Similar Classes of Convex and Close-to-Convex Meromorphic Functions Obtained Through Integral Operators

Abstract

We define new classes of meromorphic p-valent convex functions, respectively, meromorphic close-to-convex functions, by using an extension of Wanas operator in order to study the preservation properties of these classes, when a well-known integral operator is used. We find the conditions which allow this operator to preserve the classes mentioned above, and we will remark the symmetry between these classes.

1. Introduction and Preliminaries

This paper could be included in the well-known Geometric Function Theory, which is a very beautiful field of Complex Analysis. The Geometric Function Theory deals with univalent functions, starlike functions, convex and close-to-convex functions, p-valent functions, meromorphic functions, meromorphic starlike (convex, close-to-convex) functions, harmonic functions, etc. We recommend, for someone interested to start studying GFT, the monograph [1].

The study of operators plays an important role in mathematics, so all kind of operators (integral, differential, convolution) are used to obtain new subclasses and to study their properties. In this work, we introduce new classes of meromorphic p-valent functions, by using an extension of Wanas operator, and we study some general properties, together with the important preserving properties of these classes. For preserving properties, we will use an integral operator, defined on the class of meromorphic p-valent functions some years ago in [2], which is denoted by $J_{p,\gamma }$. This operator is considered to be an easy one, and we think that it can be used to study the preserving properties of other special classes of meromorphic functions obtained from spiral-like functions, Janowski-type functions or positive-real part functions. We try to look after the symmetry between our classes, while other papers deal with the symmetry of the operators applied to the meromorphic functions, as we can see in [3] or [4].

Next, we will mention some recent papers that also used the Wanas operator: [5,6,7,8,9,10].

We prefer to work on classes of meromorphic functions because we think that new interesting results could be obtained. The literature on meromorphic functions is large, but the Geometric Function Theory of meromorphic functions may be studied more. Interesting results on this topic may also be found in the following works: [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].

We consider $U=\{z\in \mathbb{C}:|z|<1\}$ as the unit disc, $\dot{U}=U\setminus \left\{0\right\}$ and

$$H\left(U\right)=\{f:U\to \mathbb{C}:f\mathrm{is}\mathrm{holomorphic}\mathrm{in}U\}.$$

For $p\in {\mathbb{N}}^{*}=\{1,2,\dots \}$, we have ${\mathrm{\Sigma }}_p=\left\{{\displaystyle g:g\left(z\right)=\frac{a_{-p}}{z^p}+a_0+a_{1}z+\cdots ,z\in \dot{U},a_{-p}\ne 0}\right\}$, the class of meromorphic functions in U.

For our results, we will also need the definitions of the following classes:

• ${\mathrm{\Sigma }}_{p,0}=\{g\in {\mathrm{\Sigma }}_p:a_{-p}=1\}$,
• $H[a,n]=\{f\in H\left(U\right):f\left(z\right)=a+a_n{z}^n+a_{n+1}{z}^{n+1}+\dots \},$ for $a\in \mathbb{C}$, $n\in {\mathbb{N}}^{*},$
• $\mathrm{\Sigma }{K}_p\left(\alpha \right)=\left\{g\in {\mathrm{\Sigma }}_p:\mathrm{Re}{\displaystyle \left[1+\frac{z{g}^{″}\left(z\right)}{g^{\prime }\left(z\right)}\right]}<-\alpha ,z\in U\right\}$, where $\alpha <p$,
• $\mathrm{\Sigma }{K}_{p,0}\left(\alpha \right)=\mathrm{\Sigma }{K}_p\left(\alpha \right)\cap {\mathrm{\Sigma }}_{p,0},$
• $\mathrm{\Sigma }{K}_p(\alpha ,\delta )=\left\{g\in {\mathrm{\Sigma }}_p:\alpha <\mathrm{Re}{\displaystyle \left[-1-\frac{z{g}^{″}\left(z\right)}{g^{\prime }\left(z\right)}\right]}<\delta ,z\in U\right\}$, where $\alpha <p<\delta ,$
• $\mathrm{\Sigma }{K}_{p,0}(\alpha ,\delta )=\mathrm{\Sigma }{K}_p(\alpha ,\delta )\cap {\mathrm{\Sigma }}_{p,0},$
• $\mathrm{\Sigma }{\mathcal{C}}_{p,0}(\alpha ,\delta ;\phi )=\left\{g\in {\mathrm{\Sigma }}_{p,0}:\alpha <\mathrm{Re}{\displaystyle \left[\frac{g^{\prime }\left(z\right)}{{\phi }^{\prime }\left(z\right)}\right]}<\delta ,z\in U\right\}$, where $\alpha <1\le p<\delta$, and the function $\phi$ belongs to the class $\mathrm{\Sigma }{K}_{p,0}(\alpha ,\delta ).$

For $A<1<B$, $\alpha <1\le p<\delta$ and $\phi \in \mathrm{\Sigma }{K}_p(\alpha ,\delta ),$ we consider the class

• $\mathrm{\Sigma }{\mathcal{C}}_{p,0}(A,B;\phi )=\left\{g\in {\mathrm{\Sigma }}_{p,0}:A<\mathrm{Re}{\displaystyle \left[\frac{g^{\prime }\left(z\right)}{{\phi }^{\prime }\left(z\right)}\right]}<B,z\in U\right\}$.

