Convolution Properties of Meromorphic $\mathcal{P}$-Valent Functions with Coefficients of Alternating Type Defined Using $\mathfrak{q}$-Difference Operator

Abstract

Certain characteristics of univalent functions with negative coefficients of the form $f\left(z\right)=z-{\sum }_{n=1}^{\infty }{a}_{2n}{z}^{2n},a_{2n}>0$ have been studied by Silverman and Berman. Pokley, Patil and Shrigan have discovered some insights into the Hadamard product of $\mathcal{P}$-valent functions with negative coefficients. S. M. Khairnar and Meena More have obtained coefficient limits and convolution results for univalent functions lacking a alternating type coefficient. In this paper, using the $\mathfrak{q}$-Difference operator, we developed the a subclass of meromorphically $\mathcal{P}$-valent functions with alternating coefficients. Additionally, we obtained multivalent function convolution results and coefficient limits.

1. Introduction

The many applications of $\mathfrak{q}$-analysis in mathematics and physics have drawn the attention of academics in recent years. Jackson [1,2] developed the $\mathfrak{q}$-analog of the derivative and integral, and he was the first to investigate a specific application of $\mathfrak{q}$-calculus. Recently, a number of authors focused on the classes of $\mathfrak{q}$-starlike functions connected to the Janowski functions [3] from various angles in a series of works [4,5,6,7,8,9,10,11,12,13]. Scholars working on these subjects will find great interest in Srivastava’s newly released survey-cum-explanatory review study [14]. In this review study, Srivastava [14] discussed applications of the fractional $\mathfrak{q}$-derivative operator in geometric function theory and provided some mathematical background. Adding an apparently unnecessary parameter p to the same survey-cum-explanatory review study [14] revealed the silly and unimportant $(p,\mathfrak{q})$ versions of a number of known $q$-results (see [14] p. 340 for further details). In this article, motivated essentially by the above works, we shall define a new subclass of meromorphic $\mathcal{P}$-valent functions with alternating coefficients is developed through the use of $\mathfrak{q}$-difference operator. In addition, coefficient bounds and convolution results of multivalent functions were found.

Let ${\mathcal{N}}_{\mathcal{P}}$ denotes the class of $\mathcal{P}$-valently meromorphic functions of the form:

$$\zeta \left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+\sum _{n=1}^{\infty }{a}_n{\mathfrak{z}}^n\mathcal{P}\in \mathbb{N}=\{1,2,3,...\},$$

which are analytic in the punctured unit disk ${\Delta }^{*}=\Delta \setminus \left\{0\right\}=\{\mathfrak{z}:\mathfrak{z}\in \mathbb{C}:0<|\mathfrak{z}|<1\}.$

Let $T_{\mathcal{P}}$ denote the subclass of ${\mathcal{N}}_{\mathcal{P}}$ consisting the functions of the form

$$\zeta \left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}-\sum _{n=1}^{\infty }{a}_n{\mathfrak{z}}^n.$$

Let $n=\{\alpha :\alpha$ is analytic in $\Delta ,$ $\alpha \left(0\right)=0,\left|\alpha \right(\mathfrak{z}\left)\right|<1$ in $\Delta$}. Let $\mathcal{H}(\gamma ,\delta )$ denote a subclass of analytic functions in ${\Delta }^{*},$ which are of the form $\frac{1+\gamma \alpha \left(\mathfrak{z}\right)}{1+\delta \alpha \left(\mathfrak{z}\right)},-1\le \delta <\gamma \le 1,$ where $\alpha \left(\mathfrak{z}\right)\in n.$

Definition 1 

([15]). The $\mathfrak{q}$-derivative of a function $\zeta \left(\mathfrak{z}\right)$ is ${\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)$ defined as follows:

$${\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)=\left\{\begin{array}{ccc}\hfill \frac{\zeta \left(\mathfrak{z}\right)-\zeta \left(\mathfrak{q}\mathfrak{z}\right)}{\mathfrak{z}(\mathfrak{q}-1)}& if& \mathfrak{z}\ne 0\hfill \\ \\ \hfill {\zeta }^{\prime }\left(0\right)& if& \mathfrak{z}=0\hfill \\ \\ \hfill {\zeta }^{\prime }\left(\mathfrak{z}\right)& if& q\to 1^{-},\mathfrak{z}\ne 0.\hfill \end{array}\right.$$

provided ${\zeta }^{\prime }\left(0\right)$ exists.

From Definition 1, we observe that

$$\underset{\mathfrak{q}\to 1-}{lim}{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)=\underset{\mathfrak{q}\to 1-}{lim}{\displaystyle \frac{\zeta \left(\mathfrak{q}\mathfrak{z}\right)-\zeta \left(\mathfrak{z}\right)}{\mathfrak{z}(\mathfrak{q}-1)}}={\zeta }^{\prime }\left(\mathfrak{z}\right)$$

for a differentiable function $\zeta \left(\mathfrak{z}\right)$.

