Theory and Modern Applications

# Inclusion relations for Bessel functions for domains bounded by conical domains

## Abstract

In recent times, applications of Bessel differential equations have been effectively used in the theory of univalent functions. In this paper we study some subclasses of k-starlike functions, k-uniformly convex functions, and quasi-convex functions involving the Bessel function and derive their inclusion relationships. Further, certain integral preserving properties are also established with these classes. We remark here that k-starlike functions and k-uniformly convex functions are related to domains bounded by conical sections.

MSC:30C45, 30C50.

## 1 Introduction

Let us consider the following second-order linear homogeneous differential equation (see for details  and ):

${z}^{2}{\omega }^{″}\left(z\right)+bz{\omega }^{\prime }\left(z\right)+\left[c{z}^{2}-{u}^{2}+\left(1-b\right)u\right]\omega \left(z\right)=0\phantom{\rule{1em}{0ex}}\left(u,b,c\in \mathbb{C}\right).$
(1.1)

The function ${\omega }_{u,b,c}\left(z\right)$, which is called the generalized Bessel function of the first kind of order u, it is defined as a particular solution of (1.1). The function ${\omega }_{u,b,c}\left(z\right)$ has the familiar representation as

${\omega }_{u,b,c}\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-c\right)}^{n}}{n!\mathrm{\Gamma }\left(u+n+\frac{b+1}{2}\right)}{\left(\frac{z}{2}\right)}^{2n+u}\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{C}\right).$
(1.2)

Here Γ stands for the Euler gamma function. The series (1.2) permits the study of Bessel, modified Bessel, and spherical Bessel function altogether. It is worth mentioning that, in particular:

1. (1)

For $b=c=1$ in (1.2), we obtain the familiar Bessel function of the first kind of order u defined by

${J}_{u}\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-1\right)}^{n}}{n!\mathrm{\Gamma }\left(u+n+1\right)}{\left(\frac{z}{2}\right)}^{2n+u}\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{C}\right).$
(1.3)
2. (2)

For $b=1$ and $c=-1$ in (1.2), we obtain the modified Bessel function of the first kind of order u defined by

${I}_{u}\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!\mathrm{\Gamma }\left(u+n+1\right)}{\left(\frac{z}{2}\right)}^{2n+u}\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{C}\right).$
(1.4)
3. (3)

For $b=2$ and $c=1$ in (1.2), the function ${\omega }_{u,b,c}\left(z\right)$ reduces to $\frac{\sqrt{2}}{\sqrt{\pi }}{j}_{u}\left(z\right)$ where ${j}_{u}$ is the spherical Bessel function of the first kind of order u, defined by

${j}_{u}\left(z\right)=\frac{\sqrt{\pi }}{\sqrt{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-1\right)}^{n}}{n!\mathrm{\Gamma }\left(u+n+\frac{3}{2}\right)}{\left(\frac{z}{2}\right)}^{2n+u}\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{C}\right).$
(1.5)

In , the author considered the function ${\phi }_{u,b,c}\left(z\right)$ defined, in terms of the generalized Bessel function ${\omega }_{u,b,c}\left(z\right)$. From (1.2), it is clear that $\omega \left(0\right)=0$. Therefore, it follows from (1.2)

${\omega }_{u,b,c}\left(z\right)={\left[{2}^{u}\mathrm{\Gamma }\left(u+\frac{b+1}{2}\right)\right]}^{-1}{z}^{u}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{-c}{4}\right)}^{n}}{n!\mathrm{\Gamma }\left(u+n+\frac{b+1}{2}\right)}{z}^{2n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }z\in \mathbb{C}.$
(1.6)

Let us set

${\phi }_{u,b,c}\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}{b}_{n}{z}^{n},$

where

${b}_{n}=\frac{{\left(\frac{-c}{4}\right)}^{n}}{n!\mathrm{\Gamma }\left(u+n+\frac{b+1}{2}\right)}.$

Hence, (1.6) becomes

${\omega }_{u,b,c}\left(z\right)={\left[{2}^{u}\mathrm{\Gamma }\left(u+\frac{b+1}{2}\right)\right]}^{-1}{z}^{u}{\phi }_{u,b,c}\left({z}^{2}\right).$
(1.7)

By using the well-known Pochhammer symbol (or the shifted factorial) ${\left(\lambda \right)}_{\mu }$ defined, for $\lambda ,\mu \in \mathbb{C}$ and in terms of the Euler Γ function, by

$\begin{array}{r}{\omega }_{u,b,c}\left(z\right)={\left[{2}^{u}\mathrm{\Gamma }\left(u+\frac{b+1}{2}\right)\right]}^{-1}{z}^{u}{\phi }_{u,b,c}\left({z}^{2}\right),\\ {\left(\lambda \right)}_{\mu }:=\frac{\mathrm{\Gamma }\left(\lambda +\mu \right)}{\mathrm{\Gamma }\left(\lambda \right)}=\left\{\begin{array}{cc}1\hfill & \left(\mu =0;\lambda \in \mathbb{C}\setminus \left\{0\right\}\right),\hfill \\ \lambda \left(\lambda +1\right)\cdots \left(\lambda +n-1\right)\hfill & \left(\mu =n\in \mathbb{N};\lambda \in \mathbb{C}\right),\hfill \end{array}\end{array}$
(1.8)

where it is being understood conventionally that ${\left(0\right)}_{0}=1$. Therefore, we obtain the following series representation for the function ${\phi }_{u,b,c}\left(z\right)$ given by (1.6):

