Inclusion results for certain subclasses of p-valent meromorphic functions associated with a new operator

Abstract

In this paper, we introduce new subclasses of p-valent starlike, p-valent convex, p-valent close-to-convex, and p-valent quasi-convex meromorphic functions and investigate some inclusion properties of these subclasses and investigate various inclusion properties and integral-preserving properties for the p-valent meromorphic function classes.

MSC:30C45.

1 Introduction

Let ${\mathrm{\Sigma }}_p$ denote the class of functions of the form

$$f(z)=z^{-p}+\sum _{k=1}^{\mathrm{\infty }}{a}_{k-p}{z}^{k-p}(p\in \mathbb{N}=\{1,2,\dots \}),$$

(1.1)

which are analytic and p-valent in the punctured unit disc $U^{\ast }=\{z:z\in \mathbb{C}\text{ and }0<|z|<1\}$. If $f(z)$ and $g(z)$ are analytic in $U=U^{\ast }\cup \{0\}$, we say that $f(z)$ is subordinate to $g(z)$, written $f\prec g$ or $f(z)\prec g(z)$ ($z\in U$), if there exists a Schwarz function $w(z)$ in U with $w(0)=0$ and $|w(z)|<1$, such that $f(z)=g(w(z))$ ($z\in U$). Furthermore, if $g(z)$ is univalent in U, then the following equivalence relationship holds true (see [3] and [8]):

$$f(z)\prec g(z)\iff f(0)=g(0)\text{and}f(U)\subset g(U).$$

For functions $f(z)\in {\mathrm{\Sigma }}_p$, given by (1.1) and $g(z)\in {\mathrm{\Sigma }}_p$ defined by

$$g(z)=z^{-p}+\sum _{k=1}^{\mathrm{\infty }}{b}_{k-p}{z}^{k-p}(p\in \mathbb{N}),$$

(1.2)

the Hadamard product (or convolution) of $f(z)$ and $g(z)$ is given by

$$(f\ast g)(z)=z^{-p}+\sum _{k=1}^{\mathrm{\infty }}{a}_{k-p}{b}_{k-p}{z}^{k-p}=(g\ast f)(z).$$

(1.3)

Aqlan et al.[1] defined the operator $Q_{\beta ,p}^{\alpha }:{\mathrm{\Sigma }}_p\to {\mathrm{\Sigma }}_p$ by:

$$Q_{\beta ,p}^{\alpha }f(z)=\{\begin{array}{cc}{z}^{-p}+\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }(\beta )}{\sum }_{k=1}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }(k+\beta )}{\mathrm{\Gamma }(k+\beta +\alpha )}{a}_{k-p}{z}^{k-p}\hfill & (\alpha >0;\beta >-1;p\in \mathbb{N};f\in {\mathrm{\Sigma }}_p),\hfill \\ f(z)\hfill & (\alpha =0;\beta >-1;p\in \mathbb{N};f\in {\mathrm{\Sigma }}_p).\hfill \end{array}$$

(1.4)

Now, we define the operator $H_{p,\beta ,\mu }^{\alpha }:{\mathrm{\Sigma }}_p\to {\mathrm{\Sigma }}_p$ as follows:

First, put

$$G_{\beta ,p}^{\alpha }(z)=z^{-p}+\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }(\beta )}\sum _{k=1}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }(k+\beta )}{\mathrm{\Gamma }(k+\beta +\alpha )}{z}^{k-p}(p\in \mathbb{N})$$

(1.5)

and let $G_{\beta ,p,\mu }^{\alpha \ast }$ be defined by

$$G_{\beta ,p}^{\alpha }(z)\ast G_{\beta ,p,\mu }^{\alpha \ast }(z)=\frac{1}{z^p{(1-z)}^{\mu }}(\mu >0;p\in \mathbb{N}).$$

(1.6)

Then

$$H_{p,\beta ,\mu }^{\alpha }f(z)=G_{\beta ,p}^{\alpha \ast }(z)\ast f(z)(f\in {\mathrm{\Sigma }}_p).$$

