Geometric Properties for Subclasses of Multivalent Analytic Functions Associated with $\mathfrak{q}$-Calculus Operator

Abstract

This paper presents new subclasses of multivalent analytic functions defined through the $\mathfrak{q}$-derivative operator and examines their inclusion properties. By employing the Jackson $\mathfrak{q}$-derivative, we construct generalized operators that encompass numerous previously established operators and provide a framework for defining new classes with distinctive analytic features. Our main results are derived using techniques from differential subordination theory for complex-valued functions of one variable. Some applications of Bernardi operator are discussed.

1. Introduction

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

$$\mathfrak{f}\left(z\right)=z^p+\sum _{n=p+1}^{\infty }{a}_n{z}^n,\left(p\in \mathbb{N}=\{1,2,3,\dots \},z\in \mathbb{D}\right),$$

(1)

that are analytic multivalent in the open unit disc $\mathbb{D}=\{z:\left|z\right|<1\}.$ For $p=1$ the class ${\mathcal{A}}_1=\mathcal{A}$ represents the class of normalized analytic and univalent functions in $\mathbb{D}$.

Numerous fields of mathematics and science, including complex analysis, hypergeometric series, particle physics, and most importantly geometric function theory (GFT), have found extensive uses for the idea of $\mathfrak{q}$-calculus operators. Ismail and his associates’ introduction of the idea of $\mathfrak{q}$-starlike functions in 1990 [1] was a significant turning point in this direction and signaled the start of a new field of study in GFT.

Jackson [2,3] established the foundation for $\mathfrak{q}$-analogues of classical operators by introducing the $\mathfrak{q}$-differential and $\mathfrak{q}$-integral operators and proving their applicability to the study of geometric functions.

Symmetric quantum (or $\mathfrak{q}$-) calculus applies $\mathfrak{q}$-calculus concepts to define new families of multivalent functions in GFT and studies the geometric properties of analytic functions. This approach uses symmetric quantum difference operators to generate new subclasses of functions with properties like $\mathfrak{q}$-starlikeness and $\mathfrak{q}$-convexity. Research in this area focuses on using these operators to establish necessary and sufficient conditions for function classes, explore properties like compactness and coefficient bounds, and generalize existing results in GFT for analytic functions. frakq-difference equations are an important aspect of mathematical analysis, particularly in the field known as GFT. Quantum calculus is frequently used in mathematical disciplines because of its numerous possible applications in basic hypergeometric functions [4], orthogonal polynomials [5,6,7], combinatorics [8], and number theory [9]. Several fundamental ideas in frakq-calculus [10,11,12] demonstrate how it is integrated into mathematical ideas. Srivastava’s 1989 book chapter [13] offered the appropriate foundation for integrating the concepts of frakq-calculus into GFT. For generalized subclasses of analytic functions, numerous authors have examined various frakq-calculus applications; see [14,15,16,17,18].

Jackson’s $\mathfrak{q}$-difference operator ${\mathfrak{d}}_{\mathfrak{q}}:{\mathcal{A}}_p\to {\mathcal{A}}_p$ is defined by

$${\mathfrak{d}}_{\mathfrak{q},\mathfrak{p}}\mathfrak{f}\left(z\right):=\left\{\begin{array}{cc}{\displaystyle \frac{\mathfrak{f}\left(z\right)-\mathfrak{f}\left(\mathfrak{q}z\right)}{(1-\mathfrak{q})z}}\hfill & \left(z\ne 0;0<\mathfrak{q}<1\right)\hfill \\ \hfill & \\ {\mathfrak{f}}^{\prime }\left(0\right)\hfill & \left(z=0\right),\hfill \end{array}\right.$$

(2)

provided that ${\mathfrak{f}}^{\prime }\left(0\right)$ exists. From (1) and (2), we deduce that

$${\mathfrak{d}}_{\mathfrak{q},\mathfrak{p}}\mathfrak{f}\left(z\right):={\left[p\right]}_{\mathfrak{q}}{z}^{p-1}+\sum _{n=p+1}^{\infty }{\left[n\right]}_{\mathfrak{q}}{a}_n{z}^{n-1},$$

(3)

where

$$\begin{array}{ccc}\hfill {\left[n\right]}_{\mathfrak{q}}& =& {\displaystyle \frac{1-{\mathfrak{q}}^n}{1-\mathfrak{q}}}=1+\sum _{n=1}^{n-1}{\mathfrak{q}}^n,{\left[0\right]}_{\mathfrak{q}}=0,\hfill \\ \hfill {\left[n\right]}_{\mathfrak{q}}!& =& \left\{\begin{array}{ll}{\left[n\right]}_{\mathfrak{q}}{\left[n-1\right]}_{\mathfrak{q}}\dots \dots \dots {\left[2\right]}_{\mathfrak{q}}{\left[1\right]}_{\mathfrak{q}}& n=1,2,3,\dots \\ 1& n=0.\end{array}\right.\hfill \end{array}$$

We observe that

$$\underset{\mathfrak{q}\to {\mathfrak{1}}^{-}}{lim}{\mathfrak{d}}_{\mathfrak{q},\mathfrak{p}}\mathfrak{f}\left(z\right):=\underset{\mathfrak{q}\to {\mathfrak{1}}^{-}}{lim}{\displaystyle \frac{\mathfrak{f}\left(z\right)-\mathfrak{f}\left(\mathfrak{q}z\right)}{(1-\mathfrak{q})z}}={\mathfrak{f}}^{\prime }\left(z\right).$$

A new application of the convolution-based $\mathfrak{q}$-analogue integral operator established in [19] using the $\mathfrak{q}$-analogue multiplier operator and the $\mathfrak{q}$-analogue of Ruscheweyh operator is the aim of the current study in light of the results previously provided. Next, those operators are called back.

