Certain Geometrical Properties and Hardy Space of Generalized k-Bessel Functions

Abstract

In this paper, we study certain geometric properties such as the starlikeness of order $\zeta$ and the convexity of order $\zeta$ of the generalized k-Bessel function. In addition, we establish several requirements for the parameters so that the generalized k-Bessel function belongs to some subclasses of analytic functions. Furthermore, as an application of the geometric properties, we establish certain results concerning the Hardy spaces of the generalized k-Bessel function. On the other hand, we present some corollaries concerning the classical Bessel function $J_{\rho }$ and the modified Bessel function $I_{\rho }$. To support our geometric results, we present some specific examples of functions that map the open unit disk onto the symmetric domains with respect to the real axis.

1. Introduction and Preliminaries

Special functions have significant applications in statistics, economics, physics, and other applied sciences. For this reason, such functions attract the attention of mathematicians and other researchers. In particular, functions that are special solutions of some differential equations attract much more attention than other functions. Some of these functions are hypergeometric, Bessel, Struve, and Lommel functions, and so on. In the last century, a number of important studies have been carried out on the special functions mentioned above.

A particular solution of the next differential equation

$$z^2{\chi }^{″}\left(z\right)+z{\chi }^{\prime }\left(z\right)+\left(z^2-{\rho }^2\right)\chi \left(z\right)=0.$$

(1)

is known as a Bessel function of the first kind and is denoted by $J_{\rho }$. The solution of the differential Equation (1) implies that $J_{\rho }$ has the following representation:

$$J_{\rho }\left(z\right)=\sum _{n\ge 0}{\displaystyle \frac{{(-1)}^n{\left(\frac{z}{2}\right)}^{2n+\rho }}{n!\Gamma (n+\rho +1)}},$$

where $\Gamma$ represents Euler’s gamma function. For comprehensive knowledge on this function, interested readers can refer to [1]. In addition, the modified Bessel function of the first kind $I_{\rho }$ is a private solution of the differential equation

$$z^2{\chi }^{″}\left(z\right)+z{\chi }^{\prime }\left(z\right)-\left(z^2+{\rho }^2\right)\chi \left(z\right)=0.$$

Also, the function $I_{\rho }$ has the following infinite series representation:

$$I_{\rho }\left(z\right)=\sum _{n\ge 0}{\displaystyle \frac{{\left(\frac{z}{2}\right)}^{2n+\rho }}{n!\Gamma (n+\rho +1)}}.$$

Another significant property of the mentioned functions is having some natural generalizations. In the literature, there exist certain natural generalizations of the Bessel functions. One of the most important of them is the generalized k-Bessel function, and it has the following infinite series representation (see [2]):

$$W_{\rho ,c}^k\left(z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-c)}^n}{n!{\Gamma }_k(nk+\rho +k)}}{\left({\displaystyle \frac{z}{2}}\right)}^{2n+{\displaystyle \frac{\rho }{k}}}$$

for $k>0,\rho >-1\mathrm{and}c\in \mathbb{R}$. Here, ${\Gamma }_k$ stands for the k-gamma function, which is defined by (see [3])

$${\Gamma }_k\left(x\right)={\int }_0^{\infty }{t}^{x-1}{e}^{{\displaystyle \frac{-t^k}{k}}}dt$$

for $k>0$. It is known that ${\Gamma }_k$ reduces to $\Gamma$ for $k\to 1$. On the other hand, the generalized k-Bessel function reduces to $J_{\rho }$ and $I_{\rho }$ for $k=c=1$ and $k=-c=1$, respectively (see [2]). In addition, it is known from [2] that the function $W_{\rho ,c}^k$ is a solution of the following differential equation:

$$x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+x{\displaystyle \frac{dy}{dx}}+{\displaystyle \frac{1}{k^2}}(c{x}^{2}k-{\rho }^2)y=0$$

for $k>0$ and $\rho >-k$.

The Pochammer s-symbol is defined by

$${\left(\lambda \right)}_{n,s}=\lambda (\lambda +s)(\lambda +2s)\cdots (\lambda +(n-1)s)$$

for $\lambda \in \mathbb{C},s\in \mathbb{R}$ and $n\in \mathbb{N}$. For other useful properties of the Pochammer s-symbol and k-gamma function, one can refer to [3].

The outcome of this paper is as follows. In the rest of this section, certain basic concepts in geometric function theory and useful lemmas for the proofs are given. Section 2 is devoted to the geometric properties such as starlikeness and convexity generalized k-Bessel functions. Also, we determine some conditions on the parameters such that the generalized k-Bessel function belongs to the some subclasses of analytic functions defined in [4]. In Section 3, we deal with Hardy class of generalized k-Bessel functions. Also, some corollaries are indicated at the end of Section 2 and Section 3.

Let $\mathcal{E}=\{z\in \mathbb{C}:\left|z\right|<1\}$ show the open unit disk and $\mathcal{H}\left(\mathcal{E}\right)$ be the collection of all holomorphic functions on $\mathcal{E}$. Let $\mathcal{A}$ indicate the class of all holomorphic functions $f:\mathcal{E}\to \mathbb{C},$ normalized by

$$f\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n,$$

(2)

which satisfy $f\left(0\right)=f^{\prime }\left(0\right)-1=0$. Also, it is known that the function f does not take the same value twice in a set; then, f is called univalent in this set. By $\mathcal{S}$, we indicate the collection of functions belonging to class $\mathcal{A}$ that are univalent in $\mathcal{E}$. The class $\mathcal{S}$ has various subclasses consisting of functions which have nice geometric properties. The starlike functions’ class ${\mathcal{S}}^{★}$ and convex functions’ class $\mathcal{C}$ are known as famous subclasses of $\mathcal{S}$. In fact, the functions in ${\mathcal{S}}^{★}$ and $\mathcal{C}$ map $\mathcal{E}$ onto a starlike domain with respect to the origin and a convex domain, respectively. In addition, for $0\le \zeta <1$, the class ${\mathcal{S}}^{★}\left(\zeta \right)$ denotes the class of starlike functions of order $\zeta$ in $\mathcal{E}$, while the class $\mathcal{C}\left(\zeta \right)$ denotes the class of convex functions of order $\zeta$ in $\mathcal{E}$. It is well known that ${\mathcal{S}}^{★}\left(\zeta \right)$ and $\mathcal{C}\left(\zeta \right)$ are also subclasses of $\mathcal{S}$. The classes ${\mathcal{S}}^{★}\left(\zeta \right)$ and $\mathcal{C}\left(\zeta \right)$ reduce to the classes ${\mathcal{S}}^{★}$ and $\mathcal{C}$ for $\zeta =0$, respectively. Analytic characterizations of the classes ${\mathcal{S}}^{★}\left(\zeta \right)$ and $\mathcal{C}\left(\zeta \right)$ may be given as follows:

$${\mathcal{S}}^{★}\left(\zeta \right)=\left\{f\in \mathcal{A}:Re\left({\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right)>\zeta ,z\in \mathcal{E}\right\}$$

and

$$\mathcal{C}\left(\zeta \right)=\left\{f\in \mathcal{A}:Re\left(1+{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right)>\zeta ,z\in \mathcal{E}\right\}.$$

