# Properties of multivalent functions associated with the integral operator defined by the hypergeometric function

## Abstract

In this paper, we introduce a new class of multivalent functions by using a generalized integral operator defined by the hypergeometric function. Some properties such as inclusion, radius problem and integral preserving are considered.

MSC:30C45, 30C50.

## 1 Introduction and preliminaries

Let ${A}_{p}$ denote the class of functions $f\left(z\right)$ of the form

$f\left(z\right)={z}^{p}+\sum _{n=p+1}^{\mathrm{\infty }}{a}_{n}{z}^{n}\phantom{\rule{1em}{0ex}}\left(p\in \mathbb{N}=\left\{1,2,3,\dots \right\}\right),$
(1.1)

which are analytic in the open unit disc E. Also ${A}_{1}=A$, the usual class of analytic functions defined in the open unit disc $E=\left\{z:|z|<1\right\}$. A function $f\in {A}_{p}$ is a p-valent starlike function of order ρ if and only if

$Re\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}>\rho ,\phantom{\rule{1em}{0ex}}0\le \rho

This class of functions is denoted by ${S}_{p}^{\ast }\left(\rho \right)$. It is noted that ${S}_{p}^{\ast }\left(0\right)={S}_{p}^{\ast }$. Let $f\left(z\right)$ and $g\left(z\right)$ be analytic in E, we say $f\left(z\right)$ is subordinate to $g\left(z\right)$, written $f\prec g$ or $f\left(z\right)\prec g\left(z\right)$ if there exists a Schwarz function $w\left(z\right)$, $w\left(0\right)=0$ and $|w\left(z\right)|<1$ in E, then $f\left(z\right)=g\left(w\left(z\right)\right)$. In particular, if g is univalent in E, then we have the following equivalence

$f\left(z\right)\prec g\left(z\right)\phantom{\rule{1em}{0ex}}⟺\phantom{\rule{1em}{0ex}}f\left(0\right)=g\left(0\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}f\left(E\right)\subset g\left(E\right).$

For any two analytic functions $f\left(z\right)$ and $g\left(z\right)$ with

$f\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}{b}_{n}{z}^{n+1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}g\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}{c}_{n}{z}^{n+1},\phantom{\rule{1em}{0ex}}z\in E,$

the convolution (the Hadamard product) is given by

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

A function $f\in A$ is said to be in the class, denoted by $\mathit{SD}\left(k,\delta \right)$ ($0\le \delta <1$), if and only if

$Re\left\{\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right\}>k|\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}-1|+\delta ,\phantom{\rule{1em}{0ex}}k\ge 0,z\in E.$
(1.2)

Similarly, a function $f\in A$ is said to be in the class, denoted by $\mathit{CD}\left(k,\delta \right)$ of k-uniformly convex of order δ ($0\le \delta <1$), if

$Re\left\{1+\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}\right\}>k|\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}|+\delta ,\phantom{\rule{1em}{0ex}}k\ge 0,z\in E.$
(1.3)

Geometric interpretation The functions $f\in \mathit{SD}\left(k,\delta \right)$ and $f\in \mathit{CD}\left(k,\delta \right)$ if and only if $\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}$ and $\frac{z{f}^{″}\left(z\right)}{f\left(z\right)}+1$, respectively, take all the values in the conic domain ${\mathrm{\Omega }}_{k,\delta }$ defined by

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

with $p\left(z\right)=\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}$ or $p\left(z\right)=\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}+1$ and considering the functions which map E onto the conic domain ${\mathrm{\Omega }}_{k,\delta }$ such that $1\in {\mathrm{\Omega }}_{k,\delta }$. One may rewrite the conditions (1.2) or (1.3) in the form

$p\left(z\right)\prec {q}_{k,\delta }\left(z\right).$

The function ${q}_{k,\delta }\left(z\right)$ plays the role of extremal for these classes and is given by

${q}_{k,\delta }\left(z\right)=\left\{\begin{array}{cc}\frac{1+\left(1-2\delta \right)z}{1-z},\hfill & k=0,\hfill \\ 1+\frac{2\delta \gamma }{{\pi }^{2}}{\left(log\frac{1+\sqrt{z}}{1-\sqrt{z}}\right)}^{2},\hfill & k=1,\hfill \\ 1+\frac{2\delta }{1-{k}^{2}}{sinh}^{2}\left[\left(\frac{2}{\pi }arccosk\right)arctanh\sqrt{z}\right],\hfill & 01.\hfill \end{array}$
(1.4)

For $f\left(z\right)$ in ${A}_{p}$, the operator ${D}^{\mu +p-1}:{A}_{p}⟶{A}_{p}$ is defined by