${\phi }_{u,b,c}\left(z\right)=z+\sum _{n=1}^{\mathrm{\infty }}\frac{{\left(-c\right)}^{n}{z}^{n+1}}{{4}^{n}{\left(\kappa \right)}_{n}n!}\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{C}\right),$
(1.9)

where $\kappa =u+\frac{b+1}{2}\notin {\mathbb{Z}}_{0}^{-}$, $\mathbb{N}=\left\{1,2,\dots \right\}$ and ${\mathbb{Z}}_{0}^{-}=\left\{0,-1,-2,\dots \right\}$, and therefore

${\phi }_{u,b,c}\left(z\right)=z+\sum _{n=2}^{\mathrm{\infty }}\frac{{\left(-c/4\right)}^{n-1}{z}^{n}}{{\left(\kappa \right)}_{n-1}\left(n-1\right)!}\phantom{\rule{1em}{0ex}}\left(\kappa :=u+\frac{b+1}{2}\notin {\mathbb{Z}}_{0}^{-}\right),$
(1.10)

where $\mathbb{N}:=\left\{1,2,3,\dots \right\}$ and ${\mathbb{Z}}_{0}^{-}:=\left\{0,-1,-2,\dots \right\}$. The function ${\phi }_{u,b,c}$ is called the generalized and ‘normalized’ Bessel function of the first kind of order u. We note that by the ratio test, the radius of convergence of the series ${\phi }_{u,b,c}\left(z\right)$ is infinity. Moreover, the function ${\phi }_{u,b,c}$ is analytic in and satisfies the differential equation $4{z}^{2}{\phi }^{″}\left(z\right)+4\kappa z{\phi }^{\prime }\left(z\right)+cz\phi \left(z\right)=0$. For convenience, we write ${\phi }_{\kappa ,c}\left(z\right)={\phi }_{u,b,c}\left(z\right)$. Let $\mathcal{A}$ denote the class of functions of the form

$f\left(z\right)=z+\sum _{n=2}^{\mathrm{\infty }}{a}_{n}{z}^{n},$
(1.11)

which are analytic in the open unit disk $\mathbb{U}=\left\{z\in \mathbb{U}:|z|<1\right\}$. Let $\mathcal{S}$ be the subclass of $\mathcal{A}$ consisting of univalent functions in $\mathbb{U}$ with the normalized condition $f\left(0\right)=0={f}^{\prime }\left(0\right)-1$. A function $f\in \mathcal{A}$ is said to be starlike of order η if it satisfies $\mathrm{\Re }\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\eta$ ($z\in \mathbb{U}$) for some η ($0\le \eta <1$) and we denote the class of functions which are starlike of order η in $\mathbb{U}$ as ${\mathcal{S}}^{\ast }\left(\eta \right)$. Also, a function $f\in \mathcal{A}$ is said to be convex of order η if it satisfies $\mathrm{\Re }\left(1+\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}\right)>\eta$ ($z\in \mathbb{U}$) for some η ($0\le \eta <1$) and we denote by $\mathcal{C}\left(\eta \right)$ the class of all convex functions of order η in $\mathbb{U}$. It follows by the Alexander relation that $f\in \mathcal{C}\left(\eta \right)⇔z{f}^{\prime }\in {\mathcal{S}}^{\ast }\left(\eta \right)$. The classes ${\mathcal{S}}^{\ast }\left(\eta \right)$ and $\mathcal{C}\left(\eta \right)$ were introduced by Robertson  (see also Srivastava and Owa ). Let $f\in \mathcal{A}$ and $g\in {\mathcal{S}}^{\ast }\left(\eta \right)$. Then f is said to be close to convex of order γ and type η if and only if $\mathrm{\Re }\left(\frac{z{f}^{\prime }\left(z\right)}{g\left(z\right)}\right)>\gamma$ ($z\in \mathbb{U}$) where $0\le \gamma <1$ and $0\le \eta <1$. The classes $\mathcal{K}\left(\gamma ,\eta \right)$ were introduced by Libera  (see also Noor and Al-Kharsani , Silverman  and Shanmugam and Ramachandran ). Furthermore, we denote by $k\text{-}\mathcal{UCV}$ and $k\text{-}\mathcal{ST}$ ($0\le k<\mathrm{\infty }$), two interesting subclasses of $\mathbb{S}$ consisting, respectively, of functions which are k-uniformly convex and k-starlike in $\mathbb{U}$ defined for $0\le k<\mathrm{\infty }$ by

$k\text{-}\mathcal{UCV}:=\left\{f\in \mathcal{S}:\mathrm{\Re }\left(1+\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}\right)>k|\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}|\phantom{\rule{0.25em}{0ex}}\left(z\in \mathbb{U}\right)\right\}$

and

$k\text{-}\mathcal{ST}:=\left\{f\in \mathcal{S}:\mathrm{\Re }\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>k|\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}-1|\phantom{\rule{0.25em}{0ex}}\left(z\in \mathbb{U}\right)\right\}.$

The class $k\text{-}\mathcal{UCV}$ was introduced by Kanas and Wiśniowska in , where its geometric definition and connections with the conic domains were considered. The class $k\text{-}\mathcal{ST}$ was investigated in . In fact, it is related to the class $k\text{-}\mathcal{UCV}$ by means of the well-known Alexander equivalence between the usual classes of convex and starlike functions (see also the work of Kanas and Srivastava  for further developments involving each of the classes $k\text{-}\mathcal{UCV}$ and $k\text{-}\mathcal{ST}$). In particular, when $k=1$, we obtain $k\text{-}\mathcal{UCV}\equiv \mathcal{UCV}$ and $k\text{-}\mathcal{ST}=\mathcal{SP}$, where $\mathcal{UCV}$ and $\mathcal{SP}$ are the familiar classes of uniformly convex functions and parabolic starlike functions in $\mathbb{U}$, respectively. We remark here that the classes $k\text{-}\mathcal{UCV}\equiv \mathcal{UCV}$ and $k\text{-}\mathcal{ST}=\mathcal{SP}$ are related to the domain bounded by conical sections. Motivated by works of Kanas and Wiśniowska  and , Al-Kharsani and Al-Hajiry  introduced the classes k-uniformly convex functions and k-starlike functions of order η ($0\le \eta <1$) as below:

$k\text{-}\mathcal{UCV}\left(\eta \right):=\left\{f\in \mathcal{S}:\mathrm{\Re }\left(1+\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}-\eta \right)>k|\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}|\phantom{\rule{0.25em}{0ex}}\left(z\in \mathbb{U}\right)\right\}$
(1.12)

and

$k\text{-}\mathcal{ST}\left(\eta \right):=\left\{f\in \mathcal{S}:\mathrm{\Re }\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}-\eta \right)>k|\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}-1|\phantom{\rule{0.25em}{0ex}}\left(z\in \mathbb{U}\right)\right\}.$
(1.13)

In the case when $k=0$ the inequalities (1.12) and (1.13) reduce to the well-known classes of starlike and convex functions of order η, respectively. Further, as mentioned earlier, for the special choices of $\eta =0$ and $k=1$ the class $k\text{-}\mathcal{UCV}\left(\eta \right)$ reduces to the class of uniformly convex functions introduced by Goodman  and the class $k\text{-}\mathcal{UCV}\left(\eta \right)$ reduces to the class of parabolic starlike functions studied extensively by Rønning  (see also the work of Ma and Minda ). If f and g are analytic in $\mathbb{U}$, then we say that the function f is subordinate to g, if there exists a Schwarz function $w\left(z\right)$, analytic in $\mathbb{U}$ with $w\left(0\right)=0$ and $|w\left(z\right)|<1$ ($z\in \mathbb{U}$), such that $f\left(z\right)=g\left(w\left(z\right)\right)$ ($z\in \mathbb{U}$). We denote this subordination by $f\prec g$ or $f\left(z\right)\prec g\left(z\right)$ ($z\in \mathbb{U}$). In view of the earlier works studied by Kanas and Kanas et al. [1012, 1722], Sim et al.  and Al-Kharsani  defined the domain ${\mathrm{\Omega }}_{k,\eta }$ for $0\le k<\mathrm{\infty }$ as

${\mathrm{\Omega }}_{k,\eta }=\left\{u+iv:{\left(u-\eta \right)}^{2}>{k}^{2}{\left(u-1\right)}^{2}+{k}^{2}{v}^{2}\right\}.$

Note that, for $0,

${\mathrm{\Omega }}_{k,\eta }=\left\{u+iv:{\left(\frac{u+\frac{{k}^{2}-\eta }{1-{k}^{2}}}{k\left(\frac{1-\eta }{1-{k}^{2}}\right)}\right)}^{2}-{\left(\frac{v}{\frac{1-\eta }{\sqrt{1-{k}^{2}}}}\right)}^{2}>1\right\},$

for $k>1$,

${\mathrm{\Omega }}_{k,\eta }=\left\{u+iv:{\left(\frac{u+\frac{{k}^{2}-\eta }{{k}^{2}-1}}{k\left(\frac{1-\eta }{{k}^{2}-1}\right)}\right)}^{2}+{\left(\frac{v}{\frac{1-\eta }{\sqrt{{k}^{2}-1}}}\right)}^{2}<1\right\}.$

The explicit form of the extremal function that maps $\mathbb{U}$ onto the conic domain ${\mathrm{\Omega }}_{k,\eta }$ is given by

${Q}_{k,\eta }\left(z\right)=\left\{\begin{array}{ll}\frac{1+\left(1-2\eta \right)z}{1-z}& k=0,\\ 1+\frac{2\left(1-\eta \right)}{{\pi }^{2}}{log}^{2}\left(\frac{1+\sqrt{z}}{1-\sqrt{z}}\right),& k=1,\\ 1+\frac{2\left(1-\eta \right)}{1-{k}^{2}}{sinh}^{2}\left(A\left(k\right)arctanh\sqrt{z}\right),& 01,\end{array}$

where $A\left(k\right)=\frac{2}{\pi }arccosk$, $\mathcal{F}\left(\omega ,t\right)$ is the Legendre elliptic integral of the first kind

$\mathcal{F}\left(\omega ,t\right)={\int }_{0}^{\omega }\frac{dx}{\sqrt{1-{x}^{2}}\sqrt{1-{t}^{2}{x}^{2}}},\phantom{\rule{1em}{0ex}}\kappa \left(t\right)=\mathcal{F}\left(1,t\right)$

and $t\in \left(0,1\right)$ is chosen such that $k=cosh\frac{\pi {\kappa }^{\prime }\left(t\right)}{4\kappa \left(t\right)}$. In view of the definition of subordination and the extremal function ${Q}_{k,\eta }\left(z\right)$,

$f\in k\text{-}\mathcal{ST}\left(\eta \right)\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\prec {Q}_{k,\eta }\left(z\right)$
(1.14)

and

$f\in k\text{-}\mathcal{UCV}\left(\eta \right)\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}1+\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}\prec {Q}_{k,\eta }\left(z\right).$
(1.15)

Therefore,

$\mathrm{\Re }\left(p\left(z\right)\right)>\mathrm{\Re }\left({Q}_{k,\eta }\left(z\right)\right)>\frac{k+\eta }{k+1}.$
(1.16)

Define $\mathcal{UCC}\left(k,\eta ,\beta \right)$ as the family of functions $f\in \mathcal{A}$ such that

(1.17)

Similarly, we define $\mathcal{UQC}\left(k,\eta ,\beta \right)$ as the family of functions $f\in \mathcal{A}$ such that

(1.18)