In addition to the above significant function classes, there exist certain subclasses of the classes $\mathcal{H}\left(\mathcal{E}\right)$ and $\mathcal{A}$ in the literature. Two of them are the classes $\mathcal{P}\left(\zeta \right)$ and $\mathcal{R}\left(\zeta \right)$, and they are defined in [4] as follows:

$$\mathcal{P}\left(\zeta \right)=\{p\in \mathcal{H}\left(\mathcal{E}\right):\exists \iota \in \mathbb{R}\mathrm{such}\mathrm{that}p\left(0\right)=1,Re\left[e^{i\iota }(p\left(z\right)-\zeta )\right]>0,z\in \mathcal{E}\}$$

and

$$\mathcal{R}\left(\zeta \right)=\{f\in \mathcal{A}:\exists \iota \in \mathbb{R}\mathrm{such}\mathrm{that}Re\left[e^{i\iota }(f^{\prime }\left(z\right)-\zeta )\right]>0,z\in \mathcal{E}\}.$$

The classes $\mathcal{P}\left(\zeta \right)$ and $\mathcal{R}\left(\zeta \right)$ reduce to ${\mathcal{P}}_0\left(\zeta \right)$ and ${\mathcal{R}}_0\left(\zeta \right)$ for $\iota =0$, respectively. Also, we simplify ${\mathcal{P}}_0\left(\zeta \right)$ and ${\mathcal{R}}_0\left(\zeta \right)$ by $\mathcal{P}$ and $\mathcal{R}$ for $\zeta =0$, respectively.

The Hadamard product of the series

$${\mathfrak{f}}_1\left(z\right)=z+\sum _{n\ge 2}{a}_n{z}^n\mathrm{and}{\mathfrak{f}}_2\left(z\right)=z+\sum _{n\ge 2}{b}_n{z}^n,$$

is defined by

$$({\mathfrak{f}}_1\ast {\mathfrak{f}}_2)\left(z\right)=z+\sum _{n\ge 2}{a}_n{b}_n{z}^n.$$

The Hardy class of all holomorphic functions inside $\mathcal{E}$ is shown by ${\mathcal{H}}^p$$(0<p\le \infty )$. A regular function f in $\mathcal{E}$ is said to belong to the Hardy class ${\mathcal{H}}^p$ for $0<p\le \infty ,$ if the set $\{M_p(r,f):r\in \left[0,1\right)\}$ is bounded, where

$$M_p(r,f)=\left\{\begin{array}{c}{\left({\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }{\left|f\left(r{e}^{i\vartheta }\right)\right|}^{p}d\vartheta \right)}^{{\displaystyle \frac{1}{p}}},\mathrm{if}0<p<\infty \hfill \\ \underset{0\le \vartheta <2\pi }{sup}\left|f\left(r{e}^{i\vartheta }\right)\right|,\mathrm{if}p=\infty .\hfill \end{array}\right.$$

It is known from [5] that ${\mathcal{H}}^p$ is a Banach space with the norm defined by

$${\left|\left|f\right|\right|}_p=\underset{r\to 1^{-}}{lim}{M}_p(r,f)$$

for $1\le p\le \infty$. Furthermore, it is known that ${\mathcal{H}}^{\infty }$ denotes the family of bounded analytic functions in $\mathcal{E}$, and the set ${\mathcal{H}}^2$ shows the collection of power series $\sum a_n{z}^n$ such that $\sum {\left|a_n\right|}^2<\infty .$ Additionally, ${\mathcal{H}}^q$ is a subset of ${\mathcal{H}}^p$ for $0<p\le q\le \infty$; see [5]. Also, for the Hardy space ${\mathcal{H}}^p$, the following is a significant result (see [5]):

$$Re{f}^{\prime }\left(z\right)>0\Rightarrow \left\{\begin{array}{c}{f}^{\prime }\in {\mathcal{H}}^q,\forall q<1\hfill \\ f\in {\mathcal{H}}^{{\displaystyle \frac{q}{1-q}}},\forall q\in (0,1).\hfill \end{array}\right.$$

(3)

In the last fifty years, several scholars proved certain important results about some geometric properties of the functions in the class $\mathcal{A}$; see, for example, [6,7,8]. These geometric properties include univalence, starlikeness, convexity and close to convexity. Then, the same geometrical properties of certain special functions like Bessel, hypergeometric, Mittag–Leffler, and Wright are investigated by researchers using the results of [6,7,8]. Additionally, some requirements for the convex, starlike, and close-to-convex functions are determined by Eenigenburg and Keogh in [9] such that these functions belong to the Hardy space ${\mathcal{H}}^p$. Utilizing the results in [6,7,8,9], the Hardy space of the hypergeometric function [10], generalized Bessel function [4], Struve functions [11,12], Lommel function [13], Mittag–Leffler function [14], Wright function [15], Dini function [16], hyper-Bessel function [17], q-Bessel function [18], Rabotnov fractional exponential function [19], and so on are studied by the researchers.

Inspired by the above studies, the main objective of this paper is to present certain conditions on the parameters such that the generalized k-Bessel function

$$h_{\rho ,c}^k\left(z\right)=2^{{\displaystyle \frac{\rho }{k}}}{\Gamma }_k(\rho +k)z^{1-{\displaystyle \frac{\rho }{2k}}}{W}_{\rho ,c}^k\left(z^{1/2}\right)\in \mathcal{A}$$

is starlike of order $\zeta$ and convex of order $\zeta$ on disk $\mathcal{E}$, respectively. It is important to emphasize here that specific examples of the given functions map the open unit disk onto some domains consisting of symmetric points with respect to the real axis. Further, we determine certain conditions for the Hadamard product $h_{\rho ,c}^k\left(z\right)\ast f\left(z\right)$ such that $h_{\rho ,c}^k\left(z\right)\ast f\left(z\right)\in {\mathcal{H}}^{\infty }\cap \mathcal{R}$ and $h_{\rho ,c}^k\left(z\right)\ast f\left(z\right)\in {\mathcal{R}}_0\left(\gamma \right)$, where f is an regular function in $\mathcal{R}$. In addition, we show that the function $h_{\rho ,c}^k\left(z\right)/z$ is in the class $\mathcal{P}$ under certain conditions. Moreover, we show that the function $h_{\rho ,c}^k\left(z\right)$ is in the Hardy space ${\mathcal{H}}^{{\displaystyle \frac{1}{1-2\zeta }}}$ and bounded analytic function class ${\mathcal{H}}^{\infty }$ under some conditions, respectively. Finally, at the end of the every subsection, we indicate certain corollaries for normalized Bessel and modified Bessel functions.

Some Key Preliminary Lemmas in Geometric Function Theory

The following useful lemmas will be used in order to prove our main results:

Lemma 1

([7]). Let $0\le \zeta <1$ and $f\in \mathcal{A}$. If

$$\sum _{n=2}^{\infty }\left(n-\zeta \right)\left|a_n\right|\le 1-\zeta ,$$

then $f\left(z\right)\in {\mathcal{S}}^{★}\left(\zeta \right)$.

