Region of variablity for Bazilevic functions

1.   Introduction and definitions

For a fixed point $\xi$ in $\mathbb{U},$ the set of values of $\log \left(\frac{f\left(\xi \right) }{\xi}\right)$ as $f$ ranging on the family of injective or univalent mappings is always a closed disk. This useful and important fact was proved by Grunsky ^[6]. The region of variability for various subfamilies of univalent functions became a fascinating area of the current research. Several authors have studied about such regions for certain subfamilies of analytic functions. These regions for the functions of bounded derivative were discussed by Yanagihara in 2005 as seen in ^[21]. Functions in such a subfamily satisfy the conditions $\left \vert f^{\prime }(z)\right \vert \leq1$ and $\operatorname{Re}(f^{\prime}\left(z\right)) > 0.$ Later on, Ponnusamy discussed the region of variability for the Kaplan family of functions $\mathcal{K}$ as described in ^[9]. Moreover, Yanagihara found the range of values for a subfamily of convex function, as seen in ^[23]$.$ Ponnusamy discussed these problems for subfamilies of $\mathcal{S}^{\ast}$ and $\mathcal{K},$ as found in ^[10]. In 2008, see ^[15], Ponnusamy again considered these aspects for the spirallike functions. Vasudevarao discussed similar results when $f$ is ranging over the functions with positive real parts, for detail see ^[11].

Most of the authors have introduced new subfamilies and studied the region of variability for these subfamilies of analytic and univalent functions. In 2010, Chen and Aiwu discussed these problems for functions with bounded Mocanu variations as seen in ^[2]. In 2011, Ponnusamy et al. ^[12] investigated these regions for the families of exponentially convex functions.

More useful and interesting results on these regions are also discussed by many authors. In 2014, Sunil Varma et al. ^[18], also worked out these issues for the related subfamilies along with Bappaditya who utilized the idea of subordination in his work as found in ^[1]. Also some related findings are seen in ^{[10,14,16,19,20]}.

Let $\mathcal{H}$ be the family of analytic functions defined in $\mathbb{U = }\left \{ z:|z| < 1\right \}$ the open unit disk with the center at origin $O$ included in the $z$-plane and $\mathcal{A\subset H}$. Then any function $f\in \mathcal{A}\ $takes the form:

From (1.1), we note that $f(0) = f^{^{\prime}}(0)-1 = 0$. The subfamily of univalent or injective mappings is represented by $\mathcal{S}$ and obviously $\mathcal{S\subset A}$. A holomorphic or analytic mapping is called univalent as it never takes the same value twice. For example, in the unit disk $\mathbb{U}$, $f\left(z\right) = 1+2z+z^{2}\in \mathcal{S}$. Formally, we have:

A mapping $f\in \mathcal{A}$ is univalent or injective in $\mathbb{U}$, if from $f(z_{1}) = f(z_{2}),$ we have $z_{1} = z_{2}$. On the other hand, for a mapping $f\in \mathcal{S}$ in $\mathbb{U}$, from $z_{1}\neq z_{2}$ we have $f(z_{1})\neq f(z_{2})$. The Koebe function\ $k\left(z\right)$ as defined by

and the mapping $\mathcal{M}\left(z\right) = \dfrac{\eta z+\beta}{\nu z+\mu }, \eta, \nu, \beta, \mu \in \mathbb{C}$ $:$ $\nu \beta-\eta \mu \neq0$, are univalent. The family $\mathcal{S}$ is preserved under basic transformations, for reference, see ^[5]. Furthermore, from the condition $f^{\prime}\left(z_{0}\right) \neq0$, we have local univalence of $f$ at $z_{0}$.

Let $\mathcal{P}$ be the family of holomorphic mappings $p:p\left(\mathbb{U}\right)$ is the right half $\omega$-plane having series form

Obviously $p\left(0\right) = 1$ and $\operatorname{Re}p(z) > 0$. This and other related families have a significant role in the recent development of the subject. Many subfamilies of $\mathcal{S}$ are connected with the class $\mathcal{P}$. For $p\in \mathcal{P}$, it may not be necessary that $p\in \mathcal{S}$. For example $p\left(z\right) = 1+z^{j}\in \mathcal{P}$, but $p(z)\notin \mathcal{S}$ for $j\geq2.$ Obviously, the mapping $L_{0}\left(z\right) \ $so that

