. The corresponding values for $\sigma =1/2$ are provided in Table 15, Table 16, and Table 17, respectively.

Table 12. The $S{P}_p(\alpha ,0)$-radius of ${\mathcal{N}}_1({\displaystyle \frac{1}{2}},z)$.

	$\mathit{c}=0,\mathit{b}=1$			$\mathit{c}=0,\mathit{a}=1$			$\mathit{a}=0,\mathit{b}=1$
	$\mathit{a}=\mathbf{2}$	$\mathit{a}=\mathbf{3}$	$\mathit{a}=\mathbf{4}$	$\mathit{b}=\mathbf{2}$	$\mathit{b}=\mathbf{3}$	$\mathit{b}=\mathbf{4}$	$\mathit{c}=\mathbf{2}$	$\mathit{c}=\mathbf{3}$	$\mathit{c}=\mathbf{4}$
$S{P}_p\left(\frac{\pi }{2.1},0\right)$	2.52026	2.36178	2.2766	0.0649205	0.0786036	0.0865199	0.135511	0.153047	0.165599
$S{P}_p\left(\frac{\pi }{3},0\right)$	2.52026	2.36178	2.2766	0.10568	0.137733	0.158201	0.330987	0.419367	0.495314
$S{P}_p\left(\frac{\pi }{4},0\right)$	5.05758	4.81257	4.68628	0.130244	0.174741	0.204071	0.481686	0.642428	0.790578
$S{P}_p\left(\frac{\pi }{6},0\right)$	5.55031	5.28314	5.14604	0.150222	0.204957	0.241582	0.60851	0.83303	1.04652
$S{P}_p(\left(\frac{\pi }{12},0\right)$	5.81388	5.53491	5.39202	0.162969	0.224154	0.265337	0.687746	0.951552	1.20532
$S{P}_p(0,0)$	5.89629	5.61366	5.46897	0.167316	0.230675	0.273385	0.714196	0.990886	1.25782

Table 13. The $S{P}_p(\alpha ,0)$-radius of ${\mathcal{N}}_2({\displaystyle \frac{1}{2}},z)$.

	$\mathit{c}=0,\mathit{b}=1$			$\mathit{c}=0,\mathit{a}=1$			$\mathit{a}=0,\mathit{b}=1$
	$\mathit{a}=\mathbf{2}$	$\mathit{a}=\mathbf{3}$	$\mathit{a}=\mathbf{4}$	$\mathit{b}=\mathbf{2}$	$\mathit{b}=\mathbf{3}$	$\mathit{b}=\mathbf{4}$	$\mathit{c}=\mathbf{2}$	$\mathit{c}=\mathbf{3}$	$\mathit{c}=\mathbf{4}$
$S{P}_p(\pi /2.1,0)$	2.0151	1.96268	1.93474	0.25642	0.292449	0.313271	2.53313	0.50467	0.547181
$S{P}_p(\pi /3,0)$	2.06857	2.0168	1.98932	0.325085	0.371124	0.397745	2.56975	0.647586	0.703785
$S{P}_p(\pi /4,0)$	2.09264	2.04106	2.01373	0.350673	0.400499	0.429317	2.58695	0.702216	0.763971
$S{P}_p(\pi /6,0)$	2.11031	2.05883	2.03159	0.367976	0.42038	0.450695	2.59985	0.739628	0.805306
$S{P}_p(\pi /12,0)$	2.12108	2.06965	2.04246	0.377996	0.431899	0.463085	2.60784	0.761474	0.829489
$S{P}_p(0,0)$	2.12469	2.07328	2.0461	0.381277	0.435672	0.467144	2.60784	0.761474	0.829489

Table 14. The $S{P}_p(\alpha ,0)$-radius of ${\mathcal{N}}_3({\displaystyle \frac{1}{2}},z)$.

	$\mathit{c}=0,\mathit{b}=1$			$\mathit{c}=0,\mathit{a}=1$			$\mathit{a}=0,\mathit{b}=1$
	$\mathit{a}=\mathbf{2}$	$\mathit{a}=\mathbf{3}$	$\mathit{a}=\mathbf{4}$	$\mathit{b}=\mathbf{2}$	$\mathit{b}=\mathbf{3}$	$\mathit{b}=\mathbf{4}$	$\mathit{c}=\mathbf{2}$	$\mathit{c}=\mathbf{3}$	$\mathit{c}=\mathbf{4}$
$S{P}_p(\pi /2.1,0)$	2.04255	1.99051	1.96283	0.294134	0.335635	0.359625	0.518295	0.582544	0.632373
$S{P}_p(\pi /3,0)$	2.06857	2.0168	1.98932	0.325085	0.371124	0.397745	0.575314	0.647586	0.703785
$S{P}_p(\pi /4,0)$	2.08076	2.02909	2.00169	0.338358	0.386358	0.414116	0.599988	0.675823	0.73487
$S{P}_p(\pi /6,0)$	2.0899	2.0383	2.01096	0.347887	0.397298	0.425876	0.617785	0.696227	0.757364
$S{P}_p(\pi /12,0)$	2.09556	2.044	2.01669	0.353613	0.403875	0.432947	0.628515	0.708545	0.770957
$S{P}_p(0,0)$	2.09747	2.04593	2.01862	0.355523	0.406069	0.435306	0.632099	0.712662	0.775502

Table 15. The $S{P}_p(\alpha ,{\displaystyle \frac{1}{2}})$-radius of ${\mathcal{N}}_1({\displaystyle \frac{1}{2}},z)$.

	$\mathit{c}=0,\mathit{b}=1$			$\mathit{c}=0,\mathit{a}=1$			$\mathit{a}=0,\mathit{b}=1$
	$\mathit{a}=\mathbf{2}$	$\mathit{a}=\mathbf{3}$	$\mathit{a}=\mathbf{4}$	$\mathit{b}=\mathbf{2}$	$\mathit{b}=\mathbf{3}$	$\mathit{b}=\mathbf{4}$	$\mathit{c}=\mathbf{2}$	$\mathit{c}=\mathbf{3}$	$\mathit{c}=\mathbf{4}$
$S{P}_p(\pi /3.1,1/2)$	2.35501	2.19904	2.11492	0.0614616	0.0738296	0.0809027	0.123316	0.137964	0.148282
$S{P}_p(\pi /4,1/2)$	3.04112	2.87159	2.78146	0.0759593	0.0941503	0.105033	0.179218	0.208885	0.231265
$S{P}_p(\pi /6,1/2)$	3.71571	3.52561	3.42579	0.0912294	0.116294	0.131865	0.251511	0.30647	0.350994
$S{P}_p(\pi /12,1/2)$	4.13864	3.93276	3.82538	0.10188	0.132061	0.151209	0.309204	0.387962	0.454676
$S{P}_p(0,1/2)$	4.27899	4.06749	3.95741	0.10568	0.137733	0.158201	0.330987	0.419367	0.495314

Table 16. The $S{P}_p(\alpha ,{\displaystyle \frac{1}{2}})$-radius of ${\mathcal{N}}_2({\displaystyle \frac{1}{2}},z)$.