We note that $\mathcal{UCC}\left(0,\eta ,\beta \right)$ is the class of close to convex univalent functions of order η and type β and $\mathcal{UQC}\left(0,\eta ,\beta \right)$ is the class of quasi-convex univalent functions of order η and type β. For $f\in \mathcal{A}$ given by (1.11) and $g\left(z\right)$ given by $g\left(z\right)=z+{\sum }_{n=1}^{\mathrm{\infty }}{b}_{n+1}{z}^{n+1}$, the Hadamard product (or convolution) of $f\left(z\right)$ and $g\left(z\right)$ is given by

$\left(f\ast g\right)\left(z\right)=z+\sum _{n=1}^{\mathrm{\infty }}{a}_{n+1}{b}_{n+1}{z}^{n+1}=\left(g\ast f\right)\left(z\right)\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{U}\right).$

Note that $f\ast g\in \mathcal{A}$. For ${\alpha }_{j}\in \mathbb{C}$ ($j=1,2,\dots ,q$) and ${\beta }_{j}\in \mathbb{C}\mathrm{\setminus }{\mathbb{Z}}_{0}^{-}$ ($j=1,2,\dots ,s$), the generalized hypergeometric function ${}_{q}F_{s}\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{q};{\beta }_{1},{\beta }_{2},\dots ,{\beta }_{s};z\right)$ is defined by the following infinite series (see the work of  and  for details):

${}_{q}F_{s}\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{q};{\beta }_{1},{\beta }_{2},\dots ,{\beta }_{s};z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left({\alpha }_{1}\right)}_{n}\cdots {\left({\alpha }_{q}\right)}_{n}{z}^{n}}{{\left({\beta }_{1}\right)}_{n}\cdots {\left({\beta }_{s}\right)}_{n}n!}$

($q\le s+1$; $q,s\in {\mathbb{N}}_{0}=\mathbb{N}\cup \left\{0\right\}$). Dziok and Srivastava  (also see ) considered the linear operator

$H\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{q};{\beta }_{1},{\beta }_{2},\dots ,{\beta }_{s}\right):\mathcal{A}\to \mathcal{A}$

$H\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{q};{\beta }_{1},{\beta }_{2},\dots ,{\beta }_{s}\right)f\left(z\right)={z}_{q}{F}_{s}\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{q};{\beta }_{1},{\beta }_{2},\dots ,{\beta }_{s};z\right)\ast f\left(z\right)$
(1.19)

($q\le s+1$; $q,s\in {\mathbb{N}}_{0}=\mathbb{N}\cup \left\{0\right\}$, $z\in \mathbb{U}$). If $f\in \mathcal{A}$ is given by (1.11), then we have

$H\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{q};{\beta }_{1},{\beta }_{2},\dots ,{\beta }_{s}\right)f\left(z\right)=z+\sum _{n=1}^{\mathrm{\infty }}\frac{{\left({\alpha }_{1}\right)}_{n}\cdots {\left({\alpha }_{q}\right)}_{n}}{{\left({\beta }_{1}\right)}_{n}\cdots {\left({\beta }_{s}\right)}_{n}}\frac{1}{n!}{a}_{n+1}{z}^{n+1}\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{U}\right).$

Now, by using the above idea of Dziok and Srivastava , Deniz  introduced the ${B}_{\kappa }^{c}$-operator as follows:

${B}_{\kappa }^{c}f\left(z\right)={\phi }_{\kappa ,c}\ast f\left(z\right)=z+\sum _{n=1}^{\mathrm{\infty }}\frac{{\left(-c\right)}^{n}{a}_{n+1}{z}^{n+1}}{{4}^{n}{\left(\kappa \right)}_{n}n!}.$
(1.20)

It easy to verify from the definition (1.20) that

$z{\left[{B}_{\kappa +1}^{c}f\left(z\right)\right]}^{\prime }=\kappa {B}_{\kappa }^{c}f\left(z\right)-\left(\kappa -1\right){B}_{\kappa +1}^{c}f\left(z\right),$
(1.21)

where $\kappa =u+\frac{b+1}{2}\notin {\mathbb{Z}}_{0}^{-}$. In fact, the function ${B}_{\kappa }^{c}$ given by (1.20) is an elementary transformation of the generalized hypergeometric function. That is, it is easy to see that ${B}_{\kappa }^{c}f\left(z\right)={z}_{0}{F}_{1}\left(\kappa ;\frac{-c}{4}z\right)\ast f\left(z\right)$ and also ${\phi }_{\kappa ,c}\left(\frac{-c}{4}z\right)={z}_{0}{F}_{1}\left(\kappa ;z\right)$. In special cases of the ${B}_{\kappa }^{c}$-operator we obtain the following operators related to the Bessel function:

1. (1)

Choosing $b=c=1$ in (1.20) or (1.21), we obtain the operator ${\mathcal{J}}_{u}:\mathcal{A}\to \mathcal{A}$ related with Bessel function, defined by

$\begin{array}{rl}{\mathcal{J}}_{u}f\left(z\right)& ={\phi }_{u,1,1}\left(z\right)\ast f\left(z\right)=\left[{2}^{u}\mathrm{\Gamma }\left(u+1\right){z}^{\frac{1-u}{2}}{J}_{u}\left(\sqrt{z}\right)\right]\ast f\left(z\right)\\ =z+\sum _{n=1}^{\mathrm{\infty }}\frac{{\left(-1\right)}^{n}{a}_{n+1}{z}^{n+1}}{{4}^{n}{\left(u+1\right)}_{n}n!}\end{array}$
(1.22)

and its recursive relation

$z{\left[{\mathcal{J}}_{u+1}f\left(z\right)\right]}^{\prime }=\left(u+1\right){\mathcal{J}}_{u}f\left(z\right)-u{\mathcal{J}}_{u+1}f\left(z\right).$
1. (2)