In [20] the multiplier $\mathfrak{q}$-analogue Cătas operator $I_{\mathfrak{q},\mathfrak{p}}^{\mathfrak{r}}(\mathfrak{s},\rho ):{\mathcal{A}}_p\to {\mathcal{A}}_p$$(\mathfrak{r}\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},\rho ,\mathfrak{s}\ge 0$, $0<\mathfrak{q}<1,p\in \mathbb{N})$ was considered to be

$${\mathcal{I}}_{\mathfrak{q},\mathfrak{p}}^{\mathfrak{r}}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)=z^p+\sum _{n=p+1}^{\infty }{\left({\displaystyle \frac{{\left[p+\rho \right]}_{\mathfrak{q}}+\mathfrak{s}({\left[n+\rho \right]}_{\mathfrak{q}}-{\left[p+\rho \right]}_{\mathfrak{q}})}{{\left[p+\rho \right]}_{\mathfrak{q}}}}\right)}^{\mathfrak{r}}{a}_n{z}^n.$$

(4)

In [21] the extended $\mathfrak{q}$-derivative operator ${\Re }_{\mathfrak{q}}^{\mu +p-1}:{\mathcal{A}}_p\to {\mathcal{A}}_p$ for p-valent analytic functions was given as

$$\begin{array}{ccc}\hfill {\Re }_{\mathfrak{q}}^{\mu +p-1}\mathfrak{f}\left(z\right)& =& Q_p(\mathfrak{q},\mu +p;z)\ast \mathfrak{f}\left(z\right)(\mu >-p),\hfill \\ & =& z^p+\sum _{n=p+1}^{\infty }{\displaystyle \frac{{[\mu +p,\mathfrak{q}]}_{n-p}}{[n-p,\mathfrak{q}]!}}{a}_n{z}^n.\hfill \end{array}$$

(5)

Definition 1

([19]). Considering (4) and (5), the operator ${\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ):{\mathcal{A}}_p\to {\mathcal{A}}_p$ is defined by the convolution of the $\mathfrak{q}$-analogue of Ruscheweyh operator and the $\mathfrak{q}$-analogue Cătas operator as

$${\mathfrak{G}}_{\mathfrak{q},\mathfrak{s},\rho }^{\mathfrak{r},p}\left(z\right)=z^p+\sum _{n=p+1}^{\infty }{\left({\displaystyle \frac{{\left[p+\rho \right]}_{\mathfrak{q}}+\mathfrak{s}({\left[n+\rho \right]}_{\mathfrak{q}}-{\left[p+\rho \right]}_{\mathfrak{q}})}{{\left[p+\rho \right]}_{\mathfrak{q}}}}\right)}^{\mathfrak{r}}{z}^n.$$

we define a new function ${\mathfrak{G}}_{\mathfrak{q},\mathfrak{p},\mathfrak{s},\rho }^{\mathfrak{r},\mu }\left(z\right)$ in terms of convolution by

$${\mathfrak{G}}_{\mathfrak{q},\mathfrak{s},\rho }^{\mathfrak{r},p}\left(z\right)\ast {\mathfrak{G}}_{\mathfrak{q},\mathfrak{p},\mathfrak{s},\rho }^{\mathfrak{r},\mu }\left(z\right)=z^p+\sum _{n=p+1}^{\infty }{\displaystyle \frac{{[\mu +p,\mathfrak{q}]}_{n-p}}{[n-p,\mathfrak{q}]!}}{z}^n(p\in \mathbb{N}).$$

if we consider

$${\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)={\mathfrak{G}}_{\mathfrak{q},\mathfrak{p},\mathfrak{s},\rho }^{\mathfrak{r},\mu }\left(z\right)\ast \mathfrak{f}\left(z\right)$$

then

$${\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)=z^p+\sum _{n=p+1}^{\infty }{\left({\displaystyle \frac{{\left[p+\rho \right]}_{\mathfrak{q}}}{{\left[p+\rho \right]}_{\mathfrak{q}}+\mathfrak{s}({\left[n+\rho \right]}_{\mathfrak{q}}-{\left[p+\rho \right]}_{\mathfrak{q}})}}\right)}^{\mathfrak{r}}{\displaystyle \frac{{[\mu +p]}_{n-p,\mathfrak{q}}}{{[n-p]}_{\mathfrak{q}}!}}{a}_n{z}^n.$$

(6)

We also observe that

$\left(i\right)I_{\mathfrak{q},\mu }^{\mathfrak{r},p}(1,0)\mathfrak{f}($z$)=I_{\mathfrak{q},\mu }^{\mathfrak{r},p}\mathfrak{f}\left(z\right)$

$$\begin{array}{ccc}\hfill \mathfrak{f}\left(z\right)& \in & {\mathcal{A}}_p:{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}\mathfrak{f}\left(z\right)=z^p+\sum _{n=p+1}^{\infty }{\left({\displaystyle \frac{{\left[p\right]}_{\mathfrak{q}}}{{\left[n\right]}_{\mathfrak{q}}}}\right)}^{\mathfrak{r}}{\displaystyle \frac{{[\mu +p]}_{n-p,\mathfrak{q}}}{{[n-p]}_{\mathfrak{q}}!}}{a}_n{z}^n,\hfill \\ \hfill (\mathfrak{r}& \in & {\mathbb{N}}_0,\mu \ge 0,0<\mathfrak{q}<1,p\in \mathbb{N},z\in \mathbb{D}).\hfill \end{array}$$