For $\zeta \left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{a}_n{\mathfrak{z}}^n,$ we can see that

$${\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)={[-\mathcal{P}]}_{\mathfrak{q}}{\mathfrak{z}}^{-\mathcal{P}-1}+\sum _{n=1}^{\infty }{\left[n\right]}_{\mathfrak{q}}{a}_n{\mathfrak{z}}^{n-1}(\mathfrak{z}\ne 0),$$

where ${[-\mathcal{P}]}_{\mathfrak{q}}=\frac{1-{\mathfrak{q}}^{-\mathcal{P}}}{1-\mathfrak{q}}=-{\mathfrak{q}}^{-\mathcal{P}}{\left[\mathcal{P}\right]}_{\mathfrak{q}}$ and

$${\left[\mathcal{P}\right]}_q=\left\{\begin{array}{ccc}\frac{1-q^{\mathcal{P}}}{1-q}\hfill & if& \mathcal{P}\in \mathbb{C}/\left\{0\right\}\hfill \\ q^{\mathcal{P}-1}+q^{\mathcal{P}-2}+...+q+1=\sum _{j=0}^{\mathcal{P}-1}{q}^i\hfill & if& \mathcal{P}\in \mathbb{N}\hfill \\ 1\hfill & if& q^{\to 0^{+}},\mathcal{P}\in \mathbb{C}/\left\{0\right\}\hfill \\ \hfill \mathcal{P}& if& q^{\to 1^{-}},\mathcal{P}\in \mathbb{C}/\left\{0\right\}\hfill \end{array}\right.$$

In this work, we examine the functions of the class $M^{*}$ of $T_{\mathcal{P}}.$

$$\zeta \left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+\sum _{n=1}^{\infty }{(-1)}^{n-1}{a}_n{\mathfrak{z}}^n.$$

S. M. Khairnar and Meena More [16] studied some properties of convolution for the class $M^{*}(\gamma ,\delta )$ and $C^{\prime }(\gamma ,\delta )$ as defined by Silverman and Berman but with missing second coefficient of alternating series see to references [17,18,19,20,21,22,23].

Analogous to the classes $M^{*}(\gamma ,\delta ,),$ $C^{\prime }(\gamma ,\delta )$ and $K^{*}(\gamma ,\delta ,\mathcal{P}),$ we define the following.

$$S^{*}(\gamma ,\delta ,\mathcal{P})=\left\{\zeta :\zeta \in {\mathcal{N}}_{\mathcal{P}}and{\displaystyle \frac{\mathfrak{z}{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}}\prec \mathcal{H}(\gamma ,\delta )\right\}$$

(1)

$$C^{\prime }(\gamma ,\delta ,\mathcal{P})=\left\{\zeta :\zeta \in M^{*}and{\displaystyle \frac{\mathfrak{z}{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}}\prec \mathcal{H}(\gamma ,\delta )\right\}$$

(2)

and

$$K^{*}(\gamma ,\delta ,\mathcal{P})=\left\{\zeta :\zeta \in M^{*}and{\displaystyle \frac{{\mathfrak{D}}_{\mathfrak{q}}\left[\mathfrak{z}{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)\right]}{{[-\mathcal{P}]}_{\mathfrak{q}}{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}}\prec \mathcal{H}(\gamma ,\delta )\right\}.$$

(3)

If $\zeta \left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{a}_n{\mathfrak{z}}^n$ and $g\left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{b}_n{\mathfrak{z}}^n,$ then their convolution is defined by

$$h\left(\mathfrak{z}\right)=\zeta \left(\mathfrak{z}\right)*g\left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+\sum _{n=1}^{\infty }{(-1)}^{n-1}{a}_n{b}_n{\mathfrak{z}}^n.$$

Remark 1 

([24]). We list the following subclasses by specialising the parameters $\mathcal{P},$ $\mathfrak{q}$, γ and δ:

(i) 

$S_{\mathfrak{q}}^{*}(1-ϵ,-1,\mathcal{P})=S^{*}(ϵ,\mathcal{P})=\{\zeta :\zeta \in {\mathcal{N}}_{\mathcal{P}}:\Re \left({\displaystyle \frac{\mathfrak{z}{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}}\right)>ϵ;0\le ϵ<1,\mathfrak{z}\in {\Delta }^{*}\}$ the subclass of $\mathcal{P}$-valent meromorphic $\mathfrak{q}$-starlike functions, and $K_{\mathfrak{q}}^{*}(1-ϵ,-1,\mathcal{P})=\{\zeta :\zeta \in {\mathcal{N}}_{\mathcal{P}}:\Re \left({\displaystyle \frac{{\mathfrak{D}}_{\mathfrak{q}}\left[\mathfrak{z}{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)\right]}{{[-\mathcal{P}]}_{\mathfrak{q}}{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}}\right)>ϵ;0\le ϵ<1,\mathfrak{z}\in {\Delta }^{*}\}$ the subclass of $\mathcal{P}$-valent meromorphic $\mathfrak{q}$-convex functions;

(ii) 