For $p\in {\mathbb{N}}^{*},n\in \mathbb{N}$ and $0<q<1$, we consider the extension of the Wanas operator for meromorphic functions, denoted by $W_q^n$ and introduced for the first time in [27]. Of course, we have $W_q^n:{\mathrm{\Sigma }}_p\to {\mathrm{\Sigma }}_p$, and

$$W_q^n\left(g\right)\left(z\right)={\displaystyle \frac{a_{-p}}{z^p}}+\sum _{k=0}^{\infty }{\left({\displaystyle \frac{1-q^{k+1}}{1-q}}\right)}^n{a}_k{z}^k,z\in U,$$

where $g\in {\mathrm{\Sigma }}_p$ is of the form $g\left(z\right)={\displaystyle \frac{a_{-p}}{z^p}+\sum _{k=0}^{\infty }{a}_k{z}^k,\mathrm{with}z\in U,a_{-p}\ne 0.}$

We know that, for $W_q^n$, we have the following six basic proprieties:

(1)

${\displaystyle W_q^{0}g\left(z\right)=g\left(z\right)}$;

(2)

${\displaystyle W_q^{1}g\left(z\right)=\frac{a_{-p}}{z^p}+\sum _{k=0}^{\infty }(1+q+\dots +q^k)a_k{z}^k}$;

(3)

${\displaystyle W_q^n\left(\alpha g_1+\beta g_2\right)\left(z\right)=\alpha W_q^n\left(g_1\right)\left(z\right)+\beta W_q^n\left(g_2\right)\left(z\right),\alpha ,\beta \in \mathbb{C},g_1,g_2\in {\mathrm{\Sigma }}_p}$;

(4)

${\displaystyle W_q^n\left(W_q^m\left(g\right)\left(z\right)\right)=W_q^m\left(W_q^n\left(g\right)\left(z\right)\right)=W_q^{n+m}\left(g\right)\left(z\right)}$;

(5)

${\displaystyle W_q^n\left(z{g}^{\prime }\left(z\right)\right)=z·{\left(W_q^{n}g\left(z\right)\right)}^{\prime }}$;

(6)

$J_{p,\gamma }\left(W_q^n\left(g\right)\right)=W_q^n\left(J_{p,\gamma }\left(g\right)\right)$, where $J_{p,\gamma }\left(g\right)\left(z\right)={\displaystyle \frac{\gamma -p}{z^{\gamma }}{\int }_0^{z}g\left(t\right)t^{\gamma -1}dt}.$

To prove the results mentioned in the next paragraph, we need the following theorems, which were already proved in [2]:

Theorem 1. 

Let $p\in {\mathbb{N}}^{*},\gamma \in \mathbb{C}$ with $\mathrm{Re}\gamma >p$ and let $\alpha <p<\delta \le \mathrm{Re}\gamma$. If $g\in \mathrm{\Sigma }{K}_p(\alpha ,\delta )$ and $z^{p+1}{J}_{p,\gamma }^{\prime }\left(g\right)\left(z\right)\ne 0,z\in U$, then

$$J_{p,\gamma }\left(g\right)\in \mathrm{\Sigma }{K}_p(\alpha ,\delta ).$$

Theorem 2. 

Let $p\in {\mathbb{N}}^{*},\alpha \in \mathbb{R},\gamma \in \mathbb{C}$ with $\alpha <p<\mathrm{Re}\gamma$.

If $g\in \mathrm{\Sigma }{K}_p\left(\alpha \right)$ and $z^{p+1}{J}_{p,\gamma }^{\prime }\left(g\right)\left(z\right)\ne 0,z\in U$, then

$$J_{p,\gamma }\left(g\right)\in \mathrm{\Sigma }{K}_p\left(\alpha \right).$$

Theorem 3. 

Let $p\in {\mathbb{N}}^{*},\gamma \in \mathbb{C}$ with $\mathrm{Re}\gamma >p$, and $\alpha <1\le p<\delta \le \mathrm{Re}\gamma$. Let φ be a function in $\mathrm{\Sigma }{K}_{p,0}(\alpha ,\delta )$ and $g\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}(\alpha ,\delta ;\phi )$, such that $z^{p+1}{J}_{p,\gamma }^{\prime }\left(\phi \right)\ne 0,z\in U$; then,

$$J_{p,\gamma }\left(g\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}(\alpha ,\delta ;\mathrm{\Phi }),$$

where $\mathrm{\Phi }=J_{p,\gamma }\left(\phi \right).$

2. Main Results

Definition 1. 