$\left(ii\right)$$I_{\mathfrak{q},\mu }^{\mathfrak{r},p}(1,\rho )\mathfrak{f}\left(z\right)=I_{\mathfrak{q},\mu }^{\mathfrak{r},p,\rho }\mathfrak{f}\left(z\right)$

$$\begin{array}{ccc}\hfill \mathfrak{f}\left(z\right)& \in & {\mathcal{A}}_p:{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p,\rho }\mathfrak{f}\left(z\right)=z^p+\sum _{n=p+1}^{\infty }{\left({\displaystyle \frac{{\left[p+\rho \right]}_{\mathfrak{q}}}{{\left[n+\rho \right]}_{\mathfrak{q}}}}\right)}^{\mathfrak{r}}{\displaystyle \frac{{[\mu +p]}_{n-p,\mathfrak{q}}}{{[n-p]}_{\mathfrak{q}}!}}{a}_n{z}^n,\hfill \\ \hfill (\mathfrak{r}& \in & {\mathbb{N}}_0,\rho >0,\mu \ge 0,0<\mathfrak{q}<1,p\in \mathbb{N},z\in \mathbb{D}).\hfill \end{array}$$

$\left(iii\right)$${\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},0)\mathfrak{f}\left(z\right)={\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p,\mathfrak{s}}\mathfrak{f}\left(z\right)$

$$\begin{array}{ccc}\hfill \mathfrak{f}\left(z\right)& \in & {\mathcal{A}}_p:{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p,\mathfrak{s}}\mathfrak{f}\left(z\right)=z^p+\sum _{n=p+1}^{\infty }{\left({\displaystyle \frac{{\left[p\right]}_{\mathfrak{q}}}{{\left[p\right]}_{\mathfrak{q}}+\mathfrak{s}({\left[n\right]}_{\mathfrak{q}}-{\left[p\right]}_{\mathfrak{q}})}}\right)}^{\mathfrak{r}}{\displaystyle \frac{{[\mu +p]}_{n-p,\mathfrak{q}}}{{[n-p]}_{\mathfrak{q}}!}}{a}_n{z}^n,\hfill \\ \hfill (\mathfrak{r}& \in & {\mathbb{N}}_0,\mathfrak{s}>0,\mu \ge 0,0<\mathfrak{q}<1,p\in \mathbb{N},z\in \mathbb{D}).\hfill \end{array}$$

The following lists the new classes created for this study using the operator displayed in Definition 1.

Definition 2.

1. We denote by ${\wp }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\alpha ;\beta )$ the subclass of ${\mathcal{A}}_p$ consisting of functions $\mathfrak{f}\in {\mathcal{A}}_p$ satisfying

$$Re\left\{(1-\alpha ){\displaystyle \frac{z{\left({\mathcal{I}}_{q,\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{{\mathcal{I}}_{q,\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}}+\alpha \left(1+{\displaystyle \frac{z{\left({\mathcal{I}}_{q,\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{″}}{{\left({\mathcal{I}}_{q,\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}}\right)\right\}>\beta ,\in \mathbb{D},$$

where $\mathfrak{r}\in {\mathbb{N}}_0,\rho ,\mathfrak{s}\ge 0,\mu >-p,0<\mathfrak{q}<1,p\in \mathbb{N},\alpha \in \mathbb{R}$ and $\beta <1$.

2. Let ${\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\varrho ,\vartheta ;\gamma )$ be the subclass of ${\mathcal{A}}_p$ consisting of $\mathfrak{f}\in {\mathcal{A}}_p$ that satisfies

$${\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right){\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{z^{2p-1}}}\ne 0,z\in \mathbb{D},$$

(7)

and

$$Re\left\{{\left({\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mathfrak{u}}^{p,\alpha }(\mathfrak{a},\rho |f){\left(z\right)}^{\prime }}{p{z}^{p-1}}}\right)}^{\mathfrak{a}}{\left[{\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mathfrak{u}}^{p,\alpha }(\mathfrak{a},\rho |f)\left(z\right)}{z^p}}\right]}^{\mathfrak{b}}\right\}>\gamma ,z\in \mathbb{D},$$

where $\mathfrak{r}\in {\mathbb{N}}_0,\rho ,\mathfrak{s}\ge 0,\mu >-p,0<\mathfrak{q}<1,p\in \mathbb{N},\varrho ,\vartheta \in \mathbb{R}$ and $\gamma <1$.

It should be noted that the values of the complicated powers listed above are assumed to be their major values in this article.

Remark 1.

The families ${\wp }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\alpha ;\beta )$ and ${\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\varrho ,\vartheta ;\gamma )$ contain many well-known classes of analytic multivalent functions.

(i) For $\varrho =1$, $\vartheta =0$ the class ${\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\varrho ,\vartheta ;\gamma )$ reduce to ${\Theta }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\gamma )$, which satisfies

$$Re\left\{{\displaystyle \frac{{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^{p-1}}}\right\}>\gamma ,z\in \mathbb{D}$$

(8)

where $\mathfrak{r}\in {\mathbb{N}}_0,\rho ,\mathfrak{s}\ge 0,\mu >-p,0<\mathfrak{q}<1,p\in \mathbb{N},$ and $\gamma <1$.