$S_{\mathfrak{q}}^{*}(1-ϵ,-1,1)=S^{*}(ϵ,1)=\{\zeta :\zeta \in {\mathcal{N}}_1:\Re \left(-{\displaystyle \frac{\mathfrak{z}{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}{\zeta \left(\mathfrak{z}\right)}}\right)>ϵ;0\le ϵ<1,\mathfrak{z}\in {\Delta }^{*}\}$ the subclass of meromorphic $\mathfrak{q}$-starlike functions, and $K_{\mathfrak{q}}^{*}(1-ϵ,-1,1)=\{\zeta :\zeta \in {\mathcal{N}}_1:\Re \left(-{\displaystyle \frac{{\mathfrak{D}}_{\mathfrak{q}}\left[\mathfrak{z}{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)\right]}{{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}}\right)>ϵ;0\le ϵ<1,\mathfrak{z}\in {\Delta }^{*}\}$ the subclass of meromorphic $\mathfrak{q}$-convex functions;

(iii) 

${lim}_{\mathfrak{q}\to 1^{-}}{S}_{\mathfrak{q}}^{*}(\gamma ,\delta ,1)=S^{*}(\gamma ,\delta ,1)=\{\zeta :\zeta \in {\mathcal{N}}_1:{\displaystyle \frac{-\mathfrak{z}{\zeta }^{\prime }\left(\mathfrak{z}\right)}{\zeta \left(\mathfrak{z}\right)}}\prec \frac{1+\gamma \alpha \left(\mathfrak{z}\right)}{1+\delta \alpha \left(\mathfrak{z}\right)},-1\le \delta <\gamma \le 1,\alpha \left(\mathfrak{z}\right)\in n\}$, and ${lim}_{\mathfrak{q}\to 1^{-}}{K}_{\mathfrak{q}}^{*}(\gamma ,\delta ,1)=K^{*}(\gamma ,\delta ,1)=\{\zeta :\zeta \in {\mathcal{N}}_1:-\left(1+{\displaystyle \frac{\mathfrak{z}{\zeta }^{″}\left(\mathfrak{z}\right)}{{\zeta }^{\prime }\left(\mathfrak{z}\right)}}\right)\prec \frac{1+\gamma \alpha \left(\mathfrak{z}\right)}{1+\delta \alpha \left(\mathfrak{z}\right)},-1\le \delta <\gamma \le 1,\alpha \left(\mathfrak{z}\right)\in n\},$ were introduced and studied by Ali and Ravichandran [25];

(iv) 

${lim}_{\mathfrak{q}\to 1^{-}}{S}_{\mathfrak{q}}^{*}(1-ϵ,-1,1)=S^{*}(ϵ,1)=\{\zeta :\zeta \in {\mathcal{N}}_1:\Re \left(-{\displaystyle \frac{\mathfrak{z}{\zeta }^{\prime }\left(\mathfrak{z}\right)}{\zeta \left(\mathfrak{z}\right)}}\right)>ϵ;0\le ϵ<1,\mathfrak{z}\in {\Delta }^{*}\}$, and ${lim}_{\mathfrak{q}\to 1^{-}}{K}_{\mathfrak{q}}^{*}(1-ϵ,-1,1)=K^{*}(ϵ,1)=\{\zeta :\zeta \in {\mathcal{N}}_1:-\Re \left(1+{\displaystyle \frac{{\mathfrak{D}}_{\mathfrak{q}}\left[\mathfrak{z}{\zeta }^{″}\left(\mathfrak{z}\right)\right]}{{\zeta }^{\prime }\left(\mathfrak{z}\right)}}\right)>ϵ;0\le ϵ<1,\mathfrak{z}\in {\Delta }^{*}\}$ were introduced and studied by Kaczmarski [26];

(v) 

${lim}_{\mathfrak{q}\to 1^{-}}{S}_{\mathfrak{q}}^{*}(1,-1,1)=S^{*}$, and ${lim}_{\mathfrak{q}\to 1^{-}}{K}_{\mathfrak{q}}^{*}(1,-1,1)=K^{*}$, which are well-known function classes of meromorphic starlike and meromorphic convex functions, respectively; see Pommerenke [27], Clunie [28] and Miller [29] for more details.

In this work, we use the $\mathfrak{q}$-difference operator to explain certain features of convolution for the classes $C^{\prime }(\gamma ,\delta ,\mathcal{P})$ and $K^{*}(\gamma ,\delta ,\mathcal{P})$.

2. Coefficient Estimate

Here, we establish the validity of two lemmas that are necessary for further examination of our convolution findings.

Lemma 1. 

A function $\zeta \left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{a}_n{\mathfrak{z}}^n,$ is in $C^{\prime }(\gamma ,\delta ,\mathcal{P})$ if and only if

$$\sum _{n=1}^{\infty }\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]a_n\le 1.$$

Proof. 