(1.7)

Using (1.5)-(1.7), we have

$$H_{p,\beta ,\mu }^{\alpha }f(z)=z^{-p}+\frac{\mathrm{\Gamma }(\beta )}{\mathrm{\Gamma }(\alpha +\beta )}\sum _{k=1}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }(k+\beta +\alpha ){(\mu )}_k}{\mathrm{\Gamma }(k+\beta ){(1)}_k}{a}_{k-p}{z}^{k-p},$$

(1.8)

where $f\in {\mathrm{\Sigma }}_p$ is in the form (1.1) and ${(\nu )}_n$ denotes the Pochhammer symbol given by

$${(\nu )}_n=\frac{\mathrm{\Gamma }(\nu +n)}{\mathrm{\Gamma }(\nu )}=\{\begin{array}{cc}1\hfill & (n=0),\hfill \\ \nu (\nu +1)\cdots (\nu +n-1)\hfill & (n\in \mathbb{N}).\hfill \end{array}$$

It is readily verified from (1.8) that

$$z{(H_{p,\beta ,\mu }^{\alpha }f(z))}^{\mathrm{\prime }}=(\alpha +\beta )H_{p,\beta ,\mu }^{\alpha +1}f(z)-(\alpha +\beta +p)H_{p,\beta ,\mu }^{\alpha }f(z)$$

(1.9)

and

$$z{(H_{p,\beta ,\mu }^{\alpha }f(z))}^{\mathrm{\prime }}=\mu H_{p,\beta ,\mu +1}^{\alpha }f(z)-(\mu +p)H_{p,\beta ,\mu }^{\alpha }f(z).$$

(1.10)

It is noticed that, putting $\mu =1$ in (1.8), we obtain the operator

$$H_{p,\beta ,1}^{\alpha }f(z)=z^{-p}+\frac{\mathrm{\Gamma }(\beta )}{\mathrm{\Gamma }(\alpha +\beta )}\sum _{k=1}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }(k+\alpha +\beta )}{\mathrm{\Gamma }(k+\beta )}{a}_{k-p}{z}^{k-p}.$$

(1.11)

Let M be the class of analytic functions $h(z)$ with $h(0)=1$, which are convex and univalent in U and satisfy $Re\{h(z)\}>0$ ($z\in U$).

For $0\le \eta$, $\gamma <p$, we denote by ${\mathrm{\Sigma }}_{p}S(\eta )$, ${\mathrm{\Sigma }}_{p}K(\eta )$, ${\mathrm{\Sigma }}_{p}C(\eta ,\gamma )$, and ${\mathrm{\Sigma }}_p{C}^{\ast }(\eta ,\gamma )$, the subclasses of ${\mathrm{\Sigma }}_p$ consisting of all p-valent meromorphic functions which are, respectively, starlike of order η, convex of order η, close-to-convex functions of order γ and type η, and quasi-convex functions of order γ and type η in U.

Making use of the principle of subordination between analytic functions, we introduce the subclasses ${\mathrm{\Sigma }}_{p}S(\eta ;\varphi )$, ${\mathrm{\Sigma }}_{p}C(\eta ;\varphi )$, ${\mathrm{\Sigma }}_{p}K(\eta ,\gamma ;\varphi ,\psi )$, and ${\mathrm{\Sigma }}_p{K}^{\ast }(\eta ,\gamma ;\varphi ,\psi )$ ($0\le \eta$, $\gamma <p$ and $\varphi ,\psi \in M$) of the class ${\mathrm{\Sigma }}_p$ which are defined by:

and

From these definitions, we can obtain some well-known subclasses of ${\mathrm{\Sigma }}_p$ by special choices of the functions ϕ and ψ as well as special choices of η, γ, and p (see [2, 5], and [10]).

