Subclasses of p-Valent Functions Associated with Linear q-Differential Borel Operator

Abstract

The aim of the present paper is to introduce and study some new subclasses of p-valent functions by making use of a linear q-differential Borel operator.We also deduce some properties, such as inclusion relationships of the newly introduced classes and the integral operator ${\mathcal{J}}_{\mu ,p}$.

1. Introduction

Let ${\mathcal{A}}_p$ denote the class of functions of the form:

$$\mathcal{F}\left(\varsigma \right)={\varsigma }^p+\sum _{j=p+1}^{\infty }{a}_j{\varsigma }^j(p\in \mathbb{N}=\{1,2,...\}),$$

(1)

which are analytic in the open unit disc $\Delta =\left\{\varsigma \in \mathbb{C}:\left|\varsigma \right|<1\right\}.$

Let ${\mathcal{P}}_{p,k}\left(\alpha \right)$ be the class of functions $h\left(\varsigma \right)$ analytic in $\Delta$ satisfying the properties $h\left(0\right)=p$ and

$$\underset{0}{\overset{2\pi }{\int }}\left|\frac{\Re \left\{h\left(\varsigma \right)\right\}-\alpha }{p-\alpha }\right|d\theta \le k\pi ,$$

(2)

where $\varsigma =r{e}^{i\theta },k\ge 2$ and $0\le \alpha <p.$ This class was introduced by (Aouf [1] with $\lambda =0$).

We note that

(i)

${\mathcal{P}}_{1,k}\left(\alpha \right)={\mathcal{P}}_k\left(\alpha \right)$$\left(k\ge 2,0\le \alpha <1\right)$ (see Padmanabhan and Parvatham [2]);

(ii)

${\mathcal{P}}_{1,k}\left(0\right)={\mathcal{P}}_k\left(k\ge 2\right)$(see Pinchuk [3] and Robertson [4]);

(iii)

${\mathcal{P}}_{p,2}\left(\alpha \right)=\mathcal{P}\left(p,\alpha \right)\left(0\le \alpha <p,p\in \mathbb{N}\right),$ where $\mathcal{P}\left(p,\alpha \right)$ is the class of functions with a positive real part greater than $\alpha$(see [1]);

(iv)

${\mathcal{P}}_{p,2}\left(0\right)=\mathcal{P}\left(p\right)\left(p\in \mathbb{N}\right),$ where $\mathcal{P}\left(p\right)$ is the class of functions with a positive real part (see [1]).

From (2), we have $h\left(\varsigma \right)\in {\mathcal{P}}_{p,k}\left(\alpha \right)$ if and only if there exists $h_1,h_2\in {\mathcal{P}}_p\left(\alpha \right)$ such that

$$\mathcal{G}\left(\varsigma \right)=\left(\frac{k}{4}+\frac{1}{2}\right)h_1\left(\varsigma \right)-\left(\frac{k}{4}-\frac{1}{2}\right)h_2\left(\varsigma \right)\left(\varsigma \in \Delta \right).$$

(3)

For two functions $\mathcal{F}\left(\varsigma \right)$ given by (1) and $\mathcal{H}\left(\varsigma \right)$ given by

$$\mathcal{H}\left(\varsigma \right)={\varsigma }^p+\sum _{j=p+1}^{\infty }{b}_j{\varsigma }^j$$

the Hadamard product (or convolution) is defined by

$$(\mathcal{F}\ast \mathcal{H})\left(\varsigma \right)={\varsigma }^p+\sum _{j=p+1}^{\infty }{a}_j{b}_j{\varsigma }^j=(\mathcal{H}\ast \mathcal{F})\left(\varsigma \right).$$

(4)

Define here a Borel distribution with parameter $\lambda$, which is a discrete random variable denoted by $\chi$. This variable takes the values $1,2,3,...$ with the probabilities $\frac{e^{-\lambda }}{1!},\frac{2\lambda e^{-2\lambda }}{2!},\frac{9{\lambda }^2{e}^{-3\lambda }}{3!},...$, respectively.

Wanas and Khuttar [5] recently introduced the Borel distribution (BD) whose probability mass function is (see [6,7])

$$P(\chi =\rho )=\frac{{\left(\rho \lambda \right)}^{\rho -1}{e}^{-\lambda \rho }}{\rho !},\rho =1,2,3,...$$

Wanas and Khuttar studied a series $\mathcal{M}(\lambda ;\varsigma )$ whose coefficients are probabilities of the Borel distribution (BD)

$$\begin{array}{ccc}\hfill {\mathcal{M}}_p(\lambda ;\varsigma )& =& {\varsigma }^p+\sum _{j=p+1}^{\infty }\frac{{\left[\lambda \left(j-p\right)\right]}^{j-p-1}{e}^{-\lambda \left(j-p\right)}}{\left(j-p\right)!}{\varsigma }^j,\left(0<\lambda \le 1\right),\hfill \\ & =& {\varsigma }^p+\sum _{j=p+1}^{\infty }{\varphi }_{j,p}\left(\lambda \right){\varsigma }^k,\left(0<\lambda \le 1\right),\hfill \end{array}$$

(5)

where

$${\varphi }_{j,p}\left(\lambda \right)=\frac{{\left[\lambda \left(j-p\right)\right]}^{j-p-1}{e}^{-\lambda \left(j-p\right)}}{\left(j-p\right)!}$$

(6)

We propose a linear operator $\mathcal{D}(p,\lambda ;\varsigma )\mathcal{F}:{\mathcal{A}}_p\to {\mathcal{A}}_p$ as follows

$$\begin{array}{ccc}\hfill \mathcal{D}(p,\lambda ;\varsigma )\mathcal{F}\left(\varsigma \right)& =& {\mathcal{M}}_p(\lambda ;\varsigma )\ast \mathcal{F}\left(\varsigma \right)\hfill \\ & =& {\varsigma }^p+\sum _{j=p+1}^{\infty }\frac{{\left[\lambda \left(j-p\right)\right]}^{j-p-1}{e}^{-\lambda \left(j-p\right)}}{\left(j-p\right)!}{a}_j{\varsigma }^j,\left(0<\lambda \le 1\right).\hfill \end{array}$$