We have $\zeta \left(\mathfrak{z}\right)\in C^{\prime }(\gamma ,\delta ,\mathcal{P})$ if and only if

$$\frac{\mathfrak{z}{\mathfrak{D}}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}\zeta \left(\mathfrak{z}\right)}\in \mathcal{H}(\gamma ,\delta ).$$

$$\frac{{[-\mathcal{P}]}_{\mathfrak{q}}{\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{\left[n\right]}_{\mathfrak{q}}{a}_n{\mathfrak{z}}^n}{{[-\mathcal{P}]}_{\mathfrak{q}}\left({\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{a}_n{\mathfrak{z}}^n\right)}=\frac{1+\gamma \alpha \left(\mathfrak{z}\right)}{1+\delta \alpha \left(\mathfrak{z}\right)}.$$

On simplification, we get

$$\alpha \left(\mathfrak{z}\right)=\frac{{\sum }_{n=1}^{\infty }{(-1)}^{n-1}\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)a_n{\mathfrak{z}}^{n+\mathcal{P}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}(\gamma {[-\mathcal{P}]}_{\mathfrak{q}}-\delta {\left[n\right]}_{\mathfrak{q}})a_n{\mathfrak{z}}^{n+\mathcal{P}}}.$$

Assuming $\left|\mathfrak{z}\right|=r\to 1$, we obtain

$$\sum _{n=1}^{\infty }\left(({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})-\gamma {[-\mathcal{P}]}_{\mathfrak{q}}+\delta {\left[n\right]}_{\mathfrak{q}}\right)a_n\le \left(\gamma -\delta \right){[-\mathcal{P}]}_{\mathfrak{q}},$$

or equivalently,

$$\sum _{n=1}^{\infty }\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]a_n\le 1.$$

□

Lemma 2. 

A function $\zeta \left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{a}_n{\mathfrak{z}}^n,$ is in $K^{*}(\gamma ,\delta ,\mathcal{P})$ if and only if

$$\sum _{n=1}^{\infty }\left[\frac{{\left[n\right]}_{\mathfrak{q}}\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+\delta )-{[-\mathcal{P}]}_{\mathfrak{q}}{\left[n\right]}_{\mathfrak{q}}(\gamma -\delta \right)}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}^2}\right]a_n\le 1.$$

Proof. 

The proof of the lemma can be obtained by applying Equation (3) and executing the identical procedures as outlined in Lemma 1. □

Theorem 1. 

A function $\zeta \left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{a}_n{\mathfrak{z}}^n\in C^{\prime }(\gamma ,\delta ,\mathcal{P})$ and $g\left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{b}_n{\mathfrak{z}}^n\in C^{\prime }(\gamma ,\delta ,\mathcal{P})$. Then

$$h\left(\mathfrak{z}\right)=\zeta \left(\mathfrak{z}\right)\ast g\left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+\sum _{n=1}^{\infty }{(-1)}^{n-1}{a}_n{b}_n{\mathfrak{z}}^n\in C^{\prime }({\gamma }_1,{\delta }_1,\mathcal{P})$$

with $-1\le {\gamma }_1<{\delta }_1\le 1$ where ${\gamma }_1\ge -1,{\delta }_1\le \frac{1+{\gamma }_1-\mathcal{P}-2v}{\mathcal{P}+2v}$ and these bounds for ${\gamma }_1$ and ${\delta }_1$ are sharp.

Proof. 

According to Lemma 1, in order to establish the theorem, it is sufficient to identify ${\gamma }_1$ and $B_1$ such that

$$\sum _{n=1}^{\infty }\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+{\delta }_1)-{({\gamma }_1-\delta )}_1{[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}\right]a_n{b}_n\le 1,$$

(4)

for $-1\le {\gamma }_1<{\delta }_1\le 1.$

Or equivalently,

$$\sum _{n=1}^{\infty }{\mu }_1{a}_n{b}_n\le 1,$$

where

$${\mu }_1=\frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)-({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}.$$

By Cauchy-Schwarz inequality,

$$\sum _{n=1}^{\infty }\sqrt{\mu a_n\mu b_n}\le {\left(\sum _{n=1}^{\infty }\mu a_n\right)}^{1/2}{\left(\sum _{n=1}^{\infty }\mu b_n\right)}^{1/2}\le 1,$$

(5)

where

$$\mu =\sum _{n=1}^{\infty }\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}.$$

Equation (4) is true if

$${\mu }_1\sqrt{a_n{b}_n}\le \mu .$$

Also from (5), we have

$$\mu \sqrt{a_n{b}_n}\le 1\mathrm{for}n=2,3,\dots .$$

Consequently, finding ${\mu }_1$ such that

$$\frac{1}{\mu }\le \frac{\mu }{{\mu }_1}\Rightarrow {\mu }_1\le {\mu }^2$$

$$\frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)-({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_q}\le {\left[\frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_q}\right]}^2={\mu }^2.$$

That is

$$({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)-({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}\le {\mu }^2\left(({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_q\right),$$

$$({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)+{\delta }_1{[-\mathcal{P}]}_{\mathfrak{q}}(1+{\mu }^2)\le {\gamma }_1{[-\mathcal{P}]}_{\mathfrak{q}}(1+{\mu }^2).$$

(6)

Take ${\delta }_1=1$ in (6), we get

$${\gamma }_1\ge \frac{2\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)+{[-\mathcal{P}]}_{\mathfrak{q}}\left(1+{\mu }^2\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}(1+{\mu }^2)},$$

where $\mu$ is

$$\mu =\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}.$$

□

Theorem 2. 