Now, by using the linear operator $H_{p,\beta ,\mu }^{\alpha }$ ($\alpha \ge 0$, $\mu >0$, $\beta >-1$; $p\in \mathbb{N}$) and for $\varphi ,\psi \in M$, $0\le \eta$, $\gamma <p$, we define new subclasses of meromorphic functions of ${\mathrm{\Sigma }}_p$ by:

and

$${\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha \ast }(\eta ,\gamma ;\varphi ,\psi )=\{f\in {\mathrm{\Sigma }}_p:H_{p,\beta ,\mu }^{\alpha }f\in {\mathrm{\Sigma }}_p{C}^{\ast }(\eta ,\gamma ;\varphi ,\psi )\}.$$

We also note that

$$f(z)\in {\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )\iff -\frac{z{f}^{\mathrm{\prime }}(z)}{p}\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi ),$$

(1.12)

and

$$f(z)\in {\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha \ast }(\eta ,\gamma ;\varphi ,\psi )\iff -\frac{z{f}^{\mathrm{\prime }}(z)}{p}\in {\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha }(\eta ,\gamma ;\varphi ,\psi ).$$

(1.13)

In particular, we set

$${\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\frac{1+Az}{1+Bz})={\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;A,B)(-1<B<A\le 1)$$

and

$${\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(\eta ;\frac{1+Az}{1+Bz})={\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(\eta ;A,B)(-1<B<A\le 1).$$

In this paper, we investigate several inclusion properties of the classes ${\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$, ${\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$, ${\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha }(\eta ,\gamma ;\varphi ,\psi )$, and ${\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha \ast }(\eta ,\gamma ;\varphi ,\psi )$ associated with the operator $H_{p,\beta ,\mu }^{\alpha }$. Some applications involving integral operators are also considered.

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

Lemma 1[4]

Let ς and υ be complex constants and let$h(z)$be convex (univalent) in U with$h(0)=1$and$Re\{\varsigma h(z)+\upsilon \}>0$. If

$$q(z)=1+q_{1}z+\cdots$$

(1.14)

is analytic in U, then

$$q(z)+\frac{z{q}^{\mathrm{\prime }}(z)}{\varsigma q(z)+\upsilon }\prec h(z)(z\in U),$$

implies

$$q(z)\prec h(z)(z\in U).$$

Lemma 2[7]

Let$h(z)$be convex (univalent) in U and$\psi (z)$be analytic in U with$Re\{\psi (z)\}\ge 0$. If q is analytic in U and$q(0)=h(0)$, then

$$q(z)+\psi (z)z{q}^{\mathrm{\prime }}(z)\prec h(z)(z\in U)$$

implies

$$q(z)\prec h(z)(z\in U).$$

2 Some inclusion results

In this section, we give some inclusion properties for meromorphic function classes, which are associated with the operator $H_{p,\beta ,\mu }^{\alpha }$. Unless otherwise mentioned, we assume that $\alpha \ge 1$, $\beta >-1$, $\mu >0$, $0\le \gamma$, $\eta <p$ and $p\in \mathbb{N}$.

Theorem 1 For$f(z)\in {\mathrm{\Sigma }}_p$, $\varphi \in M$with

$$\underset{z\in U}{max}(Re\{\varphi (z)\})<\underset{z\in U}{min}[\frac{p+\mu -\eta }{p-\eta },\frac{\alpha +\beta +p-\eta }{p-\eta }],$$

then we have

$${\mathrm{\Sigma }}_p{S}_{\beta ,\mu +1}^{\alpha }(\eta ,\varphi )\subset {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ,\varphi )\subset {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha -1}(\eta ,\varphi ).$$

(2.1)

Proof We will first show that

$${\mathrm{\Sigma }}_p{S}_{\beta ,\mu +1}^{\alpha }(\eta ,\varphi )\subset {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ,\varphi ).$$

(2.2)

Let $f\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu +1}^{\alpha }(\eta ;\varphi )$ and put

$$q(z)=\frac{1}{p-\eta }(-\frac{z{(H_{p,\beta ,\mu }^{\alpha }f(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu }^{\alpha }f(z)}-\eta ),$$