Lemma 2 

([7]). Let $0\le \zeta <1$ and $f\in \mathcal{A}$. If

$$\sum _{n=2}^{\infty }n\left(n-\zeta \right)\left|a_n\right|\le 1-\zeta ,$$

then $f\left(z\right)\in \mathcal{C}\left(\zeta \right).$

Lemma 3 

([9]). Let $\zeta \in \left[0,1\right)$. If $f\in \mathcal{C}\left(\zeta \right)$ is not of the form

$$\left\{\begin{array}{c}f\left(z\right)=\mu +\psi z{\left(1-z{e}^{i\vartheta }\right)}^{2\zeta -1},\zeta \ne {\displaystyle \frac{1}{2}}\hfill \\ f\left(z\right)=\mu +\psi log\left(1-z{e}^{i\vartheta }\right),\zeta ={\displaystyle \frac{1}{2}}\hfill \end{array}\right.$$

for some $\mu ,\psi \in \mathbb{C}$ and $\vartheta \in \mathbb{R},$ then the following statements hold:

a. 

There exist $\chi =\chi \left(f\right)>0$ such that $f^{\prime }\in {\mathcal{H}}^{\chi +{\displaystyle \frac{1}{2(1-\zeta )}}}$.

b. 

If $\zeta \in \left[0,{\displaystyle \frac{1}{2}}\right),$ then there exist $\phi =\phi \left(f\right)>0$ such that $f\in {\mathcal{H}}^{\phi +{\displaystyle \frac{1}{1-2\zeta }}}$.

c. 

If $\zeta \ge {\displaystyle \frac{1}{2}},$ then $f\in {\mathcal{H}}^{\infty }$.

Lemma 4 

([20]). Suppose that $\varsigma =1-2(1-\zeta )(1-\beta )$. Then, ${\mathcal{P}}_0\left(\zeta \right)\ast {\mathcal{P}}_0\left(\beta \right)\subset {\mathcal{P}}_0\left(\varsigma \right)$. The value of ς is the best possible.

Additionally, in order to prove the assertions, the following series sums

$$\sum _{k\ge 2}{\iota }^{k-1}={\displaystyle \frac{\iota }{1-\iota }},(\left|\iota \right|<1)$$

(4)

$$\sum _{k\ge 2}k{\iota }^{k-1}={\displaystyle \frac{\iota (2-\iota )}{{(1-\iota )}^2}},(\left|\iota \right|<1)$$

(5)

$$\sum _{k\ge 2}{k}^2{\iota }^{k-1}={\displaystyle \frac{\iota ({\iota }^2-3\iota +4)}{{(1-\iota )}^3}}(\left|\iota \right|<1)$$

(6)

and

$$\sum _{k\ge 2}{\displaystyle \frac{{\iota }^k}{k}}=log\left({\displaystyle \frac{1}{1-\iota }}\right)-\iota (\left|\iota \right|<1)$$

(7)

will be used. It is important to note that the series sums given by (5)–(7) can easily be obtained by the derivation or integration of (4). Also, the following inequalities

$$\begin{array}{c}\hfill 2^{n-2}\le (n-1)!,(n\ge 1)\end{array}$$

(8)

$$\begin{array}{c}\hfill {(\rho +k)}^{n-1}\le {(\rho +k)}_{n-1,k}(n\ge 1,k\in \mathbb{N})\end{array}$$

(9)

$$\begin{array}{c}\hfill \left|{\omega }_1+{\omega }_2\right|\le \left|{\omega }_1\right|+\left|{\omega }_2\right|,({\omega }_1,{\omega }_2\in \mathbb{C})\end{array}$$

(10)

will be useful in the proofs.

2. Certain Properties of Generalized k-Bessel Functions

In this part of the paper, we present certain geometrical properties of the function $h_{\rho ,c}^k\left(z\right)$.

Theorem 1.

Let $k>0$, $\rho >-1$, $\rho +k>0$, $c\in \mathbb{R}$, $3\left|c\right|<8(\rho +k)$, and $0\le \zeta <1$. If

$$\zeta \le {\displaystyle \frac{2\left|c\right|\left[16(\rho +k)-\left|c\right|\right]-{\left[8(\rho +k)-\left|c\right|\right]}^2}{\left[8(\rho +k)-\left|c\right|\right]\left[3\left|c\right|-8(\rho +k)\right]}}$$

(11)

then $h_{\rho ,c}^k\left(z\right)\in {\mathcal{S}}^{★}\left(\zeta \right)$.

Proof of Theorem 1.

Suppose that the hypothesis of the theorem holds true. Due to Lemma 1, the function $h_{\rho ,c}^k\left(z\right)\in \mathcal{A}$ is starlike of order $\zeta$ if

$$\sum _{n=2}^{\infty }(n-\zeta )\left|{\displaystyle \frac{{(-c)}^{n-1}}{4^{n-1}(n-1)!{(\rho +k)}_{n-1,k}}}\right|\le 1-\zeta .$$

Using the sums (4) and (5) and the inequalities given by (8)–(10), one can check that

$$\begin{array}{cc}\hfill \sum _{n=2}^{\infty }(n-\zeta )\left|{\displaystyle \frac{{(-c)}^{n-1}}{4^{n-1}(n-1)!{(\rho +k)}_{n-1,k}}}\right|& =\sum _{n=2}^{\infty }(n-\zeta ){\left({\displaystyle \frac{\left|c\right|}{4}}\right)}^{n-1}{\displaystyle \frac{1}{(n-1)!{(\rho +k)}_{n-1,k}}}\hfill \\ & \le \sum _{n=2}^{\infty }(n-\zeta ){\left({\displaystyle \frac{\left|c\right|}{4}}\right)}^{n-1}{\displaystyle \frac{1}{2^{n-2}{(\rho +k)}^{n-1}}}\hfill \\ & =2\sum _{n=2}^{\infty }n{\left({\displaystyle \frac{\left|c\right|}{8(\rho +k)}}\right)}^{n-1}-2\zeta {\left({\displaystyle \frac{\left|c\right|}{8(\rho +k)}}\right)}^{n-1}\hfill \\ & ={\displaystyle \frac{2\left|c\right|\left(16(\rho +k)-\left|c\right|\right)}{{\left(8(\rho +k)-\left|c\right|\right)}^2}}-{\displaystyle \frac{2\zeta \left|c\right|}{8(\rho +k)-\left|c\right|}}.\hfill \end{array}$$

(12)

The expression (12) is bounded above by $1-\zeta$ under the condition (11). Thus, the proof follows. □

Replacing $k=\left|c\right|=1$ in Theorem 1, we deduce the following, respectively.

Corollary 1.

Suppose that $\rho >-{\displaystyle \frac{5}{8}}$, $0\le \zeta <1$ and the inequality

$$\zeta \le {\displaystyle \frac{64{\rho }^2+80\rho +19}{(8\rho +7)(8\rho +5)}},$$

holds true.

i. 

If $k=c=1$, then

$$z\to h_{\rho ,1}^1\left(z\right)=2^{\rho }\Gamma (\rho +1)z^{1-{\displaystyle \frac{\rho }{2}}}{J}_{\rho }\left(z^{1/2}\right)\in {\mathcal{S}}^{★}\left(\zeta \right),$$

ii. 