${D}^{\mu +p-1}f\left(z\right)=\frac{{z}^{p}}{{\left(1-z\right)}^{\mu +p}}\ast f\left(z\right)\phantom{\rule{1em}{0ex}}\left(\mu >-p\right),$

or equivalently

${D}^{\mu +p-1}f\left(z\right)=\frac{{z}^{p}{\left({z}^{\mu -1}f\left(z\right)\right)}^{\mu +p-1}}{\left(\mu +p-1\right)!},$
(1.5)

where μ is any integer greater than −p. If $f\left(z\right)$ is given by (1.1), then it follows that

${D}^{\mu +p-1}f\left(z\right)={z}^{p}+\sum _{n=p+1}^{\mathrm{\infty }}\frac{\left(\mu +n-1\right)!}{\left(n-p\right)!\left(\mu +p-1\right)!}{a}_{n}{z}^{n}.$

The symbol ${D}^{\mu +p-1}$ when $p=1$, was introduced by Ruscheweyh  and ${D}^{\mu +p-1}$ is called the $\left(\mu +p-1\right)$th order Ruscheweyh derivative. We now introduce a function ${\left({z}_{2}^{p}{F}_{1}\left(a,b,c;z\right)\right)}^{-1}$ given by

$\left(z^{p}{}_{2}{F}_{1}\left(a,b,c;z\right)\right)\ast {\left(z^{p}{}_{2}{F}_{1}\left(a,b,c;z\right)\right)}^{-1}=\frac{{z}^{p}}{{\left(1-z\right)}^{\mu +p}}\phantom{\rule{1em}{0ex}}\left(\mu >-p\right),$

and the following linear operator

${I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)={\left(z^{p}{}_{2}{F}_{1}\left(a,b,c;z\right)\right)}^{-1}\ast f\left(z\right),$
(1.6)

where a, b, c are real or complex numbers other than $0,-1,-2,\dots$ , $\mu >-p$, $z\in E$ and $f\left(z\right)\in {A}_{p}$. This operator was recently introduced in . In particular, for $p=1$, this operator is studied by Noor . For $b=1$, this operator reduces to the well-known Cho-Kwon-Srivastava operator ${I}_{\mu ,p}\left(a,c\right)$, which was studied by Cho et al. , and for $\mu =1$, $b=c$, $a=n+p$, see . For $a=n+p$, $b=c=1$, this operator was investigated by Liu  and Liu and Noor .

Simple computations yield

${I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)={z}^{p}+\sum _{n=p+1}^{\mathrm{\infty }}\frac{{\left(c\right)}_{n}{\left(\mu +p\right)}_{n}}{{\left(a\right)}_{n}{\left(b\right)}_{n}}{a}_{n}{z}^{n}.$

From (1.6), we note that

$\begin{array}{r}{I}_{\mu ,1}\left(a,b,c\right)f\left(z\right)={I}_{\mu }\left(a,b,c\right)f\left(z\right)\phantom{\rule{1em}{0ex}}\text{(see )},\\ {I}_{0,p}\left(a,p,a\right)f\left(z\right)=f\left(z\right),{I}_{1,p}\left(a,p,a\right)f\left(z\right)=\frac{z{f}^{\prime }\left(z\right)}{p}.\end{array}$

Also, it can be easily seen that

$z{\left({I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)\right)}^{\prime }=\left(\mu +p\right){I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)-\mu {I}_{\mu ,p}\left(a,b,c\right)f\left(z\right),$
(1.7)

and

$z{\left({I}_{\mu ,p}\left(a+1,b,c\right)f\left(z\right)\right)}^{\prime }=a{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)-\left(a-p\right){I}_{\mu ,p}\left(a,b,c\right)f\left(z\right).$

We define the following class of multivalent analytic functions by using the operator ${I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)$ above.

Definition 1.1 Let $f\in {A}_{p}$ for $p\in \mathbb{N}$. Then $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;\gamma ,k,\delta \right)$ for $a,b,c\in \mathbb{R}\mathrm{\setminus }{\mathbb{Z}}_{0}^{-}$, $\mu >-p$, $\alpha >0$, $k\ge 0$, $0\le \delta <1$ and $\gamma >0$ if and only if

$\begin{array}{r}\left(1-\gamma \right){\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }+\gamma \frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}{\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha -1}\\ \phantom{\rule{1em}{0ex}}\prec {q}_{k,\delta }\left(z\right),\end{array}$
(1.8)

where $g\in {A}_{p}$ is such that

$q\left(z\right)=\frac{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\in P\left(\rho \right),\phantom{\rule{1em}{0ex}}\rho =\frac{k+\delta }{k+1},z\in E.$
(1.9)

Furthermore, for different choices of parameters being involved, we obtain many other well-known subclasses of the class ${A}_{p}$ and A as special cases.

