Certain Geometric Investigations of Three Normalized Bessel-Type Functions of a Complex Variable

Abstract

We recall the normalized forms for the three Bessel-type functions; these functions are the Bessel function, Lommel function, and Struve function of the first kind. By using convolution, we define normalized forms. The essential purpose is to introduce necessary and sufficient bounds of these normalized functions so these functions are starlike and convex of order $\gamma$ and type $\delta$.

1. Introduction

We use the symbol $\Omega$ to refer to the family of all functions that are analytic and univalent, with the normalization ${\varphi }^{\prime }\left(0\right)-1=0=\varphi \left(0\right)$; that is $\varphi$ is expressed as

$$\varphi \left(\xi \right)=\xi +\sum _{n=2}^{\infty }{c}_n{\xi }^n,$$

(1)

and the domain is the open unit disk $\Delta :=\{\xi :\xi \in \mathbb{C},|\xi |<1\}$. Also, let $T\subset \Omega$, consisting of functions with negative coefficients, which have the form

$$\varphi \left(\xi \right)=\xi -\sum _{n=2}^{\infty }{c}_n{\xi }^n(c_n\ge 0).$$

(2)

The Hadamard product (convolution) of ${\varphi }_1,{\varphi }_2\in \Omega$ will also be used; for this purpose it is defined by

$$\left({\varphi }_1\ast {\varphi }_2\right)\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{a}_n{b}_n{\xi }^n=\left({\varphi }_2\ast {\varphi }_1\right)\left(\xi \right).$$

(3)

Definition 1. 

For $0\le \gamma <1$ and $0<\delta \le 1$, let $S^{*}(\gamma ,\delta )\subset \Omega$ be the set of all functions ϕ (ϕ has the form (1)) satisfying the inequality

$$\left|{\displaystyle \genfrac{}{}{1.0pt}{}{{\displaystyle \frac{\xi {\varphi }^{\prime }\left(\xi \right)}{\varphi \left(\xi \right)}}-1}{{\displaystyle \frac{\xi {\varphi }^{\prime }\left(\xi \right)}{\varphi \left(\xi \right)}}+\left(1-2\gamma \right)}}\right|<\delta ;\xi \in \Delta .$$

(4)

Also, take $K(\gamma ,\delta )\subset \Omega$ as the set of all functions ϕ (ϕ has the form (1)) that satisfy the inequality

$$\left|{\displaystyle \genfrac{}{}{1.0pt}{}{{\displaystyle \frac{\xi {\varphi }^{″}\left(\xi \right)}{{\varphi }^{\prime }\left(\xi \right)}}}{{\displaystyle \frac{\xi {\varphi }^{″}\left(\xi \right)}{{\varphi }^{\prime }\left(\xi \right)}}+2\left(1-\gamma \right)}}\right|<\delta ,\left(\xi \in \Delta \right).$$

(5)

$K(\gamma ,\delta )$ and $S^{*}(\gamma ,\delta )$ are the well-known subclasses of convex and starlike functions of order γ and type δ, respectively, introduced in [1].

• It is recognized that

$$\varphi \in K(\gamma ,\delta )⟺\tilde{\varphi }\in S^{*}(\gamma ,\delta ),\tilde{\varphi }\left(\xi \right)=\xi {\varphi }^{\prime }\left(\xi \right).$$

(6)

Moreover, let $T^{*}(\gamma ,\delta )$ and $C(\gamma ,\delta )$ be two subclasses of T defined by

$$T^{*}(\gamma ,\delta )=S^{*}(\gamma ,\delta )\cap T\mathrm{and}C(\gamma ,\delta )=K(\gamma ,\delta )\cap T.$$

We observe that $K(\gamma ,1)=K\left(\gamma \right)$ and $S^{*}(\gamma ,1)=S^{*}\left(\gamma \right)$, which are the subclasses of convex and starlike functions of order $\gamma (0\le \gamma <1),$ respectively, which were explored by Robertson in [2]; see also Schild [3] and MacGregor [4]. Moreover, $K(0,1)=K$ and $S^{*}(0,1)=S^{*}$, see [5], which are the subclasses of convex and starlike functions. Additionally, $C(\gamma ,1)=C\left(\gamma \right)$ and $T^{*}(\gamma ,1)=T^{*}\left(\gamma \right)$, which are the subclasses of convex and starlike functions of order $\gamma$ with negative coefficients established by Silverman [6]. To determine the desired results, we need some additional lemmas, which are listed below.

Lemma 1. 

For function ϕ defined by (1), the inequality

$$\sum _{n=2}^{\infty }\left[n-1+\delta \left(n+1-2\gamma \right)\right]\left|c_n\right|\le 2\delta \left(1-\gamma \right),$$

(7)

suffices to ensure that $\varphi \in S^{*}(\gamma ,\delta )$.

Proof. 

It is sufficient to show that

$$\left|{\displaystyle \frac{\xi {\varphi }^{\prime }\left(\xi \right)-\varphi \left(\xi \right)}{\xi {\varphi }^{\prime }\left(\xi \right)+\left(1-2\gamma \right)\varphi \left(\xi \right)}}\right|<\delta ;\xi \in \Delta .$$

$$\begin{array}{ccc}\hfill \left|{\displaystyle \frac{\xi {\varphi }^{\prime }\left(\xi \right)-\varphi \left(\xi \right)}{\xi {\varphi }^{\prime }\left(\xi \right)+\left(1-2\gamma \right)\varphi \left(\xi \right)}}\right|& =& \left|{\displaystyle \frac{{\displaystyle \sum _{n=2}^{\infty }}(n-1)c_n{\xi }^n}{2(1-\gamma )\xi +{\displaystyle \sum _{n=2}^{\infty }}(n+1-2\gamma )c_n{\xi }^n}}\right|\hfill \\ & \le & {\displaystyle \frac{{\displaystyle \sum _{n=2}^{\infty }}(n-1)\left|c_n\right|{\left|\xi \right|}^n}{2(1-\gamma )\left|\xi \right|-{\displaystyle \sum _{n=2}^{\infty }}(n+1-2\gamma )\left|c_n\right|{\left|\xi \right|}^n}}\hfill \\ & \le & {\displaystyle \frac{{\displaystyle \sum _{n=2}^{\infty }}(n-1)\left|c_n\right|}{2(1-\gamma )-{\displaystyle \sum _{n=2}^{\infty }}(n+1-2\gamma )\left|c_n\right|}}.\hfill \end{array}$$

The last expression is bounded above by δ if the inequality (7) holds true, and hence the proof is completed. □

Lemma 2 ([1]). 