	$\mathit{c}=0,\mathit{b}=1$			$\mathit{c}=0,\mathit{a}=1$			$\mathit{a}=0,\mathit{b}=1$
	$\mathit{a}=\mathbf{2}$	$\mathit{a}=\mathbf{3}$	$\mathit{a}=\mathbf{4}$	$\mathit{b}=\mathbf{2}$	$\mathit{b}=\mathbf{3}$	$\mathit{b}=\mathbf{4}$	$\mathit{c}=\mathbf{2}$	$\mathit{c}=\mathbf{3}$	$\mathit{c}=\mathbf{4}$
$S{P}_p(\pi /3.1,1/2)$	2.00899	1.95648	1.92847	0.247093	0.281779	0.301822	0.432891	0.485624	0.52639
$S{P}_p(\pi /4,1/2)$	2.03237	1.98019	1.95242	0.280848	0.320414	0.343283	0.494026	0.554945	0.602145
$S{P}_p(\pi /6,1/2)$	2.05234	2.00041	1.97281	0.306237	0.349508	0.374524	0.54051	0.60785	0.660128
$S{P}_p(\pi /12,1/2)$	2.0645	2.01269	1.98518	0.320483	0.365844	0.392073	0.566792	0.637845	0.693075
$S{P}_p(0,1/2)$	2.06857	2.0168	1.98932	0.325085	0.371124	0.397745	0.575314	0.647586	0.703785

Table 17. The $S{P}_p(\alpha ,{\displaystyle \frac{1}{2}})$-radius of ${\mathcal{N}}_2({\displaystyle \frac{1}{2}},z)$.

	$\mathit{c}=0,\mathit{b}=1$			$\mathit{c}=0,\mathit{a}=1$			$\mathit{a}=0,\mathit{b}=1$
	$\mathit{a}=\mathbf{2}$	$\mathit{a}=\mathbf{3}$	$\mathit{a}=\mathbf{4}$	$\mathit{b}=\mathbf{2}$	$\mathit{b}=\mathbf{3}$	$\mathit{b}=\mathbf{4}$	$\mathit{c}=\mathbf{2}$	$\mathit{c}=\mathbf{3}$	$\mathit{c}=\mathbf{4}$
$S{P}_p(\pi /3.1,1/2)$	2.03967	1.98759	1.95988	0.290448	0.331412	0.355091	0.511551	0.574869	0.623963
$S{P}_p(\pi /4,1/2)$	2.0508	1.99885	1.97124	0.304374	0.347371	0.372229	0.537082	0.603943	0.65584
$S{P}_p(\pi /6,1/2)$	2.06052	2.00868	1.98114	0.315918	0.360608	0.386448	0.558353	0.628208	0.682483
$S{P}_p(\pi /12,1/2)$	2.06654	2.01475	1.98726	0.322799	0.368501	0.394928	0.571079	0.642745	0.698461
$S{P}_p(0,1/2)$	2.06857	2.0168	1.98932	0.325085	0.371124	0.397745	0.575314	0.647586	0.703785

Remark 2.

1. 

When the condition $\cos \alpha >\sigma$ is not satisfied, the roots either become negative or approach values close to zero, as shown in Table 6 and Table 7. This behavior aligns with the theoretical implications of Remark 1, since the function ${\varphi }_{.}$ is negative when $\cos \alpha \le \sigma$. Consequently, the radius loses its meaningful interpretation in such cases, reinforcing the necessity of the given condition.

2. 

For a fixed ν, the radius decreases as $(\cos \alpha -\sigma )$ approaches zero, demonstrating its strong dependence on this parameter. Even small variations in α and σ can lead to noticeable changes in the radius.

3.5. Functions Associated with the Normalized Struve Functions

The Struve function, denoted by ${\mathtt{S}}_{\nu }$, is a special function that is a solution of the non-homogeneous Bessel differential equations

$$z^2{y}^{″}\left(z\right)+z{y}^{\prime }\left(z\right)+(z^2-{\nu }^2)y\left(z\right)={\displaystyle \frac{4}{\sqrt{\pi }\mathsf{\Gamma }\left(\nu +\frac{3}{2}\right)}}{\left({\displaystyle \frac{z}{2}}\right)}^{\nu +1}.$$

If ${\mathtt{h}}_{\nu ,n}$ denotes the nth positive zero of ${\mathtt{S}}_{\nu }$, then (see [19]) for $\left|\nu \right|\le 1/2$, the function ${\mathtt{S}}_{\nu }$ can be expressed as

$$\begin{array}{c}\hfill {\mathtt{S}}_{\nu }\left(z\right)={\displaystyle \frac{z^{\nu +1}}{2^{\nu }\sqrt{\pi }\mathsf{\Gamma }\left(\nu +\frac{3}{2}\right)}}\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{{\mathtt{h}}_{\nu ,n}^2}}\right).\end{array}$$

(55)

It is noteworthy that ${\mathtt{h}}_{\nu ,n}>{\mathtt{h}}_{\nu ,1}>1$ for $\left|\nu \right|<1/2$. For further details, see [19,20].

From Equation (55), we can identify the following three normalized forms of the Struve functions:

$$\begin{array}{cc}\hfill {\left(\sqrt{\pi }{2}^{\nu }\mathsf{\Gamma }\left(\nu +{\displaystyle \frac{3}{2}}\right){\mathtt{S}}_{\nu }\left(z\right)\right)}^{{\displaystyle \frac{1}{ \nu +1}}}& =z\prod _{n=1}^{\infty }{\left(1-{\displaystyle \frac{z^2}{{\mathtt{h}}_{\nu ,n}^2}}\right)}^{{\displaystyle \frac{1}{\nu +1}}};\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill \sqrt{\pi }{2}^{\nu }\mathsf{\Gamma }\left(\nu +{\displaystyle \frac{3}{2}}\right)z^{1-\nu }{\mathtt{S}}_{\nu }\left(z\right)& =z\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{{\mathtt{h}}_{\nu ,n}^2}}\right);\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill \sqrt{\pi }{2}^{\nu }\mathsf{\Gamma }\left(\nu +{\displaystyle \frac{3}{2}}\right)z^{{\displaystyle \frac{1-\nu }{2}}}{\mathtt{S}}_{\nu }\left(\sqrt{z}\right)& =z\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z}{{\mathtt{h}}_{\nu ,n}^2}}\right).\hfill \end{array}$$

(58)

Furthermore, this representation leads to

Example 7.