1. (i)

$a=c$, $b=1$, $k=0$, $\mu =m\in {\mathbb{N}}_{0}$, we have ${B}_{m,p}^{\alpha }\left(\gamma ,\delta \right)$ studied in .

2. (ii)

$a=c=b=p=\gamma =1$, $k=\mu =0$, $g\left(z\right)=z$, the class $U{B}_{\mu ,p}^{\alpha }\left(a,b,c;\gamma ,k,\delta \right)$ reduces to the class

${B}^{\alpha }\left(\delta \right)=\left\{f\in A\left(1\right):\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}{\left(\frac{f\left(z\right)}{z}\right)}^{\alpha }\in P\left(\delta \right)\right\}$

studied in .

3. (iii)

$a=c=b=p=\gamma =1$, $k=\mu =0$, $g\left(z\right)=z$, the $U{B}_{\mu ,p}^{\alpha }\left(a,b,c;\gamma ,k,\delta \right)$ reduces to $B\left(\alpha \right)$ is the class of Bazilevich functions investigated by Singh .

4. (iv)

$a=c=b=p=\alpha =1$, $\gamma =0$, $k=\mu =0$, $g\left(z\right)=z$, the class $U{B}_{\mu ,p}^{\alpha }\left(a,b,c;\gamma ,k,\delta \right)$ reduces to the class

${B}_{\delta }=\left\{f\in A\left(1\right):\frac{f\left(z\right)}{z}\in P\left(\delta \right)\right\},$

the class studied by Chen .

Let $f\in {A}_{p\cdot }$ and ${F}_{\eta ,p}:{A}_{p\cdot }\to {A}_{p\cdot }$ be defined by

${F}_{\eta ,p}\left(z\right)=\frac{\left(\eta +p\right)}{{z}^{\eta }}{\int }_{0}^{z}{t}^{\eta -1}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{1em}{0ex}}\eta >-p.$
(1.10)

We need the following lemmas which will be used in our main results.

Lemma 1.2 

Let $u={u}_{1}+i{u}_{2}$ and $v={v}_{1}+i{v}_{2}$, and let $\psi :D\subset {\mathbb{C}}^{2}\to \mathbb{C}$ be a complex-valued function satisfying the conditions:

1. (i)

$\psi \left(u,v\right)$ is continuous in a domain $D\subset {\mathbb{C}}^{2}$,

2. (ii)

$\left(1,0\right)\in D$ and $\psi \left(1,0\right)>0$,

3. (iii)

$Re\psi \left(i{u}_{2},{v}_{1}\right)\le 0$, whenever $\left(i{u}_{2},{v}_{1}\right)\in D$ and ${v}_{1}\le -\frac{1}{2}\left(1+{u}_{2}^{2}\right)$.

If $h\left(z\right)=1+{c}_{1}z+{c}_{2}{z}^{2}+\cdots$ is analytic in E such that $\left(h,z{h}^{\prime }\right)\in D$ and $Re\psi \left(h\left(z\right),z{h}^{\prime }\left(z\right)\right)>0$ for $z\in E$, then $Reh\left(z\right)>0$.

Lemma 1.3 

Let h be convex in the unit disc E, and let $A\ge 0$. Suppose that $B\left(z\right)$ is analytic in E with $ReB\left(z\right)\ge A$. If g is analytic in E and $g\left(0\right)=h\left(0\right)$. Then

$A{z}^{2}{g}^{″}\left(z\right)+B\left(z\right)z{g}^{\prime }\left(z\right)+g\left(z\right)\prec h\left(z\right)\phantom{\rule{1em}{0ex}}\mathit{\text{implies that}}\phantom{\rule{1em}{0ex}}g\left(z\right)\prec h\left(z\right).$

Lemma 1.4 

Let F be analytic and convex in E. If $f,g\in {A}_{p}$ and $f,g\prec F$. Then

$\sigma f+\left(1-\sigma \right)g\prec F,\phantom{\rule{1em}{0ex}}0\le \sigma \le 1.$

Lemma 1.5 

Let h be convex in E with $h\left(0\right)=a$ and $\beta \in \mathbb{C}$ such that $Re\beta \ge 0$. If $p\in H\left[a,n\right]$ and

$p\left(z\right)+\frac{z{p}^{\prime }\left(z\right)}{\beta }\prec h\left(z\right),$

then $p\left(z\right)\prec q\left(z\right)\prec h\left(z\right)$, where

$q\left(z\right)=\frac{\beta }{n{z}^{\beta /n}}{\int }_{0}^{z}h\left(t\right){t}^{\beta /n-1}\phantom{\rule{0.2em}{0ex}}dt$

and $q\left(z\right)$ is the best dominant.