For ϕ given in (2), the condition

$$\sum _{n=2}^{\infty }\left[n-1+\delta \left(n+1-2\gamma \right)\right]c_n\le 2\delta \left(1-\gamma \right),$$

(8)

is necessary and sufficient to say that $\varphi \in T^{*}(\gamma ,\delta )$.

Lemma 3. 

For function ϕ defined by (1), the inequality

$$\sum _{n=2}^{\infty }n\left[n-1+\delta \left(n+1-2\gamma \right)\right]\left|c_n\right|\le 2\delta \left(1-\gamma \right).$$

(9)

suffices to ensure that $\varphi \in K(\gamma ,\delta )$.

Proof. 

Since the proof is a direct consequence of Lemma 1 and the equivalence relation 6, we omit the details. □

Lemma 4 ([1]). 

For function ϕ given in (2), the condition

$$\sum _{n=2}^{\infty }n\left[n-1+\delta \left(n+1-2\gamma \right)\right]c_n\le 2\delta \left(1-\gamma \right).$$

(10)

is necessary and sufficient to obtain that $\varphi \in C(\gamma ,\delta )$.

Recently, many researchers have considered specific categories of analytic functions, $\mathcal{F}\subset \Omega$, that include particular functions. to determine the circumstances under which the members of $\mathcal{F}$ exhibit certain geometric features, such as convexity, univalence, or starlikeness in $\Delta$. In this area, many studies about generalized hypergeometric functions are accessible in the literature [7,8]. In the field of geometric theory of functions of a complex variable, special functions like Bessel, Struve, and Lommel functions of the first kind have drawn a lot of attention. Motivated by certain previous advances, the geometric features of these special functions were recently studied. See [9,10,11,12,13], in which the univalence and starlikeness of Bessel functions of the first kind were taken into consideration. The radii of univalence, starlikeness, and convexity for the normalized forms of Bessel, Struve, and Lommel functions of the first kind have been determined in recent years; for example see [14,15,16,17,18,19,20,21,22,23,24].

Our study aims to provide some new conclusions about the convexity and starlikeness of the first-kind normalized Bessel, Struve, and Lommel functions. Now, we recall the first-kind Bessel function ${\mathbb{J}}_p$, the first-kind Struve function ${\mathbb{H}}_p,$ and first-kind Lommel function ${\mathbb{S}}_{p,q}$ as follows, respectively.

$${\mathbb{J}}_p\left(\xi \right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-1)}^n}{\Gamma \left(n+1\right)\Gamma \left(n+p+1\right)}}{\left({\displaystyle \frac{\xi }{2}}\right)}^{2n+p}$$

(11)

$$\left(\xi ,p\in \mathbb{C};p\notin {\mathbb{Z}}^{-}:=\left\{-1,-2,\dots \right\}\right),$$

$${\mathbb{H}}_p\left(\xi \right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-1)}^n}{\Gamma \left(n+\frac{3}{2}\right)\Gamma \left(n+p+\frac{3}{2}\right)}}{\left({\displaystyle \frac{\xi }{2}}\right)}^{2n+p+1}\left(p\in \mathbb{C};p+{\displaystyle \frac{1}{2}}\notin {\mathbb{Z}}^{-}\right),$$

(12)

and

$${\mathbb{S}}_{p,q}\left(\xi \right)={\displaystyle \frac{{\xi }^{q+1}}{4}}\sum _{n=0}^{\infty }{\displaystyle \frac{{(-1)}^n\Gamma \left(\frac{q-p+1}{2}\right)\Gamma \left(\frac{q+p+1}{2}\right)}{\Gamma \left(n+\frac{q-p+3}{2}\right)\Gamma \left(n+\frac{q+p+3}{2}\right)}}{\left({\displaystyle \frac{\xi }{2}}\right)}^{2n}\left(p,q\in \mathbb{C};{\displaystyle \frac{q\pm p+1}{2}}\notin {\mathbb{Z}}^{-}\right).$$

(13)

Furthermore, we are aware that (see [25] (p. 217) and [26]) the homogeneous Bessel differential equation

$${\xi }^{2}w″\left(\xi \right)+\xi w^{\prime }\left(\xi \right)+\left({\xi }^2-p^2\right)w\left(\xi \right)=0,$$

has a solution in the form of the Bessel function ${\mathbb{J}}_p\left(\xi \right).$

• And the Struve function ${\mathbb{H}}_p\left(\xi \right)$ is a particular solution of

$${\xi }^{2}w″\left(\xi \right)+\xi w^{\prime }\left(\xi \right)+\left({\xi }^2-p^2\right)w\left(\xi \right)={\displaystyle \frac{{\left(\frac{\xi }{2}\right)}^{p-1}}{\sqrt{\pi }\Gamma \left(p+\frac{1}{2}\right)}}.$$

Also, the Lommel function ${\mathbb{S}}_{p,q}\left(\xi \right)$ is a particular solution of

$${\xi }^{2}w″\left(\xi \right)+\xi w^{\prime }\left(\xi \right)+\left({\xi }^2-p^2\right)w\left(\xi \right)={\xi }^{q+1}.$$