We take $\mathcal{B}$ as a family of analytic function $\omega :\mathbb{U\longrightarrow U}$ with $\left \vert \omega(z)\right \vert < 1$ and $\omega(0) = 0.$ A function $f$ is subordinate to a function $F$ and we write $f\prec F$, if there exists $\omega \in \mathcal{B}$ such that $f(z) = F(\omega (z)).$ Particularly, if $F\in \mathcal{S}$, then $f\prec F$ can be equivalently reformulated as $f(0) = F(0)$ and $f(\mathbb{U})\subset F(\mathbb{U}).$ For $A\in \mathbb{C},$ $B\in \lbrack-1, 0),$ the function $p\in$ $\mathcal{P} _{\gamma}[A; B]$ can be written as:

where $\omega \in \mathcal{B}$. Moreover, we can defined the family $\mathcal{P}_{\gamma}[\xi, A;B]$ of analytic functions as:

where $p\in \mathcal{P}_{\gamma}[A, B]$ defined above by (1.3), $\gamma \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right), A\in \mathbb{C}, B\in \lbrack-1, 0), \xi = \omega^{\prime}\left(0\right) \in \overline{\mathbb{U}}$ and $z\in \mathbb{U}$.

The family $\mathcal{S}^{\ast}$ contains starlike functions $f$ such that $f\left(\mathbb{U}\right)$ is starlike about $O$ in the $w$-plane. This family has been extensively studied in the literature, as seen in ^[5].

Spacek ^[17] extended the family $\mathcal{S}^{\ast}$ by using the logarithmic spirals besides the line segments. Let $\gamma \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. The curve $\vartheta_{\gamma }:\mathbb{R}\rightarrow \mathbb{C}$ $:$ $\vartheta_{\gamma}\left(t\right) = te^{i\gamma}, t\in \mathbb{R}$ and its rotation $e^{i\theta}\vartheta_{\gamma }(t), \theta \in \mathbb{R}$ are called $\gamma$-spirals. A domain $\mathbb{D} \subset \mathbb{C}$ is known as $\gamma$-spirallike about the origin, if the spiral has initial point at the origin and terminal of the spiral is any other point of $\mathbb{D}$. A function $f\in \mathcal{A}$ is spirallike, if $f\left(\mathbb{U}\right)$ is spirallike about the origin. The family of spirallike functions represented by $\mathcal{S}_{\gamma}$ is defined by

For details, we refer ^[5]. A normalized analytic or holomorphic function $f$ $\in \mathcal{C}_{\gamma}$ iff $zf^{\prime}\in \mathcal{S}_{\gamma}^{\ast }.$ For details see ^[16]. For $\beta$ with $\operatorname{Re}\beta > 0,$ let $\mathcal{B}(\psi, \beta)$ be the family of functions $f$ so that

These Bazilevic functions are obviously univalent in $\mathbb{U}$. Let $\mathcal{B}_{\gamma}(\psi, \beta)$ be the family of holomorphic functions $f$ in $\mathbb{U}$ such that

These functions are called spirallike Bazilevic functions. Let $A\in \mathbb{C}$, $B\in \lbrack-1, 0)$, $\beta > 0$ and $\gamma \in \left(-\frac{\pi} {2}, \frac{\pi}{2}\right).$ Then $f\in \mathcal{B}_{\gamma}\left[\psi, \xi, \beta, A;B\right]$ if

where $\psi$ is a starlike function.

In view of Herglotz form of Janowski functions, we write $f\in \mathcal{B} _{\gamma}\left[\psi, \xi, \beta, A;B\right]$ as

or we note that

which can further take the form

Integrating on both sides to have

which leads to

Suppose that

where

then for $\omega \in \mathcal{B},$ we see that

We can also write

which leads to

or we see that

which on simplifications yields

In view of (1.6), on differentiating (1.7), we note that

which on simplifications proves that

Then by using classical Schwarz lemma ^[5] and a related result as seen in ^[4], we note that $\left \vert \omega^{\prime}(0)\right \vert \leq1$. So we obtain

Again on differentiating (1.9), we can write

where

and

Also we see that

Now, we let

This shows that

Using Schwarz lemma for $\left \vert \xi \right \vert < 1,$ we write $\left \vert g(z)\right \vert < \left \vert z\right \vert$ and $\left \vert g^{\prime}\left(0\right) \right \vert < 1.$ For equality, we see that $g(z) = e^{i\varepsilon}z$, $\varepsilon \in \mathbb{R}$. Now $\left \vert g^{\prime}\left(0\right) \right \vert < 1$ shows that there is $l\in \overline{\mathbb{U}\text{ } }:g^{\prime}\left(0\right) = l.$ Thus by using (1.10) and (1.11), we find that

which shows that

It follows from (1.5) that for each fixed $z_{0}\in \mathbb{U}$, the region of variability $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ is a set defined as:

when $f$ ranges over the family $\mathcal{B}_{\gamma}\left[\psi, \xi, \beta, A;B\right].$