## 2 Main results

Theorem 2.1 Let $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;\gamma ,k,\delta \right)$ for $a,b,c\in \mathbb{R}\mathrm{\setminus }{\mathbb{Z}}_{0}^{-}$, $\mu >-p$, $p\in \mathbb{N}$, $\alpha >0$, $k\ge 0$, $0\le \delta <1$ and $\gamma >0$. Then $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;0,k,\delta \right)$.

Proof Consider

$h\left(z\right)={\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha },$
(2.1)

where h is analytic in E with $h\left(0\right)=1$, and $g\in {A}_{p}$ satisfies condition (1.9). Differentiating (2.1) logarithmically and using (1.7), we have

$\frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}=\alpha \left(\mu +p\right)\left\{\left(\frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}-\frac{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)\right\}.$

Using (2.1) and simplifying, we obtain

$\begin{array}{rcl}h\left(z\right)+\frac{\gamma z{h}^{\prime }\left(z\right)}{\alpha \left(\mu +p\right)q\left(z\right)}& =& \left(1-\gamma \right){\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }\\ +\gamma \frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}{\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha -1}.\end{array}$

Since $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;\gamma ,k,\delta \right)$, therefore, we can write

$h\left(z\right)+\frac{\gamma z{h}^{\prime }\left(z\right)}{\alpha \left(\mu +p\right)q\left(z\right)}\prec {q}_{k,\delta }\left(z\right),\phantom{\rule{1em}{0ex}}z\in E.$

Now using Lemma 1.3 for $A=0$ and $B\left(z\right)=\frac{\gamma }{\alpha \left(\mu +p\right)q\left(z\right)}$ with $Req\left(z\right)>0$, we have $ReB\left(z\right)\ge 0$, therefore, $h\left(z\right)\prec {q}_{k,\delta }\left(z\right)$. Hence $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;0,k,\delta \right)$. □

Theorem 2.2 Let $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;\gamma ,0,\delta \right)$ for $a,b,c\in \mathbb{R}\mathrm{\setminus }{\mathbb{Z}}_{0}^{-}$, $\mu >-p$, $p\in \mathbb{N}$. Then $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;0,0,{\delta }_{1}\right)$, where

${\delta }_{1}=\frac{2\alpha \delta \left(p+\mu \right)|q\left(z\right){|}^{2}+\gamma \rho }{2\alpha \left(p+\mu \right)|q\left(z\right){|}^{2}+\gamma \rho }.$

Proof Consider

$h\left(z\right)=\frac{1}{\left(1-{\delta }_{1}\right)}\left\{{\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }-{\delta }_{1}\right\},$
(2.2)

where h is analytic in E with $h\left(0\right)=1$, and $g\in {A}_{p}$ satisfies condition (1.9). Differentiating (2.2), we have

$\begin{array}{rcl}\frac{\left(1-{\delta }_{1}\right)}{\alpha }{h}^{\prime }\left(z\right)& =& {\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha -1}\\ ×\left\{\frac{{\left({I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)\right)}^{\prime }}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}-\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\frac{{\left({I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)\right)}^{\prime }}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right\}.\end{array}$

Using (1.7) and simplifying, we obtain

$\begin{array}{r}\left(1-\gamma \right){\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }+\gamma \frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}{\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha -1}\\ \phantom{\rule{1em}{0ex}}=\left(1-{\delta }_{1}\right)h\left(z\right)+{\delta }_{1}+\frac{\gamma \left(1-{\delta }_{1}\right)z{h}^{\prime }\left(z\right)}{\alpha \left(p+\mu \right)q\left(z\right)}.\end{array}$

Since $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;\gamma ,0,\delta \right)$, therefore we have

$\left(1-{\delta }_{1}\right)h\left(z\right)+{\delta }_{1}+\frac{\gamma \left(1-{\delta }_{1}\right)z{h}^{\prime }\left(z\right)}{\alpha \left(p+\mu \right)q\left(z\right)}\prec \frac{1+\left(1-2\delta \right)z}{1-z},\phantom{\rule{1em}{0ex}}0\le \delta <1,z\in E.$

This implies that

$\frac{1}{1-\delta }\left\{\left(1-{\delta }_{1}\right)h\left(z\right)+{\delta }_{1}-\delta +\frac{\gamma \left(1-{\delta }_{1}\right)z{h}^{\prime }\left(z\right)}{\alpha \left(p+\mu \right)q\left(z\right)}\right\}\in {q}_{0,0}\left(E\right)=P.$