For $\left|\nu \right|\le 1/2$,

(i) 

${\mathcal{S}}_1(\nu ,z)=\sqrt{\pi }{2}^{\nu }\mathsf{\Gamma }\left(\nu +{\displaystyle \frac{3}{2}}\right)z^{{\displaystyle \frac{1-\nu }{2}}}{\mathtt{S}}_{\nu }\left(\sqrt{z}\right)\in {\mathbb{H}}_1,$ with $a_n\left(\nu \right)={\mathtt{h}}_{\nu ,n}$;

(ii) 

${\mathcal{S}}_2(\nu ,z)=\sqrt{\pi }{2}^{\nu }\mathsf{\Gamma }\left(\nu +{\displaystyle \frac{3}{2}}\right)z^{1-\nu }{\mathtt{S}}_{\nu }\left(z\right)\in {\mathbb{H}}_2$ with $b_n\left(\nu \right)={\mathtt{h}}_{\nu ,n}$;

(iii) 

${\mathcal{S}}_3(\nu ,z)={\left(\sqrt{\pi }{2}^{\nu }\mathsf{\Gamma }\left(\nu +{\displaystyle \frac{3}{2}}\right){\mathtt{S}}_{\nu }\left(z\right)\right)}^{{\displaystyle \frac{1}{ \nu +1}}}\in {\mathbb{H}}_3$ with $d_n\left(\nu \right)={\mathtt{h}}_{\nu ,n}$ and $\mu \left(\nu \right):=1/(\nu +1)$.

We have the following results from Section 2 and Section 3.

Theorem 4.

Suppose that for some $\left|\alpha \right|<\pi /2$ and $\sigma \in [0,1)$, the inequality $\cos \alpha >\sigma$ holds. Then,

(i) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{S}}_1(\nu ,z)$ is the smallest positive root of the equation

$$\sqrt{r}{\mathtt{S}}_{\nu }^{\prime }\left(\sqrt{r}\right)+(\cos \left(\alpha \right)-1-\nu -\sigma ){\mathtt{S}}_{\nu }\left(\sqrt{r}\right)=0,$$

in $(0,h_1^2\left(\nu \right))$.

(ii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{S}}_2(\nu ,z)$ is the smallest positive root of the equation

$$2r{\mathtt{S}}_{\nu }^{\prime }\left(r\right)+(\cos \left(\alpha \right)-2\nu -\sigma ){\mathtt{S}}_{\nu }\left(r\right)=0,$$

in $(0,h_1\left(\nu \right))$.

(iii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{S}}_3(\nu ,z)$ is the smallest positive root of the equation

$$2r{\mathtt{S}}_{\nu }^{\prime }\left(r\right)+(\nu +1)(\cos \left(\alpha \right)-2-\sigma ){\mathtt{S}}_{\nu }\left(r\right)=0$$

in $(0,h_1\left(\nu \right))$.

For $\nu =1/2$, it follows that

$$\begin{array}{cc}\hfill {\mathcal{S}}_1(1/2,z)& =2\left(1-\cos \left(\sqrt{z}\right)\right),{\mathcal{S}}_2(1/2,z)=2(1-\cos \left(z\right)),\mathrm{and}\hfill \\ \hfill {\mathcal{S}}_3(1/2,z)& =2^{2/3}{\left({\displaystyle \frac{1-\cos \left(z\right)}{\sqrt{z}}}\right)}^{2/3}.\hfill \end{array}$$

Furthermore, for $\nu =-1/2$, we have

$$\begin{array}{cc}\hfill {\mathcal{S}}_1(-1/2,z)& =\sqrt{z}sin\left(\sqrt{z}\right),{\mathcal{S}}_2(-1/2,z)=zsin\left(z\right),\mathrm{and}\hfill \\ \hfill {\mathcal{S}}_3(-1/2,z)& ={\displaystyle \frac{{sin}^2\left(z\right)}{z}}.\hfill \end{array}$$

These, together with Theorem 4, lead to the following two corollaries:

Corollary 4.

Suppose that for some $\left|\alpha \right|<\pi /2$ and $\sigma \in [0,1)$, the inequality $\cos \alpha >\sigma$ holds. Then,

(i) 

The $S{P}_p(\alpha ,\sigma )$-radius of $2\left(1-\cos \left(\sqrt{z}\right)\right)$ is the smallest positive root of the equation

$$\left(\cos \left(\sqrt{r}\right)-1\right)\left(-4\sqrt{r}\cos \left(\alpha \right)+\sqrt{r}(4\sigma +6)+1\right)+2\sqrt{r}sin\left(\sqrt{r}\right)=0,$$

in $(0,4{\pi }^2)$.

(ii) 

The $S{P}_p(\alpha ,\sigma )$-radius of $2(1-\cos (z\left)\right)$ is the smallest positive root of the equation

$$\left(\cos \right(r)-1)(-\cos (\alpha )+\sigma +2)+2rsin\left(r\right)=0,$$

in $(0,2\pi )$.

(iii) 

The $S{P}_p(\alpha ,\sigma )$-radius of $2^{2/3}{\left({\displaystyle \frac{1-\cos \left(z\right)}{\sqrt{z}}}\right)}^{2/3}$ is the smallest positive root of the equation

$$\left(\cos \right(r)-1)(-3\cos (\alpha )+3\sigma +8)+4rsin\left(r\right)=0,$$

in $(0,2\pi )$.

(iv) 

The $S{P}_p(\alpha ,\sigma )$-radius of $\sqrt{z}sin\left(\sqrt{z}\right)$ is the smallest positive root of the equation

$$2\sqrt{r}\left(sin\left(\sqrt{r}\right)(2\cos \left(\alpha \right)-2\sigma -1)+\cos \left(\sqrt{r}\right)\right)-sin\left(\sqrt{r}\right)=0,$$

in $(0,4{\pi }^2)$.

(v) 

The $S{P}_p(\alpha ,\sigma )$-radius of $zsin\left(z\right)$ is the smallest positive root of the equation

$$sin\left(r\right)\left(\cos \right(\alpha )-\sigma )+2r\cos \left(r\right)=0,$$

in $(0,2\pi )$.

(vi) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${sin}^2\left(z\right)/z$ is the smallest positive root of the equation

$$sin\left(r\right)\left(\cos \right(\alpha )-\sigma -4)+4r\cos \left(r\right)=0,$$

in $(0,2\pi )$.