If $k=-c=1$, then

$$z\to h_{\rho ,-1}^1\left(z\right)=2^{\rho }\Gamma (\rho +1)z^{1-{\displaystyle \frac{\rho }{2}}}{I}_{\rho }\left(z^{1/2}\right)\in {\mathcal{S}}^{★}\left(\zeta \right),$$

where $J_{\rho }$ and $I_{\rho }$ denote the classical Bessel function of the first kind order ρ and the modified Bessel function of the first kind order ρ, respectively.

It is well known that the functions $J_{\rho }\left(z\right)$ and $I_{\rho }\left(z\right)$ generalize trigonometric and hyperbolic functions for some private values of $\rho$, respectively. For $\rho =\pm {\displaystyle \frac{1}{2}}$ and $\rho ={\displaystyle \frac{3}{2}}$, we have the following (see [21]):

$$J_{-{\displaystyle \frac{1}{2}}}\left(z\right)=\sqrt{{\displaystyle \frac{2}{\pi z}}}cosz,J_{{\displaystyle \frac{1}{2}}}\left(z\right)=\sqrt{{\displaystyle \frac{2}{\pi z}}}sinz,J_{{\displaystyle \frac{3}{2}}}\left(z\right)=\sqrt{{\displaystyle \frac{2}{\pi z}}}\left({\displaystyle \frac{sinz}{z}}-cosz\right)$$

and

$$I_{-{\displaystyle \frac{1}{2}}}\left(z\right)=\sqrt{{\displaystyle \frac{2}{\pi z}}}coshz,I_{{\displaystyle \frac{1}{2}}}\left(z\right)=\sqrt{{\displaystyle \frac{2}{\pi z}}}sinhz,I_{{\displaystyle \frac{3}{2}}}\left(z\right)=-\sqrt{{\displaystyle \frac{2}{\pi z}}}\left({\displaystyle \frac{sinhz}{z}}-coshz\right).$$

Utilizing this fact, we can give the following examples regarding Corollary 1.

Example 1.

i. 

If $\rho ={\displaystyle \frac{1}{2}}$, $k=\left|c\right|=1$, then $\zeta \in \left[0,{\displaystyle \frac{25}{33}}\right)$. Also, both $f_1\left(z\right)=z^{1/2}sin{z}^{1/2}\in {\mathcal{S}}^{★}\left(\zeta \right)$ and $f_2\left(z\right)=z^{1/2}sinh{z}^{1/2}\in {\mathcal{S}}^{★}\left(\zeta \right)$.

ii. 

If $\rho ={\displaystyle \frac{3}{2}}$, $k=\left|c\right|=1$, then, $\zeta \in \left[0,{\displaystyle \frac{283}{323}}\right)$. Also, both $f_3\left(z\right)={\displaystyle \frac{3\left(sin{z}^{1/2}-z^{1/2}cos{z}^{1/2}\right)}{z^{1/2}}}\in {\mathcal{S}}^{★}\left(\zeta \right)$ and $f_4\left(z\right)={\displaystyle \frac{3\left(z^{1/2}cosh{z}^{1/2}-sinh{z}^{1/2}\right)}{z^{1/2}}}\in {\mathcal{S}}^{★}\left(\zeta \right)$.

Theorem 2.

Let $k>0$, $\rho >-1$, $\rho +k>0$, $c\in \mathbb{R}$, $\left|c\right|<8(\rho +k)$, and $0\le \zeta <1$. If

$$\begin{array}{cc}\hfill & {\displaystyle \frac{c^2-8(\rho +k)\left[3\left|c\right|-32(\rho +k)\right]-{\left(8(\rho +k)-\left|c\right|\right)}^3}{8(\rho +k)-\left|c\right|}}\hfill \\ & \le \zeta \left[16(\rho +k)-\left|c\right|-{\left(8(\rho +k)-\left|c\right|\right)}^2\right]\hfill \end{array}$$

(13)

then $h_{\rho ,c}^k\left(z\right)\in \mathcal{C}\left(\zeta \right)$.

Proof of Theorem 2.

Assume that the hypothesis of the theorem holds true. To show that $h_{\rho ,c}^k\left(z\right)\in \mathcal{A}$ is convex of order $\zeta$, it is enough to prove that

$$\sum _{n=2}^{\infty }n(n-\zeta )\left|{\displaystyle \frac{{(-c)}^{n-1}}{4^{n-1}(n-1)!{(\rho +k)}_{n-1,k}}}\right|\le 1-\zeta .$$

For this purpose, considering the sums (5) and (6) and the inequalities (8)–(10), one can obtain that

$$\begin{array}{cc}\hfill \sum _{n=2}^{\infty }n(n-\zeta )& \left|{\displaystyle \frac{{(-c)}^{n-1}}{4^{n-1}(n-1)!{(\rho +k)}_{n-1,k}}}\right|=\sum _{n=2}^{\infty }n(n-\zeta ){\left({\displaystyle \frac{\left|c\right|}{4}}\right)}^{n-1}{\displaystyle \frac{1}{(n-1)!{(\rho +k)}_{n-1,k}}}\hfill \\ & \le \sum _{n=2}^{\infty }n(n-\zeta ){\left({\displaystyle \frac{\left|c\right|}{4}}\right)}^{n-1}{\displaystyle \frac{1}{2^{n-2}{(\rho +k)}^{n-1}}}\hfill \\ & =2\sum _{n=2}^{\infty }{n}^2{\left({\displaystyle \frac{\left|c\right|}{8(\rho +k)}}\right)}^{n-1}-2\zeta \sum _{n=2}^{\infty }n{\left({\displaystyle \frac{\left|c\right|}{8(\rho +k)}}\right)}^{n-1}\hfill \\ & ={\displaystyle \frac{2\left|c\right|\left\{c^2-24(\rho +k)\left|c\right|+256{(\rho +k)}^2\right\}}{{\left(8(\rho +k)-\left|c\right|\right)}^3}}-{\displaystyle \frac{2\zeta \left|c\right|\left\{16(\rho +k)-\left|c\right|\right\}}{{\left(8(\rho +k)-\left|c\right|\right)}^2}}.\hfill \end{array}$$

(14)

But the expression (14) is bounded above by $1-\zeta$ under the assumption given by (13). So, the proof is completed by applying Lemma 2. □

Taking $k=\left|c\right|=1$ in Theorem 2 we deduce the following, respectively.

Corollary 2.

Suppose that $\rho >-{\displaystyle \frac{7}{8}}$, $\zeta \in \left[0,1\right)$ and

$${\displaystyle \frac{512{\rho }^3+1088{\rho }^2+688\rho +110}{8\rho +7}}\ge \zeta \left(64{\rho }^2+96\rho +34\right).$$

(15)

i. 

If $k=c=1$, then

$$z\to h_{\rho ,1}^1\left(z\right)=2^{\rho }\Gamma (\rho +1)z^{1-{\displaystyle \frac{\rho }{2}}}{J}_{\rho }\left(z^{1/2}\right)\in \mathcal{C}\left(\zeta \right),$$

ii. 

If $k=-c=1$, then

$$z\to h_{\rho ,-1}^1\left(z\right)=2^{\rho }\Gamma (\rho +1)z^{1-{\displaystyle \frac{\rho }{2}}}{I}_{\rho }\left(z^{1/2}\right)\in \mathcal{C}\left(\zeta \right).$$

Putting certain values of $\rho$ in Corollary 2, we obtain the following examples:

Example 2.

i. 