It is important to note that the hypergeometric function ${}_{1}F_2$ is used to explicitly define the functions ${\mathbb{J}}_p\left(\xi \right),{\mathbb{H}}_p\left(\xi \right)$, and ${\mathbb{S}}_{p,q}\left(\xi \right)$ as follows:

$${\mathbb{J}}_p\left(\xi \right)={\displaystyle \frac{{\left(\frac{\xi }{2}\right)}^p}{\Gamma \left(p+1\right)}}\begin{array}{c}{}_{1}F_2\left(1;1,p+1;-{\displaystyle \frac{{\xi }^2}{4}}\right)\end{array};p\notin {\mathbb{Z}}^{-},$$

(14)

$${\mathbb{H}}_p\left(\xi \right)={\displaystyle \frac{{\left(\frac{\xi }{2}\right)}^{p+1}}{\sqrt{\frac{\pi }{4}}\Gamma \left(p+\frac{3}{2}\right)}}\begin{array}{c}{}_{1}F_2\left(1;{\displaystyle \frac{3}{2}},p+{\displaystyle \frac{3}{2}};-{\displaystyle \frac{{\xi }^2}{4}}\right)\end{array};p+\frac{1}{2}\notin {\mathbb{Z}}^{-},$$

(15)

and

$${\mathbb{S}}_{p,q}\left(\xi \right)={\displaystyle \frac{{\xi }^{q+1}}{\left(q-p+1\right)\left(q+p+1\right)}}\begin{array}{c}{}_{1}F_2\left(1;{\displaystyle \frac{q-p+3}{2}},{\displaystyle \frac{q+p+3}{2}};-{\displaystyle \frac{{\xi }^2}{4}}\right)\end{array};\frac{q\pm p+1}{2}\notin {\mathbb{Z}}^{-}.$$

(16)

Our study is mainly interested in the normalized forms, which are as follows:

$${\mathbf{J}}_p\left(\xi \right):=\Gamma \left(p+1\right){\xi }^{1-{\displaystyle \frac{p}{2}}}{\mathbb{J}}_p\left(2\sqrt{\xi }\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-1)}^n}{{\left(1\right)}_n{\left(p+1\right)}_n}}{\xi }^{n+1};p\notin {\mathbb{Z}}^{-},$$

(17)

$${\mathbf{H}}_p\left(\xi \right):=\Gamma \left(\frac{3}{2}\right)\Gamma \left(p+\frac{3}{2}\right){\xi }^{1-{\displaystyle \frac{p}{2}}}{\mathbb{H}}_p\left(2\sqrt{\xi }\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-1)}^n}{{\left(\frac{3}{2}\right)}_n{\left(p+\frac{3}{2}\right)}_n}}{\xi }^{n+1};p+\frac{1}{2}\notin {\mathbb{Z}}^{-},$$

(18)

and

$${\mathbf{S}}_{p,q}\left(\xi \right):=\left(q-p+1\right)\left(q+p+1\right){\xi }^{-q}{\mathbb{S}}_{p,q}\left(2\sqrt{\xi }\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-1)}^n}{{\left(\frac{q-p+3}{2}\right)}_n{\left(\frac{q+p+3}{2}\right)}_n}}{\xi }^{n+1};\frac{q\pm p+1}{2}\notin {\mathbb{Z}}^{-}.$$

(19)

Note that

$${\mathbf{S}}_{p,p}\left(\xi \right)={\mathbf{H}}_p\left(\xi \right)\mathrm{and}{\mathbf{S}}_{p,p-1}\left(\xi \right)={\mathbf{J}}_p\left(\xi \right).$$

(20)

This work is significant as it discusses geometric characteristics of several of the most popular special functions in mathematics (Lommel, Struve, and Bessel), such as convexity and starlikeness. We use convolution techniques for a class of univalent analytic functions to show these geometric features. After introducing the basic theorems and the required and sufficient criteria, we provide several interesting instances. We have provided figure calculations using the Maple tool.

We provide necessary and sufficient circumstances under which the first-kind normalized Lommel ${\mathbf{S}}_{p,q}$, first-kind normalized Struve ${\mathbf{H}}_p$, and first-kind normalized Bessel functions ${\mathbf{J}}_p$ fall into certain families of analytic functions.

2. Results Regarding Lommel Functions

For the sake of simplicity or brevity, we shall use that $0\le \gamma <1,$ $0<\delta \le 1,$ and $\xi \in \Delta$ throughout this article until otherwise noted. If

$${\varphi }_0\left(\xi \right)={\displaystyle \frac{\xi }{1+\xi }}={\displaystyle \sum _{n=0}^{\infty }}{\left(-1\right)}^n{\xi }^{n+1}\left(\xi \in \Delta \right),$$

(21)

and applying convolution, then we introduce ${\mathcal{S}}_{p,q}\left(\xi \right)$, which is defined by

$${\mathcal{S}}_{p,q}\left(\xi \right):={\mathbf{S}}_{p,q}\left(\xi \right)\ast {\varphi }_0\left(\xi \right)=\sum _{n=0}^{\infty }{\displaystyle \frac{1}{{\left(\frac{q-p+3}{2}\right)}_n{\left(\frac{q+p+3}{2}\right)}_n}}{\xi }^{n+1}\left(\frac{q\pm p+1}{2}\notin {\mathbb{Z}}^{-}\right),$$

(22)

which is a modified normalized Lommel-type function. Actually, we have defined this form by convolution so that the function ${\mathcal{S}}_{p,q}\in \Omega$, for which the first investigation is introduced.

Theorem 1. 