The region of variability problems provide accurate information about the family $\mathcal{S},$ than theorems about bounds on functions, their derivatives and rotation theorems. It typically refers to a range or interval within which a certain variable can vary or fluctuate. It is commonly used in research to describe the extent variation of region in a particular set. It is important to note that the region of variability can vary depending on the context and the specific variable being analyzed. Different methods and techniques can be used to determine and characterize the region of variability for different types of sets. Here, we find the region of variability $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ for $f(z_{0}) = \left(\beta { \int \limits_{0}^{z_{0}}} q(t)dt\right) ^{\frac{1}{\beta}}$, where $f$ ranges over the family $\mathcal{B}_{\gamma}\left[\psi, \xi, \beta, A;B\right]$ is defined above by (1.7)and

where $q$ is described as a part of (1.5). As special cases, region of variability $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ can also described for different choices of parameters. As described above, Mohsan et al. ^[16] found $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ for a certain related subfamily of holomorphic functions. Ponnusamy et al. ^[8,9] studied these regions for $f\in \mathcal{S}$. Yanagihara ^[22] determined such region for the family of convex functions. For work on such regions, see ^{[10,11,13,14,15,16,19,21]} and others.

For any $p\in \mathbb{N}$, we let $\mathcal{S}_{p}^{\ast} = \left \{ f\left(z\right) = \left[f_{0}\left(z\right) \right] ^{p}:f_{0}\in \mathcal{S}^{\ast}\right \}.$ Also $f\in \mathcal{C}_{p}\Longleftrightarrow zf^{\prime}\in \mathcal{S} _{p}^{\ast}.$

Lemma 1.1. Let $f:f(z) = z^{p}+...\in \mathcal{A}_{p}\subset \mathcal{H}, \mathcal{A}_{1} = \mathcal{A}.$ Then $f\in \mathcal{S}_{p}^{\ast}$ iff

In the case that $p = 1,$ $f$ is starlike univalent. We refer to ^[21,22] for the Lemma 1.2 given subsequently.

Lemma 1.2. For $f$ $:$ $f(z) = z^{p}+...$ such that $\operatorname{Re}\left(\frac{zf^{\prime \prime}(z)}{f^{\prime}(z)}\right) > -1$, we have $f\in \mathcal{S}_{p}^{\ast}.$

2.   Main results

We initiate our investigations by studying certain properties of the family $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ such as compactness and convexity.

Theorem 2.1. $(i)$ $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]\mathcal{\ }$is compact in $\mathbb{C}$. $(ii)$ $\mathcal{V}_{\gamma }[\psi, z_{0}, \xi, \beta, A;B]\mathcal{\ }$is convex in $\mathbb{C}.$ $(iii)$ If $\left \vert \xi \right \vert = 1$ or $z_{0} = 0,$ then

$(iv)$ If $\left \vert \xi \right \vert < 1$ and $z_{0}\neq0,$ then

has an interior point.

Proof. $(i)$ Since $\mathcal{P}_{\gamma}[\xi, A;B]$ is a compact set of $\mathbb{C}$. This shows that $\mathcal{B}_{\gamma}\left[\psi, \xi, \beta, A;B\right]$ is compact, which leads to the compactness of the set $\mathcal{V}_{\gamma} [\psi, z_{0}, \xi, \beta, A;B]$. $(ii)$ Now, we show that $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ is convex in $\mathbb{C}.$ For this end in view, we take $q_{1},$ $q_{2} \in \mathcal{P}_{\gamma}[\xi, A;B],$ $t\in \left[0, 1\right]$ and note that