(ii) For $\varrho =1$, $\vartheta =-1$, $\mathfrak{r}=0$, and $\mu =0$, we obtain the family of p-valent starlike functions of order γ, $0\le \gamma <1$, denoted by ${\mathcal{S}}_p\left(\gamma \right)$;

(iii) For $\alpha =1$, $\beta =\gamma$, $\mathfrak{r}=0$, and $\mu =0$, we obtain the family of p-valent convex functions of order γ, $0\le \gamma <1$, denoted by $K_p\left(\gamma \right)$. We mention that the classes ${\mathcal{S}}_p\left(\gamma \right)$ and $K_p\left(\gamma \right)$ were introduced by Patil and Thakare [22] and Owa [23].

Our analysis deals with certain disparities for a differential operator ${\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\varrho ,\vartheta ):{\mathcal{A}}_p\to {\mathcal{A}}_p$ defined by

$${\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\varrho ,\vartheta )f\left(z\right):=\vartheta {\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}}+\varrho \left(1+{\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{″}}{{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}}\right),$$

with $\varrho ,\vartheta \in \mathbb{R}$ since $f\in {\wp }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\alpha ;\beta )$ if and only if $Re{\Omega }_{q,\mu }^{\tau ,p}(\mathfrak{s},\rho ,\alpha ,1-\left(\alpha \right)f\left(z\right)>\beta ,z\in \mathbb{D}$.

To prove our main results that generalize the recent results obtained by Irmak et al. [24], we need the following lemmas. More general forms of these lemmas that are very useful in the theory of differential subordinations are provided by Miller and Mocanu [25] (Theorem 2.3h., Theorem 2.3i.)

For $m\in \mathbb{N}$ and $a\in \mathbb{C}$, we denote by $\mathcal{H}[a,m]$ the class of all functions p that are analytic in the unit disc $\mathbb{D}$ with the power series expansion of the form

$$\mathrm{p}\left(z\right)=a+a_m{z}^m+a_{m+1}{z}^{m+1}+\dots ,z\in \mathbb{D}.$$

Lemma 1

([25]). Let $\Omega \subset \mathbb{C}$ and suppose the function $\psi :{\mathbb{C}}^2\times \mathbb{D}\to \mathbb{C}$ satisfies $\psi (N{e}^{i\theta },K{e}^{i\theta };z)\notin \Omega$ for all $K\ge mN$, $\theta \in \mathbb{R}$ and $z\in \mathbb{D}$. If $\mathrm{p}\in \mathcal{H}[0,m]$ and $\psi \left(\mathrm{p}\left(z\right),z{\mathrm{p}}^{\prime }\left(z\right);z\right)\in \Omega$ for all $z\in \mathbb{D}$, then $\left|\mathrm{p}\right(z\left)\right|<N$, $z\in \mathbb{D}$.

Lemma 2

([25]). Let $\Omega \subset \mathbb{C}$ and suppose the function $\psi :{\mathbb{C}}^2\times \mathbb{D}\to \mathbb{C}$ satisfies $\psi (ix,y;z)\notin \Omega$ for all $x,y\in \mathbb{R}$, $y\le -m(1+x^2)/2$ and $z\in \mathbb{D}$. If $\mathrm{p}\in \mathcal{H}[1,m]$ and $\psi \left(\mathrm{p}\left(z\right),z{\mathrm{p}}^{\prime }\left(z\right);z\right)\in \Omega$ for all $z\in \mathbb{D}$, then $Re\mathrm{p}\left(z\right)>0$, $z\in \mathbb{D}$.

2. Some Properties for the Classes ${\wp }_{\mathfrak{q},\mathit{\mu }}^{\mathfrak{r},\mathit{p}}(\mathfrak{s},\mathit{\rho },\mathit{\alpha };\mathit{\beta })$ and ${\Xi }_{\mathfrak{q},\mathit{\mu }}^{\mathfrak{r},\mathit{p}}(\mathfrak{s},\mathit{\rho };\mathit{\varrho },\mathit{\vartheta };\mathit{\gamma })$

Now we will prove each of our main results given by the following theorems.

Theorem 1.

Let $f\in {\mathcal{A}}_p$ such that (7) holds, and let $\varrho ,\vartheta \in \mathbb{R}$. If

$$Re{\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\varrho ,\vartheta )f\left(z\right)<p(\varrho +\vartheta )+{\displaystyle \frac{N}{N+1}},z\in \mathbb{D},$$

(9)

with $N\ge 1$, then

$$\left|{\left[{\displaystyle \frac{{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^{p-1}}}\right]}^{\varrho }{\left[{\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}{z^p}}\right]}^{\vartheta }-1\right|<N,z\in \mathbb{D},$$

(10)

and $f\in {\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\varrho ,\vartheta ;1-N)$.

Proof. 

Let $\mathrm{p}$ be defined by

$$\mathrm{P}\left(z\right)={\left[{\displaystyle \frac{{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^{p-1}}}\right]}^{\varrho }{\left[{\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}{z^p}}\right]}^{\vartheta }-1,z\in \mathbb{D}.$$

From the assumption (7) it follows that the function $\mathrm{p}$ is analytic in $\mathbb{D}$, and $\mathrm{p}\left(0\right)=0$, that is, $\mathrm{p}\in \mathcal{H}[0,1]$. A simple computation shows that

$${\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\varrho ,\vartheta )f\left(z\right)=p(\varrho +\vartheta )+{\displaystyle \frac{z{\mathrm{p}}^{\prime }\left(z\right)}{\mathrm{p}\left(z\right)+1}},z\in \mathbb{D}.$$

Now, letting

$$\begin{array}{cc}\hfill \psi (r,\mathfrak{r};z)& :=p(\varrho +\vartheta )+{\displaystyle \frac{\mathfrak{r}}{r+1}},\hfill \\ \hfill \Omega & :=\left\{w\in \mathbb{C}:\mathrm{Re}w<p(\varrho +\vartheta )+{\displaystyle \frac{N}{N+1}}\right\},\hfill \end{array}$$

the assumption (9) is equivalent to $\psi \left(\mathrm{p}\left(z\right),z{\mathrm{p}}^{\prime }\left(z\right);z\right)={\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\varrho ,\vartheta )f\left(z\right)\in \Omega$ all $z\in \mathbb{D}$.