Choosing $b=1$ and $c=-1$ in (1.20) or (1.21), we obtain the operator ${\mathcal{I}}_{u}:\mathcal{A}\to \mathcal{A}$ related with the modified Bessel function, defined by

$\begin{array}{rl}{\mathcal{I}}_{u}f\left(z\right)& ={\phi }_{u,1,-1}\left(z\right)\ast f\left(z\right)=\left[{2}^{u}\mathrm{\Gamma }\left(u+1\right){z}^{\frac{1-u}{2}}{I}_{u}\left(\sqrt{z}\right)\right]\ast f\left(z\right)\\ =z+\sum _{n=1}^{\mathrm{\infty }}\frac{{a}_{n+1}{z}^{n+1}}{{4}^{n}{\left(u+1\right)}_{n}n!}\end{array}$
(1.23)

and its recursive relation

$z{\left[{\mathcal{I}}_{u+1}f\left(z\right)\right]}^{\prime }=\left(u+1\right){\mathcal{I}}_{u}f\left(z\right)-u{\mathcal{I}}_{u+1}f\left(z\right).$
1. (3)

Choosing $b=2$ and $c=1$ in (1.20) or (1.21), we obtain the operator ${\mathcal{S}}_{u}:\mathcal{A}\to \mathcal{A}$ related with the spherical Bessel function, defined by

${\mathcal{S}}_{u}f\left(z\right)=\left[{\pi }^{\frac{-1}{2}}{2}^{\frac{u+1}{2}}\mathrm{\Gamma }\left(\frac{u+3}{2}\right){z}^{\frac{1-u}{2}}{J}_{u}\left(\sqrt{z}\right)\right]\ast f\left(z\right)=z+\sum _{n=1}^{\mathrm{\infty }}\frac{{\left(-1\right)}^{n}{a}_{n+1}{z}^{n+1}}{{4}^{n}{\left(\frac{u+3}{2}\right)}_{n}n!}$
(1.24)

and its recursive relation

$z{\left[{\mathcal{S}}_{u+1}f\left(z\right)\right]}^{\prime }=\left(\frac{u+3}{2}\right){\mathcal{S}}_{u}f\left(z\right)-\left(\frac{u+1}{2}\right){\mathcal{S}}_{u+1}f\left(z\right).$

Finally we recall the generalized Bernardi-Libera-Livingston integral operator, which is defined by

${L}_{\gamma }\left(f\right)={L}_{\gamma }\left(f\left(z\right)\right)=\frac{\gamma +1}{{z}^{\gamma }}{\int }_{0}^{z}{t}^{\gamma -1}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{1em}{0ex}}\gamma >-1.$

## 2 Preliminaries

In proving our main results, we need the following lemmas.

Lemma 2.1 

Let h be convex univalent in $\mathbb{U}$ with $h\left(0\right)=1$ and $\mathrm{\Re }\left(\nu h\left(z\right)+\mu \right)>0$ ($\nu ,\mu \in \mathbb{C}$). If p is analytic in $\mathbb{U}$ with $p\left(0\right)=1$ then

$p\left(z\right)+\frac{z{p}^{\prime }\left(z\right)}{\nu p\left(z\right)+\mu }\prec h\left(z\right)\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{U}\right)\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}p\left(z\right)\prec h\left(z\right)\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{U}\right).$
(2.1)

Lemma 2.2 

Let h be convex in the open unit disk $\mathbb{U}$ and let $E\ge 0$. Suppose $B\left(z\right)$ ($z\in \mathbb{U}$) is analytic in $\mathbb{U}$ with $\mathrm{\Re }\left(B\left(z\right)\right)>0$. If $g\left(z\right)$ is analytic in $\mathbb{U}$ and $h\left(0\right)=g\left(0\right)$. Then

$E{z}^{2}{g}^{″}\left(z\right)+B\left(z\right)g\left(z\right)\prec h\left(z\right)\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}g\left(z\right)\prec h\left(z\right).$
(2.2)

## 3 Main results

We study certain inclusion relationships for some subclasses of k-starlike functions, k-uniformly convex functions, and quasi-convex functions involving the Bessel equation. We reiterate that these classes of k-starlike functions and k-uniformly convex functions are related to domains bounded by conical sections.

Theorem 3.1 Let $c\ge 1$, and h be convex univalent in $\mathbb{U}$ with $h\left(0\right)=1$ and $\mathrm{\Re }\left(h\left(z\right)\right)>0$. If a function $f\in \mathcal{A}$ satisfies the condition

$\frac{1}{1-\eta }\left[\frac{z{\left({B}_{\kappa }^{c}f\left(z\right)\right)}^{\prime }}{{B}_{\kappa }^{c}f\left(z\right)}-\eta \right]\prec h\left(z\right)\phantom{\rule{1em}{0ex}}\left(0\le \eta <1;z\in \mathbb{U}\right),$
(3.1)

then

$\frac{1}{1-\eta }\left[\frac{z{\left({B}_{\kappa +1}^{c}f\left(z\right)\right)}^{\prime }}{{B}_{\kappa +1}^{c}f\left(z\right)}-\eta \right]\prec h\left(z\right)\phantom{\rule{1em}{0ex}}\left(0\le \eta <1;z\in \mathbb{U}\right).$
(3.2)

Proof Let

$p\left(z\right)=\frac{1}{1-\eta }\left[\frac{z{\left({B}_{\kappa +1}^{c}f\left(z\right)\right)}^{\prime }}{{B}_{\kappa +1}^{c}f\left(z\right)}-\eta \right]\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{U}\right),$
(3.3)

where p is an analytic function in $\mathbb{U}$ with $p\left(0\right)=1$. By using (1.21), we get

$\left(1-\eta \right)p\left(z\right)+\eta =\kappa \frac{z{B}_{\kappa }^{c}f\left(z\right)}{{B}_{\kappa +1}^{c}f\left(z\right)}-\left(\kappa -1\right).$