3.6. Functions Associated with Wright Functions

Wright [21] introduced the following function related to the asymptotic theory of partitions:

$$\begin{array}{c}\hfill \varphi (\rho ,\beta ,z):=\sum _{n=0}^{\infty }{\displaystyle \frac{z^n}{n!\mathsf{\Gamma }(n\rho +\delta )}}\end{array}$$

(59)

for $\rho >0$ and $\delta ,z\in \mathbb{C}$. The Wright function is valid for $\rho >-1$ and is an entire function of z in this region. For more details, see [21,22]. Its geometric properties are discussed in [23,24,25,26,27,28,29,30]. We denote the nth positive zero of ${\lambda }_{\rho ,\delta }\left(z\right)=\varphi (\rho ,\delta ,-z^2)$ by ${\lambda }_{\rho ,\delta ,n}$, and the nth positive zero of ${\mathsf{\Psi }}_{\rho ,\delta }^{\prime }$, where $\mathsf{\Psi }\left(z\right)=z^{\delta }\varphi (\rho ,\delta ,-z^2)$ by ${\eta }_{\rho ,\delta ,n}^{\prime }$.

For $\rho >0$ and $\delta >0$, it has been proven in [24] that the function $z\mapsto {\lambda }_{\rho ,\delta }=\varphi (\rho ,\delta ,-z^2)$ possesses infinitely many real zeros. Additionally, it has the Weierstrass decomposition given by

$$\begin{array}{c}\hfill \mathsf{\Gamma }\left(\delta \right)\varphi (\rho ,\delta ,-z^2)=\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{{\lambda }_{\rho ,\delta ,n}^2}}\right).\end{array}$$

(60)

The product converges uniformly on a compact subset of the complex plane. The positive zeros of ${\lambda }_{\rho ,\delta }$ interlace with the zeros of ${\mathsf{\Psi }}_{\rho ,\delta }$, resulting in the following inequalities:

$${\eta }_{\rho ,\delta ,1}^{\prime }<{\lambda }_{\rho ,\delta ,1}<{\eta }_{\rho ,\delta ,2}^{\prime }<{\lambda }_{\rho ,\delta ,2}<\dots .$$

We have the following example related to $\varphi (\rho ,\delta ,.)$.

Example 8

(The normalized Wright function). For $\rho ,\delta >0$, denote ${\lambda }_{\rho ,\delta ,n}$ as the n-th zero of the function ${\lambda }_{\rho ,\delta }\left(z\right)$. Then,

$$\begin{array}{cc}\hfill {\mathcal{W}}_1(\rho ,\delta ,z)& =z\mathsf{\Gamma }\left(\delta \right){\lambda }_{\rho ,\delta }\left(\sqrt{z}\right)=z\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z}{{\lambda }_{\rho ,\delta ,n}^2}}\right)\in {\mathbb{H}}_1;\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {\mathcal{W}}_2(\rho ,\delta ,z)& =z\mathsf{\Gamma }\left(\delta \right){\lambda }_{\rho ,\delta }\left(z\right)=z\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{{\lambda }_{\rho ,\delta ,n}^2}}\right)\in {\mathbb{H}}_2;\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill {\mathcal{W}}_3(\rho ,\delta ,z)& ={\left(z^{\delta }\mathsf{\Gamma }\left(\delta \right){\lambda }_{\rho ,\delta }\left(z\right)\right)}^{{\displaystyle \frac{1}{ \delta }}}=z\prod _{n=1}^{\infty }{\left(1-{\displaystyle \frac{z^2}{{\lambda }_{\rho ,\delta ,n}^2}}\right)}^{\frac{1}{\delta }}\in {\mathbb{H}}_3.\hfill \end{array}$$

(63)

We will now outline the radius problem for the Wright function.

Theorem 5.

Suppose that for some $\left|\alpha \right|<\pi /2$ and $\sigma \in [0,1)$, the inequality $\cos \alpha >\sigma$ holds. Then,

(i) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{W}}_1(\rho ,\delta ,z)$ is the smallest positive root of the equation

$$\sqrt{r}{\lambda }_{\rho ,\delta }^{\prime }\left(\sqrt{r}\right)+(\cos \left(\alpha \right)-\sigma ){\lambda }_{\rho ,\delta }\left(\sqrt{r}\right)=0$$

in $(0,{\lambda }_{\rho ,\delta ,1}^2)$.

(ii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{W}}_2(\rho ,\delta ,z)$ is the smallest positive root of the equation

$$2r{\lambda }_{\rho ,\delta }^{\prime }\left(r\right)+(\cos \left(\alpha \right)-\sigma ){\lambda }_{\rho ,\delta }\left(r\right)=0$$

in $(0,{\lambda }_{\rho ,\delta ,1})$.

(iii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{W}}_3(\rho ,\delta ,z)$ is the smallest positive root of the equation

$$2r{\lambda }_{\rho ,\delta }^{\prime }\left(r\right)+\delta (\cos \left(\alpha \right)-\sigma ){\lambda }_{\rho ,\delta }\left(r\right)=0$$

in $(0,{\lambda }_{\rho ,\delta ,1})$.

The Wright function generalizes a transformation of the classical Bessel function of the first kind of $\nu$. This relationship can be expressed as follows:

$${\lambda }_{1,1+\nu }\left(z\right)=\varphi (1,1+\nu ,-z^2)=z^{-\nu }{J}_{\nu }\left(2z\right).$$

We now have the following special cases of Theorem 5:

Corollary 5.

Suppose that for some $\left|\alpha \right|<\pi /2$ and $\sigma \in [0,1)$, the inequality $\cos \alpha >\sigma$ holds. Let $\nu >-1$. Then,

(i) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{W}}_1(1,1+\nu ,z)=\mathsf{\Gamma }(\nu +1)z^{1-{\displaystyle \frac{\nu }{2}}}{J}_{\nu }\left(2\sqrt{z}\right)$ is the smallest positive root of the equation

$$J_{\nu }\left(2\sqrt{r}\right)\left(2\sqrt{r}\cos \left(\alpha \right)-2\sqrt{r}\sigma -\nu \right)+\sqrt{r}{J}_{\nu -1}\left(2\sqrt{r}\right)-\sqrt{r}{J}_{\nu +1}\left(2\sqrt{r}\right)=0$$

in $(0,{\lambda }_{1,1+\nu ,1}^2)$.

(ii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{W}}_2(1,1+\nu ,z)=\mathsf{\Gamma }(\nu +1)z^{1-\nu }{J}_{\nu }\left(2z\right)$ is the smallest positive root of the equation

$$J_{\nu }\left(2r\right)(\cos \left(\alpha \right)-2\nu -\sigma )+2r{J}_{\nu -1}\left(2r\right)-2r{J}_{\nu +1}\left(2r\right)=0$$

in $(0,{\lambda }_{1,1+\nu ,1})$.