For any $\theta \in \mathbb{R}$, $K\ge N$ and $z\in \mathbb{D}$, since $N\ge 1$ we obtain that

$$Re\psi \left(N{e}^{i\theta },K{e}^{i\theta };z\right)=p(\varrho +\vartheta )+KRe{\displaystyle \frac{1}{N+e^{-i\theta }}}\ge p(\varrho +\vartheta )+{\displaystyle \frac{N}{N+1}},$$

which shows that $\psi (N{e}^{i\theta },K{e}^{i\theta };z)\notin \Omega$ whenever $\theta \in \mathbb{R}$, $K\ge N$ and $z\in \mathbb{D}$. Therefore, according to Lemma 1 we obtain $\left|\mathrm{p}\left(z\right)\right|<N$ for all $z\in \mathbb{D}$, that is, (10) holds. □

For the special case $N=\gamma +1$, the above theorem reduces to the next result, which represents a sufficient condition for a function $f\in {\mathcal{A}}_p$ to be in the class ${\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\varrho ,\vartheta ;-\gamma )$:

Corollary 1.

Let $f\in {\mathcal{A}}_p$ such that (7) holds, and let $\varrho ,\vartheta \in \mathbb{R}$. If

$$Re{\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\varrho ,\vartheta )f\left(z\right)<p(\varrho +\vartheta )+{\displaystyle \frac{\gamma +1}{\gamma +2}},z\in \mathbb{D},$$

with $\gamma \ge 0$, then

$$\left|{\left[{\displaystyle \frac{{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^{p-1}}}\right]}^{\varrho }{\left[{\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}{z^p}}\right]}^{\vartheta }-1\right|<\gamma +1,z\in \mathbb{D},$$

therefore $f\in {\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\varrho ,\vartheta ;-\gamma )$.

If we set $\varrho =\alpha$ and $\vartheta =1-\alpha$ in the above corollary, we get

Corollary 2.

Let $f\in {\mathcal{A}}_p$ such that (7) holds, and let $\alpha \in \mathbb{R}$. If

$$Re{\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\alpha ,1-\alpha )f\left(z\right)<p+{\displaystyle \frac{\gamma +1}{\gamma +2}},z\in \mathbb{D},$$

with $\gamma \ge 0$, then

$$\left|{\left[{\displaystyle \frac{{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^{p-1}}}\right]}^{\alpha }{\left[{\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}{z^p}}\right]}^{1-\alpha }-1\right|<\gamma +1,z\in \mathbb{D},$$

therefore $f\in {\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\alpha ,1-\alpha ;-\gamma )$.

For $\alpha =1$ and $\alpha =0$, the above corollary reduces to the following examples, respectively:

Example 1.

Let $\gamma \ge 0$ and $f\in {\mathcal{A}}_p$.

$\left(i\right)$ If

$$Re\left(1+{\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{″}}{{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}}\right)<p+{\displaystyle \frac{\gamma +1}{\gamma +2}},z\in \mathbb{D},$$

then

$$\left|{\displaystyle \frac{{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^{p-1}}}-1\right|<\gamma +1,z\in \mathbb{D},$$

hence $f\in {\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;1,0;-\gamma )$.

$\left(ii\right)$ If

$$Re{\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}}<p+{\displaystyle \frac{\gamma +1}{\gamma +2}},z\in \mathbb{D},$$

then

$$\left|{\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}{z^p}}-1\right|<\gamma +1,z\in \mathbb{D},$$

hence $f\in {\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;0,1;-\gamma )$.

Theorem 2.

Let $f\in {\mathcal{A}}_p$ such that (7) holds, and let $\varrho ,\vartheta \in \mathbb{R}$ and $\gamma \in [0,1)$. If

$$Re{\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\varrho ,\vartheta )f\left(z\right)>\xi (p,\varrho ,\vartheta ;\gamma ),z\in \mathbb{D},$$

(11)

where

$$\xi (p,\varrho ,\vartheta ;\gamma ):=\left\{\begin{array}{ccc}p(\varrho +\vartheta )-{\displaystyle \frac{\gamma }{2(1-\gamma )}},\hfill & \mathit{if}\hfill & \gamma \in \left[0,{\displaystyle \frac{1}{2}}\right],\hfill \\ p(\varrho +\vartheta )-{\displaystyle \frac{1-\gamma }{2\gamma }},\hfill & \mathit{if}\hfill & \gamma \in \left[{\displaystyle \frac{1}{2}},1\right),\hfill \end{array}\right.$$

(12)

then

$$Re\left\{{\left[{\displaystyle \frac{{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^{p-1}}}\right]}^{\varrho }{\left[{\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}{z^p}}\right]}^{\vartheta }\right\}>\gamma ,z\in \mathbb{D},$$

(13)

that is, $f\in {\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\varrho ,\vartheta ;\gamma )$.

Proof. 