In a recent paper, Srivastava [8] studied various types of operators regarding q-calculus. We recall further some important definitions and notations. The q-shifted factorial is defined for $\lambda ,q\in \mathbb{C}$ and $n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ as follows

$${(\mu ;q)}_j=\left\{\begin{array}{cc} 1\hfill & j=0,\hfill \\ \left(1-\mu \right)\left(1-\mu q\right)...\left(1-\mu q^{j-1}\right)\hfill & j\in \mathbb{N}.\hfill \end{array}\right.$$

By using the q-gamma function ${\mathsf{\Gamma }}_q\left(\varsigma \right),$ we get

$${\left(q^{\mu };q\right)}_j=\frac{{\left(1-q\right)}^j{\mathsf{\Gamma }}_q\left(\mu +j\right)}{{\mathsf{\Gamma }}_q\left(\mu \right)},\left(j\in {\mathbb{N}}_0\right),$$

where (see [9])

$${\mathsf{\Gamma }}_q\left(\varsigma \right)={\left(1-q\right)}^{1-\varsigma }\frac{{\left(q;q\right)}_{\infty }}{{\left(q^{\varsigma };q\right)}_{\infty }},\left(\left|q\right|<1\right).$$

Furthermore, we note that

$${\left(\mu ;q\right)}_{\infty }=\prod _{j=0}^{\infty }\left(1-\mu q^j\right),\left(\left|q\right|<1\right),$$

and, the q-gamma function ${\mathsf{\Gamma }}_q\left(\varsigma \right)$ is known

$${\mathsf{\Gamma }}_q(\varsigma +1)={\left[j\right]}_q{\mathsf{\Gamma }}_q\left(\varsigma \right),$$

where ${\left[j\right]}_q$ denotes the basic q-number defined as follows

$${\left[j\right]}_q:=\left\{\begin{array}{c}\frac{1-q^j}{1-q},j\in \mathbb{C},\\ 1+{\displaystyle \sum _{i=1}^{j-1}}{q}^i,j\in \mathbb{N}.\end{array}\right.$$

(7)

Using the definition from (7), we have the next two products:

(i)

For a non negative integer j, the q-shifted factorial is defined by

$${\left[j\right]}_q!:=\left\{\begin{array}{ccc}1,\hfill & \mathrm{if}\hfill & j=0,\hfill \\ {\displaystyle \prod _{n=1}^j}{\left[n\right]}_q,\hfill & \mathrm{if}\hfill & j\in \mathbb{N}.\hfill \end{array}\right.$$

(ii)

For a positive number r, the q-generalized Pochhammer symbol is given by

$${\left[r\right]}_{q,j}:=\left\{\begin{array}{ccc}1,\hfill & \mathrm{if}\hfill & j=0,\hfill \\ {\displaystyle \prod _{n=r}^{r+j-1}}{\left[n\right]}_q,\hfill & \mathrm{if}\hfill & j\in \mathbb{N}.\hfill \end{array}\right.$$

In terms of the classical (Euler’s) gamma function $\mathsf{\Gamma }\left(\varsigma \right)$, we have

$${\mathsf{\Gamma }}_q\left(\varsigma \right)\to \mathsf{\Gamma }\left(\varsigma \right)\mathrm{as}q\to 1^{-}.$$

Furthermore, we notice that

$$\underset{q\to 1^{-}}{lim}\left\{\frac{{\left(q^{\mu };q\right)}_j}{{\left(1-q\right)}^j}\right\}={\left(\mu \right)}_j.$$

2. Preliminaries

In order to establish our new results, we have to recall the construct of a q-derivative operator. Considering $0<q<1$, the q-derivative operator [10] (see also other specific and generalized results [11,12,13,14,15]) for $\mathcal{D}(p,\lambda ;\varsigma )\mathcal{F}$ is defined by

$$\begin{array}{ccc}\hfill {\mathcal{D}}_q\left(\mathcal{D}(p,\lambda ;\varsigma )\mathcal{F}\left(\varsigma \right)\right)& :& =\frac{\mathcal{D}(p,\lambda ;\varsigma )\mathcal{F}\left(\varsigma \right)-\mathcal{D}(p,\lambda ;\varsigma )\mathcal{F}\left(q\varsigma \right)}{\varsigma (1-q)}\hfill \\ & =& {\left[p\right]}_q{\varsigma }^{p-1}+\sum _{j=p+1}^{\infty }{\left[j\right]}_q\frac{{\left[\lambda \left(j-p\right)\right]}^{j-p-1}{e}^{-\lambda \left(j-p\right)}}{\left(j-p\right)!}{a}_j{\varsigma }^{j-1},\hfill \end{array}$$

where ${\left[j\right]}_q$ is defined in (7)

For $\alpha >-1$ and $0<q<1$, we obtain the linear operator ${\mathcal{D}}_{p,\lambda }^{\mu ,q}\mathcal{F}:{\mathcal{A}}_p\to {\mathcal{A}}_p$ by

$${\mathcal{D}}_{p,\lambda }^{\mu ,q}\mathcal{F}\left(\varsigma \right)\ast {\mathcal{N}}_{p,\mu +1}^q\left(\varsigma \right)=\frac{\varsigma }{{\left[p\right]}_q}{\mathcal{D}}_q\left(\mathcal{D}(p,\lambda ;\varsigma )\mathcal{F}\left(\varsigma \right)\right),\varsigma \in \Delta ,$$

where the function ${\mathcal{N}}_{p,\alpha +1}^q$ is given by

$${\mathcal{N}}_{p,\mu +1}^q\left(\varsigma \right):={\varsigma }^p+\sum _{j=p+1}^{\infty }\frac{{[\mu +1]}_{q,j-p}}{{[j-1]}_q!}{\varsigma }^j,\varsigma \in \Delta .$$

A simple computation shows that

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{p,\lambda }^{\mu ,q}\mathcal{F}\left(\varsigma \right)& :& ={\varsigma }^p+\sum _{j=p+1}^{\infty }\frac{{\left[j\right]}_q!{\left[\lambda \left(j-p\right)\right]}^{j-p-1}{e}^{-\lambda \left(j-p\right)}}{{\left[p\right]}_q{[\mu +1]}_{q,j-p}\left(j-p\right)!}{a}_j{\varsigma }^j\hfill \\ & =& z^p+\sum _{j=p+1}^{\infty }{\varphi }_j{a}_j{\varsigma }^j(0<\lambda \le 1,\mu >p,0<q<1,\varsigma \in \Delta ).\hfill \end{array}$$