(2.3)

where $q(z)$ is analytic in U with $q(0)=1$. Applying (1.10) in (2.3), we have

$$-\mu \frac{H_{p,\beta ,\mu +1}^{\alpha }f(z)}{H_{p,\beta ,\mu }^{\alpha }f(z)}=(p-\eta )q(z)+\eta -(p+\mu ).$$

(2.4)

Differentiating (2.4) logarithmically with respect to z, we have

$$\frac{1}{p-\eta }(-\frac{z{(H_{p,\beta ,\mu +1}^{\alpha }f(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu +1}^{\alpha }f(z)}-\eta )=q(z)+\frac{z{q}^{\mathrm{\prime }}(z)}{(p+\mu )-\eta -(p-\eta )q(z)}(z\in U).$$

(2.5)

Since

$$\underset{z\in U}{max}(Re\{\varphi (z)\})<\underset{z\in U}{min}\frac{p+\mu -\eta }{p-\eta },$$

we see that

$$Re\{(p+\mu )-\eta -(p-\eta )\varphi (z)\}>0(z\in U).$$

Applying Lemma 1 to (2.5), it follows that $q\prec \varphi$, that is, that $f\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$. The proof of the second part will follow by using arguments similar to those used in the first part with $f\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$ and using the identity (1.9) instead of (1.10). This completes the proof of Theorem 1. □

Theorem 2 For$f(z)\in {\mathrm{\Sigma }}_p$, $\varphi \in M$with

Proof Applying (1.10) and using Theorem 1, we have

$$\begin{array}{rcl}f(z)\in {\mathrm{\Sigma }}_p{K}_{\beta ,\mu +1}^{\alpha }(\eta ;\varphi )& \iff & H_{p,\beta ,\mu +1}^{\alpha }f(z)\in {\mathrm{\Sigma }}_{p}K(\eta ;\varphi )\\ \iff & -\frac{z{(H_{p,\beta ,\mu +1}^{\alpha }f(z))}^{\mathrm{\prime }}}{p}\in {\mathrm{\Sigma }}_{p}S(\eta ;\varphi )\\ \iff & H_{p,\beta ,\mu +1}^{\alpha }(-\frac{z{f}^{\mathrm{\prime }}(z)}{p})\in {\mathrm{\Sigma }}_{p}S(\eta ;\varphi )\\ \iff & -\frac{z{f}^{\mathrm{\prime }}(z)}{p}\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu +1}^{\alpha }(\eta ;\varphi )\\ \Rightarrow & -\frac{z{f}^{\mathrm{\prime }}(z)}{p}\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )\\ \iff & H_{p,\beta ,\mu }^{\alpha }(-\frac{z{f}^{\mathrm{\prime }}(z)}{p})\in {\mathrm{\Sigma }}_{p}S(\eta ;\varphi )\\ \iff & -\frac{z{(H_{p,\beta ,\mu }^{\alpha }f(z))}^{\mathrm{\prime }}}{p}\in {\mathrm{\Sigma }}_{p}S(\eta ;\varphi )\\ \iff & H_{p,\beta ,\mu }^{\alpha }f(z)\in {\mathrm{\Sigma }}_{p}K(\eta ;\varphi )\\ \iff & f(z)\in {\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(\eta ;\varphi ).\end{array}$$

Also,

$$\begin{array}{rcl}f(z)\in {\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )& \iff & -\frac{z{f}^{\mathrm{\prime }}(z)}{p}\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )\\ \Rightarrow & -\frac{z{f}^{\mathrm{\prime }}(z)}{p}\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha -1}(\eta ;\varphi )\\ \iff & -\frac{z{(H_{p,\beta ,\mu }^{\alpha -1}f(z))}^{\mathrm{\prime }}}{p}\in {\mathrm{\Sigma }}_{p}S(\eta ;\varphi )\\ \iff & H_{p,\beta ,\mu }^{\alpha -1}f(z)\in {\mathrm{\Sigma }}_{p}K(\eta ;\varphi )\\ \iff & f(z)\in {\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha -1}(\eta ;\varphi ).\end{array}$$

This completes the proof of Theorem 2. □

Taking

$$\varphi (z)=\frac{1+Az}{1+Bz}(-1<B<A\le 1;z\in U)$$

in Theorem 1 and Theorem 2, we have the following corollary.