If $\zeta \left(\mathfrak{z}\right)\in C^{\prime }(\gamma ,\delta ,\mathcal{P})$ and $g\left(\mathfrak{z}\right)\in C^{\prime }({\gamma }^{\prime },{\delta }^{\prime },\mathcal{P})$. Then $\zeta \left(\mathfrak{z}\right)\ast g\left(\mathfrak{z}\right)\in C^{\prime }({\gamma }_1,{\delta }_1,\mathcal{P})$ where

$${\gamma }_1\ge -1,{\delta }_1\le \frac{{\gamma }_1-({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})v}{1+({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})v},$$

with

$$v=\frac{\left[(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right]\left[({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}\right]}{{[-\mathcal{P}]}_{\mathfrak{q}}\left[\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}{)(1+\delta )-(\gamma -\delta )[-\mathcal{P}]}_{\mathfrak{q}}\right)\left(({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }^{\prime })-({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}\right)+\left((\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right)\left(({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}\right)\right]}.$$

Proof. 

Following the same procedure as in Theorem 1, we need

$$\frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)-({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_q}\le$$

$$\left(\frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_q}\right)\left(\frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }^{\prime })-({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_q}\right)=\lambda$$

$$\begin{array}{cc}\hfill \frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)-({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_q}& \le \lambda \hfill \\ \hfill \frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}-1& \le \lambda \hfill \\ \hfill \frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}& \le 1+\lambda \hfill \\ \hfill \frac{(1+{\delta }_1)}{({\gamma }_1-{\delta }_1)}& \le \frac{{[-\mathcal{P}]}_{\mathfrak{q}}(1+\lambda )}{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})}.\hfill \end{array}$$

We take the reciprocal of both sides of the equation and perform some calculations, and we will obtain.

$$\frac{(1+{\gamma }_1)-(1+{\delta }_1)}{(1+{\delta }_1)}\ge \frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})}{{[-\mathcal{P}]}_{\mathfrak{q}}(1+\lambda )},$$

with doing some calculations

$$\frac{(1+{\gamma }_1)}{(1+{\delta }_1)}\ge 1+\frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})}{{[-\mathcal{P}]}_{\mathfrak{q}}\left(1+\left(\frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right)\left(\frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }^{\prime })-({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_q}\right)\right)}$$

$$\frac{1+{\gamma }_1}{1+{\delta }_1}\ge 1+({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})v,$$

where

$$v=\frac{\left[(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right]\left[({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}\right]}{{[-\mathcal{P}]}_{\mathfrak{q}}\left[\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}{)(1+\delta )-(\gamma -\delta )[-\mathcal{P}]}_{\mathfrak{q}}\right)\left(({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }^{\prime })-({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}\right)+\left((\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right)\left(({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}\right)\right]}$$

$$\frac{1+{\gamma }_1}{1+({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})v}\ge 1+{\delta }_1.$$

Therefore, we have

$${\delta }_1\le \frac{{\gamma }_1-({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})v}{1+({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})v}.$$

But ${\delta }_1\ge -1,$ we have $\frac{{\gamma }_1-({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})v}{1+({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})v}\ge -1\Rightarrow {\gamma }_1\ge -1.$ □

Theorem 3. 

If $\zeta \left(\mathfrak{z}\right)\in C^{\prime }(\gamma ,\delta ,\mathcal{P})$ and $g\left(\mathfrak{z}\right)\in C^{\prime }({\gamma }^{\prime },{\delta }^{\prime },\mathcal{P})$. Then $\zeta \left(\mathfrak{z}\right)\ast g\left(\mathfrak{z}\right)\in K^{\prime }({\gamma }_1,{\delta }_1,\mathcal{P})$ where

$${\gamma }_1\ge -1and{\delta }_1\le \frac{{\gamma }_1-\left({\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})\right)v}{1+\left({\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})\right)v}$$

with

$$\begin{array}{cc}\hfill v=& \frac{\left((\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)\left(({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}^2\left[\left({\left[n\right]}_{\mathfrak{q}}\left(({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+\delta )-{[-\mathcal{P}]}_{\mathfrak{q}}(\gamma -\delta )\right)\right)\left({\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }^{\prime })-{[-\mathcal{P}]}_{\mathfrak{q}}({\gamma }^{\prime }-{\delta }^{\prime })\right)\right]+}\hfill \\ \hfill & {\left[n\right]}_{\mathfrak{q}}{[-\mathcal{P}]}_{\mathfrak{q}}\left[\left((\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)\left(({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)\right]\hfill \end{array}$$

and the result is best possible.

Proof. 