Differentiating logarithmically with respect to z and multiplying by z, we obtain

$p\left(z\right)+\frac{z{p}^{\prime }\left(z\right)}{\left(1-\eta \right)p\left(z\right)+\eta +\kappa -1}=\frac{1}{1-\eta }\left[\frac{z{B}_{\kappa }^{c}f\left(z\right)}{{B}_{\kappa +1}^{c}f\left(z\right)}-\eta \right].$

The proof of the theorem follows now by an application of Lemma 2.1. □

Theorem 3.2 Let $f\in \mathcal{A}$. If ${B}_{\kappa }^{c}f\left(z\right)\in k\text{-}\mathcal{ST}\left(\eta \right)$, then ${B}_{\kappa +1}^{c}f\left(z\right)\in k\text{-}\mathcal{ST}\left(\eta \right)$.

Proof Let

$s\left(z\right)=\frac{z{\left({B}_{\kappa +1}^{c}f\left(z\right)\right)}^{\prime }}{{B}_{\kappa +1}^{c}f\left(z\right)}.$

From (1.21), we can write

$\kappa \frac{{B}_{\kappa }^{c}f\left(z\right)}{{B}_{\kappa +1}^{c}f\left(z\right)}=s\left(z\right)+\kappa -1.$

Taking logarithmic differentiation and multiplying by z, we get

$\frac{z{\left({B}_{\kappa }^{c}f\left(z\right)\right)}^{\prime }}{{B}_{\kappa }^{c}f\left(z\right)}=s\left(z\right)+\frac{z{s}^{\prime }\left(z\right)}{s\left(z\right)+\kappa -1}\prec {Q}_{k,\eta }\left(z\right).$

Since ${Q}_{k,\eta }\left(z\right)$ is convex univalent in $\mathbb{U}$ and

$\mathrm{\Re }\left({Q}_{k,\eta }\left(z\right)\right)>\frac{k+\eta }{k+1},$

the proof of the theorem follows by Theorem 3.1 and condition (1.14). □

Theorem 3.3 Let $f\in \mathcal{A}$. If ${B}_{\kappa }^{c}f\left(z\right)\in k\text{-}\mathcal{UCV}\left(\eta \right)$, then ${B}_{\kappa +1}^{c}f\left(z\right)\in k\text{-}\mathcal{UCV}\left(\eta \right)$.

Proof By virtue of (1.12), (1.13), and Theorem 3.2, we obtain

$\begin{array}{rl}{B}_{\kappa }^{c}f\left(z\right)\in k\text{-}\mathcal{UCV}\left(\eta \right)& \phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}z{\left({B}_{\kappa }^{c}f\left(z\right)\right)}^{\prime }\in k\text{-}\mathcal{ST}\left(\eta \right)\\ \phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}{B}_{\kappa }^{c}z{f}^{\prime }\left(z\right)\in k\text{-}\mathcal{ST}\left(\eta \right)\\ \phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}{B}_{\kappa +1}^{c}z{f}^{\prime }\left(z\right)\in k\text{-}\mathcal{ST}\left(\eta \right)\\ \phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}{B}_{\kappa +1}^{c}f\left(z\right)\in k\text{-}\mathcal{UCV}\left(\eta \right)\end{array}$

and hence the proof is complete. □

Theorem 3.4 Let $f\in \mathcal{A}$. If ${B}_{\kappa }^{c}f\left(z\right)\in \mathcal{UCC}\left(k,\eta ,\beta \right)$, then ${B}_{\kappa +1}^{c}f\left(z\right)\in \mathcal{UCC}\left(k,\eta ,\beta \right)$.

Proof Since

For $g\left(z\right)$ such that ${B}_{\kappa }^{c}g\left(z\right)=k\left(z\right)$ we have

$\frac{z{\left({B}_{\kappa }^{c}f\left(z\right)\right)}^{\prime }}{{B}_{\kappa }^{c}g\left(z\right)}\prec {Q}_{k,\eta }\left(z\right).$
(3.4)

Letting

$h\left(z\right)=\frac{z{\left({B}_{\kappa +1}^{c}f\left(z\right)\right)}^{\prime }}{{B}_{\kappa +1}^{c}g\left(z\right)}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}H\left(z\right)=\frac{z{\left({B}_{\kappa +1}^{c}g\left(z\right)\right)}^{\prime }}{{B}_{\kappa +1}^{c}g\left(z\right)}.$

We observe that $h\left(z\right)$ and $H\left(z\right)$ are analytic in $\mathbb{U}$ and $h\left(0\right)=H\left(0\right)=1$.

Now, by Theorem 3.2,

${B}_{\kappa +1}^{c}g\left(z\right)\in k\text{-}\mathcal{ST}\left(\beta \right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{\Re }\left(H\left(z\right)\right)>\frac{k+\beta }{k+1}.$

Also note that

$z{\left({B}_{\kappa +1}^{c}f\left(z\right)\right)}^{\prime }=\left({B}_{\kappa +1}^{c}g\left(z\right)\right)h\left(z\right).$
(3.5)

Differentiating both sides of (3.5), we obtain

$\frac{z{\left(z{\left({B}_{\kappa +1}^{c}f\left(z\right)\right)}^{\prime }\right)}^{\prime }}{{B}_{\kappa +1}^{c}g\left(z\right)}=z\frac{{\left({B}_{\kappa +1}^{c}g\left(z\right)\right)}^{\prime }}{{B}_{\kappa +1}^{c}g\left(z\right)}h\left(z\right)+z{h}^{\prime }\left(z\right)=H\left(z\right)\cdot h\left(z\right)+z{h}^{\prime }\left(z\right).$
(3.6)