(8)

where

$${\varphi }_j=\frac{{\left[j\right]}_q!{\left[\lambda \left(j-p\right)\right]}^{j-p-1}{e}^{-\lambda \left(j-p\right)}}{{\left[p\right]}_q{[\mu +1]}_{q,j-p}\left(j-p\right)!}.$$

(9)

For $\delta \ge 0,$ with the aid of the operator ${\mathcal{D}}_{p,\lambda }^{\mu ,q}$ one can defined the linear q-differential Borel operator ${\mathcal{A}}_p\to {\mathcal{A}}_p$ as follows:

$$\begin{array}{ccc}\hfill {\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,0}\mathcal{F}\left(\varsigma \right)& :& ={\mathcal{D}}_{p,\lambda }^{\mu ,q}\mathcal{F}\left(\varsigma \right)\hfill \\ \hfill {\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,1}\mathcal{F}\left(\varsigma \right)& :& =\left(1-\delta \right){\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,0}\mathcal{F}\left(\varsigma \right)+\delta \frac{\varsigma }{p}{\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,0}\mathcal{F}\left(\varsigma \right)\right)}^{\prime }\hfill \\ & =& {\varsigma }^p+\sum _{j=p+1}^{\infty }\frac{{\left[j\right]}_q!{\left[\lambda \left(j-p\right)\right]}^{j-p-1}{e}^{-\lambda \left(j-p\right)}}{{\left[p\right]}_q{[\mu +1]}_{q,j-p}\left(j-p\right)!}\left[1+\delta \left(\frac{j}{p}-1\right)\right]a_j{\varsigma }^j\hfill \\ \hfill {\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,2}\mathcal{F}\left(\varsigma \right)& :& =\left(1-\delta \right){\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,1}\mathcal{F}\left(\varsigma \right)+\delta \frac{\varsigma }{p}{\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,1}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}\hfill \\ & =& {\varsigma }^p+\sum _{j=p+1}^{\infty }\frac{{\left[j\right]}_q!{\left[\lambda \left(j-p\right)\right]}^{j-p-1}{e}^{-\lambda \left(j-p\right)}}{{\left[p\right]}_q{[\mu +1]}_{q,j-p}\left(j-p\right)!}{\left[1+\delta \left(\frac{j}{p}-1\right)\right]}^2{a}_j{\varsigma }^j\hfill \\ & & .\hfill \\ & & .\hfill \\ & & .\hfill \\ \hfill {\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)& :& ={\varsigma }^p+\sum _{j=p+1}^{\infty }\frac{{\left[j\right]}_q!{\left[\lambda \left(j-p\right)\right]}^{j-p-1}{e}^{-\lambda \left(j-p\right)}}{{\left[p\right]}_q{[\mu +1]}_{q,j-p}\left(j-p\right)!}{\left[1+\delta \left(\frac{j}{p}-1\right)\right]}^m{a}_j{\varsigma }^j,\hfill \\ & & \left(m\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},\delta \ge 0,0<\lambda \le 1,\mu >p,0<q<1\right).\hfill \end{array}$$

(10)

From the relation (10), we can easily deduce that the next relations held for all $\mathcal{F}\in {\mathcal{A}}_p$:

$$\left(\mathrm{i}\right)\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)\right)}^{\prime }=\mu {\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)-\left(\mu -p\right){\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right),$$

(11)

and

$$\left(\mathrm{ii}\right)\delta \varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)\right)}^{\prime }=p{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m+1}\mathcal{F}\left(\varsigma \right)-p\left(1-\delta \right){\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)$$

(12)

Remark 1. 

By particularizing the parameters $p$ and $m,$ we derive the following operators based on Borel distribution:

(1)

Letting $p=1,$ we obtain that ${\mathcal{G}}_{1,q,\lambda ,\delta }^{\mu ,m}=:{\mathcal{I}}_{q,\lambda ,\delta }^{\mu ,m}$, where the operator ${\mathcal{I}}_{q,\lambda ,\delta }^{\mu ,m}$ is defined as follows:

$${\mathcal{I}}_{q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right):=\varsigma +\sum _{j=2}^{\infty }\frac{{\left[j\right]}_q!{\left[\lambda \left(j-1\right)\right]}^{j-2}{e}^{-\lambda \left(j-1\right)}}{{[\mu +1]}_{q,j-1}\left(j-1\right)!}{\left[1+\delta \left(j-1\right)\right]}^m{a}_j{\varsigma }^j;$$

(2)

Letting $p=1$ and $m=0,$ we deduce that ${\mathcal{G}}_{1,q,\lambda ,\delta }^{\mu ,0}=:{\mathcal{B}}_{\lambda }^{\mu ,q}$, where the operator ${\mathcal{B}}_{\lambda }^{\mu ,q}$, introduced by El-Deeb and Murugusundaramoorthy [16];

(3)

Letting $q\to 1^{-}$ and $p=1$, we deduce that $\underset{q\to 1^{-}}{lim}{\mathcal{G}}_{1,q,\lambda ,\delta }^{\mu ,m}:={\mathcal{R}}_{\lambda ,\delta }^{\mu ,m},$ where the operator ${\mathcal{R}}_{\lambda ,\delta }^{\mu ,m}$ is defined as follows

$${\mathcal{R}}_{\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right):=\varsigma +\sum _{j=2}^{\infty }\frac{j{\left[\lambda \left(j-1\right)\right]}^{j-2}{e}^{-\lambda \left(j-1\right)}}{{(\mu +1)}_{j-1}}{\left[1+\delta \left(j-1\right)\right]}^m{a}_j{\varsigma }^j;$$

(4)

Putting $q\to 1^{-},$$p=1$ and $m=0$, we obtain that $\underset{q\to 1^{-}}{lim}{\mathcal{G}}_{1,q,\lambda ,\delta }^{\mu ,0}:={\mathcal{M}}_{\lambda }^{\mu },$ where the operator ${\mathcal{M}}_{\lambda }^{\mu }$, studied byEl-Deeb and Murugusundaramoorthy [16].