To obtain our desired result, we show that $h\in P$, for $z\in E$. Let $u={u}_{1}+i{u}_{2}$, $v={v}_{1}+i{v}_{2}$, and let $\mathrm{\Psi }:D\subset {\mathbb{C}}^{2}\to \mathbb{C}$ be a complex-valued function such that $u=h\left(z\right)$, $v=z{h}^{\prime }\left(z\right)$. Then

$\mathrm{\Psi }\left(u,v\right)=\left(1-{\delta }_{1}\right)u+{\delta }_{1}-\delta +\frac{\gamma \left(1-{\delta }_{1}\right)v}{\alpha \left(p+\mu \right)q\left(z\right)}.$

The first two conditions of Lemma 1.2 are easily verified. To verify the third condition, we consider

$\begin{array}{r}Re\mathrm{\Psi }\left(i{u}_{2},{v}_{1}\right)\\ \phantom{\rule{1em}{0ex}}=Re\left\{\left(1-{\delta }_{1}\right)i{u}_{2}+{\delta }_{1}-\delta +\frac{\gamma \left(1-{\delta }_{1}\right){v}_{1}}{\alpha \left(p+\mu \right)q\left(z\right)}\right\}\\ \phantom{\rule{1em}{0ex}}={\delta }_{1}-\delta +Re\frac{\gamma \left(1-{\delta }_{1}\right){v}_{1}}{\alpha \left(p+\mu \right)q\left(z\right)}\\ \phantom{\rule{1em}{0ex}}\le {\delta }_{1}-\delta -Re\frac{\gamma \left(1-{\delta }_{1}\right)\left(1+{u}_{2}^{2}\right)\overline{q\left(z\right)}}{2\alpha \left(p+\mu \right)|q\left(z\right){|}^{2}}\\ \phantom{\rule{1em}{0ex}}\le {\delta }_{1}-\delta -\frac{\gamma \left(1-{\delta }_{1}\right)\left(1+{u}_{2}^{2}\right)\rho }{2\alpha \left(p+\mu \right)|q\left(z\right){|}^{2}}=\frac{A+B{u}^{2}}{C},\end{array}$

where $A=2\alpha \left(p+\mu \right)\left({\delta }_{1}-\delta \right)|q\left(z\right){|}^{2}-\gamma \rho \left(1-{\delta }_{1}\right)$, $B=-\gamma \rho \left(1-{\delta }_{1}\right)\le 0$ if $0\le {\delta }_{1}<1$ and $C=2\alpha \left(p+\mu \right)|q\left(z\right){|}^{2}>0$. From the relation ${\delta }_{1}=\frac{2\alpha \delta \left(p+\mu \right)|q\left(z\right){|}^{2}+\gamma \rho }{2\alpha \left(p+\mu \right)|q\left(z\right){|}^{2}+\gamma \rho }$, we have $A\le 0$. This implies that $Re\mathrm{\Psi }\left(i{u}_{2},{v}_{1}\right)\le 0$. Using Lemma 1.2, we have $h\in P$ for $z\in E$. This completes the proof. □

Theorem 2.3 Let $a,b,c\in \mathbb{R}\mathrm{\setminus }{\mathbb{Z}}_{0}^{-}$, $\mu >-p$, $p\in \mathbb{N}$, $\alpha >0$, $k\ge 0$ and $0\le \delta <1$. Then

$U{B}_{\mu ,p}^{\alpha }\left(a,b,c;{\gamma }_{2},k,\delta \right)\subset U{B}_{\mu ,p}^{\alpha }\left(a,b,c;{\gamma }_{1},k,\delta \right),\phantom{\rule{1em}{0ex}}0\le {\gamma }_{1}<{\gamma }_{2},z\in E.$

Proof Since $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;{\gamma }_{2},k,\delta \right)$, therefore, we have

$\begin{array}{r}\left(1-{\gamma }_{2}\right){\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }+{\gamma }_{2}\frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}{\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha -1}\\ \phantom{\rule{1em}{0ex}}={h}_{1}\left(z\right)\prec {q}_{k,\delta }\left(z\right),\end{array}$
(2.3)

where $g\in {A}_{p}$ satisfies condition (1.9). From Theorem 2.1, we write

${\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }={h}_{2}\left(z\right)\prec {q}_{k,\delta }\left(z\right),\phantom{\rule{1em}{0ex}}z\in E.$
(2.4)