If $\rho ={\displaystyle \frac{1}{2}}$, $k=\left|c\right|=1$, then $\zeta \in \left[0,{\displaystyle \frac{395}{539}}\right)$. Also, both $f_1\left(z\right)=z^{1/2}sin{z}^{1/2}\in \mathcal{C}\left(\zeta \right)$ and $f_2\left(z\right)=z^{1/2}sinh{z}^{1/2}\in \mathcal{C}\left(\zeta \right)$.

ii. 

If $\rho ={\displaystyle \frac{3}{2}}$, $k=\left|c\right|=1$, then $\zeta \in \left[0,{\displaystyle \frac{2659}{3059}}\right)$. Also, both $f_3\left(z\right)={\displaystyle \frac{3\left(sin{z}^{1/2}-z^{1/2}cos{z}^{1/2}\right)}{z^{1/2}}}\in \mathcal{C}\left(\zeta \right)$ and $f_4\left(z\right)={\displaystyle \frac{3\left(z^{1/2}cosh{z}^{1/2}-sinh{z}^{1/2}\right)}{z^{1/2}}}\in \mathcal{C}\left(\zeta \right)$.

Figure 1 shows the mappings of functions $f_1-f_4$ over $\mathcal{E}$ provided in Examples 1–4. It is clear that the image domains shown here are symmetrical with respect to the real axis.

Theorem 3.

Let $k>0$, $\rho >-1$, $\rho +k>0$, $c\in \mathbb{R}$, $\left|c\right|<8(\rho +k)$ and $0\le \zeta <1$. If

$$\zeta <{\displaystyle \frac{8(\rho +k)-3\left|c\right|}{8(\rho +k)-\left|c\right|}},$$

(16)

then the function ${\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}\in {\mathcal{P}}_0\left(\zeta \right)$.

Proof of Theorem 3.

In order to show that ${\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}\in {\mathcal{P}}_0\left(\zeta \right)$, we need to prove that $Re\left[{\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}\right]>\zeta$ for $\zeta <1$ and $z\in \mathcal{E}$. For this purpose, considering the function $\mathcal{T}\left(z\right)={\displaystyle \frac{1}{1-\zeta }}\left[{\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}-\zeta \right]$, we will show that $\left|\mathcal{T}\left(z\right)-1\right|<1$. Then, using the inequalities (8) and (9) and geometric series sum, one can easily deduce

$$\begin{array}{cc}\hfill \left|\mathcal{T}\left(z\right)-1\right|& =\left|{\displaystyle \frac{1}{1-\zeta }}\left[{\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}-\zeta \right]-1\right|\hfill \\ & ={\displaystyle \frac{1}{1-\zeta }}\left|\sum _{n=2}^{\infty }{\displaystyle \frac{{(-c)}^{n-1}{z}^{n-1}}{4^{n-1}(n-1)!{(\rho +k)}_{n-1,k}}}\right|\hfill \\ & ={\displaystyle \frac{1}{1-\zeta }}\sum _{n=2}^{\infty }{\left({\displaystyle \frac{\left|c\right|}{4}}\right)}^{n-1}{\displaystyle \frac{1}{(n-1)!{(\rho +k)}_{n-1,k}}}\hfill \\ & \le {\displaystyle \frac{1}{1-\zeta }}\sum _{n=2}^{\infty }{\left({\displaystyle \frac{\left|c\right|}{4}}\right)}^{n-1}{\displaystyle \frac{1}{2^{n-2}{(\rho +k)}^{n-1}}}\hfill \\ & ={\displaystyle \frac{2}{1-\zeta }}\sum _{n=2}^{\infty }{\left({\displaystyle \frac{\left|c\right|}{8(\rho +k)}}\right)}^{n-1}\hfill \end{array}$$

$$\begin{array}{c}\hfill ={\displaystyle \frac{2\left|c\right|}{(1-\zeta )\left[8(\rho +k)-\left|c\right|\right]}}.\end{array}$$

(17)

But under the assumption given by (16), the expression given in (17) is bounded above by 1. This means that ${\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}\in {\mathcal{P}}_0\left(\zeta \right)$. □

Figure 1. (a) $f_1\left(\mathcal{E}\right)$, (b) $f_2\left(\mathcal{E}\right)$, (c) $f_3\left(\mathcal{E}\right)$, (d) $f_4\left(\mathcal{E}\right)$.

If we replace $\zeta =0$ and $\zeta ={\displaystyle \frac{1}{2}}$ in Theorem 3, we have the following results, respectively.

Corollary 3.

Assume that $k>0$, $\rho >-1$, $\rho +k>0$, $c\in \mathbb{R}$ and $\left|c\right|<8(\rho +k)$.

i. 

If $8(\rho +k)-3\left|c\right|>0$, then ${\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}\in {\mathcal{P}}_0\left(0\right)=\mathcal{P}$.

ii. 

If $8(\rho +k)-5\left|c\right|>0$, then ${\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}\in {\mathcal{P}}_0\left({\displaystyle \frac{1}{2}}\right)$.

Example 3.

Certain special values of ρ in Corollary 3 imply the following:

i. 

Let $\rho =-{\displaystyle \frac{1}{2}}$. If $k=c=1$, then $g_1\left(z\right)=cos{z}^{1/2}\in {\mathcal{P}}_0\left(0\right)=\mathcal{P}.$

ii. 

Let $\rho ={\displaystyle \frac{1}{2}}$. If $k=c=1$, then $g_3\left(z\right)={\displaystyle \frac{sin{z}^{1/2}}{z^{1/2}}}\in {\mathcal{P}}_0\left(0\right)=\mathcal{P},$ while $g_4\left(z\right)={\displaystyle \frac{sinh{z}^{1/2}}{z^{1/2}}}\in {\mathcal{P}}_0\left({\displaystyle \frac{1}{2}}\right)$ for $k=-c=1$.

iii. 

Let $\rho ={\displaystyle \frac{3}{2}}$. If $k=c=1$, then $g_5\left(z\right)={\displaystyle \frac{3\left(sin{z}^{1/2}-z^{1/2}cos{z}^{1/2}\right)}{z{z}^{1/2}}}\in {\mathcal{P}}_0\left(0\right)=\mathcal{P},$ while $g_6\left(z\right)={\displaystyle \frac{3\left(z^{1/2}cosh{z}^{1/2}-sinh{z}^{1/2}\right)}{z{z}^{1/2}}}\in {\mathcal{P}}_0\left({\displaystyle \frac{1}{2}}\right)$ for $k=-c=1$.

Considering $k=\left|c\right|=1$ in Theorem 3, we obtain the following.

Corollary 4.

Let $\rho >-{\displaystyle \frac{7}{8}}$, $\zeta \in [0,1)$, $z\in \mathcal{E}$ and

$$\zeta <{\displaystyle \frac{8\rho +5}{8\rho +7}}.$$

i. 

If $k=c=1$, then

$$z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{J}_{\rho }\left(z^{1/2}\right)\in {\mathcal{P}}_0\left(\zeta \right).$$

ii. 