Now we introduce the following classes ${\mathcal{S}}_p^k\left(\alpha \right),{\mathcal{C}}_p^k\left(\alpha \right)$ and ${\mathcal{K}}_p^k\left(\beta ,\alpha \right)$ of the class ${\mathcal{A}}_p$ for $0\le \alpha ,\beta <p,p\in \mathbb{N}$ and $k\ge 2$ as follows:

$${\mathcal{S}}_p^k\left(\alpha \right)=\left\{\mathcal{F}:\mathcal{F}\in {\mathcal{A}}_p\mathrm{and}\frac{\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}{\mathcal{F}\left(\varsigma \right)}\in {\mathcal{P}}_{p,k}\left(\alpha \right),\varsigma \in \Delta \right\},$$

$${\mathcal{C}}_p^k\left(\alpha \right)=\left\{\mathcal{F}:\mathcal{F}\in {\mathcal{A}}_p\mathrm{and}1+\frac{\varsigma {\mathcal{F}}^{{}^{{}^{\prime \prime }}}\left(\varsigma \right)}{{\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}\in {\mathcal{P}}_{p,k}\left(\alpha \right),\varsigma \in \Delta \right\},$$

and

$${\mathcal{K}}_p^k\left(\beta ,\alpha \right)=\left\{\mathcal{F}:\mathcal{F}\in {\mathcal{A}}_p,g\in {\mathcal{S}}_p^2\left(\alpha \right)\mathrm{and}\frac{\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}{g\left(\varsigma \right)}\in {\mathcal{P}}_{p,k}\left(\beta \right),\varsigma \in \Delta \right\}.$$

Obviously, we know that

$$\mathcal{F}\left(\varsigma \right)\in {\mathcal{C}}_p^k\left(\alpha \right)\iff \frac{\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}{p}\in {\mathcal{S}}_p^k\left(\alpha \right).$$

(13)

Remark 2. 

By particularizing the parameter $k,$ we obtain the following classes:

(i)

${\mathcal{S}}_p^2\left(\alpha \right)={\mathcal{S}}_p^{\ast }\left(\alpha \right)$$\left(0\le \alpha <p,p\in \mathbb{N}\right),$ where ${\mathcal{S}}_p^{\ast }\left(\alpha \right)$ is the well-known class of $p-$valently starlike functions of order α and was studied by Patil and Thakare [17];

(ii)

${\mathcal{C}}_p^2\left(\alpha \right)={\mathcal{C}}_p\left(\alpha \right)$$\left(0\le \alpha <p,p\in \mathbb{N}\right),$ where ${\mathcal{C}}_p\left(\alpha \right)$ is the well-known class of $p-$valently convex functions of order α and was studied by Owa [18];

(iii)

${\mathcal{K}}_p^2\left(\beta ,\alpha \right)={\mathcal{K}}_p(\beta ,\alpha )$$\left(0\le \alpha <p,p\in \mathbb{N}\right),$ where ${\mathcal{K}}_p(\beta ,\alpha )$ is the class of all $p-$valently close-to-convex functions of order β and type $\alpha$ and was introduced by Aouf [19].

Next, by making use of the operator defined by (10), we obtain the following subclasses ${\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)$, ${\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)$ and ${\mathcal{K}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\beta ,\alpha \right)$ of the class ${\mathcal{A}}_p$ as follows:

$${\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)=\left\{\mathcal{F}:\mathcal{F}\in {\mathcal{A}}_p\mathrm{and}{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)\in {\mathcal{S}}_p^k\left(\alpha \right),\varsigma \in \Delta \right\},$$

(14)

$${\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)=\left\{\mathcal{F}:\mathcal{F}\in {\mathcal{A}}_p\mathrm{and}{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)\in {\mathcal{C}}_p^k\left(\alpha \right),\varsigma \in \Delta \right\},$$

(15)

and

$${\mathcal{K}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\beta ,\alpha \right)=\left\{\mathcal{F}:\mathcal{F}\in {\mathcal{A}}_p\mathrm{and}{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)\in {\mathcal{K}}_p^k\left(\beta ,\alpha \right),\varsigma \in \Delta \right\}.$$

(16)

We can easily see that

$$\mathcal{F}\left(\varsigma \right)\in {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)\iff \frac{\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}{p}\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)$$

(17)

In order to establish our main results, we will require the following lemmas.

Lemma 1 

([20,21]). Let $\mathsf{\Phi }(r,s)$ be complex valued function, $\mathsf{\Phi }:$$\mathcal{D}\to \mathbb{C},$$\mathcal{D}\subset \mathbb{C}\times \mathbb{C}$$(\mathbb{C}$ is the complex plane) and let $r=r_1+i{r}_2,$ $s=s_1+i{s}_2.$ Suppose that $\mathsf{\Phi }(r,s)$ satisfies the following conditions:

(i)

$\mathsf{\Phi }(r,s)$ is continuous in a domain $\mathcal{D};$

(ii)

$(1,0)\in \mathcal{D}$ and $\Re \left\{\mathsf{\Phi }(1,0)\right\}>0;$

(iii)

$\Re \left\{\mathsf{\Phi }(i{r}_2,s_1)\right\}\le 0$ for all $(i{r}_2,s_1)\in \mathcal{D}$ and such that $s_1\le -{\displaystyle \frac{1}{2}}(1+r_2^2).$

Let $h\left(\varsigma \right)=1+{\displaystyle \sum _{m=1}^{\infty }}{c}_m{\varsigma }^m,$ be regular in Δ such that $(h\left(\varsigma \right),\varsigma h^{{}^{\prime }}\left(\varsigma \right))\in \mathcal{D}$ for all $\varsigma \in \Delta .$ If

$$\Re \left\{\mathsf{\Phi }(h\left(\varsigma \right),\varsigma h^{{}^{\prime }}\left(\varsigma \right))\right\}>0(\varsigma \in \Delta ),$$

then

$$\Re \left\{h\left(\varsigma \right)\right\}>0(\varsigma \in \Delta ).$$

Lemma 2 

([22]). Let Φ be convex and $\mathcal{F}$ be starlike in Δ. Then, for Υ analytic in Δ with $\mathsf{{\rm Y}}\left(0\right)=1$, $\frac{\mathsf{\Phi }\ast \mathsf{{\rm Y}}\mathcal{F}}{\mathsf{\Phi }\ast \mathcal{F}}$ is contained in the convex hull of $\mathsf{{\rm Y}}(\Delta )$.