If $q\ne \pm p-3$, then the inequality

$$\left(1+\delta \right){\mathcal{S}}_{p,q+2}^{\prime }\left(1\right)+2\delta \left(1-\gamma \right){\mathcal{S}}_{p,q+2}\left(1\right)\le {\displaystyle \frac{8\delta (1-\gamma )}{\left(q-p+3\right)\left(q+p+3\right)}},$$

(23)

suffices to ensure that ${\mathbf{S}}_{p,q}\left(\xi \right)\in S^{*}(\gamma ,\delta )$, where ${\mathbf{S}}_{p,q}$ and ${\mathcal{S}}_{p,q}$ are defined in (19) and (22), respectively. Additionally, ${\mathcal{S}}_{p,q}\left(1\right)$ refers to the sum of the series in (22) evaluated at $\xi =1$.

Proof. 

Here

$${\mathbf{S}}_{p,q}\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{\displaystyle \frac{{(-1)}^{n-1}}{{\left(\frac{q-p+3}{2}\right)}_{n-1}{\left(\frac{q+p+3}{2}\right)}_{n-1}}}{\xi }^n;\frac{q\pm p+1}{2}\notin {\mathbb{Z}}^{-}.$$

By using Lemma 1, it is sufficient to give that

$$\sum _{n=2}^{\infty }\left[n-1+\delta \left(n+1-2\gamma \right)\right]{\displaystyle \frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n-1}{\left(\frac{q+p+3}{2}\right)}_{n-1}}}\le 2\delta (1-\gamma ).$$

We have

$$\begin{array}{cc}\hfill & \mathcal{L}(\gamma ,\delta ;p,q)\begin{array}{c}:=\end{array}\sum _{n=2}^{\infty }\left[n-1+\delta \left(n+1-2\gamma \right)\right]{\displaystyle \frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n-1}{\left(\frac{q+p+3}{2}\right)}_{n-1}}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\left[n+1+\delta \left(n+3-2\gamma \right)\right]{\displaystyle \frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n+1}{\left(\frac{q+p+3}{2}\right)}_{n+1}}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\left[\left(n+1\right)\left(1+\delta \right)+2\delta \left(1-\gamma \right)\right]{\displaystyle \frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n+1}{\left(\frac{q+p+3}{2}\right)}_{n+1}}}\hfill \\ \hfill & =\left(1+\delta \right)\sum _{n=0}^{\infty }\left(n+1\right)\frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n+1}{\left(\frac{q+p+3}{2}\right)}_{n+1}}+2\delta \left(1-\gamma \right)\sum _{n=0}^{\infty }\frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n+1}{\left(\frac{q+p+3}{2}\right)}_{n+1}}\hfill \\ \hfill & ={\displaystyle \frac{\left(1+\delta \right)}{\left(\frac{q-p+3}{2}\right)\left(\frac{q+p+3}{2}\right)}}\sum _{n=0}^{\infty }\left(n+1\right){\displaystyle \frac{1}{{\left(\frac{q-p+5}{2}\right)}_n{\left(\frac{q+p+5}{2}\right)}_n}}\hfill \\ \hfill & +{\displaystyle \frac{2\delta \left(1-\gamma \right)}{\left(\frac{q-p+3}{2}\right)\left(\frac{q+p+3}{2}\right)}}\sum _{n=0}^{\infty }{\displaystyle \frac{1}{{\left(\frac{q-p+5}{2}\right)}_n{\left(\frac{q+p+5}{2}\right)}_n}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{\left(1+\delta \right)}{\left(\frac{q-p+3}{2}\right)\left(\frac{q+p+3}{2}\right)}}\sum _{n=0}^{\infty }\left(n+1\right){\displaystyle \frac{1}{{\left(\frac{q+2-p+3}{2}\right)}_n{\left(\frac{q+2+p+3}{2}\right)}_n}}\hfill \\ \hfill & +{\displaystyle \frac{2\delta \left(1-\gamma \right)}{\left(\frac{q-p+3}{2}\right)\left(\frac{q+p+3}{2}\right)}}\sum _{n=0}^{\infty }{\displaystyle \frac{1}{{\left(\frac{q+2-p+3}{2}\right)}_n{\left(\frac{q+2+p+3}{2}\right)}_n}}\hfill \\ \hfill & ={\displaystyle \frac{\left(1+\delta \right)}{\left(\frac{q-p+3}{2}\right)\left(\frac{q+p+3}{2}\right)}}\sum _{n=0}^{\infty }\left(n+1\right){\displaystyle \frac{1}{{\left(\frac{q+2-p+3}{2}\right)}_n{\left(\frac{q+2+p+3}{2}\right)}_n}}\hfill \\ \hfill & +{\displaystyle \frac{2\delta \left(1-\gamma \right)}{\left(\frac{q-p+3}{2}\right)\left(\frac{q+p+3}{2}\right)}}\sum _{n=0}^{\infty }{\displaystyle \frac{1}{{\left(\frac{q+2-p+3}{2}\right)}_n{\left(\frac{q+2+p+3}{2}\right)}_n}}.\hfill \end{array}$$

Then

$$\mathcal{L}(\gamma ,\delta ;p,q)=\left(\frac{q-p+3}{2}\right)\left(\frac{q+p+3}{2}\right)\left[\left(1+\delta \right){\mathcal{S}}_{p,q+2}^{\prime }\left(1\right)+2\delta \left(1-\gamma \right){\mathcal{S}}_{p,q+2}\left(1\right)\right].$$

(24)

$\mathcal{L}(\gamma ,\delta ;p,q)$ is bounded above by $2\delta (1-\gamma )$ if inequality (23) is satisfied. This completes the proof. □

The following result introduce a necessary and sufficient condition. For this purpose let

$${\varphi }_1\left(\xi \right):=\xi \left(2-{\displaystyle \frac{1}{1+\xi }}\right)=\xi +\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{\xi }^{n+1}\left(\xi \in \Delta \right),$$

(25)

Also, by applying convolution, the function ${\mathfrak{S}}_{p,q}$ is given by

$${\mathfrak{S}}_{p,q}\left(\xi \right):={\mathbf{S}}_{p,q}\left(\xi \right)\ast {\varphi }_1\left(\xi \right).$$

(26)

Indeed, we have defined this form by convolution so that the function ${\mathfrak{S}}_{p,q}\in T$, where T is the subclass of functions with negative real coefficients defined in (2).