For $p\in {\mathbb{N}}^{*},n\in \mathbb{N}$, $0<q<1$ and $\alpha <p<\delta$, let

$$\mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )=\left\{g\in {\mathrm{\Sigma }}_p:\alpha <\mathrm{Re}\left[-1-{\displaystyle \frac{z{\left(W_q^{n}g\left(z\right)\right)}^{″}}{{\left(W_q^{n}g\left(z\right)\right)}^{\prime }}}\right]<\delta ,z\in U.\right\},$$

(1)

$$\mathrm{\Sigma }{K}_{p,q}^n\left(\alpha \right)=\left\{g\in {\mathrm{\Sigma }}_p:\mathrm{Re}\left[-1-{\displaystyle \frac{z{\left(W_q^{n}g\left(z\right)\right)}^{″}}{{\left(W_q^{n}g\left(z\right)\right)}^{\prime }}}\right]>\alpha ,z\in U.\right\},$$

(2)

and $\mathrm{\Sigma }{K}_{p,q}^n=\mathrm{\Sigma }{K}_{p,q}^n\left(0\right).$

We remark that for $n=0$, we have $\mathrm{\Sigma }{K}_{p,q}^0(\alpha ,\delta )=\mathrm{\Sigma }{K}_p(\alpha ,\delta )$ and $\mathrm{\Sigma }{K}_{p,q}^0\left(\alpha \right)=\mathrm{\Sigma }{K}_p\left(\alpha \right)$, classes which were already studied in [2].

Our first remark presents the link between the sets $\mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )$ and $\mathrm{\Sigma }{K}_{p,q}^{n-1}(\alpha ,\delta )$.

Remark 1. 

Let us consider the function $g\in {\mathrm{\Sigma }}_p$ and the numbers $p\in {\mathbb{N}}^{*},n\in \mathbb{N}$, $q\in (0,1)$, $\alpha <p<\delta$. Then,

$$g\in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )\iff W_q\left(g\right)\in \mathrm{\Sigma }{K}_{p,q}^{n-1}(\alpha ,\delta ).$$

Proof. 

We know that $g\in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )$ if and only if $W_q^{n}g\in \mathrm{\Sigma }{K}_p(\alpha ,\delta )$.

As $W_q^{n}g=W_q^{n-1}\left(W_{q}g\right)$, we have $W_q^{n-1}\left(W_{q}g\right)\in \mathrm{\Sigma }{K}_p(\alpha ,\delta )$. This means that

$$W_q\left(g\right)\in \mathrm{\Sigma }{K}_{p,q}^{n-1}(\alpha ,\delta ).$$

□

Theorem 4. 

Let us consider the function $g\in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )$ (where $p\in {\mathbb{N}}^{*},n\in \mathbb{N}$, $q\in (0,1)$, $\alpha <p<\delta$) and $\gamma \in \mathbb{C}$, with $\mathrm{Re}\gamma \ge \delta$. If $G\left(z\right)=J_{p,\gamma }\left(g\right)\left(z\right)={\displaystyle \frac{\gamma -p}{z^{\gamma }}{\int }_0^{z}g\left(t\right)t^{\gamma -1}dt}$ with $z^p{W}_q^n\left(z{G}^{\prime }\left(z\right)\right)\ne 0$, then $G\in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta ).$

Proof. 

Since $g\in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )$, we have $W_q^n\left(g\right)\in \mathrm{\Sigma }{K}_p(\alpha ,\delta )$. Let us denote $W_q^n\left(g\right)=h$. We want to apply Theorem 13 from [2] (see Theorem 1) to the new function h. Since $h\in \mathrm{\Sigma }{K}_p(\alpha ,\delta )$, we need to verify the condition

$$z^{p+1}{J}_{p,\gamma }^{\prime }\left(h\right)\left(z\right)\ne 0,z\in U.$$

We have

$$J_{p,\gamma }\left(h\right)=H\iff (\gamma -p){\int }_0^{z}h\left(t\right)t^{\gamma -1}dt=z^{\gamma }H\left(z\right),$$

and after differentiating, we get that

$$(\gamma -p)h\left(z\right)z^{\gamma -1}=\gamma z^{\gamma -1}H\left(z\right)+z^{\gamma }{H}^{\prime }\left(z\right).$$

This means that we have

$$z{H}^{\prime }\left(z\right)=(\gamma -p)h\left(z\right)-\gamma H\left(z\right).$$

(3)

Hence, $z^{p+1}{H}^{\prime }\left(z\right)=(\gamma -p)z^{p}h\left(z\right)-\gamma z^{p}H\left(z\right),$ which is equivalent to

$$z^{p+1}{\left[J_{p,\gamma }\left(h\right)\right]}^{\prime }\left(z\right)=(\gamma -p)z^{p}h\left(z\right)-\gamma z^p{J}_{p,\gamma }\left(h\right)\left(z\right).$$

On the other hand,

$$\begin{array}{c}(\gamma -p)z^{p}h\left(z\right)-\gamma z^p{J}_{p,\gamma }\left(h\right)\left(z\right)=(\gamma -p)z^p{W}_q^n\left(g\right)\left(z\right)-\gamma z^p{J}_{p,\gamma }\left(W_q^n\left(g\right)\right)\left(z\right)=\\ =(\gamma -p)z^p{W}_q^n\left(g\right)\left(z\right)-\gamma z^p{W}_q^n\left(J_{p,\gamma }\left(g\right)\right)\left(z\right)=z^p{W}_q^n\left((\gamma -p)g\left(z\right)-\gamma J_{p,\gamma }\left(g\right)\left(z\right)\right).\end{array}$$

For the last equality, we used the fact that the function $W_q^n$ is linear.