Now, for ${\gamma }_{1}\ge 0$, we obtain

$\begin{array}{r}\left(1-{\gamma }_{1}\right){\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }+{\gamma }_{1}\frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}{\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha -1}\\ \phantom{\rule{1em}{0ex}}=\left(1-\frac{{\gamma }_{1}}{{\gamma }_{2}}\right){\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }\\ \phantom{\rule{2em}{0ex}}+\frac{{\gamma }_{1}}{{\gamma }_{2}}\left\{\left(1-{\gamma }_{2}\right){\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }+{\gamma }_{2}\frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}{\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha -1}\right\}\\ \phantom{\rule{1em}{0ex}}=\frac{{\gamma }_{1}}{{\gamma }_{2}}{h}_{1}\left(z\right)+\left(1-\frac{{\gamma }_{1}}{{\gamma }_{2}}\right){h}_{2}\left(z\right).\end{array}$

Using the convexity of the class of the function ${q}_{k,\delta }\left(z\right)$ and Lemma 1.4, we write

$\frac{{\gamma }_{1}}{{\gamma }_{2}}{h}_{1}\left(z\right)+\left(1-\frac{{\gamma }_{1}}{{\gamma }_{2}}\right){h}_{2}\left(z\right)\prec {q}_{k,\delta }\left(z\right),\phantom{\rule{1em}{0ex}}z\in E,$

where ${h}_{1}$ and ${h}_{2}$ are given by (2.3) and (2.4), respectively. This implies that $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;{\gamma }_{1},k,\delta \right)$. Hence the proof of the theorem is completed. □

Theorem 2.4 Let $f\in U{B}_{\mu ,p}^{1}\left(a,b,c;\gamma ,k,\delta \right)$, $a,b,c\in \mathbb{R}\mathrm{\setminus }{\mathbb{Z}}_{0}^{-}$, $\mu >-p$, $p\in \mathbb{N}$, $\alpha >0$, $k\ge 0$, $\gamma \ge 1$ and $0\le \delta <1$. Then $f\in U{B}_{\mu +1,p}^{1}\left(a,b,c;1,k,\delta \right)$.

Proof Since $f\in U{B}_{\mu ,p}^{1}\left(a,b,c;\gamma ,k,\delta \right)$, therefore, we have

$\left(1-\gamma \right)\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)+\gamma \frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}\prec {q}_{k,\delta }\left(z\right).$

Now, consider

$\begin{array}{rcl}\gamma \frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}& =& \left(1-\gamma \right)\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)+\gamma \frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}\\ +\left(\gamma -1\right)\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right).\end{array}$

This implies that

$\begin{array}{rcl}\frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}& =& \frac{1}{\gamma }\left\{\left(1-\gamma \right)\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)+\gamma \frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}\right\}\\ +\left(1-\frac{1}{\gamma }\right)\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right).\end{array}$

Using Theorem 2.1, Lemma 1.4 and the convexity of ${q}_{k,\delta }\left(z\right)$, we have the required result. □

Now, using the operators ${I}_{\mu ,p}\left(a,b,c\right)$ and ${F}_{\eta ,p}$ defined by (1.7) and (1.10), respectively, we have

$z{\left({I}_{p}^{\mu }\left(a,b,c\right){F}_{\eta ,p}\left(f\right)\left(z\right)\right)}^{\prime }=\left(p+\eta \right){I}_{p}^{\mu }\left(a,b,c\right)f\left(z\right)-\eta {I}_{p}^{\mu }\left(a,b,c\right){F}_{\eta ,p}\left(f\right)\left(z\right),\phantom{\rule{1em}{0ex}}\eta >-p.$
(2.5)

Theorem 2.5 Let $f\in {A}_{p}$ and ${F}_{\eta ,p}$ be given by (1.10). If

$\left(1-\gamma \right)\frac{{I}_{\mu ,p}\left(a,b,c\right){F}_{\eta ,p}\left(f\right)\left(z\right)}{{z}^{p}}+\gamma \frac{{I}_{\mu ,p}\left(a,b,c\right)\left(f\left(z\right)\right)}{{z}^{p}}\prec {q}_{k,\delta }\left(z\right),\phantom{\rule{1em}{0ex}}z\in E,$
(2.6)

with $a,b,c\in \mathbb{R}\mathrm{\setminus }{\mathbb{Z}}_{0}^{-}$, $\mu ,\eta >-p$, $p\in \mathbb{N}$, $\gamma >0$, then