(iii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${\mathcal{W}}_3(1,1+\nu ,z)={\left(\mathsf{\Gamma }(\nu +1)z{J}_{\nu }\left(2z\right)\right)}^{ 1/1+\nu }$ is the smallest positive root of the equation

$$((\nu +1)\cos \left(\alpha \right)-\nu (\sigma +2)-\sigma )J_{\nu }\left(2r\right)+2r{J}_{\nu -1}\left(2r\right)-2r{J}_{\nu +1}\left(2r\right))=0$$

in $(0,{\lambda }_{1,1+\nu ,1})$.

We note that Corollary 5 can be simplified further as a radius problem involving trigonometric functions by selecting $\nu =-1/2$ and $\nu =1/2$, for example.

1.

Corollary 5(i) leads to $S{P}_p(\alpha ,\sigma )$-radius of $z\cos \left(2\sqrt{z}\right)$ and ${\displaystyle \frac{1}{2}}\sqrt{z}sin\left(2\sqrt{z}\right)$.

2.

Corollary 5(ii) leads to $S{P}_p(\alpha ,\sigma )$-radius of $z\cos \left(2z\right)$ and ${\displaystyle \frac{1}{2}}sin\left(2z\right)$.

3.

Corollary 5(iii) leads to $S{P}_p(\alpha ,\sigma )$-radius of $z{\cos }^2\left(2z\right)$ and ${\displaystyle \frac{{\left(\sqrt{z}sin\left(2z\right)\right)}^{2/3}}{2^{2/3}}}$.

Therefore, the careful selection of $\alpha$, $\sigma$, $\rho$, and $\delta$ leads to various special cases, and numerical values of the $S{P}_p(\alpha ,\sigma )$-radius can be obtained, as demonstrated in earlier examples. We omit those calculation for this example.

3.7. Functions Involving q-Bessel Functions

This section addresses the radius problem for Jackson and Hahn–Exton q-Bessel functions, ${\mathtt{J}}_{\nu }^{\left(2\right)}(z;q)$ and ${\mathtt{J}}_{\nu }^{\left(3\right)}(z;q)$. For $z\in \mathbb{C}$, $\nu >-1$, and $q\in (0,1)$, both functions are defined by series expansions as follows:

$$\begin{array}{cc}\hfill {\mathtt{J}}_{\nu }^{\left(2\right)}(z;q)& :={\displaystyle \frac{{\left(q^{\nu +1};q\right)}_{\infty }}{{\left(q;q\right)}_{\infty }}}\sum _{n\ge 0}{\displaystyle \frac{{(-1)}^n{\left(\frac{z}{2}\right)}^{2n+\nu }}{{\left(q;q\right)}_{\infty }{\left(q^{\nu +1};q\right)}_n}}{q}^{n(n+\nu )}\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill {\mathtt{J}}_{\nu }^{\left(3\right)}(z;q)& :={\displaystyle \frac{{\left(q^{\nu +1};q\right)}_{\infty }}{{\left(q;q\right)}_{\infty }}}\sum _{n\ge 0}{\displaystyle \frac{{(-1)}^n{z}^{2n+\nu }}{{\left(q;q\right)}_{\infty }{\left(q^{\nu +1};q\right)}_n}}{q}^{{\displaystyle \frac{n(n+1)}{2}}}.\hfill \end{array}$$

(65)

Here, we define the q-Pochhammer symbol as follows:

$$\begin{array}{cc}\hfill {(a;q)}_0& =1,\hfill \\ \hfill {(a;q)}_n& =\prod _{k=1}^n\left(1-a{q}^{k-1}\right),\hfill \\ \hfill {(a;q)}_{\infty }& =\prod _{k\ge 1}\left(1-a{q}^{k-1}\right).\hfill \end{array}$$

For a fixed z and as q approaches 1, the q-Bessel functions relate to the classical Bessel function $J_{\nu }$ as follows: ${\mathtt{J}}_{\nu }^{\left(2\right)}((1-z)q;q)\to J_{\nu }\left(z\right)$ and ${\mathtt{J}}_{\nu }^{\left(3\right)}((1-z)q;q)\to J_{\nu }\left(2z\right)$.

The q-extension of Bessel functions has been explored by several authors, notably, [31,32,33,34,35,36], among others. Additionally, the geometric properties of q-Bessel functions have been discussed in [13,37,38,39,40].

Now, let us recall the Hadamard factorization for the normalized q-Bessel functions.

$$\begin{array}{c}\hfill z\to {\mathcal{J}}_{\nu }^{\left(2\right)}(z;q)=2^{\nu }{c}_{\nu }\left(q\right)z^{-\nu }{\mathtt{J}}_{\nu }^{\left(2\right)}(z;q)\mathrm{and}z\to {\mathcal{J}}_{\nu }^{\left(3\right)}(z;q)=c_{\nu }\left(q\right)z^{-\nu }{\mathtt{J}}_{\nu }^{\left(3\right)}(z;q),\end{array}$$

where $c_{\nu }\left(q\right)={\left(q;q\right)}_{\infty }/{\left(q^{\nu +1};q\right)}_{\infty }.$

Lemma 2

([37]). For $\nu >-1$, the functions $z\to {\mathcal{J}}_{\nu }^{\left(2\right)}(z;q)$ and $z\to {\mathcal{J}}_{\nu }^{\left(3\right)}(z;q)$ are entire functions of order zero and pose the Hadamard factorization for the form 

$$\begin{array}{c}\hfill {\mathcal{J}}_{\nu }^{\left(2\right)}(z;q)=\prod _{n\ge 1}\left(1-{\displaystyle \frac{z^2}{j_{\nu ,n}^2\left(q\right)}}\right),{\mathcal{J}}_{\nu }^{\left(3\right)}(z;q)=\prod _{n\ge 1}\left(1-{\displaystyle \frac{z^2}{l_{\nu ,n}^2\left(q\right)}}\right)\end{array}$$

(66)

where $j_{\nu ,n}\left(q\right)$ and $l_{\nu ,n}\left(q\right)$ represent the $n^{th}$ positive zeros of the functions ${\mathcal{J}}_{\nu }^{\left(2\right)}(.;q)$ and ${\mathcal{J}}_{\nu }^{\left(3\right)}(.;q)$, respectively.

Now, we are ready to set our example for the ${\mathbb{H}}_1$, ${\mathbb{H}}_2$, and ${\mathbb{H}}_3$ classes involving q-Bessel functions.

Example 9.