Since we have $G\left(z\right)=J_{p,\gamma }\left(g\right)\left(z\right),z\in U$, we may use Equality (3) for g instead of h, and we can obtain

$$z{G}^{\prime }\left(z\right)=(\gamma -p)g\left(z\right)-\gamma G\left(z\right),z\in U.$$

Therefore,

$$z^{p+1}{\left[J_{p,\gamma }\left(h\right)\right]}^{\prime }\left(z\right)=(\gamma -p)z^{p}h\left(z\right)-\gamma z^p{J}_{p,\gamma }\left(h\right)\left(z\right)=z^p{W}_q^n\left(z{G}^{\prime }\left(z\right)\right).$$

(4)

From the hypothesis, we have $z^p{W}_q^n\left(z{G}^{\prime }\left(z\right)\right)\ne 0,z\in U.$ Therefore, Equality (4) implies that

$$z^{p+1}{\left[J_{p,\gamma }\left(h\right)\right]}^{\prime }\left(z\right)\ne 0,z\in U.$$

Hence, from Theorem 1, we get $J_{p,\gamma }\left(h\right)\in \mathrm{\Sigma }{K}_p(\alpha ,\delta )$. This means that

$$J_{p,\gamma }\left(W_q^n\left(g\right)\right)\in \mathrm{\Sigma }{K}_p(\alpha ,\delta ).$$

We use now the fact that

$$J_{p,\gamma }\left(W_q^n\left(g\right)\right)=W_q^n\left(J_{p,\gamma }\left(g\right)\right),\left(\mathrm{see}\mathrm{property}\left(6\right)\right)$$

and we obtain the following conclusion:

$$W_q^n\left(J_{p,\gamma }\left(g\right)\right)\in \mathrm{\Sigma }{K}_p(\alpha ,\delta ).$$

Therefore, $G=J_{p,\gamma }\left(g\right)\in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta ).$ □

If we consider (in the above theorem) that $n=0$, we obtain the following:

Corollary 1. 

Let us consider the function $g\in \mathrm{\Sigma }{K}_p(\alpha ,\delta )$ (where $p\in {\mathbb{N}}^{*},\alpha <p<\delta$) and $\gamma \in \mathbb{C}$, with $\mathrm{Re}\gamma \ge \delta$. If $G\left(z\right)=J_{p,\gamma }\left(g\right)\left(z\right)={\displaystyle \frac{\gamma -p}{z^{\gamma }}{\int }_0^{z}g\left(t\right)t^{\gamma -1}dt}$ with $z^{p+1}{G}^{\prime }\left(z\right))\ne 0$, then $G\in \mathrm{\Sigma }{K}_p(\alpha ,\delta ).$

Proof. 

The proof is pretty obvious since we have $\mathrm{\Sigma }{K}_{p,q}^0(\alpha ,\delta )=\mathrm{\Sigma }{K}_p(\alpha ,\delta )$ and $W_q^0\left(g\right)=g$ (for any $q\in (0,1),g\in {\mathrm{\Sigma }}_p$). We would like to mention that what follows condition $z^p{W}_q^n\left(z{G}^{\prime }\left(z\right)\right)\ne 0,z\in U,$ from Theorem 4, is that (when $n=0$) $z^{p+1}{G}^{\prime }\left(z\right)\ne 0,z\in U,$ which is the condition which we have met in Theorem 1. □

The result of Corollary 1 was also found in [2].

It is easy to see that a proof similar to that of Theorem 4 can be made, now using Theorem 2, to obtain the result that follows.

Theorem 5. 

Consider the function $g\in \mathrm{\Sigma }{K}_{p,q}^n\left(\alpha \right)$ (where $p\in {\mathbb{N}}^{*},n\in \mathbb{N}$, $q\in (0,1)$, $\alpha <p$) and $\gamma \in \mathbb{C}$, with $\mathrm{Re}\gamma \ge p$. If $G\left(z\right)=J_{p,\gamma }\left(g\right)\left(z\right)={\displaystyle \frac{\gamma -p}{z^{\gamma }}{\int }_0^{z}g\left(t\right)t^{\gamma -1}dt}$ with $z^p{W}_q^n\left(z{G}^{\prime }\left(z\right)\right)\ne 0$, then $G\in \mathrm{\Sigma }{K}_{p,q}^n\left(\alpha \right).$

Next, we define two new classes of meromorphic functions, which will generalize the class of close-to-convex meromorphic functions. These classes are obtained through the well-known condition of close-to-convexity combined with the Wanas operator. Thus, we have the following definitions:

Definition 2. 

For $A<1<B$, $p\in {\mathbb{N}}^{*},n\in \mathbb{N}$, $0<q<1$, $\alpha <1\le p<\delta$ and the function $\phi \in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )\cap {\mathrm{\Sigma }}_{p,0},$ let us define

$$\mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n(A,B;\phi )=\left\{g\in {\mathrm{\Sigma }}_{p,0}:A<\mathrm{Re}{\displaystyle \left[\frac{{\left(W_q^{n}g\right)}^{\prime }\left(z\right)}{{\left(W_q^n\phi \right)}^{\prime }\left(z\right)}\right]}<B,z\in U\right\}.$$

Definition 3. 