If we put

$$\mathrm{p}\left(z\right)={\displaystyle \frac{1}{1-\gamma }}\left({\left[{\displaystyle \frac{{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^{p-1}}}\right]}^{\varrho }{\left[{\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}{z^p}}\right]}^{\vartheta }-\gamma \right),z\in \mathbb{D},$$

(14)

from the assumption (7) we deduce that $\mathrm{p}$ is analytic in $\mathbb{D}$, with $\mathrm{p}\left(0\right)=1$, hence $\mathrm{p}\in \mathcal{H}[1,1]$. From the definition relation (14) it is easy to check that

$${\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\varrho ,\vartheta )f\left(z\right)=p(\varrho +\vartheta )+{\displaystyle \frac{(1-\gamma )z{\mathrm{p}}^{\prime }\left(z\right)}{(1-\gamma )\mathrm{p}\left(z\right)+\gamma }},z\in \mathbb{D}.$$

Denoting

$$\begin{array}{ccc}& & \psi (r,\mathfrak{r};z):=p(\varrho +\vartheta )+{\displaystyle \frac{(1-\gamma )\mathfrak{r}}{(1-\gamma )r+\gamma }},\hfill \\ & & \Omega :=\left\{w\in \mathbb{C}:Rew>\xi (p,\varrho ,\vartheta ;\gamma )\right\},\hfill \end{array}$$

the assumption (11) is equivalent to $\psi \left(\mathrm{p}\left(z\right),z{\mathrm{p}}^{\prime }\left(z\right);z\right)={\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\varrho ,\vartheta )f\left(z\right)\in \Omega$ for all $z\in \mathbb{D}$.

Also, for any $x,y\in \mathbb{R}$ with $y\le -(1+x^2)/2$ and $z\in \mathbb{D}$, a simple computation shows that

$$\begin{array}{ccc}& & Re\psi (ix,y;z)=p(\varrho +\vartheta )+{\displaystyle \frac{\gamma (1-\gamma )y}{{(1-\gamma )}^2{x}^2+{\gamma }^2}}\hfill \\ & \le & p(\varrho +\vartheta )-{\displaystyle \frac{\gamma (1-\gamma )}{2}}{\displaystyle \frac{x^2+1}{{(1-\gamma )}^2{x}^2+{\gamma }^2}}=:h\left(x\right)\hfill \\ & \le & sup\left\{h\left(x\right):x\in \mathbb{R}\right\}=\left\{\begin{array}{ccc}\underset{x\to +\infty }{lim}h\left(x\right),\hfill & \mathrm{if}\hfill & \gamma \in \left[0,{\displaystyle \frac{1}{2}}\right],\hfill \\ h\left(0\right),\hfill & \mathrm{if}\hfill & \gamma \in \left[{\displaystyle \frac{1}{2}},1\right)\hfill \end{array}\right.=\xi (p,\varrho ,\vartheta ;\gamma ),\hfill \end{array}$$

where $\xi (p,\varrho ,\vartheta ;\gamma )$ is given by (12). Therefore, $\psi (ix,y;z)\notin \Omega$ if $x,y\in \mathbb{R}$ with $y\le -(1+x^2)/2$, and $z\in \mathbb{D}$. Using Lemma 2 we conclude that $Re\mathrm{p}\left(z\right)>0$ for all $z\in \mathbb{D}$, that is, the conclusion (13) holds. □

If we set $\varrho :=\alpha$ and $\vartheta :=1-\alpha$ in Theorem 2, we obtain the following special case:

Corollary 3.

Let the function $f\in {\mathcal{A}}_p$ such that the condition (7) holds, and let $\alpha \in \mathbb{R}$ and $\gamma \in [0,1)$. If

$$Re\left\{{\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\alpha ,1-\alpha )f\left(z\right)\right\}>\tilde{\xi }(p,\alpha ,1-\alpha ;\gamma ):=\xi (p,\alpha ,1-\alpha ;\gamma ),z\in \mathbb{D},$$

where $\xi (p,\mathfrak{s},\vartheta ;\gamma )$ is defined by (12), then

$$Re\left\{{\left[{\displaystyle \frac{{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^{p-1}}}\right]}^{\alpha }{\left[{\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}{z^p}}\right]}^{1-\alpha }\right\}>\gamma ,z\in \mathbb{D},$$

that is, $f\in {\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\alpha ,1-\alpha ;\gamma )$.

Remark 2.

The corollary above could be expressed as follows:

Let $f\in {\mathcal{A}}_p$ such that the condition (7) holds, and let $\alpha \in \mathbb{R}$ and $\gamma \in [0,1)$. If $f\in {\wp }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\alpha ;\xi (p,\alpha ,1-\alpha ;\gamma ))$ where $\xi (p,\alpha ,1-\alpha ;\gamma )$ is given by (12), then $f\in {\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,\alpha ,$$1-\alpha ;\gamma )$.

Setting $\varrho =-1$ and $\vartheta =1$ in Theorems 1 and 2, we next get the following corollaries, respectively.

Corollary 4.

Let $f\in {\mathcal{A}}_p$ such that (7) holds, and let $N\ge 1$. Then

$$Re\left\{{\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,-1,1)f\left(z\right)\right\}<{\displaystyle \frac{N}{N+1}},z\in \mathbb{D},$$

implies

$$\left|{\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}}-{\displaystyle \frac{1}{p}}\right|<{\displaystyle \frac{N}{p}},z\in \mathbb{D},$$

or equivalently

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}}+{\displaystyle \frac{p}{N^2-1}}\right|>{\displaystyle \frac{pN}{N^2-1}},z\in \mathbb{D},\mathit{if}N>1,\hfill \\ & & Re{\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}}>{\displaystyle \frac{p}{2}},z\in \mathbb{D},\mathit{if}N=1.\hfill \end{array}$$

Corollary 5.