$\zeta \left(\mathfrak{z}\right)\ast g\left(\mathfrak{z}\right)\in K^{\prime }({\gamma }_1,{\delta }_1,\mathcal{P})$ if

$$\begin{array}{cc}\hfill & \frac{{\left[n\right]}_{\mathfrak{q}}\left(({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)-{[-\mathcal{P}]}_{\mathfrak{q}}({\gamma }_1-{\delta }_1)\right)}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}^2}\le \hfill \\ & \left(\frac{{\left[n\right]}_{\mathfrak{q}}\left(({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+\delta )-{[-\mathcal{P}]}_{\mathfrak{q}}(\gamma -\delta )\right)}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}^2}\right)\left(\frac{{\left[n\right]}_{\mathfrak{q}}\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }^{\prime })-{[-\mathcal{P}]}_{\mathfrak{q}}({\gamma }^{\prime }-{\delta }^{\prime })\right)}{({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}^2}\right)=\lambda .\hfill \end{array}$$

Following the same procedure as in Theorem 2, we need

$$\begin{array}{cc}\hfill \frac{{\left[n\right]}_{\mathfrak{q}}\left(({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)-{[-\mathcal{P}]}_{\mathfrak{q}}({\gamma }_1-{\delta }_1)\right)}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}^2}& \le \lambda \hfill \\ \hfill \frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)-{[-\mathcal{P}]}_{\mathfrak{q}}({\gamma }_1-{\delta }_1\right)}{({\gamma }_1-{\delta }_1)}& \le {\displaystyle \frac{{[-\mathcal{P}]}_{\mathfrak{q}}^2}{{\left[n\right]}_{\mathfrak{q}}}}\lambda \hfill \\ \hfill \frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)}{({\gamma }_1-{\delta }_1)}-{[-\mathcal{P}]}_{\mathfrak{q}}& \le {\displaystyle \frac{{[-\mathcal{P}]}_{\mathfrak{q}}^2}{{\left[n\right]}_{\mathfrak{q}}}}\lambda \hfill \\ \hfill \frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)}{({\gamma }_1-{\delta }_1)}& \le {\displaystyle \frac{{[-\mathcal{P}]}_{\mathfrak{q}}^2\lambda +{[-\mathcal{P}]}_{\mathfrak{q}}{\left[n\right]}_{\mathfrak{q}}}{{\left[n\right]}_{\mathfrak{q}}}}\hfill \\ \hfill \frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }_1)}{({\gamma }_1-{\delta }_1)}& \le {\displaystyle \frac{{[-\mathcal{P}]}_{\mathfrak{q}}\left({[-\mathcal{P}]}_{\mathfrak{q}}\lambda +{\left[n\right]}_{\mathfrak{q}}\right)}{{\left[n\right]}_{\mathfrak{q}}}}\hfill \\ \hfill \frac{(1+{\delta }_1)}{({\gamma }_1-{\delta }_1)}& \le {\displaystyle \frac{{[-\mathcal{P}]}_{\mathfrak{q}}\left({[-\mathcal{P}]}_{\mathfrak{q}}\lambda +{\left[n\right]}_{\mathfrak{q}}\right)}{{\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})}}.\hfill \end{array}$$

We take the reciprocal of both sides of the equation and perform some calculations, and we will obtain.

$$\frac{(1+{\gamma }_1)-(1+{\delta }_1)}{(1+{\delta }_1)}\ge {\displaystyle \frac{\left({\left[n\right]}_{\mathfrak{q}}\right)({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})}{{[-\mathcal{P}]}_{\mathfrak{q}}\left({[-\mathcal{P}]}_{\mathfrak{q}}\lambda +{\left[n\right]}_{\mathfrak{q}}\right)}},$$

with doing some calculations

$$\frac{(1+{\gamma }_1)}{(1+{\delta }_1)}\ge 1+{\displaystyle \frac{{\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})}{{[-\mathcal{P}]}_{\mathfrak{q}}^2\left(\frac{{\left[n\right]}_{\mathfrak{q}}\left(({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+\delta )-{[-\mathcal{P}]}_{\mathfrak{q}}(\gamma -\delta )\right)}{(\gamma -\delta ){\left({[-\mathcal{P}]}_{\mathfrak{q}}\right)}^2}\right)\left(\frac{{\left[n\right]}_{\mathfrak{q}}\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }^{\prime })-{[-\mathcal{P}]}_{\mathfrak{q}}({\gamma }^{\prime }-{\delta }^{\prime })\right)}{({\gamma }^{\prime }-{\delta }^{\prime }){\left({[-\mathcal{P}]}_{\mathfrak{q}}\right)}^2}\right)+{\left[n\right]}_{\mathfrak{q}}{[-\mathcal{P}]}_{\mathfrak{q}}}}$$