Corollary 1 Let $f(z)\in {\mathrm{\Sigma }}_p$ and

$$\frac{1+A}{1+B}<\underset{z\in U}{min}(\frac{p+\mu -\eta }{p-\eta },\frac{\alpha +\beta +p-\eta }{p-\eta })(-1<B<A\le 1).$$

Then we have

$${\mathrm{\Sigma }}_p{S}_{\beta ,\mu +1}^{\alpha }(\eta ;A,B)\subset {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;A,B)\subset {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha -1}(\eta ;A,B)$$

and

$${\mathrm{\Sigma }}_p{K}_{\beta ,\mu +1}^{\alpha }(\eta ;A,B)\subset {\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(\eta ;A,B)\subset {\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha -1}(\eta ;A,B).$$

Now, using Lemma 2, we obtain similar inclusion relations for the subclass ${\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha }(\eta ;\gamma ;\varphi ,\psi )$.

Theorem 3 Let $f(z)\in {\mathrm{\Sigma }}_p$ and

$$\underset{z\in U}{max}Re\{\varphi (z)\}<\underset{z\in U}{min}(\frac{p+\mu -\eta }{p-\eta },\frac{\alpha +\beta +p-\eta }{p-\eta }).$$

Then we have

Proof First, we will prove that

$${\mathrm{\Sigma }}_p{C}_{\beta ,\mu +1}^{\alpha }(\eta ;\gamma ;\varphi ,\psi )\subset {\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha }(\eta ;\gamma ;\varphi ,\psi ).$$

Let $f\in {\mathrm{\Sigma }}_p{C}_{\beta ,\mu +1}^{\alpha }(\eta ;\gamma ;\varphi ,\psi )$. Then, from the definition of the class ${\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha }(\eta ;\gamma ;\varphi ,\psi )$, there exists a function $g\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$ such that

$$\frac{1}{p-\gamma }(-\frac{z{(H_{p,\beta ,\mu +1}^{\alpha }f(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu +1}^{\alpha }g(z)}-\gamma )\prec \psi (z)(z\in U).$$

(2.6)

Now, let

$$q(z)=\frac{1}{p-\gamma }(-\frac{z{(H_{p,\beta ,\mu }^{\alpha }f(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu }^{\alpha }g(z)}-\gamma ),$$

(2.7)

where $q(z)$ is analytic in U with $q(0)=1$. Applying (1.10) in (2.6), we have

(2.8)

Since, by Theorem 1,

$$g(z)\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu +1}^{\alpha }(\eta ;\varphi )\subset {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi ),$$

set

$$h(z)=\frac{1}{p-\eta }(-\frac{z{(H_{p,\beta ,\mu }^{\alpha }g(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu }^{\alpha }g(z)}-\eta ),$$

where $h\prec \varphi$ in U, and $\varphi \in M$. Then, using (2.7) and (2.8), we have

$$H_{p,\beta ,\mu }^{\alpha }(-\frac{z{f}^{\mathrm{\prime }}(z)}{p})=[(p-\gamma )q(z)+\gamma ]H_{p,\beta ,\mu }^{\alpha }g(z)$$

(2.9)

and

(2.10)

Differentiating both sides of (2.9) with respect to z and multiplying by z, we have

$$\frac{z{(H_{p,\beta ,\mu }^{\alpha }(-\frac{z{f}^{\mathrm{\prime }}(z)}{p}))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu }^{\alpha }g(z)}=(p-\gamma )z{q}^{\mathrm{\prime }}(z)-[(p-\gamma )q(z)+\gamma ][(p-\eta )h(z)+\eta ].$$

(2.11)