For $p\in {\mathbb{N}}^{*},n\in \mathbb{N}$, $0<q<1$, $\alpha <1\le p<\delta$ and $\phi \in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )\cap {\mathrm{\Sigma }}_{p,0},$ let us define

$$\mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n(\alpha ,\delta ;\phi )=\left\{g\in {\mathrm{\Sigma }}_{p,0}:\alpha <\mathrm{Re}{\displaystyle \left[\frac{{\left(W_q^{n}g\right)}^{\prime }\left(z\right)}{{\left(W_q^n\phi \right)}^{\prime }\left(z\right)}\right]}<\delta ,z\in U\right\}.$$

We remark that for $n=0$, we have $\mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^0(\alpha ,\delta ;\phi )=\mathrm{\Sigma }{\mathcal{C}}_{p,0}(\alpha ,\delta ;\phi )$. This class was studied in [2].

Our next remark presents the link between the sets $\mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n(\alpha ,\delta ;\phi )$ and $\mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^{n-1}(\alpha ,\delta ;W_q\left(\phi \right))$.

Remark 2. 

Consider the function $g\in {\mathrm{\Sigma }}_{p,0}$ and the numbers $p\in {\mathbb{N}}^{*},n\in \mathbb{N}$, $q\in (0,1)$, $\alpha <1\le p<\delta$. Then,

$$g\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n(\alpha ,\delta ;\phi )\iff W_q\left(g\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^{n-1}(\alpha ,\delta ;W_q\left(\phi \right)).$$

Proof. 

We know that $g\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n(\alpha ,\delta ;\phi )$ if and only if $g\in {\mathrm{\Sigma }}_{p,0}$ and

$$\alpha <\mathrm{Re}{\displaystyle \left[\frac{{\left(W_q^{n}g\right)}^{\prime }\left(z\right)}{{\left(W_q^n\phi \right)}^{\prime }\left(z\right)}\right]}<\delta ,z\in U,$$

where $\phi \in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )\cap {\mathrm{\Sigma }}_{p,0}$.

As $W_q^{n}g=W_q^{n-1}\left(W_{q}g\right)$, we have

$${\displaystyle \frac{{\left(W_q^{n}g\right)}^{\prime }\left(z\right)}{{\left(W_q^n\phi \right)}^{\prime }\left(z\right)}}={\displaystyle \frac{{\left(W_q^{n-1}\left(W_{q}g\right)\right)}^{\prime }\left(z\right)}{{\left(W_q^{n-1}\left(W_q\phi \right)\right)}^{\prime }\left(z\right)}},$$

so

$$\alpha <\mathrm{Re}{\displaystyle \left[\frac{{\left(W_q^{n-1}\left(W_{q}g\right)\right)}^{\prime }\left(z\right)}{{\left(W_q^{n-1}\left(W_q\phi \right)\right)}^{\prime }\left(z\right)}\right]}<\delta ,z\in U.$$

On the other hand, since $\phi \in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta ),$ we $W_q\phi \in \mathrm{\Sigma }{K}_{p,q}^{n-1}(\alpha ,\delta )$ from Remark 1.

It is obvious from the expression of $W_q$ that, for $g,\phi \in {\mathrm{\Sigma }}_{p,0},$ we have $W_{q}g,W_q\phi \in {\mathrm{\Sigma }}_{p,0}.$ Hence, from the above results, we may write the following:

$$W_q\left(g\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^{n-1}(\alpha ,\delta ;W_q\left(\phi \right)).$$

□

The following result will study the preservation of the class $\mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n(\alpha ,\delta ;\phi )$, when the integral operator $J_{p,\gamma }$ will be applied to a function g from the above-mentioned class.

Theorem 6. 

Let $p\in {\mathbb{N}}^{*},\alpha ,\delta \in \mathbb{R},\gamma \in \mathbb{C}$ with $\alpha <1\le p<\delta \le \mathrm{Re}\gamma$.

We consider $\phi \in \mathrm{\Sigma }{K}_{p,q,0}^n(\alpha ,\delta )$ and $\varphi =J_{p,\gamma }\left(\phi \right)$, such that $z^p{W}_q^n\left((\gamma -p)\phi -\gamma \varphi \right)\ne 0,z\in U.$

Then,

$$J_{p,\gamma }\left(g\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n(\alpha ,\delta ;\varphi ),$$

when $g\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n(\alpha ,\delta ;\phi ).$

Proof. 

From the definition of the class $\mathrm{\Sigma }{K}_{p,q,0}^n(\alpha ,\delta )$, we have $\phi \in \mathrm{\Sigma }{K}_{p,q,0}^n(\alpha ,\delta )$ if and only if $W_q^n\phi \in \mathrm{\Sigma }{K}_{p,0}(\alpha ,\delta )$. Using this equivalence and the definition of the class $\mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n(\alpha ,\delta ;\phi )$, we may say that the function $g\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n(\alpha ,\delta ;\phi )$ if and only if we have $W_q^{n}g\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}(\alpha ,\delta ;W_q^n\phi ).$

We will denote $W_q^n\phi$ by ${\phi }_1$ and $W_q^{n}g$ by $g_1$.