$\frac{{I}_{\mu ,p}\left(a,b,c\right){F}_{\eta ,p}\left(f\left(z\right)\right)}{{z}^{p}}\prec h\left(z\right)\prec {q}_{k,\delta }\left(z\right),\phantom{\rule{1em}{0ex}}z\in E,$

where

$h\left(z\right)=\frac{p+\eta }{\gamma {z}^{\left(p+\eta \right)/\gamma }}{\int }_{0}^{z}{q}_{k,\delta }\left(z\right){t}^{{}^{\left(p+\eta \right)/\gamma }-1}\phantom{\rule{0.2em}{0ex}}dt.$

Proof Let

$\frac{{I}_{\mu ,p}\left(a,b,c\right){F}_{\eta ,p}\left(f\left(z\right)\right)}{{z}^{p}}={h}_{1}\left(z\right),\phantom{\rule{1em}{0ex}}z\in E,$

where ${h}_{1}$ is analytic in E with ${h}_{1}\left(0\right)=1$. Then

$z{\left({I}_{\mu ,p}\left(a,b,c\right){F}_{\eta ,p}\left(f\left(z\right)\right)\right)}^{\prime }=p{z}^{p}{h}_{1}\left(z\right)+{z}^{p+1}{h}_{1}^{\prime }\left(z\right).$

Using (2.5), we have

$\gamma \left(p+\eta \right)\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{z}^{p}}-\gamma \eta \frac{{I}_{\mu ,p}\left(a,b,c\right){F}_{\eta ,p}\left(f\right)\left(z\right)}{{z}^{p}}=p\gamma {h}_{1}\left(z\right)+\gamma z{h}_{1}^{\prime }\left(z\right).$

Thus,

$\left(1-\gamma \right)\frac{{I}_{\mu ,p}\left(a,b,c\right){F}_{\eta ,p}\left(f\left(z\right)\right)}{{z}^{p}}+\gamma \frac{{I}_{\mu ,p}\left(a,b,c\right)\left(f\left(z\right)\right)}{{z}^{p}}={h}_{1}\left(z\right)+\gamma \frac{z{h}_{1}^{\prime }\left(z\right)}{p+\eta }.$
(2.7)

From (2.7), it follows that

${h}_{1}\left(z\right)+\gamma \frac{z{h}_{1}^{\prime }\left(z\right)}{p+\eta }\prec {q}_{k,\delta }\left(z\right),\phantom{\rule{1em}{0ex}}z\in E.$

Using Lemma 1.5, for ${\beta }_{1}=\frac{p+\eta }{\gamma }$, $n=1$ and $a=1$, we obtain ${h}_{1}\left(z\right)\prec h\left(z\right)\prec {q}_{k,\delta }\left(z\right)$. That is, $\frac{{I}_{\mu ,p}\left(a,b,c\right){F}_{\eta ,p}\left(f\left(z\right)\right)}{{z}^{p}}\prec {q}_{k,\delta }\left(z\right)$. □

Theorem 2.6 Let $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;0,0,\delta \right)$ for $a,b,c\in \mathbb{R}\mathrm{\setminus }{\mathbb{Z}}_{0}^{-}$, $\mu >-p$, $p\in \mathbb{N}$, $\alpha ,\gamma >0$, $0\le \delta <1$. Then $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;\gamma ,0,\delta \right)$, for $|z|<{r}_{0}$, where

${r}_{0}=\frac{\alpha \left(p}{+}\mu \right)+\gamma -\sqrt{{\gamma }^{2}+2\alpha \gamma \left(p+\mu \right)}\alpha \left(p+\mu \right).$
(2.8)

Proof Let $f\in U{B}_{\mu ,p}^{\alpha }\left(a,b,c;0,0,\delta \right)$. Then we have

${\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }=\left(1-\delta \right)h\left(z\right)+\delta ,$
(2.9)

where $g\in {A}_{p}$ satisfies the condition

$q\left(z\right)=\frac{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\in P,\phantom{\rule{1em}{0ex}}z\in E$

and $h\in P$. Differentiating (2.9) and then using (1.7), we obtain

$\gamma \frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}{\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha -1}-\gamma {\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }=\frac{\gamma \left(p-\delta \right)z{h}^{\prime }\left(z\right)}{\alpha \left(p+\mu \right)q\left(z\right)}.$

This implies that

$\begin{array}{r}\frac{1}{1-\delta }\left\{\left(1-\gamma \right){\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha }+\gamma \frac{{I}_{\mu +1,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu +1,p}\left(a,b,c\right)g\left(z\right)}{\left(\frac{{I}_{\mu ,p}\left(a,b,c\right)f\left(z\right)}{{I}_{\mu ,p}\left(a,b,c\right)g\left(z\right)}\right)}^{\alpha -1}-\delta \right\}\\ \phantom{\rule{1em}{0ex}}=h\left(z\right)+\frac{\gamma z{h}^{\prime }\left(z\right)}{\alpha \left(p+\mu \right)q\left(z\right)},\phantom{\rule{1em}{0ex}}z\in E.\end{array}$
(2.10)