$$\frac{1+{\gamma }_1}{1+{\delta }_1}\ge 1+\frac{{\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})}{{[-\mathcal{P}]}_{\mathfrak{q}}^2\left(\frac{{\left[n\right]}_{\mathfrak{q}}\left(({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+\delta )-{[-\mathcal{P}]}_{\mathfrak{q}}(\gamma -\delta )\right)}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}^2}\right)\left(\frac{{\left[n\right]}_{\mathfrak{q}}\left(({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }^{\prime })-{[-\mathcal{P}]}_{\mathfrak{q}}({\gamma }^{\prime }-{\delta }^{\prime })\right)}{({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}^2}\right)+{\left[n\right]}_{\mathfrak{q}}{[-\mathcal{P}]}_{\mathfrak{q}}\left[{\displaystyle \frac{\left((\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)\left(({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)}{\left((\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)\left(({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)}}\right]}$$

$$\frac{1+{\gamma }_1}{1+{\delta }_1}\ge 1+\left({\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})\right)v,$$

where

$$\begin{array}{cc}\hfill v=& \frac{\left((\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)\left(({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}^2\left[\left({\left[n\right]}_{\mathfrak{q}}\left(({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+\delta )-{[-\mathcal{P}]}_{\mathfrak{q}}(\gamma -\delta )\right)\right)\left({\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+{\delta }^{\prime })-{[-\mathcal{P}]}_{\mathfrak{q}}({\gamma }^{\prime }-{\delta }^{\prime })\right)\right]+}\hfill \\ \hfill & {\left[n\right]}_{\mathfrak{q}}{[-\mathcal{P}]}_{\mathfrak{q}}\left[\left((\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)\left(({\gamma }^{\prime }-{\delta }^{\prime }){[-\mathcal{P}]}_{\mathfrak{q}}^2\right)\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill \frac{1+{\gamma }_1}{1+\left({\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})\right)v}& \ge 1+{\delta }_1\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\delta }_1& \le \frac{1+{\gamma }_1-1-\left({\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})\right)v}{1+\left({\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})\right)v}\hfill \\ \hfill {\delta }_1& \le \frac{{\gamma }_1-\left({\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})\right)v}{1+\left({\left[n\right]}_{\mathfrak{q}}({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})\right)v}.\hfill \end{array}$$

But ${\delta }_1\ge -1,$ and this implies ${\gamma }_1\ge -1.$

This makes it quite evident that our boundaries are the best available. □

Theorem 4. 

If $\zeta \left(\mathfrak{z}\right),g\left(\mathfrak{z}\right)\in C^{\prime }(\gamma ,\delta ,\mathcal{P})$. Then

$$h\left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+\sum _{n=1}^{\infty }{(-1)}^{n-1}\left({\gamma }_n^2+b_n^2\right){\mathfrak{z}}^n\in C^{\prime }({\gamma }_1,{\delta }_1,\mathcal{P}),$$

where ${\gamma }_1\ge -1and{\delta }_1\le \frac{{\gamma }_1-2\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)v}{1+2\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)v},$ with

$$v=\frac{{\left[(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right]}^2}{{[-\mathcal{P}]}_{\mathfrak{q}}\left(2{\left[(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right]}^2+{\left[\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right]}^2\right)},$$

and the result is the best possible.

Proof. 

Since $\zeta \left(\mathfrak{z}\right),g\left(\mathfrak{z}\right)\in C^{\prime }(\gamma ,\delta ,\mathcal{P})$, we have

$$\sum _{n=1}^{\infty }\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]a_n\le 1,$$

and

$$\sum _{n=1}^{\infty }\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]b_n\le 1.$$

Note that

$$\sum _{n=1}^{\infty }{\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]}^2{a}_n^2\le \sum _{n=1}^{\infty }{\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]}^2\le 1.$$

(7)

Similarly,

$$\sum _{n=1}^{\infty }{\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]}^2{b}_n^2\le \sum _{n=1}^{\infty }{\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]}^2\le 1.$$

(8)

Adding (7) and (8)

$$\sum _{n=1}^{\infty }{\left[\frac{\left({\left(n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]}^2(a_n^2+b_n^2)\le 2.$$

(9)

Now $h\left(\mathfrak{z}\right)\in C^{\prime }({\gamma }_1,{\delta }_1,\mathcal{P}),$ if

$$\sum _{n=1}^{\infty }\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+{\delta }_1)-({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}\right](a_n^2+b_n^2)\le 1$$

(9) implies that it is enough to show that

$$\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+{\delta }_1)-({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}\right]\le \frac{1}{2}{\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]}^2=\frac{{\mu }^2}{2}$$