3. Inclusion Properties Involving the Operator ${\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}$

Further, we assume throughout this paper that $k\ge 2,p\in \mathbb{N},m\in {\mathbb{N}}_0,\delta \ge 0,0<\lambda \le 1,0<q<1,\varsigma \in \Delta$ and the power are the principal values.

Theorem 1. 

For $0\le \zeta \le \alpha <p$ and $\mu >p,$ then

$${\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu -1,m,k}\left(\alpha \right)\subset {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\zeta \right),$$

where ζ is given by

$$\zeta =\frac{2\left[p-2\alpha \left(p-\mu \right)\right]}{\sqrt{{\left(2\mu -2p-2\alpha +1\right)}^2+8\left(p-2\alpha \left(p-\mu \right)\right)}+\left(2\mu -2p-2\alpha +1\right)}.$$

(18)

Proof. 

Assume that $\mathcal{F}\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu -1,m,k}\left(\alpha \right)$ and let

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)}=M\left(\varsigma \right)=(p-\zeta )h\left(\varsigma \right)+\zeta .$$

(19)

where

$$h\left(\varsigma \right)=\left(\frac{k}{4}+\frac{1}{2}\right)h_1\left(\varsigma \right)-\left(\frac{k}{4}-\frac{1}{2}\right)h_2\left(\varsigma \right)$$

(20)

and $h_i\left(z\right)$ $\left(i=1,2\right)$ are analytic in $\Delta$ with $h_i\left(0\right)=1,i=1,2.$ Using (11) and (19), we have

$$\mu \frac{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)}=(p-\zeta )h\left(\varsigma \right)+\zeta -\mu +p.$$

(21)

By computing the logarithmical derivative of (21) with respect to $\varsigma$, we have

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)}-\alpha =\zeta -\alpha +(p-\zeta )h\left(\varsigma \right)+\frac{(p-\zeta )\varsigma h^{{}^{\prime }}\left(\varsigma \right)}{(p-\zeta )h\left(\varsigma \right)+\zeta -\mu +p}.$$

(22)

Now we show that $M\left(\varsigma \right)\in {\mathcal{P}}_{p,k}\left(\alpha \right)$ or $h_i\left(\varsigma \right)\in \mathcal{P},i=1,2.$ From (20) and (22), we have

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)}-\alpha =\left(\frac{k}{4}+\frac{1}{2}\right)\left\{\zeta -\alpha +(p-\zeta )h_1\left(\varsigma \right)+\frac{(p-\zeta )\varsigma h_1^{{}^{\prime }}\left(\varsigma \right)}{(p-\zeta )h_1\left(\varsigma \right)+\zeta -\mu +p}\right\}$$

$$-\left(\frac{k}{4}-\frac{1}{2}\right)\left\{\zeta -\alpha +(p-\zeta )h_2\left(\varsigma \right)+\frac{(p-\zeta )\varsigma h_2^{{}^{\prime }}\left(\varsigma \right)}{(p-\zeta )h_2\left(\varsigma \right)+\zeta -\mu +p}\right\}$$

and this implies that

$$\Re \left\{\zeta -\alpha +(p-\zeta )h_i\left(\varsigma \right)+\frac{(p-\zeta )\varsigma h_i^{{}^{\prime }}\left(\varsigma \right)}{(p-\zeta )h_i\left(\varsigma \right)+\zeta -\mu +p}\right\}>0\left(\varsigma \in \Delta ;i=1,2\right).$$

We form the function $\mathsf{\Phi }(r,s)$ by choosing $r=h_i\left(\varsigma \right)$ and $s=\varsigma h_i^{{}^{\prime }}\left(\varsigma \right).$ Thus

$$\mathsf{\Phi }(r,s)=\zeta -\alpha +(p-\zeta )r+\frac{(p-\zeta )s}{(p-\zeta )r+\zeta -\mu +p}.$$

Then, we have

(i)

$\mathsf{\Phi }(r,s)$ is continuous function in $\mathcal{D}=\left(\mathbb{C}\backslash \frac{\zeta -\mu +p}{\zeta -p}\right)\times \mathbb{C};$

(ii)

$(1,0)\in \mathcal{D}$ and $\Re \left\{\mathsf{\Phi }(1,0)\right\}=p-\alpha >0;$

(iii)

$$\begin{array}{ccc}\hfill \Re \left\{\mathsf{\Phi }(i{r}_2,s_1)\right\}& =& \Re \left\{\zeta -\alpha +(p-\zeta )i{r}_2+\frac{(p-\zeta )s_1}{(p-\zeta )i{r}_2+\zeta -\mu +p}\right\}\hfill \\ & =& \zeta -\alpha +\frac{(p-\zeta )\left(\zeta -\mu +p\right)s_1}{{(p-\zeta )}^2{r}_2^2+{\left(\zeta -\mu +p\right)}^2}\hfill \\ & \le & \zeta -\alpha -\frac{(p-\zeta )\left(\zeta -\mu +p\right)(1+r_2^2)}{2\left[{(p-\zeta )}^2{r}_2^2+{\left(\zeta -\mu +p\right)}^2\right]}\hfill \\ & =& \frac{R+E{r}_2^2}{2C},\hfill \end{array}$$

for all $(i{r}_2,s_1)\in \mathcal{D}$ such that $s_1\le -{\displaystyle \frac{1}{2}}(1+r_2^2),$

where

$$R=2\left(\zeta -\alpha \right){\left(\zeta -\mu +p\right)}^2-(p-\zeta )\left(\zeta -\mu +p\right),$$

$$E=2\left(\zeta -\alpha \right){(p-\zeta )}^2-(p-\zeta )\left(\zeta -\mu +p\right),$$

$$C={(p-\zeta )}^2{r}_2^2+{\left(\zeta -\mu +p\right)}^2.$$

We note that $\Re \left\{\mathsf{\Phi }(i{r}_2,s_1)\right\}<0$, if and only if $R\le 0,E<0$ and $C>0.$ From $R\le 0$, we obtain $\zeta$ as given by (18), and from $0\le \zeta <\alpha <p,$ we have $E<0.$ By applying Lemma 1, $h_i\left(\varsigma \right)\in \mathcal{P}\left(i=1,2\right)$ and consequently $M\left(\varsigma \right)\in {\mathcal{P}}_{p,k}\left(\gamma \right)$ for $\varsigma \in \Delta .$ This completes the proof of Theorem 1. □

Theorem 2. 