Hence, $g_1\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}(\alpha ,\delta ;{\phi }_1)$ and

$${\varphi }_1=W_q^n\varphi =W_q^n\left(J_{p,\gamma }\left(\phi \right)\right)=J_{p,\gamma }\left(W_q^n\phi \right)=J_{p,\gamma }\left({\phi }_1\right).$$

Next, we will prove that the condition $z^p{W}_q^n\left((\gamma -p)\phi -\gamma \varphi \right)\ne 0$ is equivalent to $z^{p+1}{J}_{p,\gamma }^{\prime }\left({\phi }_1\right)\ne 0,z\in U,$ a condition in which it is necessary to apply Theorem 3, and it has already been proved in [2].

Since the operator $W_q^n$ is linear, the condition $z^p{W}_q^n\left((\gamma -p)\phi -\gamma \varphi \right)\ne 0,z\in U,$ is equivalent to

$$z^p\left[(\gamma -p)W_q^n\left(\phi \right)-\gamma W_q^n\left(\varphi \right)\right]\ne 0\iff z^p\left[(\gamma -p){\phi }_1-\gamma {\varphi }_1\right]\ne 0,z\in U.$$

Since we have ${\varphi }_1\left(z\right)=J_{p,\gamma }\left({\phi }_1\right)\left(z\right),z\in U$, we use Equality (3) for ${\phi }_1$ instead of h and we obtain

$$z{\varphi }_1^{\prime }\left(z\right)=(\gamma -p){\phi }_1\left(z\right)-\gamma {\varphi }_1\left(z\right),z\in \dot{U};$$

so,

$$z^{p+1}{\varphi }_1^{\prime }\left(z\right)=z^p\left[(\gamma -p){\phi }_1\left(z\right)-\gamma {\varphi }_1\left(z\right)\right],z\in U.$$

Therefore,

$$z^{p+1}{\varphi }_1^{\prime }\left(z\right)=z^{p+1}{\left[J_{p,\gamma }\left({\phi }_1\right)\right]}^{\prime }\left(z\right)\ne 0,z\in U.$$

We have now that ${\phi }_1$ is a function in $\mathrm{\Sigma }{K}_{p,0}(\alpha ,\delta )$ with $z^{p+1}{J}_{p,\gamma }^{\prime }\left({\phi }_1\right)\ne 0,z\in U$ and $g_1\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}(\alpha ,\delta ;{\phi }_1)$. Then, from Theorem 3, we obtain

$$J_{p,\gamma }\left(g_1\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}(\alpha ,\delta ;{\varphi }_1),$$

which is equivalent to

$$J_{p,\gamma }\left(W_q^n\left(g\right)\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}(\alpha ,\delta ;{\varphi }_1).$$

(5)

Since $J_{p,\gamma }\left(W_q^n\left(g\right)\right)=W_q^n\left(J_{p,\gamma }\left(g\right)\right)$, we obtain that

$$W_q^n\left(J_{p,\gamma }\left(g\right)\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}(\alpha ,\delta ;{\varphi }_1)$$

from (5), where ${\varphi }_1=J_{p,\gamma }\left({\phi }_1\right)=J_{p,\gamma }\left(W_q^n\left(\phi \right)\right)=W_q^n\left(J_{p,\gamma }\left(\phi \right)\right).$

Therefore,

$$W_q^n\left(J_{p,\gamma }\left(g\right)\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(\alpha ,\delta ;W_q^n\left(J_{p,\gamma }\left(\phi \right)\right)\right).$$

This means that

$$J_{p,\gamma }\left(g\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n\left(\alpha ,\delta ;J_{p,\gamma }\left(\phi \right)\right)=\mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n\left(\alpha ,\delta ;\varphi \right).$$

□

The next result will be used to obtain functions from $\mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n\left(\alpha ,\delta ;\phi \right)$, using functions from the class $\mathrm{\Sigma }{K}_{p,q,0}^n(\alpha ,\delta )$.

Theorem 7. 

Let $p\in {\mathbb{N}}^{*},\alpha ,\beta ,\delta \in \mathbb{R}$ with $\beta \ne 0$ and $\alpha <1\le p<\delta$. For $g\in \mathrm{\Sigma }{K}_{p,0}(\alpha ,\delta )$, we consider the function

$$f_{\beta ,g}\left(z\right)=(1+\beta p)g\left(z\right)+\beta z{g}^{\prime }\left(z\right).$$

For $\beta >0$, we have

$$f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(1+\beta (p-\delta ),1+\beta (p-\alpha );g\right),$$

and for $\beta <0$, we have

$$f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(1+\beta (p-\alpha ),1+\beta (p-\delta );g\right).$$

Proof. 

Firstly, since $g\in {\mathrm{\Sigma }}_{p,0}$, g is of the form

$${\displaystyle g\left(z\right)=\frac{1}{z^p}+\sum _{k=0}^{\infty }{a}_k{z}^k.}$$

Therefore, the function $f_{\beta ,g}$ has the form

$${\displaystyle f_{\beta ,g}\left(z\right)=(1+\beta p)g\left(z\right)+\beta z{g}^{\prime }\left(z\right)=\frac{1}{z^p}+\sum _{k=0}^{\infty }(1+\beta p+\beta k)a_k{z}^k.}$$

Hence, we have $f_{\beta ,g}\in {\mathrm{\Sigma }}_{p,0}$.