For $\nu >-1$, $q\in (0,1)$, denote $j_{\nu ,n}\left(q\right)$ as the n-th zero of the q-Bessel functions ${\mathtt{J}}_{\nu }^{\left(2\right)}(z;q)$. Then,

$$\begin{array}{cc}\hfill {}_{2}f_{\nu ,q}\left(z\right)& =z{\mathcal{J}}_{\nu }^{\left(2\right)}(\sqrt{z};q)=z\prod _{n\ge 1}\left(1-{\displaystyle \frac{z}{j_{\nu ,n}^2\left(q\right)}}\right)\in {\mathbb{H}}_1;\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill {}_{2}g_{\nu ,q}\left(z\right)& =z{\mathcal{J}}_{\nu }^{\left(2\right)}(z;q)=z\prod _{n\ge 1}\left(1-{\displaystyle \frac{z^2}{j_{\nu ,n}^2\left(q\right)}}\right)\in {\mathbb{H}}_2;\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill {}_{2}h_{\nu ,q}\left(z\right)& ={\left(z^{\nu }{\mathcal{J}}_{\nu }^{\left(2\right)}(z;q)\right)}^{\frac{1}{\nu }}=z\prod _{n\ge 1}{\left(1-{\displaystyle \frac{z^2}{j_{\nu ,n}^2\left(q\right)}}\right)}^{\frac{1}{\nu }}\in {\mathbb{H}}_3.\hfill \end{array}$$

(69)

Example 10.

For $\nu >-1$, $q\in (0,1)$, denote $l_{\nu ,n}\left(q\right)$ as the n-th zero of the q-Bessel functions ${\mathtt{J}}_{\nu }^{\left(3\right)}(z;q)$. Then,

$$\begin{array}{cc}\hfill {}_{3}f_{\nu ,q}\left(z\right)& =z{\mathcal{J}}_{\nu }^{\left(3\right)}(\sqrt{z};q)=z\prod _{n\ge 1}\left(1-{\displaystyle \frac{z}{l_{\nu ,n}^2\left(q\right)}}\right)\in {\mathbb{H}}_1;\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill {}_{3}g_{\nu ,q}\left(z\right)& =z{\mathcal{J}}_{\nu }^{\left(3\right)}(z;q)=z\prod _{n\ge 1}\left(1-{\displaystyle \frac{z^2}{l_{\nu ,n}^2\left(q\right)}}\right)\in {\mathbb{H}}_2;\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill {}_{3}h_{\nu ,q}\left(z\right)& ={\left(z^{\nu }{\mathcal{J}}_{\nu }^{\left(3\right)}(z;q)\right)}^{\frac{1}{\nu }}=z\prod _{n\ge 1}{\left(1-{\displaystyle \frac{z^2}{l_{\nu ,n}^2\left(q\right)}}\right)}^{\frac{1}{\nu }}\in {\mathbb{H}}_3.\hfill \end{array}$$

(72)

Theorem 6.

Suppose that for some $\left|\alpha \right|<\pi /2$ and $\sigma \in [0,1)$, the inequality $\cos \alpha >\sigma$ holds. Let $\nu >-1$. Then, for $s=\{2,3\}$, we have following results

(i) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${}_{s}f_{\nu ,q}$ is the smallest positive root of the equation

$$\begin{array}{c}\hfill \sqrt{r}{\displaystyle \frac{d}{dr}}\left({\mathtt{J}}_{\nu }^{\left(s\right)}(\sqrt{r};q)\right)+\left(\cos \left(\alpha \right)-\sigma \right){\mathtt{J}}_{\nu }^{\left(s\right)}(r;q)=0\end{array}$$

(ii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${}_{s}g_{\nu ,q}$ is the smallest positive root of the equation

$$\begin{array}{c}\hfill r{\displaystyle \frac{d}{dr}}\left({\mathtt{J}}_{\nu }^{\left(s\right)}(r;q)\right)+\left(\cos \left(\alpha \right)-\sigma \right){\mathtt{J}}_{\nu }^{\left(s\right)}(r;q)=0\end{array}$$

(iii) 

The $S{P}_p(\alpha ,\sigma )$-radius of ${}_{s}h_{\nu ,q}$ is the smallest positive root of the equation

$$\begin{array}{c}\hfill r{\displaystyle \frac{d}{dr}}\left({\mathtt{J}}_{\nu }^{\left(s\right)}(r;q)\right)-\nu \left(\cos \left(\alpha \right)+\sigma \right){\mathtt{J}}_{\nu }^{\left(s\right)}(r;q)=0\end{array}$$

3.8. Outline of Few More Examples

The aforementioned examples make it clear that those functions have an infinite product representation, which can be further normalized to include them in class ${\mathbb{H}}_i$, where $i=1,2,3$. Theorem 1 can therefore be used to determine the $S{P}_p(\alpha ,\sigma )$-radius for those functions. We will provide a few more functions in this section of the article without providing the radius outcomes because the statements are similar to those in the other instances above.

(I)

Ramanujan-type entire function: The Ramanujan entire function and the Stieltjes–Wigert polynomial, respectively (see [41]), denoted by $A_q\left(z\right)$ and $S_n(z;q)$, are defined by power series as

$$A_q\left(z\right)=\sum _{n\ge 0}{\displaystyle \frac{{(-1)}^n{q}^{n^2}}{{(q;q)}_n}}{z}^n\mathrm{and}{S}_n(z{q}^{{\displaystyle \frac{1}{2}}-n};q)={\displaystyle \frac{1}{{(q;q)}_n}}\sum _{k\ge 0}{\displaystyle \frac{{(q^{-n};q)}_k{q}^{k^2}}{{(q;q)}_k}}{z}^k.$$

A Ramanujan-type entire function was defined and examined by Ismail and Zhang in [42], as follows:

$$\begin{array}{c}\hfill A_q^{\left(\alpha \right)}(a;z)=\sum _{n\ge 0}{\displaystyle \frac{{(a;q)}_n{q}^{\alpha n^2}}{{(q;q)}_n}}{z}^n,z\in \mathbb{C},\end{array}$$

(73)

where $\alpha >0$, $0<q<1$, $a\in \mathbb{C}$, and

$${(a;q)}_0=1,{(a;q)}_k=\prod _{j=0}^{k-1}(1-a{q}^j),k\ge 1.$$

It can be observed that $A_q^{\left(\alpha \right)}(a;z)$ generalizes both $A_q\left(z\right)$ and $S_n(z;q)$.

For $a\ge 0$, $\alpha >0$, and $0<q<1$, Zhang [43] proved that $A_q^{\left(\alpha \right)}(-a;z)$ has an infinite number of negative zeros, and that the growth order of the entire function $A_q^{\left(\alpha \right)}(a;z)$ is zero for $\alpha >0$.