For $0\le \zeta \le \alpha <p$ and $\mu >p,$ then

$${\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu -1,m,k}\left(\alpha \right)\subset {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\zeta \right),$$

where ζ is given by (18).

Proof. 

Let

$$\begin{array}{ccc}\hfill \mathcal{F}& \in & {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu -1,m,k}\left(\alpha \right)\Rightarrow {\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)\in {\mathcal{C}}_p^k\left(\alpha \right)\hfill \\ & \Rightarrow & \frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}}{p}\in {\mathcal{S}}_p^k\left(\alpha \right)\hfill \\ & \Rightarrow & {\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\left(\frac{\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}{p}\right)\in {\mathcal{S}}_p^k\left(\alpha \right)\hfill \\ & \Rightarrow & \frac{\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}{p}\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu -1,m,k}\left(\alpha \right)\subset {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\zeta \right)\hfill \\ & \Rightarrow & {\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\left(\frac{\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}{p}\right)\in {\mathcal{S}}_p^k\left(\zeta \right)\hfill \\ & \Rightarrow & {\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)\in {\mathcal{C}}_p^k\left(\zeta \right)\hfill \\ & \Rightarrow & \mathcal{F}\in {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\zeta \right).\hfill \end{array}$$

This completes the proof of Theorem 2. □

Theorem 3. 

For $0\le \beta \le \alpha <p$ and $\mu >p,$ then

$${\mathcal{K}}_{p,q,\lambda ,\delta }^{\mu -1,m,k}\left(\beta ,\alpha \right)\subset {\mathcal{K}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\beta ,\alpha \right).$$

Proof. 

Let $\mathcal{F}\in {\mathcal{K}}_{p,q,\lambda ,\delta }^{\mu -1,m,k}\left(\beta ,\alpha \right).$ Then, there exists $G\left(\varsigma \right)\in {\mathcal{S}}_p^2\left(\alpha \right)\equiv S_p^{\ast }\left(\alpha \right)$ such that

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}}{G\left(\varsigma \right)}\in {\mathcal{P}}_{p,k}\left(\beta \right).$$

(23)

Then

$$G\left(\varsigma \right)={\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}g\left(\varsigma \right)\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu -1,m,2}\left(\alpha \right).$$

We set

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}g\left(\varsigma \right)}=\mathcal{R}\left(\varsigma \right)=\left(p-\beta \right)h\left(\varsigma \right)+\beta ,$$

(24)

where $h\left(\varsigma \right)$ is given by (20). By using (11) in (23), we get

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}g\left(\varsigma \right)}=\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\left(\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)\right)\right)}^{{}^{\prime }}+\left(\mu -p\right){\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\left(\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)\right)}{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}g\left(\varsigma \right)\right)}^{{}^{\prime }}+\left(\mu -p\right){\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}g\left(\varsigma \right)}.$$

(25)

Furthermore, $G\left(\varsigma \right)\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu -1,m,2}\left(\alpha \right)$ and by using Theorem 1, with $k=2,$ we have $G\left(\varsigma \right)\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(\alpha \right).$ Therefore, we can write

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}g\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}g\left(\varsigma \right)}={\mathcal{R}}_0\left(\varsigma \right)=\left(p-\alpha \right)q\left(\varsigma \right)+\alpha \left(q\in {\mathcal{P}}_k\right),$$

(26)

where $q\left(\varsigma \right)=1+c_1\varsigma +c_2{\varsigma }^2+...$ is analytic and $q\left(0\right)=1$ in $\Delta$. By differentiating (24) with respect to $\varsigma$, we have

$$\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\left(\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)\right)\right)}^{{}^{\prime }}=\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}g\left(\varsigma \right)\right)}^{{}^{\prime }}\mathcal{R}\left(\varsigma \right)+\varsigma {\mathcal{R}}^{{}^{\prime }}\left(\varsigma \right){\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}g\left(\varsigma \right)$$

then

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\left(\varsigma f^{{}^{\prime }}\left(\varsigma \right)\right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}g\left(\varsigma \right)}=\varsigma {\mathcal{R}}^{{}^{\prime }}\left(\varsigma \right)+{\mathcal{R}}_0\left(\varsigma \right)\mathcal{R}\left(\varsigma \right).$$

(27)

From (25) and (27), we obtain

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}g\left(\varsigma \right)}=\frac{\varsigma {\mathcal{R}}^{{}^{\prime }}\left(\varsigma \right)+{\mathcal{R}}_0\left(\varsigma \right)\mathcal{R}\left(\varsigma \right)+\left(\mu -p\right)\mathcal{R}\left(\varsigma \right)}{{\mathcal{R}}_0\left(\varsigma \right)+\left(\mu -p\right)}$$

so that

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}f\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}g\left(\varsigma \right)}=\mathcal{R}\left(\varsigma \right)+\frac{\varsigma {\mathcal{R}}^{{}^{\prime }}\left(\varsigma \right)}{{\mathcal{R}}_0\left(\varsigma \right)+\left(\mu -p\right)}.$$

(28)

Let

$$\mathcal{R}\left(\varsigma \right)=\left(\frac{k}{4}+\frac{1}{2}\right)\left\{\left(p-\beta \right)h_1\left(\varsigma \right)+\beta \right\}-\left(\frac{k}{4}-\frac{1}{2}\right)\left\{\left(p-\beta \right)h_2\left(\varsigma \right)+\beta \right\}$$

and

$${\mathcal{R}}_0\left(\varsigma \right)+\left(\mu -p\right)=\left(p-\alpha \right)q\left(\varsigma \right)+\left(\alpha +\mu -p\right).$$