It is obvious that we have

$$f_{\beta ,g}^{\prime }\left(z\right)=(1+\beta p)g^{\prime }\left(z\right)+\beta g^{\prime }\left(z\right)+\beta z{g}^{″}\left(z\right),z\in \dot{U};$$

hence,

$${\displaystyle \frac{f_{\beta ,g}^{\prime }\left(z\right)}{g^{\prime }\left(z\right)}}=1+\beta p+\beta \left(1+{\displaystyle \frac{z{g}^{″}\left(z\right)}{g^{\prime }\left(z\right)}}\right).$$

(6)

Since $g\in \mathrm{\Sigma }{K}_{p,0}(\alpha ,\delta )$, we have

$$\alpha <\mathrm{Re}{\displaystyle \left[-1-\frac{z{g}^{″}\left(z\right)}{g^{\prime }\left(z\right)}\right]}<\delta ,z\in U.$$

Therefore, for $\beta >0$, we get $\beta ·\alpha <\beta \mathrm{Re}{\displaystyle \left[-1-\frac{z{g}^{″}\left(z\right)}{g^{\prime }\left(z\right)}\right]}<\beta ·\delta$, which is equivalent to

$$1+\beta (p-\delta )<\mathrm{Re}\left[1+\beta p+\beta \left(1+{\displaystyle \frac{z{g}^{″}\left(z\right)}{g^{\prime }\left(z\right)}}\right)\right]<1+\beta (p-\alpha ).$$

(7)

For $\beta <0$, we get $\beta ·\alpha >\beta \mathrm{Re}{\displaystyle \left[-1-\frac{z{g}^{″}\left(z\right)}{g^{\prime }\left(z\right)}\right]}>\beta ·\delta$, which is equivalent to

$$1+\beta (p-\alpha )<\mathrm{Re}\left[1+\beta p+\beta \left(1+{\displaystyle \frac{z{g}^{″}\left(z\right)}{g^{\prime }\left(z\right)}}\right)\right]<1+\beta (p-\delta ).$$

(8)

It follows from (6) and (7) that

$$1+\beta (p-\delta )<\mathrm{Re}{\displaystyle \frac{f_{\beta ,g}^{\prime }\left(z\right)}{g^{\prime }\left(z\right)}}<1+\beta (p-\alpha );$$

therefore,

$$f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(1+\beta (p-\delta ),1+\beta (p-\alpha );g\right).$$

From (6) and (8),

$$1+\beta (p-\alpha )<\mathrm{Re}{\displaystyle \frac{f_{\beta ,g}^{\prime }\left(z\right)}{g^{\prime }\left(z\right)}}<1+\beta (p-\delta );$$

therefore,

$$f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(1+\beta (p-\alpha ),1+\beta (p-\delta );g\right).$$

□

We notice that for $\beta =0$, the function used in Theorem 7 is $f_{0,g}=g$.

Since $g\in \mathrm{\Sigma }{K}_{p,0}(\alpha ,\delta )$, we have $g\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(\alpha ,\delta ;g\right)$, so $f_{0,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(\alpha ,\delta ;g\right)$.

The next two corollaries follow on from Theorem 7.

Corollary 2. 

Let $p\in {\mathbb{N}}^{*},\alpha ,\beta ,\delta \in \mathbb{R}$ with $\alpha <1\le p<\delta$ and

$$max\left\{{\displaystyle \frac{1-\alpha }{\alpha -p};\frac{1-\delta }{\delta -p}}\right\}\le \beta <0.$$

Then, for $g\in \mathrm{\Sigma }{K}_{p,0}(\alpha ,\delta )$, we have the function $f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(\alpha ,\delta ;g\right).$

Proof. 

From

$$max\left\{{\displaystyle \frac{1-\alpha }{\alpha -p};\frac{1-\delta }{\delta -p}}\right\}\le \beta <0,$$

we have

$$\left\{\begin{array}{c}{\displaystyle \frac{1-\alpha }{\alpha -p}\le \beta }\hfill \\ \\ {\displaystyle \frac{1-\delta }{\delta -p}\le \beta }\hfill \end{array}\right.\iff \left\{\begin{array}{c}1-\alpha \ge \beta (\alpha -p)\hfill \\ 1-\delta \le \beta (\delta -p)\hfill \end{array}\right.\iff \left\{\begin{array}{c}1+\beta (p-\alpha )\ge \alpha \hfill \\ 1+\beta (p-\delta )\le \delta \hfill \end{array}\right..$$

We know from Theorem 7 that, for $\beta <0$, we have

$$f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(1+\beta (p-\alpha ),1+\beta (p-\delta );g\right).$$

This means that

$$1+\beta (p-\alpha )<\mathrm{Re}{\displaystyle \frac{f_{\beta ,g}^{\prime }\left(z\right)}{g^{\prime }\left(z\right)}}<1+\beta (p-\delta ).$$

Moreover, from

$$1+\beta (p-\alpha )\ge \alpha ,1+\beta (p-\delta )\le \delta ,$$

we get that

$$\alpha <\mathrm{Re}{\displaystyle \frac{f_{\beta ,g}^{\prime }\left(z\right)}{g^{\prime }\left(z\right)}}<\delta .$$

Hence, $f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(\alpha ,\delta ;g\right).$ □

Corollary 3. 