$$\begin{array}{cc}\hfill \frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+{\delta }_1)-({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}& \le \frac{{\mu }^2}{2}\hfill \\ \hfill \frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+{\delta }_1)}{({\gamma }_1-{\delta }_1)\left({[-\mathcal{P}]}_{\mathfrak{q}}\right)}-1& \le \frac{{\mu }^2}{2}\hfill \\ \hfill \frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+{\delta }_1)}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}\le 1+\frac{{\mu }^2}{2}& =\frac{2+{\mu }^2}{2}\hfill \\ \hfill \frac{(1+{\delta }_1)}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}& \le \frac{2+{\mu }^2}{2\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)}\hfill \\ \hfill \frac{({\gamma }_1-{\delta }_1)}{(1+{\delta }_1)}& \ge \frac{2\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}(2+{\mu }^2)}\hfill \\ \hfill \frac{(1+{\gamma }_1)-(1+{\delta }_1)}{(1+{\delta }_1)}& \ge \frac{2\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}(2+{\mu }^2)}\hfill \\ \hfill \frac{1+{\gamma }_1}{1+{\delta }_1}-1& \ge \frac{2\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}(2+{\mu }^2)}\hfill \\ \hfill \frac{1+{\gamma }_1}{1+{\delta }_1}& \ge 1+\frac{2\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}(2+{\mu }^2)}\hfill \end{array}$$

$$\frac{1+{\gamma }_1}{1+{\delta }_1}\ge 1+\frac{2\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)}{{[-\mathcal{P}]}_{\mathfrak{q}}\left(2{\left[\frac{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]}^2+{\left[\frac{({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}})(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}})}{(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}}\right]}^2\right)}$$

$$\frac{1+{\gamma }_1}{1+{\delta }_1}\ge 1+\frac{2\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right){\left[(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right]}^2}{{[-\mathcal{P}]}_{\mathfrak{q}}\left(2{\left[(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right]}^2+{\left[\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right]}^2\right)}$$

$$\frac{1+{\gamma }_1}{1+{\delta }_1}\ge 1+2\left[{\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right]v$$

$$\frac{1+{\delta }_1}{1+{\gamma }_1}\le \frac{1}{1+2\left[{\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right]v}$$

$${\delta }_1\le \frac{1+{\gamma }_1}{1+2\left[{\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right]v}-1$$

$${\delta }_1\le \frac{{\gamma }_1-2[{\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}]v}{1+2[{\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}]v}\mathrm{and}{\gamma }_1\ge -1\mathrm{with}$$

$$v=\frac{{\left[(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right]}^2}{{[-\mathcal{P}]}_{\mathfrak{q}}\left(2{\left[(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right]}^2+{\left[\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+\delta )-(\gamma -\delta ){[-\mathcal{P}]}_{\mathfrak{q}}\right]}^2\right)}.$$

□

Theorem 5. 

If $\zeta \left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{a}_n{\mathfrak{z}}^n,a_n\ge 0\mathrm{belongs}\mathrm{to}{C}^{\prime }(\gamma ,\delta ,\mathcal{P})$ and $g\left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{b}_n{\mathfrak{z}}^n\mathrm{for}n\ge 1,$ then

$$\zeta \left(\mathfrak{z}\right)\ast g\left(\mathfrak{z}\right)\in S^{*}(\gamma ,\delta ,\mathcal{P})$$

Proof. 

$$\sum _{n=1}^{\infty }\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+{\delta }_1)-({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }_1-{\delta }_1)\left({[-\mathcal{P}]}_{\mathfrak{q}}\right)}\right]a_n{b}_n$$

$$\le \sum _{n=1}^{\infty }\left[\frac{\left({\left[n\right]}_{\mathfrak{q}}-{[-\mathcal{P}]}_{\mathfrak{q}}\right)(1+{\delta }_1)-({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}{({\gamma }_1-{\delta }_1){[-\mathcal{P}]}_{\mathfrak{q}}}\right]a_n\left|b_n\right|\le 1.$$

(10)

According to Equation (10), it can be observed that $|b_k|\le 1$. This indicates that

$$\zeta \left(\mathfrak{z}\right)\ast g\left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+\sum _{n=1}^{\infty }{(-1)}^{n-1}{a}_n{b}_n{\mathfrak{z}}^n\in S^{*}(\gamma ,\delta ,\mathcal{P}).$$

In a similar vein, we can derive the subsequent theorem. □

Theorem 6. 

If $\zeta \left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{a}_n{\mathfrak{z}}^n\mathrm{belongs}\mathrm{to}{K}^{*}(\gamma ,\delta ,\mathcal{P})$ and $g\left(\mathfrak{z}\right)={\mathfrak{z}}^{-\mathcal{P}}+{\sum }_{n=1}^{\infty }{(-1)}^{n-1}{b}_n{\mathfrak{z}}^n,\left|b_n\right|\le 1\mathrm{for}n\ge 1,$ then

$$\zeta \left(\mathfrak{z}\right)\ast g\left(\mathfrak{z}\right)\in K^{*}(\gamma ,\delta ,\mathcal{P}).$$

3. Conclusions

We defined a new operator on the class of meromorphically $\mathcal{P}$-valent functions with alternating coefficients. We introduced the new subclasses $C^{\prime }(\gamma ,\delta ,\mathcal{P})$ and $K^{*}(\gamma ,\delta ,\mathcal{P})$. The study concentrated on convolutional results and coefficient estimates. The results reported in this paper offer new suggestions for further study, and we have opened up possibilities for researchers to extend the findings and produce innovative outcomes in geometric function theory and its applications.