We intend to show that $\mathcal{R}\in {\mathcal{P}}_{p,k}\left(\beta \right)$ or $h_i\in \mathcal{P}$ for $i=1,2.$ Then, we can say that $\Re \left\{{\mathcal{R}}_0\left(\varsigma \right)+\left(\mu -p\right)\right\}>0.$ From (24) and (28), we have

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu -1,m}g\left(\varsigma \right)}-\beta =\left(\frac{k}{4}+\frac{1}{2}\right)\left\{\left(p-\beta \right)h_1\left(\varsigma \right)+\frac{\left(p-\beta \right)\varsigma h_1^{{}^{\prime }}\left(\varsigma \right)}{\left(p-\alpha \right)q\left(\varsigma \right)+\left(\alpha +\mu -p\right)}\right\}$$

$$-\left(\frac{k}{4}-\frac{1}{2}\right)\left\{\left(p-\beta \right)h_2\left(\varsigma \right)+\frac{\left(p-\beta \right)\varsigma h_2^{{}^{\prime }}\left(\varsigma \right)}{\left(p-\alpha \right)q\left(\varsigma \right)+\left(\alpha +\mu -p\right)}\right\}$$

and this implies that

$$\Re \left\{\left(p-\beta \right)h_i\left(\varsigma \right)+\frac{\left(p-\beta \right)\varsigma h_i^{{}^{\prime }}\left(\varsigma \right)}{\left(p-\alpha \right)q\left(\varsigma \right)+\left(\alpha +\mu -p\right)}\right\}>0\left(\varsigma \in \Delta ,i=1,2\right).$$

We form the function $\mathsf{\Phi }(r,s)$ by choosing $r=h_i\left(\varsigma \right)$ and $s=\varsigma h_i^{{}^{\prime }}\left(\varsigma \right).$ Thus,

$$\mathsf{\Phi }(r,s)=(p-\beta )r+\frac{(p-\beta )s}{\left(p-\alpha \right)q\left(\varsigma \right)+\left(\alpha +\mu -p\right)}.$$

(29)

Then

(i)

$\mathsf{\Phi }(r,s)$ is continuous in $D=\mathbb{C}\times \mathbb{C};$

(ii)

$(1,0)\in D$ and $\Re \left\{\mathsf{\Phi }(1,0)\right\}=p-\beta >0;$

(iii)

$$\begin{array}{ccc}\hfill \Re \left\{\mathsf{\Phi }(i{r}_2,s_1)\right\}& =& \Re \left\{(p-\beta )i{u}_2+\frac{(p-\beta )v_1}{(p-\alpha )\left(q_1+i{q}_2\right)+\left(\alpha +\mu -p\right)}\right\}\hfill \\ & =& \frac{(p-\beta )\left[(p-\alpha )q_1+\alpha +\mu -p\right]s_1}{{\left[(p-\alpha )q_1+\alpha +\mu -p\right]}^2+{(p-\alpha )}^2{q}_2^2}\hfill \\ & \le & -\frac{(p-\beta )\left[(p-\alpha )q_1+\alpha +\mu -p\right](1+r_2^2)}{2\left\{{\left[(p-\alpha )q_1+\alpha +\mu -p\right]}^2+{(p-\alpha )}^2{q}_2^2\right\}}<0,\hfill \end{array}$$

for all $(i{r}_2,s_1)\in D$ such that $s_1\le -{\displaystyle \frac{1}{2}}(1+r_2^2).$

Byapplying Lemma 1, we have $\Re \left\{h_i\left(\varsigma \right)\right\}>0$ for $\left(i=1,2\right)$ and consequently $\mathcal{R}\left(\varsigma \right)\in {\mathcal{P}}_{p,k}\left(\beta \right)$ for $\varsigma \in \Delta .$ This completes the proof of Theorem 3. □

4. Inclusion Properties Involving the Integral Operator ${\mathcal{J}}_{\delta ,p}$

The generalized Bernardi operator is defined by (see [23])

$${\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\left(\varsigma \right)=\frac{\delta +p}{{\varsigma }^{\delta }}\underset{0}{\overset{\varsigma }{\int }}{t}^{\delta -1}\mathcal{F}\left(t\right)dt\left(\delta >-p\right),$$

(30)

which satisfies the following relationship:

$$\varsigma {\left({\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\left(\varsigma \right)\right)}^{{}^{\prime }}=\left(\delta +p\right)\mathcal{F}\left(\varsigma \right)-\delta {\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\left(\varsigma \right).$$

(31)

Theorem 4. 

If $0\le \alpha <p,$$k\ge 2$ and $\mathcal{F}\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right),$ then ${\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)\left(\delta \ge 0\right).$

Proof. 

Let

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}{\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}{\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\left(\varsigma \right)}=\mathcal{R}\left(\varsigma \right)=(p-\alpha )h\left(\varsigma \right)+\alpha ,$$

(32)

where $h\left(\varsigma \right)$, given by (20). Using (31), we have

$$\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}{\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\left(\varsigma \right)\right)}^{{}^{\prime }}=\left(\delta +p\right){\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)-\delta {\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}{\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\left(\varsigma \right).$$

(33)

From (32) and (33), we have

$$\left(\delta +p\right)\frac{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}{\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\left(\varsigma \right)}=(p-\alpha )h\left(\varsigma \right)+\alpha +\delta .$$

(34)

By computing the logarithmical derivative of (34) with respect to $\varsigma$ and multiplying by $\varsigma$, we have

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)}-\alpha =(p-\alpha )h\left(\varsigma \right)+\frac{(p-\alpha )\varsigma h^{{}^{\prime }}\left(\varsigma \right)}{(p-\alpha )h\left(\varsigma \right)+\alpha +\delta }.$$

(35)

Now, we show that $\mathcal{R}\left(\varsigma \right)\in {\mathcal{P}}_{p,k}\left(\alpha \right)$ or $h_i\in \mathcal{P}$ for $i=1,2.$ From (20) and (35), we have