Let $f\in {\mathcal{A}}_p$ such that (7) holds, and let $\gamma \in [0,1)$. If

$$Re\left\{{\Omega }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ,-1,1)f\left(z\right)\right\}>\xi (p,-1,1;\gamma ),z\in \mathbb{D},$$

where $\xi (p,\mathfrak{s},\vartheta ;\gamma )$ is defined by (12), then

$$Re{\displaystyle \frac{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}}>{\displaystyle \frac{\gamma }{p}},z\in \mathbb{D},$$

or equivalently

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}}-{\displaystyle \frac{p}{2\gamma }}\right|<{\displaystyle \frac{p}{2\gamma }},z\in \mathbb{D},\mathit{if}\gamma \in (0,1),\hfill \\ & & Re{\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{{\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)}}>0,z\in \mathbb{D},\mathit{if}\gamma =0.\hfill \end{array}$$

In conclusion, both of the aforementioned theorems provide us with basic sufficient criteria for a function $f\in {\mathcal{A}}_p$ to belong to distinct subclasses of ${\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\varrho ,\vartheta ;\gamma )$ provided that the parameters are chosen appropriately.

3. Some Applications on Bernardi Integral Operator for ${\Theta }_{\mathfrak{q},\mathit{\mu }}^{\mathfrak{r},\mathit{p}}(\mathfrak{s},\mathit{\rho };\mathit{\gamma })$

For $\mathfrak{f}\left(z\right)\in {\mathcal{A}}_p,$ the generalized Bernardi integral operator for p-valent functions ${\mathfrak{F}}^p\mathfrak{f}\left(z\right):{\mathcal{A}}_p\to {\mathcal{A}}_p$ is defined by

$${\mathfrak{F}}^p\mathfrak{f}\left(z\right)=z^p+\sum _{n=p+1}^{\infty }\left({\displaystyle \frac{p+\upsilon }{n+\upsilon }}\right)a_n{z}^n,(\upsilon >-p),$$

where the operator ${\mathfrak{F}}^p\mathfrak{f}\left(z\right)$ is the generalized Bernardi–Libera–Livingston integral operator (see [26]), and ${\mathfrak{F}}^1\mathfrak{f}\left(z\right)=\mathfrak{Ff}\left(z\right)$ was introduced by Bernardi [27].

Lemma 3

(Jack Lemma [28]). Let the (nonconstant) function $\omega \left(z\right)$ be analytic in $\mathbb{D}$, with $\omega \left(0\right)=0.$ If $\left|\omega \left(z\right)\right|$ attains its maximum value on the circle $\left|z\right|=r<1$ at a point $z_0\in \mathbb{D}$; then $z_0{\omega }^{\prime }\left(z_0\right)=\eta \omega \left(z_0\right),$ where η is real number and $\eta \ge 1.$

Theorem 3.

Let $\mathfrak{f}\in {\Theta }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\gamma )$; then

$${\mathfrak{F}}_{\upsilon }^p\left(z\right)={\displaystyle \frac{p+\upsilon }{z^{\upsilon }}}{\int }_0^z{t}^{\upsilon -1}\mathfrak{f}\left(t\right)dt\in {\Theta }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\gamma )(\upsilon >-p).$$

(15)

Proof. 

From (15) it is implied that

$$z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ){\mathfrak{F}}_{\upsilon }^p\left(z\right)\right)}^{\prime }=\left(p+\upsilon \right){\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)-\upsilon {\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ){\mathfrak{F}}_{\upsilon }^p\left(z\right).$$

(16)

Define a regular function $\omega \left(z\right)$ in $\mathbb{D}$ such that $\omega \left(0\right)=0$, $\omega \left(z\right)\ne -1$ by

$${\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ){\mathfrak{F}}_{\upsilon }^p\left(z\right)\right)}^{\prime }}{p{z}^p}}={\displaystyle \frac{1+(2\gamma -1)\omega \left(z\right)}{1+\omega \left(z\right)}}.$$

(17)

From (16) and (17) we have

$$\left(p+\upsilon \right){\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)-\upsilon {\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ){\mathfrak{F}}_{\upsilon }^p\left(z\right)=p{z}^p{\displaystyle \frac{1+(2\gamma -1)\omega \left(z\right)}{1+\omega \left(z\right)}}.$$

(18)

Differentiating (18) with respect to z and using (17), we obtain

$${\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^p}}={\displaystyle \frac{1+(2\gamma -1)\omega \left(z\right)}{1+\omega \left(z\right)}}-{\displaystyle \frac{2(1-\gamma )}{\left(p+\upsilon \right)}}{\displaystyle \frac{z{\omega }^{\prime }\left(z\right)}{{\left(1+\omega \left(z\right)\right)}^2}}.$$

(19)

We claim that $\left|\omega \left(z\right)\right|<1$ for $z\in \mathbb{D}$. Otherwise there exists a point $z_0\in \mathbb{D}$ such that $\underset{\left|z\right|\le \left|z_0\right|}{max}\left|\omega \left(z\right)\right|=\left|\omega \left(z_0\right)\right|=1.$ Applying Jack’s lemma, we have

$$z_0{\omega }^{\prime }\left(z_0\right)=\eta \omega \left(z_0\right)(\eta \ge 1).$$

(20)

From (19) and (20) we have

$${\displaystyle \frac{z_0{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z_0\right)\right)}^{\prime }}{p{z}_0^p}}={\displaystyle \frac{1+(2\gamma -1)\omega \left(z_0\right)}{1+\omega \left(z_0\right)}}-{\displaystyle \frac{2(1-\gamma )}{\left(p+\upsilon \right)}}{\displaystyle \frac{\eta \omega \left(z_0\right)}{{\left(1+\omega \left(z_0\right)\right)}^2}}.$$