Hence $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ is convex. $(iii)$ For $\left \vert \xi \right \vert = \left \vert \omega_{f}^{\prime }(0)\right \vert = 1$, using Schwarz lemma ^[3], we have $\omega_{f}(z) = \xi z.$ This leads to the function

which proves (2.1). Since $z_{0} = 0,$ so $\mathcal{V}_{\gamma} [\psi, 0, \xi, \beta, A;B] = \left \{ 0\right \}$ is trivially satisfied. $(iv)$ For $\left \vert \xi \right \vert < 1,$ $\xi \in \mathbb{U}$ and $\gamma \in \overline{\mathbb{U}}$, we can write

and also we note that

as seen by the Eq (1.5) is in the family $\mathcal{B}_{\gamma }\left[\psi, \xi, \beta, A;B\right]$. Next, we claim that $\Psi_{l, \xi}(z)$ is a nonconstant analytic function of $l$ for each fixed $z_{0}\in \mathbb{U}\backslash \{0\}$ and $\xi \in \mathbb{U}$. Put

where $\phi_{\gamma}\left(A; B\right) = A\cos \gamma+iB\sin \gamma.$ It is easy to see that

and since $\psi$ is starlike, it follows that

By Lemma 1.2, there exists a function $h_{0}\in \mathcal{S}^{\ast}$ with $h = h_{0}^{\beta+2}.$ The univalence of $h_{0}$ and $h_{0}(0) = 0$ implies that $h_{0}(z_{0})\neq0$ for all $z_{0}\in \mathbb{U}\backslash \{0\}.$ Consequently, the mapping $\mathbb{U\ni}l\rightarrow \Psi_{l, \xi}(z_{0})$ is a nonconstant function and hence it is an open mapping. Thus, $\mathcal{V} _{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ contains the open set $\left \{ \Psi_{l, \xi}(z):\left \vert \xi \right \vert < 1\right \}.$ For $l = 0$, we get

$\Psi_{0, \xi}(z_{0})$ is an interior point $\left \{ \Psi_{l, \xi} (z):l\in \mathbb{U}\right \} \subset \mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B].$ Therefore, $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ is a simple closed domain bounded by a simple closed curve $\partial \mathcal{V} _{\gamma}[\psi, z_{0}, \xi, \beta, A;B].$ □

We now prove that for $f\in \mathcal{B}_{\gamma}\left[\psi, \xi, \beta, A;B\right]$, the $\left[f(z)\right] ^{\beta}$ is contained in some closed disk with center $\chi(\xi, \gamma, z)$ and radius $R_{\xi}^{\gamma}$.

Theorem 2.2. If $f\in \mathcal{B}_{\gamma}\left[\psi, \xi, \beta, A;B\right]$, then

and

where

and

Proof. Since $f\in \mathcal{B}_{\gamma}\left[\psi, \xi, \beta, A;B\right]$. Then by using Schwarz lemma ^[3] for $\omega_{p}\in \mathcal{B}$ with $\omega _{p}^{\prime}\left(0\right) = \xi,$ we have

Now, by substituting $\omega_{p}\left(z\right)$ from (1.8) and after calculation we get

On substitution of $p(z)$ from (1.6), we write

or we obtain

Now, by letting

and

we get

After simplifying (2.7), we get

Thus, for a parametrized $C^{1}$-curve $\gamma$ defined by $z = z\left(t\right),$ $0\leq t\leq1\ :\ z(0) = 0$ and $z(1) = z_{0},$ we may write

On some calculations and simplification, we see that

and

where $\phi_{\gamma}\left(A; B\right) = A\cos \gamma+iB\sin \gamma$. The relation (2.3) occurs from (2.4) and the above relations. Equality is attained in (2.3) when $f\left(z\right) = \Psi _{e^{i\theta}, \xi}\left(z\right)$, for some $z\in \mathbb{U}$. Conversely, if equality occurs in (2.3) for some $z\in \mathbb{U}\backslash \{0\}$, then equality must hold in (2.4). Thus, by applying Schwarz lemma, for $\theta \in \mathbb{R},$ we write $\omega_{p}(z) = z\delta(e^{i\theta}z, \xi),$ $z\in \mathbb{U}$. This shows $f\left(z\right) = \Psi_{e^{i\theta}, \xi }\left(z\right)$. □

The choice of $\xi = 0$ in Theorem 2.2, leads to the following corollary:

Corollary 2.1. If $f\in \mathcal{B}_{\gamma}\left[\psi, 0, \beta, A;B\right]$, then

and

where

and

If we put $\beta = 1$ and $\psi = z\phi^{\prime},$ where $\phi \in \mathcal{C}$, the class of convex function in Theorem 2.2, we get the following result:

Corollary 2.2. If $f\in \mathcal{B}_{\gamma}\left[\psi, \xi, 1, A;B\right]$, then

and

where

and

We need the following lemma which ensures the existence of a normalized starlike function, useful in the proof of next result.