Theorem 2. 

If $q\ne \pm p-3$, then inequality (23) is the necessary and sufficient condition to have ${\mathfrak{S}}_{p,q}\left(\xi \right)\in T^{*}(\gamma ,\delta )$.

Proof. 

Simply, we can verify that

$${\mathfrak{S}}_{p,q}\left(\xi \right)=\xi -\sum _{n=2}^{\infty }{\displaystyle \frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n-1}{\left(\frac{q+p+3}{2}\right)}_{n-1}}}{\xi }^n;\frac{q\pm p+1}{2}\notin {\mathbb{Z}}^{-}.$$

(27)

using Lemma 1 in conjunction with the methods described in Theorem 1. We have already provided the proof of Theorem 2. □

Now, we introduce the convex results.

Theorem 3. 

If $q\ne \pm p-3$, then the inequality

$$\left(1+\delta \right){\mathcal{S}}_{p,q+2}^{″}\left(1\right)+2\left(1+\delta \right){\mathcal{S}}_{p,q+2}^{\prime }\left(1\right)+2\left(1-\gamma \right)\left(2\delta -1\right){\mathcal{S}}_{p,q+2}\left(1\right)\le {\displaystyle \frac{8\delta (1-\gamma )}{\left(q-p+3\right)\left(q+p+3\right)}},$$

(28)

suffices to ensure that ${\mathbf{S}}_{p,q}\left(\xi \right)\in K(\gamma ,\delta )$, where ${\mathbf{S}}_{p,q}$ and ${\mathcal{S}}_{p,q}$ are defined in (19) and (22), respectively.

Proof. 

Lemma 4 tells us that the condition

$$\sum _{n=2}^{\infty }n\left[n-1+\delta \left(n+1-2\gamma \right)\right]{\displaystyle \frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n-1}{\left(\frac{q+p+3}{2}\right)}_{n-1}}}\le 2\delta (1-\gamma ),$$

suffices to yield the required result. We have

$$\begin{array}{cc}\hfill & \mathcal{F}(\gamma ,\delta ;p,q)\begin{array}{c}:=\end{array}\sum _{n=2}^{\infty }n\left[n-1+\delta \left(n+1-2\gamma \right)\right]{\displaystyle \frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n-1}{\left(\frac{q+p+3}{2}\right)}_{n-1}}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\left(n+2\right)\left[n+1+\delta \left(n+1+2\left(1-\gamma \right)\right)\right]{\displaystyle \frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n+1}{\left(\frac{q+p+3}{2}\right)}_{n+1}}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\left[\begin{array}{c}\left(1+\delta \right)\left(n+1\right)\left(n\right)\\ +2\left(2+\delta -\gamma \right)\left(n+1\right)+2\left(1-\gamma \right)\left(2\delta -1\right)\end{array}\right]{\displaystyle \frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n+1}{\left(\frac{q+p+3}{2}\right)}_{n+1}}}\hfill \\ \hfill & =\left(1+\delta \right)\sum _{n=0}^{\infty }{\displaystyle \frac{\left(n+1\right)\left(n\right)}{{\left(\frac{q-p+3}{2}\right)}_{n+1}{\left(\frac{q+p+3}{2}\right)}_{n+1}}}+2\left(2+\delta -\gamma \right)\sum _{n=0}^{\infty }{\displaystyle \frac{n+1}{{\left(\frac{q-p+3}{2}\right)}_{n+1}{\left(\frac{q+p+3}{2}\right)}_{n+1}}}\hfill \\ \hfill & +2\left(1-\gamma \right)\left(2\delta -1\right)\sum _{n=0}^{\infty }{\displaystyle \frac{1}{{\left(\frac{q-p+3}{2}\right)}_{n+1}{\left(\frac{q+p+3}{2}\right)}_{n+1}}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{1}{\left(\frac{q-p+3}{2}\right)\left(\frac{q+p+3}{2}\right)}}\left[\left(1+\delta \right)\sum _{n=0}^{\infty }{\displaystyle \frac{\left(n+1\right)\left(n\right)}{{\left(\frac{q-p+5}{2}\right)}_n{\left(\frac{q+p+5}{2}\right)}_n}}\right.\hfill \\ \hfill & \left.+2\left(2+\delta -\gamma \right)\sum _{n=0}^{\infty }{\displaystyle \frac{n+1}{{\left(\frac{q-p+5}{2}\right)}_n{\left(\frac{q+p+5}{2}\right)}_n}}+2\left(1-\gamma \right)\left(2\delta -1\right)\sum _{n=0}^{\infty }\frac{1}{{\left(\frac{q-p+5}{2}\right)}_n{\left(\frac{q+p+5}{2}\right)}_k}\right],\hfill \end{array}$$

or

$$\begin{array}{cc}\hfill & \mathcal{F}(\gamma ,\delta ;p,q)\hfill \\ \hfill & ={\displaystyle \frac{1}{\left(\frac{q-p+3}{2}\right)\left(\frac{q+p+3}{2}\right)}}\left[\left(1+\delta \right){\mathcal{S}}_{p,q+2}^{″}\left(1\right)+2\left(1+\delta \right){\mathcal{S}}_{p,q+2}^{\prime }\left(1\right)+2\left(1-\gamma \right)\left(2\delta -1\right){\mathcal{S}}_{p,q+2}\left(1\right)\right].\hfill \end{array}$$

(29)

$\mathcal{F}(\gamma ,\delta ;p,q)$ in (29) is bounded above by $2\delta (1-\gamma )$ if (28) is satisfied. The proof is achieved. □

Similarly, we give the necessary and sufficient theorem for convexity.