If $k=-c=1$, then

$$z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{I}_{\rho }\left(z^{1/2}\right)\in {\mathcal{P}}_0\left(\zeta \right).$$

Example 4.

The following are the examples of Corollary 4 for special values of ρ:

a. 

Let $\rho =-{\displaystyle \frac{1}{2}}$. Then, $0\le \zeta <{\displaystyle \frac{1}{3}}$ and $g_1\left(z\right)=cos{z}^{1/2}\in {\mathcal{P}}_0\left(\zeta \right)$ for $k=c=1$, while $g_2\left(z\right)=cosh{z}^{1/2}\in {\mathcal{P}}_0\left(\zeta \right)$ for $k=-c=1$.

b. 

Let $\rho ={\displaystyle \frac{1}{2}}$. Then, $0\le \zeta <{\displaystyle \frac{9}{11}}$ and $g_3\left(z\right)={\displaystyle \frac{sin{z}^{1/2}}{z^{1/2}}}\in {\mathcal{P}}_0\left(\zeta \right)$ for $k=c=1$, while $g_4\left(z\right)={\displaystyle \frac{sinh{z}^{1/2}}{z^{1/2}}}\in {\mathcal{P}}_0\left(\zeta \right)$ for $k=-c=1$.

c. 

Let $\rho ={\displaystyle \frac{3}{2}}$. Then, $0\le \zeta <{\displaystyle \frac{17}{19}}$ and $g_5\left(z\right)={\displaystyle \frac{3\left(sin{z}^{1/2}-z^{1/2}cos{z}^{1/2}\right)}{z{z}^{1/2}}}\in {\mathcal{P}}_0\left(\zeta \right)$ for $k=c=1$, while $g_6\left(z\right)={\displaystyle \frac{3\left(z^{1/2}cosh{z}^{1/2}-sinh{z}^{1/2}\right)}{z{z}^{1/2}}}\in {\mathcal{P}}_0\left(\zeta \right)$ for $k=-c=1$.

Taking $\zeta =0$ and $\zeta ={\displaystyle \frac{1}{2}}$ in Corollary 4, we have the following results, respectively.

Corollary 5.

Let $\rho >-{\displaystyle \frac{7}{8}}$ and $z\in \mathcal{E}$.

i. 

If $\rho >-{\displaystyle \frac{5}{8}}$, then

$$z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{J}_{\rho }\left(z^{1/2}\right)\in {\mathcal{P}}_0\left(0\right)=\mathcal{P}$$

and

$$z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{I}_{\rho }\left(z^{1/2}\right)\in {\mathcal{P}}_0\left(0\right)=\mathcal{P}.$$

ii. 

If $\rho >-{\displaystyle \frac{3}{8}}$, then

$$z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{J}_{\rho }\left(z^{1/2}\right)\in {\mathcal{P}}_0\left({\displaystyle \frac{1}{2}}\right)$$

and

$$z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{I}_{\rho }\left(z^{1/2}\right)\in {\mathcal{P}}_0\left({\displaystyle \frac{1}{2}}\right).$$

Example 5.

The following are the examples of Corollary 5 for certain values of ρ:

a. 

Let $\rho =-{\displaystyle \frac{1}{2}}$. Then, $g_1\left(z\right)=cos{z}^{1/2}\in \mathcal{P}$ for $k=c=1$, while $g_2\left(z\right)=cosh{z}^{1/2}\in \mathcal{P}$ for $k=-c=1$.

b. 

Let $\rho ={\displaystyle \frac{1}{2}}$. Then, $g_3\left(z\right)={\displaystyle \frac{sin{z}^{1/2}}{z^{1/2}}}$ is in both classes $\mathcal{P}$ and ${\mathcal{P}}_0(1/2)$ for $k=c=1$, while $g_4\left(z\right)={\displaystyle \frac{sinh{z}^{1/2}}{z^{1/2}}}$ is in both classes $\mathcal{P}$ and ${\mathcal{P}}_0(1/2)$ for $k=-c=1$.

c. 

Let $\rho ={\displaystyle \frac{3}{2}}$. Then, $g_5\left(z\right)={\displaystyle \frac{3\left(sin{z}^{1/2}-z^{1/2}cos{z}^{1/2}\right)}{z{z}^{1/2}}}$ is in both classes $\mathcal{P}$ and ${\mathcal{P}}_0(1/2)$ for $k=c=1$, while $g_6\left(z\right)={\displaystyle \frac{3\left(z^{1/2}cosh{z}^{1/2}-sinh{z}^{1/2}\right)}{z{z}^{1/2}}}$ is in both classes $\mathcal{P}$ and ${\mathcal{P}}_0(1/2)$ for $k=-c=1$.

Theorem 4.

Let $k>0$, $\rho >-1$, $\rho +k>0$, $c\in \mathbb{R}$, $\left|c\right|<8(\rho +k)$ and $0\le \zeta <1$. If

$$\zeta <{\displaystyle \frac{64{(\rho +k)}^2-48\left|c\right|(\rho +k)+3c^2}{{\left[8(\rho +k)-\left|c\right|\right]}^2}},$$

(18)

then the function $h_{\rho ,c}^k\left(z\right)\in {\mathcal{R}}_0\left(\zeta \right)$.

Proof of Theorem 4.

To show that $h_{\rho ,c}^k\left(z\right)\in {\mathcal{R}}_0\left(\zeta \right)$, we have to prove that $Re\left[{\left(h_{\rho ,c}^k\left(z\right)\right)}^{\prime }\right]>\zeta$ for $\zeta <1$ and $z\in \mathcal{E}$. Hence, we consider the function $\mathcal{N}\left(z\right)={\displaystyle \frac{1}{1-\zeta }}\left[{\left(h_{\rho ,c}^k\left(z\right)\right)}^{\prime }-\zeta \right]$ and show that $\left|\mathcal{N}\left(z\right)-1\right|<1$. In fact, it may be easily checked that

$$\begin{array}{cc}\hfill \left|\mathcal{N}\left(z\right)-1\right|& =\left|{\displaystyle \frac{1}{1-\zeta }}\sum _{n=2}^{\infty }{\displaystyle \frac{n{(-c)}^{n-1}{z}^{n-1}}{4^{n-1}(n-1)!{(\rho +k)}_{n-1,k}}}\right|\hfill \\ & \le {\displaystyle \frac{1}{1-\zeta }}\sum _{n=2}^{\infty }{\left({\displaystyle \frac{\left|c\right|}{4}}\right)}^{n-1}{\displaystyle \frac{n}{(n-1)!{(\rho +k)}_{n-1,k}}}\hfill \\ & \le {\displaystyle \frac{1}{1-\zeta }}\sum _{n=2}^{\infty }{\left({\displaystyle \frac{\left|c\right|}{4}}\right)}^{n-1}{\displaystyle \frac{n}{2^{n-2}{(\rho +k)}^{n-1}}}\hfill \\ & ={\displaystyle \frac{2}{1-\zeta }}\sum _{n=2}^{\infty }n{\left({\displaystyle \frac{\left|c\right|}{8(\rho +k)}}\right)}^{n-1}\hfill \end{array}$$

$$\begin{array}{c}\hfill ={\displaystyle \frac{2\left|c\right|\left[16(\rho +k)-\left|c\right|\right]}{(1-\zeta ){\left[8(\rho +k)-\left|c\right|\right]}^2}}.\end{array}$$

(19)

It can be easily seen that the expression given in (19) is bounded above by 1 under the condition given by (18). This completes the proof. □

If we replace $\zeta =0$ and $\zeta ={\displaystyle \frac{1}{2}}$ in Theorem 4, we have the following results, respectively:

Corollary 6.