Now using the identity (1.21), we obtain

$\begin{array}{rl}\frac{z{\left({B}_{\kappa }^{c}f\left(z\right)\right)}^{\prime }}{{B}_{\kappa }^{c}g\left(z\right)}& =\frac{{B}_{\kappa }^{c}\left(z{f}^{\prime }\left(z\right)\right)}{{B}_{\kappa }^{c}g\left(z\right)}\\ =\frac{z{\left({B}_{\kappa +1}^{c}z{f}^{\prime }\left(z\right)\right)}^{\prime }+\left(\kappa -1\right){B}_{\kappa +1}^{c}\left(z{f}^{\prime }\left(z\right)\right)}{z{\left({B}_{\kappa +1}^{c}g\left(z\right)\right)}^{\prime }+\left(\kappa -1\right){B}_{\kappa +1}^{c}g\left(z\right)}\\ =\frac{\frac{z{\left({B}_{\kappa +1}^{c}z{f}^{\prime }\left(z\right)\right)}^{\prime }}{{B}_{\kappa +1}^{c}g\left(z\right)}+\left(\kappa -1\right)\frac{{B}_{\kappa +1}^{c}\left(z{f}^{\prime }\left(z\right)\right)}{{B}_{\kappa +1}^{c}g\left(z\right)}}{\frac{z{\left({B}_{\kappa +1}^{c}g\left(z\right)\right)}^{\prime }}{{B}_{\kappa +1}^{c}g\left(z\right)}+\kappa -1}\\ =h\left(z\right)+\frac{z{h}^{\prime }\left(z\right)}{H\left(z\right)+\kappa -1}.\end{array}$
(3.7)

From (3.4), (3.6), and the above equation, we conclude that

$h\left(z\right)+\frac{z{h}^{\prime }\left(z\right)}{H\left(z\right)+\kappa -1}\prec {Q}_{k,\eta }\left(z\right).$

On letting $E=0$ and $B\left(z\right)=\frac{1}{H\left(z\right)+\kappa -1}$, we obtain

$\mathrm{\Re }\left(B\left(z\right)\right)=\frac{\mathrm{\Re }\left(H\left(z\right)+\kappa -1\right)}{{|H\left(z\right)+\kappa -1|}^{2}}>0$

and the above inequality satisfies the conditions required by Lemma 2.2. Hence

$h\left(z\right)\prec {Q}_{k,\eta }\left(z\right)$

and so the proof is complete. □

Using a similar argument to Theorem 3.4, we can prove the following theorem.

Theorem 3.5 Let $f\in \mathcal{A}$. If ${B}_{\kappa }^{c}f\left(z\right)\in \mathcal{UQC}\left(k,\eta ,\beta \right)$, then ${B}_{\kappa +1}^{c}f\left(z\right)\in \mathcal{UQC}\left(k,\eta ,\beta \right)$.

Now we examine the closure properties of the integral operator ${L}_{\gamma }$.

Theorem 3.6 Let $\gamma >-\frac{k+\eta }{k+1}$. If ${B}_{\kappa }^{c}\in k\text{-}\mathcal{ST}\left(\eta \right)$ so is ${L}_{\gamma }\left({B}_{\kappa }^{c}\right)$.

Proof From the definition of ${L}_{\gamma }\left(f\right)$ and the linearity of the operator ${B}_{\kappa }^{c}$ we have

$z{\left({B}_{\kappa }^{c}{L}_{\gamma }\left(f\right)\right)}^{\prime }=\left(\gamma +1\right){B}_{\kappa }^{c}f\left(z\right)-\gamma {B}_{\kappa }^{c}{L}_{\gamma }\left(f\right).$
(3.8)

Substituting $\frac{z{\left({B}_{\kappa }^{c}{L}_{\gamma }\left(f\left(z\right)\right)\right)}^{\prime }}{{B}_{\kappa }^{c}v{L}_{\gamma }\left(f\left(z\right)\right)}=p\left(z\right)$ in (3.8) we may write

$p\left(z\right)=\left(\gamma +1\right)\frac{{B}_{\kappa }^{c}f\left(z\right)}{{B}_{\kappa }^{c}{L}_{\gamma }\left(f\left(z\right)\right)}-\gamma .$
(3.9)

On differentiating (3.9) we get

$\frac{z{\left({B}_{\kappa }^{c}\left(f\left(z\right)\right)\right)}^{\prime }}{{B}_{\kappa }^{c}\left(f\left(z\right)\right)}=\frac{z{\left({B}_{\kappa }^{c}{L}_{\gamma }f\left(z\right)\right)}^{\prime }}{{B}_{\kappa }^{c}{L}_{\gamma }\left(f\left(z\right)\right)}+\frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)+\gamma }=p\left(z\right)+\frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)+\gamma }.$

By Lemma 2.1, we have $p\left(z\right)\prec Q\left(k,\eta \right)$, since $\mathrm{\Re }\left(Q\left(k,\eta \right)+\gamma \right)>0$. This completes the proof of Theorem 3.6. □

By a similar argument we can prove Theorem 3.7 as below.

Theorem 3.7 Let $\gamma >-\frac{k+\eta }{k+1}$. If ${B}_{\kappa }^{c}\in k\text{-}\mathcal{UCV}\left(\eta \right)$ so is ${L}_{\gamma }\left({B}_{\kappa }^{c}\right)$.

Theorem 3.8 Let $\gamma >-\frac{k+\eta }{k+1}$. If ${B}_{\kappa }^{c}\in \mathcal{UCC}\left(k,\eta ,\beta \right)$ so is ${L}_{\gamma }\left({B}_{\kappa }^{c}\right)$.