Theorem 4. 

If $q\ne \pm p-3$, then the inequality given by (28) is the necessary and sufficient condition for ${\mathfrak{S}}_{p,q}\left(\xi \right)\in C(\gamma ,\delta )$.

Corollary 1. 

The function ${\mathfrak{S}}_{p,q}\left(\xi \right)\in T^{*}\left(\gamma \right)$ if and only if

$${\mathcal{S}}_{p,q+2}^{\prime }\left(1\right)+\left(1-\gamma \right){\mathcal{S}}_{p,q+2}\left(1\right)\le {\displaystyle \frac{4(1-\gamma )}{\left(q-p+3\right)\left(q+p+3\right)}}.$$

(30)

Corollary 2. 

The function ${\mathfrak{S}}_{p,q}\left(\xi \right)\in C\left(\gamma \right)$ if and only if

$${\mathcal{S}}_{p,q+2}^{″}\left(1\right)+2{\mathcal{S}}_{p,q+2}^{\prime }\left(1\right)+\left(1-\gamma \right){\mathcal{S}}_{p,q+2}\left(1\right)\le {\displaystyle \frac{4(1-\gamma )}{\left(q-p+3\right)\left(q+p+3\right)}}.$$

(31)

3. Results Regarding Struve Functions

Putting $q=p$ in Theorems 1–4, we obtain the matching Struve function ${\mathbf{H}}_p$ results, which are as follows.

Theorem 5. 

If $p>-{\displaystyle \frac{1}{2}}$, then the condition

$$\left(1+\delta \right){\mathcal{H}}_{p+2}^{\prime }\left(1\right)+2\delta \left(1-\gamma \right){\mathcal{H}}_{p+2}\left(1\right)\le {\displaystyle \frac{8\delta (1-\gamma )}{3\left(2p+3\right)}},$$

(32)

suffices to ensure that ${\mathbf{H}}_p\left(\xi \right)\in S^{*}(\gamma ,\delta )$, where ${\mathbf{H}}_p$ is defined in (18) and

$${\mathcal{H}}_p\left(\xi \right)\begin{array}{c}:=\end{array}{\mathbf{H}}_p\left(\xi \right)\ast {\varphi }_0\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{\displaystyle \frac{1}{{\left(\frac{3}{2}\right)}_{n-1}{\left(p+\frac{3}{2}\right)}_{n-1}}}{\xi }^n\left(p>-{\displaystyle \frac{1}{2}}\right).$$

(33)

Taking $\gamma =0$ and $\delta =1$ in Theorem 5 leads us to the following illustrative example.

Example 1. 

If ${\mathcal{H}}_{p+2}^{\prime }\left(1\right)+{\mathcal{H}}_{p+2}\left(1\right)\le {\displaystyle \frac{4}{3(2p+3)}}$, then the function ${\mathbf{H}}_p\in S^{*}$; i.e., ${\mathbf{H}}_p$ is a starlike function. We list the following cases for different values of $p>-1.5$ as follows:

• Choosing $p=-0.9$, we notice that

  $$2.89\simeq {\mathcal{H}}_{-0.9+2}^{\prime }\left(1\right)+{\mathcal{H}}_{-0.9+2}\left(1\right)\nleqq {\displaystyle \frac{4}{3\left(2\right(-0.9)+3)}}\simeq 1.11,$$

  because the function ${\mathbf{H}}_{-0.9}$ is not a starlike function; see Figure 1a below.
• Choosing $p=6$, we get that

  $$-3={\mathcal{H}}_{6+2}^{\prime }\left(1\right)+{\mathcal{H}}_{6+2}\left(1\right)<{\displaystyle \frac{4}{3\left(2\right(6)+3)}}\simeq 6.67,$$

  which suffices to ensure that ${\mathbf{H}}_6$ is a starlike function; see Figure 1b below.

Theorem 6. 

If $p>-{\displaystyle \frac{3}{2}}$, then the inequality (32) is necessary and sufficient to have ${\mathfrak{H}}_p\left(\xi \right)\in T^{*}(\gamma ,\delta )$ such that

$${\mathfrak{H}}_p\left(\xi \right)\begin{array}{c}:=\end{array}{\mathbf{H}}_p\left(\xi \right)\ast {\varphi }_1\left(\xi \right)=\xi -\sum _{n=2}^{\infty }{\displaystyle \frac{1}{{\left(\frac{3}{2}\right)}_{n-1}{\left(p+\frac{3}{2}\right)}_{n-1}}}{\xi }^n\left(p>-{\displaystyle \frac{1}{2}}\right).$$

(34)

Theorem 7. 

If $p>-{\displaystyle \frac{3}{2}}$, then the condition

$$\left(1+\delta \right){\mathcal{H}}_{p+2}^{″}\left(1\right)+2\left(1+\delta \right){\mathcal{H}}_{p+2}^{\prime }\left(1\right)+2\left(1-\gamma \right)\left(2\delta -1\right){\mathcal{H}}_{p+2}\left(1\right)\le {\displaystyle \frac{8\delta (1-\gamma )}{3\left(2p+3\right)}},$$

(35)

suffices to ensure that ${\mathbf{H}}_p\left(\xi \right)\in K(\gamma ,\delta ).$

Taking $\gamma =0$ and $\delta =1$ in Theorem 7 leads us to following illustrative example.

Example 2. 