Assume that $k>0$, $\rho >-1$, $\rho +k>0$, $c\in \mathbb{R}$ and $\left|c\right|<8(\rho +k)$. If

i. 

$64{(\rho +k)}^2-48\left|c\right|(\rho +k)+3c^2>0$, then $h_{\rho ,c}^k\left(z\right)\in {\mathcal{R}}_0\left(0\right)=\mathcal{R}$.

ii. 

$64{(\rho +k)}^2-80\left|c\right|(\rho +k)+5c^2>0$, then $h_{\rho ,c}^k\left(z\right)\in {\mathcal{R}}_0\left({\displaystyle \frac{1}{2}}\right)$.

Example 6.

The following are the examples of Corollary 6 for certain values of ρ:

a. 

Let $\rho ={\displaystyle \frac{1}{2}}$. Then, $f_1\left(z\right)=z^{1/2}sin{z}^{1/2}$ is in both classes $\mathcal{R}$ and ${\mathcal{R}}_0(1/2)$ for $k=c=1$, while $f_2\left(z\right)=z^{1/2}sinh{z}^{1/2}$ is in both classes $\mathcal{R}$ and ${\mathcal{R}}_0(1/2)$ for $k=-c=1$.

b. 

Let $\rho ={\displaystyle \frac{3}{2}}$. Then, $f_3\left(z\right)={\displaystyle \frac{3\left(sin{z}^{1/2}-z^{1/2}cos{z}^{1/2}\right)}{z^{1/2}}}$ is in both classes $\mathcal{R}$ and ${\mathcal{R}}_0(1/2)$ for $k=c=1$, while $f_4\left(z\right)={\displaystyle \frac{3\left(z^{1/2}cosh{z}^{1/2}-sinh{z}^{1/2}\right)}{z^{1/2}}}$ is in both classes $\mathcal{R}$ and ${\mathcal{R}}_0(1/2)$ for $k=-c=1$.

Considering $k=\left|c\right|=1$ in Theorem 4, we obtain the following.

Corollary 7.

Let $\rho >-{\displaystyle \frac{7}{8}}$, $\zeta \in [0,1)$, $z\in \mathcal{E}$ and

$$\zeta <{\displaystyle \frac{64{\rho }^2+80\rho +19}{{(8\rho +7)}^2}}.$$

i. 

If $k=c=1$, then $z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{J}_{\rho }\left(z^{1/2}\right)\in {\mathcal{R}}_0\left(\zeta \right).$

ii. 

If $k=-c=1$, then $z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{I}_{\rho }\left(z^{1/2}\right)\in {\mathcal{R}}_0\left(\zeta \right).$

Example 7.

The following are the examples of Corollary 7 for certain values of ρ:

a. 

Let $\rho ={\displaystyle \frac{1}{2}}$. Then, $0\le \zeta <75/121$, and both $f_1\left(z\right)=z^{1/2}sin{z}^{1/2}$ and $f_2\left(z\right)=z^{1/2}sinh{z}^{1/2}$ are in the class ${\mathcal{R}}_0\left(\zeta \right)$ for $k=c=1$ and $k=-c=1$, respectively.

b. 

Let $\rho ={\displaystyle \frac{3}{2}}$. Then, $0\le \zeta <283/361$, and both $f_3\left(z\right)={\displaystyle \frac{3\left(sin{z}^{1/2}-z^{1/2}cos{z}^{1/2}\right)}{z^{1/2}}}$ and $f_4\left(z\right)={\displaystyle \frac{3\left(z^{1/2}cosh{z}^{1/2}-sinh{z}^{1/2}\right)}{z^{1/2}}}$ are in the class ${\mathcal{R}}_0\left(\zeta \right)$ for $k=c=1$ and $k=-c=1$, respectively.

Taking $\zeta =0$ and $\zeta =1/2$ in Corollary 7, we give the following special cases.

Corollary 8.

Let $\rho >-{\displaystyle \frac{7}{8}}$ and $z\in \mathcal{E}$.

i. 

If $\rho >{\displaystyle \frac{-5+\sqrt{6}}{8}}$, then

$$z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{J}_{\rho }\left(z^{1/2}\right)\in {\mathcal{R}}_0\left(0\right)=\mathcal{R}$$

and

$$z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{I}_{\rho }\left(z^{1/2}\right)\in {\mathcal{R}}_0\left(0\right)=\mathcal{R}.$$

ii. 

If $\rho >{\displaystyle \frac{-3+2\sqrt{5}}{8}}$, then

$$z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{J}_{\rho }\left(z^{1/2}\right)\in {\mathcal{R}}_0(1/2)$$

and

$$z\to 2^{\rho }\Gamma (\rho +1)z^{-{\displaystyle \frac{\rho }{2}}}{I}_{\rho }\left(z^{1/2}\right)\in {\mathcal{R}}_0(1/2).$$

3. Hardy Space of Generalized k-Bessel Functions

Theorem 5.

Assume that $k>0$, $\rho >-1$, $\rho +k>0$, $c\in \mathbb{R}$, $\left|c\right|<8(\rho +k)$ and $0\le \zeta <1$. Also, the inequality (13) holds true:

i. 

If $0\le \zeta <{\displaystyle \frac{1}{2}}$, then $h_{\rho ,c}^k\left(z\right)\in {\mathcal{H}}^{{\displaystyle \frac{1}{1-2\zeta }}}.$

ii. 

If ${\displaystyle \frac{1}{2}}<\zeta <1$, then $h_{\rho ,c}^k\left(z\right)\in {\mathcal{H}}^{\infty }.$

Proof of Theorem 5.

It is known from Theorem 2 that the function $z\mapsto h_{\rho ,c}^k\left(z\right)\in \mathcal{C}\left(\zeta \right)$ under the hypothesis conditions. Also, from the infinite series representation of the Gauss hypergeometric function, it can be easily seen that

$$\mu +{\displaystyle \frac{\psi z}{{(1-z{e}^{i\vartheta })}^{1-2\zeta }}}=\mu +\psi \sum _{n=0}^{\infty }{\displaystyle \frac{{(1-2\zeta )}_n}{n!}}{e}^{i\vartheta n}{z}^{n+1}$$

(20)

and

$$\mu +\psi log\left(1-z{e}^{i\vartheta }\right)=\mu -\psi \sum _{n=0}^{\infty }{\displaystyle \frac{1}{n+1}}{e}^{i\vartheta (n+1)}{z}^{n+1},$$

(21)

where $\mu ,\psi \in \mathbb{C}$ and $\vartheta \in \mathbb{R}$. One can obviously see that the function $z\mapsto h_{\rho ,c}^k\left(z\right)$ is not in the forms of (20) for $\zeta \ne {\displaystyle \frac{1}{2}}$ and (21) for $\zeta ={\displaystyle \frac{1}{2}}$, respectively. Now, in view of Lemma 3, the proof is completed. □

Theorem 5 reduces to the following result for $k=\left|c\right|=1$.