Now, using the well-known distortion result for class P, we have

$|z{h}^{\prime }\left(z\right)|\le \frac{2rReh\left(z\right)}{1-{r}^{2}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}Reh\left(z\right)\ge \frac{1-r}{1+r},\phantom{\rule{1em}{0ex}}|z|

Thus, due to the applications of these inequalities, we have

$\begin{array}{rl}Re\left(h\left(z\right)+\frac{\gamma z{h}^{\prime }\left(z\right)}{\alpha \left(p+\mu \right)q\left(z\right)}\right)& \ge Reh\left(z\right)-\frac{\gamma |z{h}^{\prime }\left(z\right)|}{\alpha \left(p+\mu \right)|q\left(z\right)|}\\ \ge Reh\left(z\right)\left(1-\frac{2\gamma r}{\alpha \left(p+\mu \right){\left(1-r\right)}^{2}}\right)\\ =Reh\left(z\right)\left(\frac{\alpha \left(p+\mu \right){\left(1-r\right)}^{2}-2\gamma r}{\alpha \left(p+\mu \right){\left(1-r\right)}^{2}}\right).\end{array}$

For $|z|<{r}_{0}$, where ${r}_{0}$ is given in (2.8), the inequality above is positive. Sharpness of the result follows by taking $h\left(z\right)=\frac{1+z}{1-z}$. Hence from (2.10), we have the required result. □

## References

1. Ruscheweyh S: New criteria for univalent functions. Proc. Am. Math. Soc. 1975, 49: 109–115. 10.1090/S0002-9939-1975-0367176-1

2. Ul-Haq, W: Some Classes of Generalized Convex and Related Functions. Ph.D. thesis, COMSATS Institute of Information Technology, Islamabad, Pakistan (2011)

3. Noor KI: Integral operators defined by convolution with hypergometric functions. Appl. Math. Comput. 2006, 182: 1872–1881. 10.1016/j.amc.2006.06.023

4. Cho NE, Kwon OS, Srivastava HM: Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators. J. Math. Anal. Appl. 2004, 292: 470–483.

5. Patel J, Cho NE: Some classes of analytic functions involving Noor integral operator. J. Math. Anal. Appl. 2005, 312: 564–575. 10.1016/j.jmaa.2005.03.047

6. Liu J-L: The Noor integral and strongly starlike functions. J. Math. Anal. Appl. 2001, 261: 441–447. 10.1006/jmaa.2001.7489

7. Liu J-L, Noor KI: Some properties of Noor integral operator. J. Nat. Geom. 2002, 21: 81–90.

8. Patel J, Rout S: Properties of certain analytic functions involving Ruscheweyh’s derivatives. Math. Jpn. 1994, 39: 509–518.

9. Ponnusamy S, Karanukaran V: Differential subordination and conformal mappings. Complex Var. Theory Appl. 1988, 11: 70–86.

10. Singh R: On Bazilevič functions. Proc. Am. Math. Soc. 1973, 38: 261–271.

11. Chen MP:On the regular functions satisfying $Ref\left(z\right)/z$. Bull. Inst. Math. Acad. Sin. 1975, 3: 65–70.

12. Miller SS, Mocanu PT: Differential subordination and inequalities in the complex plane. J. Differ. Equ. 1987, 67: 199–211. 10.1016/0022-0396(87)90146-X

13. Miller SS, Mocanu PT: Second order differential inequalities in the complex plane. J. Math. Anal. Appl. 1978, 65: 289–305. 10.1016/0022-247X(78)90181-6

14. Liu M-S: On certain subclass of analytic functions. J. South China Normal Univ. Natur. Sci. Ed. 2002, 4: 15–20. (in Chinese)

15. Hallenbeck DJ, Ruscheweyh S: Subordination by convex functions. Proc. Am. Math. Soc. 1975, 52: 191–195. 10.1090/S0002-9939-1975-0374403-3

## Author information

Authors

### Corresponding author

Correspondence to Mohsan Raza.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

MR and SNM jointly discussed and presented the ideas of this article. MR made the text file and corresponded it to the journal. Both authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

Raza, M., Malik, S.N. Properties of multivalent functions associated with the integral operator defined by the hypergeometric function. J Inequal Appl 2013, 458 (2013). https://doi.org/10.1186/1029-242X-2013-458

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1029-242X-2013-458

### Keywords

• Bazilevic functions
• multivalent functions
• integral operator
• hypergeometric functions 