If ${\mathcal{H}}_{p+2}^{″}\left(1\right)+2{\mathcal{H}}_{p+2}^{\prime }\left(1\right){\mathcal{H}}_{p+2}\left(1\right)\le {\displaystyle \frac{4}{3(2p+3)}}$, then the function ${\mathbf{H}}_p\in K$; i.e., ${\mathbf{H}}_p$ is a convex function. We list the following cases for different values of $p>-1.5$ as follows:

• Choosing $p=-0.5$, we notice that

  $$4.87\simeq {\mathcal{H}}_{-0.5+2}^{″}\left(1\right)+2{\mathcal{H}}_{-0.5+2}^{\prime }\left(1\right){\mathcal{H}}_{-0.5+2}\left(1\right)\nleqq {\displaystyle \frac{4}{3\left(2\right(-0.5)+3)}}\simeq 0.67,$$

  because the function ${\mathbf{H}}_{-0.5}$ is not a convex function; see Figure 2a below.
• Choosing $p=5.8$, we get that

  $$-3.88\simeq {\mathcal{H}}_{5.8+2}^{″}\left(1\right)+2{\mathcal{H}}_{5.8+2}^{\prime }\left(1\right){\mathcal{H}}_{5.8+2}\left(1\right)<{\displaystyle \frac{4}{3\left(2\right(5.8)+3)}}\simeq 0.09,$$

  which suffices to ensure that ${\mathbf{H}}_{5.8}$ is a convex function; see Figure 2b below.

Theorem 8. 

If $p>-{\displaystyle \frac{3}{2}}$, then the inequality (35) is necessary and sufficient to have ${\mathfrak{H}}_p\left(\xi \right)\in C(\gamma ,\delta )$.

Corollary 3. 

The function ${\mathfrak{H}}_p\left(\xi \right)\in T^{*}\left(\gamma \right)$ if and only if

$${\mathcal{H}}_{p+2}^{\prime }\left(1\right)+\left(1-\gamma \right){\mathcal{H}}_{p+2}\left(1\right)\le {\displaystyle \frac{4(1-\gamma )}{3\left(2p+3\right)}}\left(p>-{\displaystyle \frac{3}{2}}\right).$$

(36)

Corollary 4. 

The function ${\mathfrak{H}}_p\left(\xi \right)\in C\left(\gamma \right)$ if and only if

$${\mathcal{H}}_{p+2}^{″}\left(1\right)+2{\mathcal{H}}_{p+2}^{\prime }\left(1\right)+\left(1-\gamma \right){\mathcal{H}}_{p+2}\left(1\right)\le {\displaystyle \frac{4(1-\gamma )}{3\left(2p+3\right)}}\left(p>-{\displaystyle \frac{3}{2}}\right).$$

(37)

4. Results Regarding Bessel Functions

Apply the restriction $q=p-1$ in Theorems 1–4; then we derive the appropriate Bessel function ${\mathbf{J}}_p$ results, which are shown below.

Theorem 9. 

If $p>-1$, then the inequality

$$\left(1+\delta \right){\mathcal{J}}_{p+1}^{\prime }\left(1\right)+2\delta \left(1-\gamma \right){\mathcal{J}}_{p+1}\left(1\right)\le {\displaystyle \frac{2\delta (1-\gamma )}{p+1}},$$

(38)

suffices to ensure that ${\mathbf{J}}_p\in S^{*}(\gamma ,\delta )$, where ${\mathbf{J}}_p$ is defined in (17) and

$${\mathcal{J}}_p\left(\xi \right)\begin{array}{c}:=\end{array}{\mathbf{J}}_p\left(\xi \right)\ast {\varphi }_0\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{\displaystyle \frac{1}{{\left(1\right)}_{n-1}{\left(p+1\right)}_{n-1}}}{\xi }^n\left(p>-1\right).$$

(39)

Taking $\gamma =0$ and $\delta =1$ in Theorem 9 leads us to following illustrative example.

Example 3. 

If ${\mathcal{J}}_{p+1}^{\prime }\left(1\right)+{\mathcal{J}}_{p+1}\left(1\right)\le {\displaystyle \frac{1}{p+1}}$, then the function ${\mathbf{J}}_p\in S^{*}$; i.e., ${\mathbf{J}}_p$ is a starlike function. We list the following cases for different values of $p>-1$ as follows:

• Choosing $p=-0.5$, we notice that

  $$4.6\simeq {\mathcal{J}}_{-0.5+1}^{\prime }\left(1\right)+{\mathcal{J}}_{-0.5+1}\left(1\right)\nleqq {\displaystyle \frac{1}{-0.5+1}}=2,$$

  because the function ${\mathbf{J}}_{-0.9}$ is not a starlike function; see Figure 3a below.
• Choosing $p=0$, we get that

  $$3.87\simeq {\mathcal{J}}_{0+1}^{\prime }\left(1\right)+{\mathcal{J}}_{0+1}\left(1\right)\nleqq {\displaystyle \frac{1}{0+1}}=1,$$

  because of function ${\mathbf{J}}_0$ is not a starlike function; see Figure 3b below.
• Choosing $p=8$, we get that

  $$-80={\mathcal{J}}_{8+1}^{\prime }\left(1\right)+{\mathcal{J}}_{8+1}\left(1\right)<{\displaystyle \frac{1}{8+1}}\simeq 0.11,$$

  which suffices to ensure that ${\mathbf{J}}_8$ is a starlike function; see Figure 3c below.

Theorem 10. 

If $p>-1$, then inequality (38) is necessary and sufficient to have ${\mathfrak{J}}_p\left(\xi \right)\in T^{*}(\gamma ,\delta )$, where

$${\mathfrak{J}}_p\left(\xi \right)\begin{array}{c}:=\end{array}{\mathbf{J}}_p\left(\xi \right)\ast {\varphi }_1\left(\xi \right)=\xi -\sum _{n=2}^{\infty }{\displaystyle \frac{1}{{\left(1\right)}_{n-1}{\left(p+1\right)}_{n-1}}}{\xi }^n\left(p>-1\right).$$

(40)

Theorem 11. 