Making use of (2.6), (2.10), and (2.11), we have

$$\frac{1}{p-\gamma }(-\frac{z{(H_{p,\beta ,\mu +1}^{\alpha }f(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu +1}^{\alpha }g(z)}-\gamma )=q(z)+\frac{z{q}^{\mathrm{\prime }}(z)}{p+\mu -\eta -(p-\eta )h(z)}\prec \psi (z),z\in U.$$

(2.12)

Since $h\prec \varphi$ in U, and

$$\underset{z\in U}{max}Re\{h(z)\}<\frac{p+\mu -\eta }{p-\eta },$$

then

$$Re\{p+\mu -\eta -(p-\eta )h(z)\}>0(z\in U).$$

(2.13)

Hence, putting

$$\chi (z)=\frac{1}{\{p+\mu -\eta -(p-\eta )h(z)\}},$$

in Eq. (2.12) and applying Lemma 2, we can show that $q\prec \psi$, that is, that $f\in {\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha }(\eta ,\gamma ;\varphi ,\psi )$. The second part can be proved by using similar arguments and using (1.9). This completes the proof of Theorem 3. □

3 Inclusion properties involving the integral operator $F_{p,\delta }$

Now, we consider the generalized Libera integral operator $F_{p,\delta }(f)$ (see [6] and [9]), defined by

$$F_{p,\delta }(f)(z)=\frac{\delta }{z^{\delta +p}}{\int }_0^z{t}^{\delta +p-1}f(t)dt=z^{-p}+\sum _{k=1}^{\mathrm{\infty }}\frac{\delta }{\delta +k}{a}_{k-p}{z}^{k-p}(\delta >-p).$$

(3.1)

From (3.1), we have

$$z{(H_{p,\beta ,\mu }^{\sigma }{F}_{p,\delta }(f)(z))}^{\mathrm{\prime }}=\delta H_{p,\beta ,\mu }^{\sigma }f(z)-(\delta +p)H_{p,\beta ,\mu }^{\sigma }{F}_{p,\delta }(f)(z).$$

(3.2)

Theorem 4 Let $\varphi \in M$ with

$$\underset{z\in U}{max}(Re\{\varphi (z)\})<\frac{\delta +p-\eta }{p-\eta }(\delta >-p).$$

If$f\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$, then$F_{p,\delta }(f)\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$.

Proof Let $f\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$ and put

$$h(z)=\frac{1}{p-\eta }(-\frac{z{(H_{p,\beta ,\mu }^{\sigma }{F}_{p,\delta }(f)(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu }^{\sigma }{F}_{p,\delta }(f)(z)}-\eta ),$$

(3.3)

where h is analytic in U with $h(0)=1$. Then, by using (3.2) and (3.3), we have

$$-\delta \frac{H_{p,\beta ,\mu }^{\sigma }f(z)}{H_{p,\beta ,\mu }^{\sigma }{F}_{p,\delta }(f)(z)}=(p-\eta )h(z)+\eta -(p+\delta ).$$

(3.4)

Differentiating (3.4) logarithmically with respect to z, we have

$$\frac{1}{p-\eta }(-\frac{z{(H_{p,\beta ,\mu }^{\sigma }f(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu }^{\sigma }f(z)}-\eta )=h(z)+\frac{z{h}^{\mathrm{\prime }}(z)}{p+\delta -\eta -(p-\eta )h(z)}(z\in U).$$

Applying Lemma 1, we conclude that $h\prec \varphi$$(z\in U)$, which implies that $F_{p,\delta }(f)\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$. □

Theorem 5 Let $\varphi \in M$ with

$$\underset{z\in U}{max}(Re\{\varphi (z)\})<\frac{\delta +p-\eta }{p-\eta }(\delta >-p).$$

If$f\in {\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$, then$F_{p,\delta }(f)\in {\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$.