Proof By definition, there exists a function

$K\left(z\right)={B}_{\kappa }^{c}g\left(z\right)\in k\text{-}\mathcal{ST}\left(\eta \right),$

so that

$\frac{z{\left({B}_{\kappa }^{c}\left(f\left(z\right)\right)\right)}^{\prime }}{{B}_{\kappa }^{c}\left(g\left(z\right)\right)}\prec {Q}_{k,\eta }\left(z\right)\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{U}\right).$
(3.10)

Now from (3.8) we have

$\begin{array}{rl}\frac{z{\left({B}_{\kappa }^{c}f\right)}^{\prime }}{{B}_{\kappa }^{c}\left(g\left(z\right)\right)}& =\frac{z{\left({B}_{\kappa }^{c}{L}_{\gamma }\left(z{f}^{\prime }\right)\right)}^{\prime }+\gamma {B}_{\kappa }^{c}{L}_{\gamma }\left(z{f}^{\prime }\left(z\right)\right)}{z{\left({B}_{\kappa }^{c}{L}_{\gamma }\left(g\left(z\right)\right)\right)}^{\prime }+\gamma {B}_{\kappa }^{c}{L}_{\gamma }\left(g\left(z\right)\right)}\\ =\frac{\frac{z{\left({B}_{\kappa }^{c}\left(z{f}^{\prime }\left(z\right)\right)\right)}^{\prime }}{{B}_{\kappa }^{c}{L}_{\gamma }\left(g\left(z\right)\right)}+\frac{\gamma {B}_{\kappa }^{c}\left(z{f}^{\prime }\left(z\right)\right)}{{B}_{\kappa }^{c}{L}_{\gamma }\left(g\left(z\right)\right)}}{\frac{z{\left({B}_{\kappa }^{c}{L}_{\gamma }\left(g\left(z\right)\right)\right)}^{\prime }}{{B}_{\kappa }^{c}{L}_{\gamma }\left(g\left(z\right)\right)}+\gamma }.\end{array}$
(3.11)

Since ${B}_{\kappa }^{c}g\in k\text{-}\mathcal{ST}\left(\eta \right)$, by Theorem 3.6, we have ${L}_{\gamma }\left({B}_{\kappa }^{c}g\right)\in k\text{-}\mathcal{ST}\left(\eta \right)$. Taking $\frac{z{\left({B}_{\kappa }^{c}{L}_{\gamma }\left(g\left(z\right)\right)\right)}^{\prime }}{{B}_{\kappa }^{c}{L}_{\gamma }\left(g\right)}=H\left(z\right)$, we note that $\mathrm{\Re }\left(H\left(z\right)\right)>\frac{k+\eta }{k+1}$. Now for $h\left(z\right)=\frac{z{\left({B}_{\kappa }^{c}{L}_{\gamma }\left(f\left(z\right)\right)\right)}^{\prime }}{{B}_{\kappa }^{c}{L}_{\gamma }\left(g\left(z\right)\right)}$ we obtain

$z{\left({B}_{\kappa }^{c}{L}_{\gamma }\left(f\left(z\right)\right)\right)}^{\prime }=h\left(z\right){B}_{\kappa }^{c}{L}_{\gamma }\left(g\left(z\right)\right).$
(3.12)

Differentiating both sides of (3.12) yields

$\begin{array}{rl}\frac{z{\left({B}_{\kappa }^{c}{\left(z{L}_{\gamma }\left(f\right)\right)}^{\prime }\right)}^{\prime }}{{B}_{\kappa }^{c}{L}_{\gamma }\left(g\right)}& =z{h}^{\prime }\left(z\right)+h\left(z\right)\frac{z{\left({B}_{\kappa }^{c}{L}_{\gamma }\left(g\right)\right)}^{\prime }}{{B}_{\kappa }^{c}{L}_{\gamma }\left(g\right)}\\ =z{h}^{\prime }\left(z\right)+H\left(z\right)h\left(z\right).\end{array}$
(3.13)

Therefore from (3.11) and (3.13) we obtain

$\frac{z{\left({B}_{\kappa }^{c}f\left(z\right)\right)}^{\prime }}{{B}_{\kappa }^{c}g}=\frac{z{h}^{\prime }\left(z\right)+H\left(z\right)h\left(z\right)+\gamma h\left(z\right)}{H\left(z\right)+\gamma }.$
(3.14)

This in conjunction with (3.10) leads to

$h\left(z\right)+\frac{z{h}^{\prime }\left(z\right)}{H\left(z\right)+\gamma }\prec Q\left(k,\eta \right)\left(z\right).$
(3.15)

Let us take $B\left(z\right)=\frac{1}{H\left(z\right)+\gamma }$ in (3.15) and observe that $\mathrm{\Re }\left(B\left(z\right)\right)>0$ as $\gamma >-\frac{k+\eta }{k+1}$. Now for $A=0$ and B as described we conclude the proof since the required conditions of Lemma 2.2 are satisfied. □

A similar argument yields the following.

Theorem 3.9 Let $\gamma >-\frac{k+\eta }{k+1}$. If ${B}_{\kappa }^{c}\in \mathcal{UQC}\left(k,\eta ,\beta \right)$ so is ${L}_{\gamma }\left({B}_{\kappa }^{c}\right)$.

## 4 Concluding remarks

As observed earlier when ${B}_{\kappa }^{c}$ was defined, all the results discussed can easily be stated for the convolution operators ${\mathcal{J}}_{u}f\left(z\right)$, ${\mathcal{I}}_{u}f\left(z\right)$, and ${\mathcal{S}}_{u}f\left(z\right)$, which are defined by (1.22), (1.23), and (1.24), respectively. However, we leave those results to the interested readers.

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## Acknowledgements

The authors sincerely thank the referee(s) for their valuable comments which essentially improved the manuscript. The work of the third author is supported by a grant from Department of Science and Technology, Government of India, vide ref: SR/FTP/MS-022/2012 under the fast track scheme.

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