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)\right)}^{{}^{\prime }}}{{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\left(\varsigma \right)}-\alpha =\left(\frac{k}{4}+\frac{1}{2}\right)\left\{(p-\alpha )h_1\left(\varsigma \right)+\frac{(p-\alpha )\varsigma h_1^{{}^{\prime }}\left(\varsigma \right)}{(p-\alpha )h_1\left(\varsigma \right)+\alpha +\delta }\right\}$$

$$-\left(\frac{k}{4}-\frac{1}{2}\right)\left\{(p-\alpha )h_2\left(\varsigma \right)+\frac{(p-\alpha )\varsigma h_2^{{}^{\prime }}\left(\varsigma \right)}{(p-\alpha )h_2\left(\varsigma \right)+\alpha +\delta }\right\}$$

and this implies that

$$\Re \left\{(p-\alpha )h_i\left(\varsigma \right)+\frac{(p-\alpha )\varsigma h_i^{{}^{\prime }}\left(\varsigma \right)}{(p-\alpha )h_i\left(\varsigma \right)+\alpha +\delta }\right\}>0\left(\varsigma \in \Delta ;i=1,2\right).$$

We form the function $\mathsf{\Phi }(r,s)$ by choosing $r=h_i\left(\varsigma \right)$ and $s=\varsigma h_i^{{}^{\prime }}\left(\varsigma \right).$ Thus

$$\mathsf{\Phi }(r,s)=(p-\alpha )r+\frac{(p-\alpha )s}{(p-\alpha )r+\alpha +\delta }.$$

(36)

Clearly, conditions (i), (ii) and (iii) of Lemma 1 are satisfied. Byapplying Lemma 1, we have $\Re \left\{h_i\left(\varsigma \right)\right\}>0$ for $\left(i=1,2\right)$ and consequently ${\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)$ for $\varsigma \in \Delta .$ This completes the proof of Theorem 1. □

Theorem 5. 

If $0\le \alpha <p,$$k\ge 2$ and $\mathcal{F}\in {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right),$ then ${\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\in {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)\left(\delta \ge 0\right).$

Proof. 

Let

$$\mathcal{F}\in {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)\iff \frac{\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}{p}\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right).$$

By applying Theorem 4, we have

$${\mathcal{J}}_{\delta ,p}\left(\frac{\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}{p}\right)\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)\iff \frac{\varsigma {\left({\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\left(\varsigma \right)\right)}^{{}^{\prime }}}{p}\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right)\iff {\mathcal{J}}_{\delta ,p}\left(\mathcal{F}\right)\left(\varsigma \right)\in {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,k}\left(\alpha \right),$$

which evidently proves Theorem 5. □

5. Inclusion Properties by Convolution

Theorem 6. 

Let Φ be a convex function and $\mathcal{F}\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(p\gamma \right),$ then $G\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(p\gamma \right),$ where $G=\mathcal{F}\ast \mathsf{\Phi }$ and $0\le \gamma <1.$

Proof. 

To show that $G=\mathcal{F}\ast \mathsf{\Phi }\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(p\gamma \right)\left(0\le \gamma <1\right),$ it sufficient to show that $\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}G\right)}^{{}^{\prime }}}{p{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}G}$ contained in the convex hull of $\mathsf{{\rm Y}}\left(\Delta \right).$ Now

$$\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}G\right)}^{{}^{\prime }}}{p{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}G}=\frac{\mathsf{\Phi }\ast \mathsf{{\rm Y}}{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}}{\mathsf{\Phi }\ast {\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}},$$

(37)

where $\mathsf{{\rm Y}}=\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}\right)}^{{}^{\prime }}}{p{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}\mathcal{F}}$ is analytic in $\Delta$ and $\mathsf{{\rm Y}}\left(0\right)=1.$ From Lemma 2, we can see that $\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}G\right)}^{{}^{\prime }}}{p{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}G}$ is contained in the convex hull of $\mathsf{{\rm Y}}\left(\Delta \right),$ since $\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}G\right)}^{{}^{\prime }}}{p{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}G}$ is analytic in $\Delta$ and

$$\mathsf{{\rm Y}}\left(\Delta \right)\subseteq \mathsf{\Omega }=\left\{w:\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}w\left(\varsigma \right)\right)}^{{}^{\prime }}}{p{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}w\left(\varsigma \right)}\in \mathcal{P}\left(\gamma \right)\right\},$$

(38)

then $\frac{\varsigma {\left({\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}G\right)}^{{}^{\prime }}}{p{\mathcal{G}}_{p,q,\lambda ,\delta }^{\mu ,m}G}$ lies in $\mathsf{\Omega },$ this implies that $G=\mathcal{F}\ast \mathsf{\Phi }\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(p\gamma \right)$. □

Theorem 7. 

Let Φ be a convex function and $\mathcal{F}\in {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(p\gamma \right),$ then $G\in {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(p\gamma \right),$ where $G=\mathcal{F}\ast \mathsf{\Phi }$ and $0\le \gamma <1.$

Proof. 

Let $\mathcal{F}\in {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(p\gamma \right),$ then, by using (13), we have

$$\frac{\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}{p}\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(p\gamma \right)$$

and hence by using Theorem 6, we get

$$\begin{array}{ccc}\hfill \frac{\varsigma {\mathcal{F}}^{{}^{\prime }}\left(\varsigma \right)}{p}\ast \mathsf{\Phi }\left(\varsigma \right)& \in & {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(p\gamma \right)\hfill \\ & \Rightarrow & \frac{\varsigma {\left(\mathcal{F}\ast \mathsf{\Phi }\right)}^{{}^{\prime }}\left(\varsigma \right)}{p}\in {\mathcal{S}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(p\gamma \right).\hfill \end{array}$$

Now applying (13) again, we obtain $G=\mathcal{F}\ast \mathsf{\Phi }\in {\mathcal{C}}_{p,q,\lambda ,\delta }^{\mu ,m,2}\left(p\gamma \right)$, which evidently proves Theorem 7. □

Remark 3. 

Particularizing the parameters $q$ and $m$ in the results of this paper, we derive various results for different operators.

6. Conclusions

In the present survey, we propose new subclasses of p-valent functions by making use of the linear q-differential Borel operator. The applications of this interesting operator are discussed. Inclusion properties and certain integral preserving relations were aimed to be our main concern.