In order to establish the inclusion relation of $A_q^{\left(\alpha \right)}(a;z)$ in the class ${\mathbb{H}}_i$, $i=1,2,3$, we need the following lemma:

Lemma 3

([44]). If $\alpha >0$, $a\ge 0$, and $0<q<1$, then the function

$$z\mapsto {\mathsf{\Psi }}_{\alpha ,q}(a;z):=A_q^{\left(\alpha \right)}(-a;-z^2)$$

has infinitely many zeros, all of which are positive. Denoting by ${\psi }_{\alpha ,q,n}\left(a\right)$ the n-th positive zero of ${\mathsf{\Psi }}_{\alpha ,q}(a;z)$, we obtain the Weierstrassian decomposition as follows:

$$\begin{array}{c}\hfill {\mathsf{\Psi }}_{\alpha ,q}(a;z)=\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{z^2}{{\psi }_{\alpha ,q,n}^2\left(a\right)}}\right),\end{array}$$

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which is valid, and this product is uniformly convergent on compact subsets of the complex plane.

Moreover, if we denote by ${\xi }_{\alpha ,q,n}\left(a\right)$ the n-th positive zero of ${\mathsf{\Phi }}_{\alpha ,q}(a;z)$, where

$${\mathsf{\Phi }}_{\alpha ,q}(a;z)=z^{\alpha }{\mathsf{\Psi }}_{\alpha ,q}(a;z),$$

then, the positive zeros of ${\mathsf{\Psi }}_{\alpha ,q}(a;z)$ are interlaced with those of ${\mathsf{\Phi }}_{\alpha ,q}(a;z)$. In other words, the zeros satisfy the following chain of inequalities:

$${\xi }_{\alpha ,q,1}\left(a\right)<{\psi }_{\alpha ,q,1}\left(a\right)<{\xi }_{\alpha ,q,2}\left(a\right)<{\psi }_{\alpha ,q,2}\left(a\right)<{\xi }_{\alpha ,q,3}\left(a\right)<\dots$$

Now, by taking the normalization of the function $A_q^{\left(\alpha \right)}(a;z)$, we have the following example:

Example 11.

For $a\ge 0$, $\alpha >0$, and $0<q<1$,

$$\begin{array}{cc}\hfill {{\mathtt{R}}^{\mathtt{f}}}_{\alpha ,q}(a;z)& =z{A}_q^{\left(\alpha \right)}(-a;-z)\in {\mathbb{H}}_1;\hfill \end{array}$$

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$$\begin{array}{cc}\hfill {{\mathtt{R}}^{\mathtt{g}}}_{\alpha ,q}(a;z)& =z{A}_q^{\left(\alpha \right)}(-a;-z^2)\in {\mathbb{H}}_2;\hfill \end{array}$$

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$$\begin{array}{cc}\hfill {{\mathtt{R}}^{\mathtt{h}}}_{\alpha ,q}(a;z)& ={\left(z^{\alpha }{A}_q^{\left(\alpha \right)}(-a;-z^2)\right)}^{1/\alpha }\in {\mathbb{H}}_3.\hfill \end{array}$$

(77)

(II)

Cross product of the Bessel and modified functions: It is shown in [45] that for $\nu >-1$ and $z\in \mathbb{C}$, the cross product of the classical Bessel and modified Bessel function ${\mathcal{W}}_{\nu }\left(z\right)=J_{\nu +1}\left(z\right)I_{\nu }\left(z\right)+J_{\nu }\left(z\right)I_{\nu +1}\left(z\right)$ exhibit the power series

$${\mathcal{W}}_{\nu }\left(z\right)=2\sum _{n\ge 0}{\displaystyle \frac{{(-1)}^n{z}^{2\nu +4n+1}}{n!\mathsf{\Gamma }(\nu +n+1)\mathsf{\Gamma }(\nu +2n+2)2^{2\nu +4n}}} ,$$

and Hadamard factorization

$$\begin{array}{c}\hfill 2^{2\nu }{z}^{-2\nu -1}\mathsf{\Gamma }(\nu +1)\mathsf{\Gamma }(\nu +2){\mathcal{W}}_{\nu }\left(z\right)=\prod _{n\ge 1}\left(1-{\displaystyle \frac{z^4}{{\gamma }_{\nu ,n}^4}}\right).\end{array}$$

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Here, ${\gamma }_{\nu ,n}$ represents the n-th positive zero of the function ${\mathcal{W}}_{\nu }\left(z\right)$. It is also shown that the zeros ${\gamma }_{\nu ,n}$ satisfy the interlacing inequalities

$$j_{\nu ,n}<{\gamma }_{\nu ,n}<j_{\nu ,n+1}\mathrm{and}{j}_{\nu ,n}<{\gamma }_{\nu ,n}<j_{\nu +1,n},$$

for $n\in \mathbb{N}$ and $\nu >-1$, where $j_{\nu ,n}$ stands for the n-th positive zero of the Bessel function $J_{\nu }$. Clearly, ${\mathcal{W}}_{\nu }\left(z\right)$ is not normalized as per our requirements. However, there are three normalized forms given in [12]. We modified two of them to make them suitable for our cases and we present the following example:

Example 12.

For $\nu >-1$, consider the following normalizations

$$\begin{array}{cc}\hfill f_{\nu }^{\mathtt{BCP}}\left(z\right)& =2^{2\nu }{z}^{-{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{3}{4}}}\mathsf{\Gamma }(\nu +1)\mathsf{\Gamma }(\nu +2){\mathcal{W}}_{\nu }\left(\sqrt[4]{z}\right),\hfill \end{array}$$

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$$\begin{array}{cc}\hfill g_{\nu }^{\mathtt{BCP}}\left(z\right)& =2^{2\nu }{z}^{-\nu +{\displaystyle \frac{1}{2}}}\mathsf{\Gamma }(\nu +1)\mathsf{\Gamma }(\nu +2){\mathcal{W}}_{\nu }\left(\sqrt{z}\right),\hfill \end{array}$$

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$$\begin{array}{cc}\hfill h_{\nu }^{\mathtt{BCP}}\left(z\right)& ={\left(2^{2\nu }{z}^{-\nu +{\displaystyle \frac{1}{2}}}\mathsf{\Gamma }(\nu +1)\mathsf{\Gamma }(\nu +2){\mathcal{W}}_{\nu }\left(\sqrt{z}\right)\right) }^{{\displaystyle \frac{1}{2\nu +1}}},\nu \ne -{\displaystyle \frac{1}{2}}.\hfill \end{array}$$

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It can be observed that $f_{\nu }^{\mathtt{BCP}}\left(z\right)\in {\mathbb{H}}_1$, $g_{\nu }^{\mathtt{BCP}}\left(z\right)\in {\mathbb{H}}_2$, and $h_{\nu }^{\mathtt{BCP}}\left(z\right)\in {\mathbb{H}}_3$.