Proof Applying Theorem 4 and (1.12), we have

$$\begin{array}{rcl}f(z)\in {\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )& \iff & -\frac{z{f}^{\mathrm{\prime }}(z)}{p}\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )\\ \Rightarrow & F_{p,\delta }(-\frac{z{f}^{\mathrm{\prime }}(z)}{p})(z)\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )\\ \iff & -\frac{z}{p}{F}_{p,\delta }^{\mathrm{\prime }}(f)(z)\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )\\ \iff & F_{p,\delta }(f)(z)\in K_{p,\lambda }^{\sigma }(\eta ;\varphi ).\end{array}$$

This completes the proof of Theorem 5. □

From Theorem 4 and Theorem 5, we have the following corollary.

Corollary 2 Suppose that

$$\frac{1+A}{1+B}<\frac{\delta +p-\eta }{p-\eta }(\delta >-p;-1<B<A\le 1).$$

Then, for the classes${\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$and${\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$, the following inclusion relations hold true:

$$f\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(A,B)\Rightarrow F_{p,\delta }(f)\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(A,B)$$

and

$$f\in {\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(A,B)\Rightarrow F_{p,\delta }(f)\in {\mathrm{\Sigma }}_p{K}_{\beta ,\mu }^{\alpha }(A,B).$$

Theorem 6 Let $\varphi ,\psi \in M$ with

$$\underset{z\in U}{max}Re\{\varphi (z)\}<\frac{\delta +p-\eta }{p-\eta }(\delta >-p).$$

If$f\in {\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha }(\eta ,\gamma ;\varphi ,\psi )$, then$F_{p,\delta }(f)\in {\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha }(\eta ,\gamma ;\varphi ,\psi )$.

Proof Let $f\in {\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha }(\eta ,\gamma ;\varphi ,\psi )$. Then, from the definition of the class ${\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha }(\eta ,\gamma ;\varphi ,\psi )$, there exists a function $g\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$ such that

$$\frac{1}{p-\gamma }(-\frac{z{(H_{p,\beta ,\mu }^{\sigma }f(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu }^{\sigma }g(z)}-\gamma )\prec \psi (z)(z\in U).$$

(3.5)

Now, let

$$h(z)=\frac{1}{p-\gamma }(-\frac{z{(H_{p,\beta ,\mu }^{\sigma }{F}_{p,\delta }(f)(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu }^{\sigma }{F}_{p,\delta }(g)(z)}-\gamma ),$$

(3.6)

where $h(z)$ is analytic in U with $h(0)=1$. Applying (3.2) in (3.6), we have

(3.7)

Since $g\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$, then by Theorem 4, we have $F_{p,\delta }(g)(z)\in {\mathrm{\Sigma }}_p{S}_{\beta ,\mu }^{\alpha }(\eta ;\varphi )$. Let

$$q(z)=\frac{1}{p-\eta }(-\frac{z{(H_{p,\beta ,\mu }^{\sigma }{F}_{p,\delta }(g)(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu }^{\sigma }{F}_{p,\delta }g(z)}-\eta ),$$

(3.8)

where $q\prec \varphi$ in U. Then, using the same techniques as in the proof of Theorem 3 and using (3.5) and (3.7), we have

$$\frac{1}{p-\gamma }(-\frac{z{(H_{p,\beta ,\mu }^{\sigma }f(z))}^{\mathrm{\prime }}}{H_{p,\beta ,\mu }^{\sigma }g(z)}-\gamma )=h(z)+\frac{z{h}^{\mathrm{\prime }}(z)}{\delta +p-\eta -(p-\eta )q(z)}\prec \psi (z).$$

(3.9)

Since $Re\{\frac{1}{\delta +p-\eta -(p-\eta )q(z)}\}>0$, then applying Lemma 2, we find that $h\prec \psi$, which yields $F_{p,\delta }(f)(z)\in {\mathrm{\Sigma }}_p{C}_{\beta ,\mu }^{\alpha }(\eta ,\gamma ;\varphi ,\psi )$. This completes the proof of Theorem 6. □

Remark Putting $\mu =1$ in the above results, we obtain the results corresponding to the operator $H_{p,\beta ,1}^{\alpha }$ defined by (1.11).

Author’s contributions

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