If $p>-1$, then the inequality

$$\left(1+\delta \right){\mathcal{J}}_{p+1}^{″}\left(1\right)+2\left(1+\delta \right){\mathcal{J}}_{p+1}^{\prime }\left(1\right)+2\left(1-\gamma \right)\left(2\delta -1\right){\mathcal{J}}_{p+1}\left(1\right)\le {\displaystyle \frac{2\delta (1-\gamma )}{p+1}},$$

(41)

suffices to ensure that ${\mathbf{J}}_p\left(\xi \right)\in K(\gamma ,\delta ).$

Taking $\gamma =0$ and $\delta =1$ in Theorem 11 leads us to following illustrative example.

Example 4. 

If ${\mathcal{J}}_{p+1}^{″}\left(1\right)+2{\mathcal{J}}_{p+1}^{\prime }\left(1\right)+{\mathcal{J}}_{p+1}\left(1\right)\le {\displaystyle \frac{1}{p+1}}$, then the function ${\mathbf{J}}_p\in K$; i.e., ${\mathbf{J}}_p$ is a convex function. We list the following cases for different values of $p>-1$ as follows:

• Choosing $p=-0.5$, we get that

  $$18.89\simeq {\mathcal{J}}_{-0.5+1}^{″}\left(1\right)+2{\mathcal{J}}_{-0.5+1}^{\prime }\left(1\right)+{\mathcal{J}}_{-0.5+1}\left(1\right)\nleqq {\displaystyle \frac{1}{-0.5+1}}=2,$$

  because the function ${\mathbf{J}}_{-0.5}$ is not a convex function; see Figure 3a.
• Choosing $p=0$, we get that

  $$7.74\simeq {\mathcal{J}}_{0+1}^{″}\left(1\right)+2{\mathcal{J}}_{0+1}^{\prime }\left(1\right)+{\mathcal{J}}_{0+1}\left(1\right)\nleqq {\displaystyle \frac{1}{0+1}}=1,$$

  because the function ${\mathbf{J}}_0$ is not a convex function; see Figure 3b.
• Choosing $p=8$, we get that

  $$-180={\mathcal{J}}_{8+1}^{″}\left(1\right)+2{\mathcal{J}}_{8+1}^{\prime }\left(1\right)+{\mathcal{J}}_{8+1}\left(1\right)<{\displaystyle \frac{1}{8+1}}\simeq 0.11,$$

  which suffices to ensure that ${\mathbf{J}}_8$ is a convex function; see Figure 3c.

Theorem 12. 

If $p>-1$, then relation (41) is necessary and sufficient to have ${\mathfrak{J}}_p\left(\xi \right)\in C(\gamma ,\delta )$.

Corollary 5. 

The function ${\mathfrak{J}}_p\left(\xi \right)\in T^{*}\left(\gamma \right)$ if and only if

$${\mathcal{J}}_{p+1}^{\prime }\left(1\right)+\left(1-\gamma \right){\mathcal{J}}_{p+1}\left(1\right)\le {\displaystyle \frac{1-\gamma }{p+1}}.$$

(42)

Corollary 6. 

The function ${\mathfrak{J}}_p\left(\xi \right)\in C\left(\gamma \right)$ if and only if

$${\mathcal{J}}_{p+1}^{″}\left(1\right)+2{\mathcal{J}}_{p+1}^{\prime }\left(1\right)+\left(1-\gamma \right){\mathcal{J}}_{p+1}\left(1\right)\le {\displaystyle \frac{1-\gamma }{p+1}}.$$

(43)

Taking $\gamma =0$ in Corollaries 5 and 6 leads us to following illustrative example.

Example 5. 

For the function ${\mathfrak{J}}_p$ we have the following equivalences:

$${\mathfrak{J}}_p\in T^{*}⟺{\mathcal{J}}_{p+1}^{\prime }\left(1\right)+{\mathcal{J}}_{p+1}\left(1\right)\le {\displaystyle \frac{1}{p+1}}$$

and

$${\mathfrak{J}}_p\in C⟺{\mathcal{J}}_{p+1}^{″}\left(1\right)+2{\mathcal{J}}_{p+1}^{\prime }\left(1\right)+\left(1\right){\mathcal{J}}_{p+1}\left(1\right)\le {\displaystyle \frac{1}{p+1}}.$$

Choosing $p=-0.5$, we notice that

$$4.6\simeq {\mathcal{J}}_{-0.5+1}^{\prime }\left(1\right)+{\mathcal{J}}_{-0.5+1}\left(1\right)\nleqq {\displaystyle \frac{1}{-0.5+1}}=2,$$

Similarly,

$$9.69\simeq {\mathcal{J}}_{-0.5+1}^{″}\left(1\right)+2{\mathcal{J}}_{-0.5+1}^{\prime }\left(1\right)+{\mathcal{J}}_{-0.5+1}\left(1\right)\nleqq {\displaystyle \frac{1}{-0.5+1}}=2,$$

because the function ${\mathfrak{J}}_{-0.5}\notin T^{*}$ and ${\mathfrak{J}}_{-0.5}\notin C$; see Figure 4 below.

5. Conclusions

The familiar Bessel function, Lommel function, and Struve function of the first kind are the three Bessel-type functions for which we utilized the normalized forms. These normalized forms were defined using the convolution operation. The main goal was to provide sufficient and necessary constraints for these normalized functions, making them starlike and convex of type $\delta$ and order $\gamma$. For future work, we recommend applying the convolution technique that has been used in this article to investigate the starlikeness and convexity of some other special functions of a complex variable, such as the Mittag–Leffler function [27], Hurwitz–Lerch Zeta function [28], and hypergeometric functions [29], and also other geometric properties, such as close to convexity. Additionally, the multivalent case of functions can be studied.