Thus, the $S{P}_p(\alpha ,\sigma )$-radius for $f_{\nu }^{\mathtt{BCP}}\left(z\right)$, $g_{\nu }^{\mathtt{BCP}}\left(z\right)$, and $h_{\nu }^{\mathtt{BCP}}\left(z\right)$ can be determined using Theorem 1.

(III)

Mittag-Leffler functions: Mittag-Leffler [46] introduced the following function:

$$\mathcal{ML}(\beta ,z):=\sum _{n\ge 0}{\displaystyle \frac{z^n}{\mathsf{\Gamma }(\beta n+1)}},$$

which is generalized by Wiman [47] by adding a parameter $\nu$ and defined as

$$\mathcal{ML}(\beta ,\gamma ,z):=\sum _{n\ge 0}{\displaystyle \frac{z^n}{\mathsf{\Gamma }(\beta n+\gamma )}},$$

In 1971, Prabhakar [48] introduced the following generalization:

$$\mathcal{ML}(\beta ,\gamma ,c,z):=\sum _{n\ge 0}{\displaystyle \frac{{\left(a\right)}_n{z}^n}{n!\mathsf{\Gamma }(\beta n+\gamma )}},$$

where $\beta ,\gamma ,c>0$. The function $\mathcal{ML}(\beta ,\gamma ,c,z)$ is known as the generalized Mittag-Leffler function.

The notion of the nature of the zeros of $\mathcal{ML}(\beta ,\gamma ,z)$ is studied in [49], and it described by the following lemma.

Lemma 4

([49]). Let there exist three transformations mapping the set $(1/\beta ,\gamma ):0<\beta <2,\gamma >0$ into itself as follows:

$$A:\left(\frac{1}{\beta },\gamma \right)\to \left({\displaystyle \frac{\beta }{2}},\gamma \right);$$

$$B:\left(\frac{1}{\beta },\gamma \right)\to \left({\displaystyle \frac{\beta }{2}},\gamma +\beta \right);$$

$$C:\left(\frac{1}{\beta },\gamma \right)\to \left\{\begin{array}{cc}\left(\frac{1}{\beta },\gamma -1\right),\hfill & \mathit{for}\gamma >1;\hfill \\ \left(\frac{1}{\beta },\gamma -1\right)\hfill & 0<\gamma \le 1.\hfill \end{array}\right.$$

Define the set

$$W_a=\left\{\left(\frac{1}{\beta },\gamma \right):1<\beta <2,\gamma \in (\beta -1,1)\cup (\beta ,2)\right\},$$

and

$$W_b=A\left(W_a\right)\cup B\left(W_a\right).$$

Denote by $W_i$ the least set containing $W_b$ and the invariant with respect to $A,B,$ and C. Then, the set $W_i$ can be represented by

$$W_i=\bigcup A^{k_{1,1}}{B}^{k_{1,2}}{C}^{k_{1,3}}\cdots A^{k_{n,1}}{B}^{k_{n,2}}{C}^{k_{n,3}}\left(W_b\right),$$

where the union has taken over all $n=1,2,3,\dots$ and over all $3n$-tuples

$$(k_{1,1},k_{1,2},k_{1,3};\dots ;k_{n,1},k_{n,2},k_{n,3})$$

of non-negative integers. Then, for $\left(\frac{1}{\beta },\gamma \right)\in W_i$, all zeros of $\mathcal{ML}(\beta ,\gamma ,z)$ are negative and simple.

It is further established that if $\left({\displaystyle \frac{1}{\beta }},\gamma \right)\in W_i$ and $a>0$, then all zeros of $\mathcal{ML}(\beta ,\gamma ,a,z)$ are real and negative. From [50], we see that if $\left({\displaystyle \frac{1}{\beta }},\gamma \right)\in W_i$ and $a>0$, then the function $\mathcal{ML}(\beta ,\gamma ,a,-z^2)$ has infinitely many zeros, which are all real and can be represented as follows:

$$\begin{array}{c}\hfill \mathsf{\Gamma }\left(\gamma \right)\mathcal{ML}(\beta ,\gamma ,a,-z^2)=\prod _{n\ge 1}\left(1-{\displaystyle \frac{z^2}{{\lambda }_{\beta ,\gamma ,a,n}^2}}\right),\end{array}$$

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where ${\lambda }_{\beta ,\gamma ,a,n}$ is the n-th positive zero of $\mathcal{ML}(\beta ,\gamma ,a,-z^2)$. Now, we finally have three normalizations of $\mathcal{ML}(\beta ,\gamma ,a,-z^2)$ which are members of ${\mathbb{H}}_i$.

Example 13.

For $\left({\displaystyle \frac{1}{\beta }},\gamma \right)\in W_i$ and $a>0$,

$$\begin{array}{cc}\hfill {\mathcal{ML}}_f(\beta ,\gamma ,a,z)& =z\mathsf{\Gamma }\left(\gamma \right)\mathcal{ML}(\beta ,\gamma ,a,-z)\in {\mathbb{H}}_1;\hfill \end{array}$$

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$$\begin{array}{cc}\hfill {\mathcal{ML}}_g(\beta ,\gamma ,a,z)& =z\mathsf{\Gamma }\left(\gamma \right)\mathcal{ML}(\beta ,\gamma ,a,-z^2)\in {\mathbb{H}}_2;\hfill \end{array}$$

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$$\begin{array}{cc}\hfill {\mathcal{ML}}_g(\beta ,\gamma ,a,z)& ={\left(z^{\gamma }\mathsf{\Gamma }\left(\gamma \right)\mathcal{ML}(\beta ,\gamma ,a,-z^2)\right)}^{1/\gamma }\in {\mathbb{H}}_3.\hfill \end{array}$$

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4. Conclusions

In this article, we focus on three classes of functions represented by a convergent infinite product of factors involving the positive zeros of the function. Through the examples provided, we demonstrate that functions with a Hadamard factorization can belong to the classes ${\mathbb{H}}_i$, $i=1,2,3$. In such cases, the $S{P}_p(\alpha ,\sigma )$-radius can be deduced from Theorem 1.

It is worth noting that most of the examples, except Example 2 and Example 3, are not entirely new. These normalized functions have been explored in previous studies in different contexts, such as the radius of starlikeness, convexity, and parabolic starlikeness. However, the present study focuses specifically on the radius of spirallikeness, with particular emphasis on the spirallikeness of order $\cos \left(\alpha \right)/2$.

Additionally, functions such as the Lommel, Dini, q-Struve-Bessel, and products of the Bessel and modified Bessel functions are within the scope of this study. However, since the nature of the radius problem closely resembles the other examples presented, they have not been explicitly included in the discussion.