Corollary 9.

Let us suppose that $\rho >-{\displaystyle \frac{7}{8}}$, and the inequality (15) holds true. Then, we have the following:

i. 

The functions $h_{\rho ,1}^1\left(z\right)$ and $h_{\rho ,-1}^1\left(z\right)\in {\mathcal{H}}^{{\displaystyle \frac{1}{1-2\zeta }}}$ for $0\le \zeta <{\displaystyle \frac{1}{2}}$.

ii. 

The functions $h_{\rho ,1}^1\left(z\right)$ and $h_{\rho ,-1}^1\left(z\right)\in {\mathcal{H}}^{\infty }$ for ${\displaystyle \frac{1}{2}}<\zeta <1$.

Example 8.

Taking certain values of ρ in Corollary 9, we obtain the following:

i. 

Let $\rho ={\displaystyle \frac{1}{2}}$. Then, $\zeta \le {\displaystyle \frac{501}{1078}}\cong 0.46475$ and the functions $f_1\left(z\right)=z^{1/2}sin{z}^{1/2}$ and $f_2\left(z\right)=z^{1/2}sinh{z}^{1/2}$ are in the class ${\mathcal{H}}^{{\displaystyle \frac{1}{1-2\zeta }}}$.

ii. 

Let $\rho ={\displaystyle \frac{3}{2}}$. Then, $\zeta \le {\displaystyle \frac{5519}{6118}}\cong 0.902092$ and the functions $f_3\left(z\right)={\displaystyle \frac{3\left(sin{z}^{1/2}-z^{1/2}cos{z}^{1/2}\right)}{z^{1/2}}}$ and $f_4\left(z\right)={\displaystyle \frac{3\left(z^{1/2}cosh{z}^{1/2}-sinh{z}^{1/2}\right)}{z^{1/2}}}$ are in the class ${\mathcal{H}}^{\infty }$.

Theorem 6.

Let suppose that $k>0$, $\rho >-1$, $\rho +k>0$, $c\in \mathbb{R}$, $\left|c\right|<8(\rho +k)$ and $0\le \zeta <1$. If the function $f\left(z\right)$ of the form (2) is in the class $\mathcal{R}$ and $8(\rho +k)-5\left|c\right|>0$, then the function $z\to h_{\rho ,c}^k\left(z\right)\ast f\left(z\right)\in {\mathcal{H}}^{\infty }\cap \mathcal{R}.$

Proof of Theorem 6.

By definition of the class $\mathcal{R}$, the hypothesis condition $f\left(z\right)\in \mathcal{R}$ implies that $f^{\prime }\left(z\right)\in \mathcal{P}$. Further, suppose that $\xi \left(z\right)=h_{\rho ,c}^k\left(z\right)\ast f\left(z\right)$, which implies ${\xi }^{\prime }\left(z\right)={\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}\ast f^{\prime }\left(z\right)$. Also, it is known from Corollary 3 that the function ${\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}\in P_0(1/2).$ Applying Lemma 4, we conclude that ${\xi }^{\prime }\left(z\right)\in \mathcal{P}$ which implies $\xi \left(z\right)\in \mathcal{R},$ and so we get $Re\left({\xi }^{\prime }\left(z\right)\right)>0.$ Now, from the relation given by (3), we deduce ${\xi }^{\prime }\left(z\right)\in {\mathcal{H}}^p$ for $p<1$ and $\xi \left(z\right)\in {\mathcal{H}}^{{\displaystyle \frac{q}{1-q}}}$ for $0<q<1$, or equivalently, $\xi \left(z\right)\in {\mathcal{H}}^p$ for all $0<p<\infty$. In addition, by using the inequalities (8) and (9) and basic calculations, we conclude

$$\left|\xi \left(z\right)\right|\le 1+2\sum _{n=2}^{\infty }{\left({\displaystyle \frac{\left|c\right|}{8(\rho +k)}}\right)}^{n-1}\left|a_n\right|.$$

Moreover, it is known from [22] that if the function $f\in \mathcal{R}$, then $\left|a_n\right|\le {\displaystyle \frac{2}{n}}$ for $n\ge 2.$ By utilizing this fact and the series sum rule given by (7), we may write that

$$\left|\xi \left(z\right)\right|\le 1+{\displaystyle \frac{32(\rho +k)}{\left|c\right|}}\left\{log{\displaystyle \frac{8(\rho +k)}{8(\rho +k)-\left|c\right|}}-{\displaystyle \frac{\left|c\right|}{8(\rho +k)}}\right\}<\infty .$$

Hence, the function $\xi \left(z\right)$ is convergent in $\left|z\right|=1$ under assumption. On the other hand, it is known from [5] that ${\xi }^{\prime }\left(z\right)\in {\mathcal{H}}^q$ implies that $\xi \left(z\right)$ is continuous on $\overline{\mathcal{E}}$ which is the closure of $\mathcal{E}$. Due to $\overline{\mathcal{E}}$ being a compact set, $\xi \left(z\right)$ is a bounded function on $\mathcal{E}$. So, the proof follows. □

Theorem 7.

Let $k>0$, $\rho >-1$, $\rho +k>0$, $c\in \mathbb{R}$, $0\le \zeta <1$, $\beta <1$ and $\gamma =1-2(1-\zeta )(1-\beta )$. If $f\left(z\right)\in {\mathcal{R}}_0\left(\beta \right)$, and the inequality (16) holds true, then $\xi \left(z\right)=h_{\rho ,c}^k\left(z\right)\ast f\left(z\right)\in {\mathcal{R}}_0\left(\gamma \right)$.

Proof of Theorem 7.

By definition of the class ${\mathcal{R}}_0\left(\beta \right)$, the assumption $f\left(z\right)\in {\mathcal{R}}_0\left(\beta \right)$ yields $f^{\prime }\left(z\right)\in {\mathcal{P}}_0\left(\beta \right)$. In addition, it is known from Theorem 3 that ${\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}\in {\mathcal{P}}_0\left(\zeta \right)$. Since ${\xi }^{\prime }\left(z\right)={\displaystyle \frac{h_{\rho ,c}^k\left(z\right)}{z}}\ast f^{\prime }\left(z\right)$, applying Lemma 4, we deduce ${\xi }^{\prime }\left(z\right)\in {\mathcal{P}}_0\left(\gamma \right)$, which implies that $\xi \left(z\right)\in {\mathcal{R}}_0\left(\gamma \right).$ □

4. Conclusions

In this paper, several geometric properties of the generalized k-Bessel function are established. As application of the convexity properties, certain sufficient conditions are established so that the mentioned function belongs to the Hardy space of analytic functions. Moreover, for the Hadamard product of the generalized k-Bessel function and the analytic function, several results are presented such that the mentioned functions are in the previously defined classes of analytic functions. Furthermore, by specializing the parameters, some direct consequences are given for the classical Bessel function and the modified Bessel function.

In the future, uniform convexity, starlikeness, and convexity associated with the exponential function, as well as Bernoulli’s lemniscate may be studied for generalized k-Bessel functions.