(21)

Since $Re\left\{{\displaystyle \frac{1+(2\gamma -1)\omega \left(z_0\right)}{1+\omega \left(z_0\right)}}\right\}=\gamma ,\eta \ge 1,$ and ${\displaystyle \frac{\eta \omega \left(z_0\right)}{{\left(1+\omega \left(z_0\right)\right)}^2}}$ is real and positive, we see that $Re\left\{{\displaystyle \frac{z_0{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z_0\right)\right)}^{\prime }}{p{z}_0^p}}\right\}<\gamma ,$ which obviously contradicts $\mathfrak{f}\in {\Theta }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\gamma ).$ Hence $\left|\omega \left(z\right)\right|<1$ for $z\in \mathbb{D}$, and it follows from (17) that ${\mathfrak{F}}_{\upsilon }^p\left(z\right)\in {\Theta }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\gamma ).$ This completes the proof. □

Theorem 4.

Let $f\left(z\right)$ be defined by (1). If ${\mathfrak{F}}_{\upsilon }^p\left(z\right)\in {\Theta }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\gamma ),$ then $\mathfrak{f}\in {\Theta }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\gamma )$ in $\left|z\right|<{\displaystyle \frac{\left(p+\upsilon \right)}{1+\sqrt{{\left(p+\upsilon \right)}^2+1}}},$ where ${\mathfrak{F}}_{\upsilon }^p\left(z\right)$ is defined by (15).

Proof. 

Since ${\mathfrak{F}}_{\upsilon }^p\left(z\right)\in {\Theta }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\gamma ),$ we can write

$${\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ){\mathfrak{F}}_{\upsilon }^p\left(z\right)\right)}^{\prime }}{p{z}^p}}=\left[\gamma +(1-\gamma )\hslash \left(z\right)\right],$$

(22)

where $\hslash \left(z\right)\in \mathbb{D},$ the class of functions with the positive real part in the unit disk $\mathbb{D}$ normalized by $\hslash \left(0\right)=1$. We can rewrite (22) as

$$\left(p+\upsilon \right){\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)-\upsilon {\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ){\mathfrak{F}}_{\upsilon }^p\left(z\right)=p{z}^p\left[\gamma +(1-\gamma )\hslash \left(z\right)\right],$$

(23)

Differentiating (23) with respect to z and using (16), we get

$$\left({\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^p}}-\gamma \right){(1-\gamma )}^{-1}=\hslash \left(z\right)+{\displaystyle \frac{z{\hslash }^{\prime }\left(z\right)}{\left(p+\upsilon \right)}}.$$

(24)

Using the well-known estimate (see Nehari [29])

$$\left|z_0{\hslash }^{\prime }\left(z_0\right)\right|\le {\displaystyle \frac{2r}{1-r^2}}Re\hslash \left(z_0\right),$$

$\left|z\right|=r,$ then (24) yields

$$Re\left\{\left({\displaystyle \frac{z{\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )\mathfrak{f}\left(z\right)\right)}^{\prime }}{p{z}^p}}-\gamma \right){(1-\gamma )}^{-1}\right\}\ge \left(1-{\displaystyle \frac{2r}{\left(p+\upsilon \right)(1-r^2)}}\right)Re\hslash \left(z\right).$$

(25)

The right-hand side of (25) is positive if

$$r<{\displaystyle \frac{\left(p+\upsilon \right)}{1+\sqrt{{\left(p+\upsilon \right)}^2+1}}}.$$

□

The result is sharp for the function $\mathfrak{f}$ defined by

$$\mathfrak{f}\left(z\right)={\displaystyle \frac{1}{\left(p+\upsilon \right)z^{\upsilon -1}}}{\left(z^{\upsilon }{\mathfrak{F}}_{\upsilon }^p\left(z\right)\right)}^{\prime },$$

where ${\mathfrak{F}}_{\upsilon }^p\left(z\right)$ is given by

$${\left({\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ){\mathfrak{F}}_{\upsilon }^p\left(z\right)\right)}^{\prime }=p{z}^{p-1}{\displaystyle \frac{1+(2\gamma -1)z}{1+z}}.$$

4. Conclusions

The newly defined $\mathfrak{q}$-p-analogue multiplier-Ruscheweyh operator ${\mathcal{I}}_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho )$ is defined utilizing the concept of a $\mathfrak{q}$-difference operator and the concept of convolution, and it extends several previously researched operators by many authors. This is what makes the above results innovative. Using classical principles of the general theory of differential subordinations, we discovered adequate requirements for a function $f\in {\mathcal{A}}_p$ to belong to the two new subclasses of functions defined by this operator. While the inquiry methods involving the two lemmas are more potent than those employed by the previous writers, these subclasses of multivalent functions could be linked to those indicated in Remark 1 and extend the classes of Remark 2. Furthermore, both of the aforementioned theorems provide us with straightforward necessary requirements for a function $f\in {\mathcal{A}}_p$ to belong to distinct subclasses of ${\Xi }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\varrho ,\vartheta ;\gamma )$ provided that the parameters are chosen appropriately. The studies focus on specific inclusion outcomes for the newly specified classes by ${\Theta }_{\mathfrak{q},\mu }^{\mathfrak{r},p}(\mathfrak{s},\rho ;\gamma )$ and the Bernardi integral operator for the p-valent function preservation feature.

The goal of this work is to inspire further research in this area by introducing further generalized q-calculus operators and creating additional generalized subclasses of multivalent functions.