Let $p\in {\mathbb{N}}^{*},\alpha ,\beta ,\delta \in \mathbb{R}$ with $\alpha <1\le p<\delta$ and

$$0<\beta \le min\left\{{\displaystyle \frac{1-\alpha }{\delta -p};\frac{1-\delta }{\alpha -p}}\right\}.$$

Then, for $g\in \mathrm{\Sigma }{K}_{p,0}(\alpha ,\delta )$, we have the function $f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(\alpha ,\delta ;g\right).$

Theorem 8. 

Let $n\in \mathbb{N},p\in {\mathbb{N}}^{*},\alpha ,\beta ,\delta \in \mathbb{R}$ and $q\in (0,1)$ with $\beta \ne 0$ and $\alpha <1\le p<\delta$. If $g\in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )\cap {\mathrm{\Sigma }}_0$, then for $\beta >0$, we have

$$f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n\left(1+\beta (p-\delta ),1+\beta (p-\alpha );g\right),$$

and for $\beta <0$, we have

$$f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n\left(1+\beta (p-\alpha ),1+\beta (p-\delta );g\right).$$

Proof. 

Since $g\in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )\cap {\mathrm{\Sigma }}_0$, we have that $W_q^n\left(g\right)\in \mathrm{\Sigma }{K}_{p,0}(\alpha ,\delta )$. Thus, from Theorem 7, we get

$$f_{\beta ,W_q^n\left(g\right)}\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(1+\beta (p-\delta ),1+\beta (p-\alpha );W_q^n\left(g\right)\right),$$

(9)

for $\beta >0$, and

$$f_{\beta ,W_q^n\left(g\right)}\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(1+\beta (p-\alpha ),1+\beta (p-\delta );W_q^n\left(g\right)\right),$$

(10)

for $\beta <0$.

On the other hand, we have

$$f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n(a,b;g)\iff W_q^n\left(f_{\beta ,g}\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}(a,b;W_q^n\left(g\right)).$$

By using the properties of the operator $W_q^n$, we see that

$$\begin{array}{c}{W}_q^n\left(f_{\beta ,g}\right)=W_q^n\left((1+\beta p)g+\beta z{g}^{\prime }\right)=\\ =(1+\beta p)W_q^n\left(g\right)+\beta W_q^n\left(z{g}^{\prime }\right)=(1+\beta p)W_q^n\left(g\right)+\beta z{\left(W_q^n\left(g\right)\right)}^{\prime }=f_{\beta ,W_q^n\left(g\right)}.\end{array}$$

Therefore, we get

$$W_q^n\left(f_{\beta ,g}\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(1+\beta (p-\delta ),1+\beta (p-\alpha );W_q^n\left(g\right)\right),$$

for $\beta >0$ and

$$W_q^n\left(f_{\beta ,g}\right)\in \mathrm{\Sigma }{\mathcal{C}}_{p,0}\left(1+\beta (p-\alpha ),1+\beta (p-\delta );W_q^n\left(g\right)\right),$$

for $\beta <0$. This means that for $\beta >0$, we have

$$f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n\left(1+\beta (p-\delta ),1+\beta (p-\alpha );g\right),$$

and for $\beta <0$, we have

$$f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n\left(1+\beta (p-\alpha ),1+\beta (p-\delta );g\right).$$

□

By combining Theorem 8 with Corollary 2 and Corollary 3, we obtain the following two results:

Corollary 4. 

Let $n\in \mathbb{N},p\in {\mathbb{N}}^{*},\alpha ,\beta ,\delta \in \mathbb{R}$ and $q\in (0,1)$ with $\alpha <1\le p<\delta$ and

$$max\left\{{\displaystyle \frac{1-\alpha }{\alpha -p};\frac{1-\delta }{\delta -p}}\right\}\le \beta <0.$$

Then, for $g\in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )\cap {\mathrm{\Sigma }}_0$, we have the function $f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n\left(\alpha ,\delta ;g\right).$

Corollary 5. 

Let $n\in \mathbb{N},p\in {\mathbb{N}}^{*},\alpha ,\beta ,\delta \in \mathbb{R}$ and $q\in (0,1)$ with $\alpha <1\le p<\delta$ and

$$0<\beta \le min\left\{{\displaystyle \frac{1-\alpha }{\delta -p};\frac{1-\delta }{\alpha -p}}\right\}.$$

Then, for $g\in \mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )\cap {\mathrm{\Sigma }}_0$, we have the function $f_{\beta ,g}\in \mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n\left(\alpha ,\delta ;g\right).$

3. Conclusions

In this paper, we define two new classes of meromorphic p-valent functions, denoted by $\mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )$, respectively $\mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n\left(\alpha ,\delta ;\phi \right)$, using the extension of the Wanas operator defined in [27]. We used the condition met at convex meromorphic functions to define the first class and the condition met at close-to-convex meromorphic functions to define the second class. Then, we provided the results with regard to the preserving conditions concerning these classes.

We used the operator $J_{p,\gamma }$, which is a well-known integral operator used to study the symmetry of new classes. The symmetrical preservation of the classes $\mathrm{\Sigma }{K}_{p,q}^n(\alpha ,\delta )$ and $\mathrm{\Sigma }{\mathcal{C}}_{p,q,0}^n\left(\alpha ,\delta ;\phi \right)$, through different integral operators, will be investigated in our future papers.