Lemma 2.1. Let $\theta$ be a real numbers and $z$ belong to unit disk $\mathbb{U}$. Then

where $H(0) = H^{\prime}\left(0\right) = 0$ and $H(z)\neq0$ elsewhere in $\mathbb{U}$. Moreover, there exists a starlike normalized univalent function $H_{0}\in \mathcal{S}^{\ast}$ in $\mathbb{U}$ $:H\left(z\right) = \frac{1} {2}e^{i\theta}H_{0}^{2}\left(z\right).$

This lemma is proved by Ponnusamy et al. as found in ^[8]. In the theorem below, we show that $\Psi_{e^{i\theta}, \xi}(z_{0})$ lies on the boundary of $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$.

Theorem 2.3. Let $z_{0}\in \mathbb{U}\backslash \left \{ 0\right \}$ and $\Psi_{e^{i\theta}, \xi}\left(z\right)$ is given by (2.2)$.$ Then for $\theta \in \left(-\pi, \pi \right]$, we have $\Psi_{e^{i\theta}, \xi }(z_{0})\in \partial \mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B].$ Furthermore, if $\frac{f^{\prime}(z)}{\left[f(z)\right]^{1-\beta}} = \frac{\Psi_{e^{i\theta }, \xi}^{\prime}(z)}{\left(\Psi_{e^{i\theta}, \xi}(z)\right)^{1-\beta}}$ for $f\in \mathcal{B}_{\gamma}\left[\psi, \xi, \beta, A;B\right]$, then $f\left(z\right) = \Psi_{e^{i\theta}, \xi}\left(z\right)$.

Proof. Now (2.2) gives that

or

On differentiating (2.9), we get

From (2.5), it follows that

Moreover, we find that

and

Therefore, we can write

Moreover, we see that

Putting $l = e^{i\theta},$ we have

Thus

where $\psi \in \mathcal{S}^{\ast}$ and $\operatorname{Re}\left(\frac {z\psi^{\prime}(z)}{\psi(z)}\right) > 0.$ By Lemma 1.2, we have $H(z) = 2^{-1}e^{i\theta}H_{0}^{2}\left(z\right) \in \mathcal{S}_{2}^{\ast},$ where $H_{0}$ is starlike in $\mathbb{U}$ with $H_{0}(0) = H_{0}^{^{\prime} }(0)-1 = 0$, for any $z_{0}\in \mathbb{U}\backslash \{0\},$ the line segment joining origin to $H_{0}(z_{0})$ lies in $H_{0}(\mathbb{U})$. Assume that $\gamma_{0}$ is defined by

It follows that $H(z(t)) = 2^{-1}e^{i\theta}(H_{0}(z(t)))^{2} = 2^{-1}e^{i\theta }(tH_{0}(z_{0}))^{2} = t^{2}H(z_{0})$. Differentiation over $t$ gives us $H(z(t)) = tH\left(z_{0}\right)$ and hence,

so that

Integrating the above equality yields the result

Or we see that

and so we have

Since

we have

Now, we find that $\left[f(z)\right]^{\beta} = \left(\Psi_{e^{i\theta}, \xi}(z)\right) ^{\beta}$ for some $f\in \mathcal{P}_{\gamma}[\xi, A;B]$ and $c\in \partial \mathbb{U}.$ Let

and

where $\gamma_{0}:z = z(t),$ $0\leq t\leq1.$ Thus $F_{1}(t)$ is a continuous function of $t.$ As in ^[7]$,$ we see from (2.10), (2.13) and (2.15) that

or we can see that

From (2.13), we obtain (2.11) and (2.12). This proves that $\frac{f^{\prime}(z)}{\left[f(z)\right]^{1-\beta}} = \frac{\Psi_{e^{i\theta}, \xi}^{\prime }(z)}{\left(\Psi_{e^{i\theta}, \xi}(z)\right) ^{1-\beta}}$ on $\gamma_{0}.$ On applying the identity theorem, we have $f\left(z\right) = \Psi _{e^{i\theta}, \xi}\left(z\right)$. □

Theorem 2.4. Let $z_{0}\in \mathbb{U}$ and $\operatorname{Re}\beta > 0.$ If $z_{0} = 0,$ then $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B] = \{0\}.$ The boundary is the closed Jordan curve defined by $\theta \in \left(-\pi, \pi \right] \mathbb{\rightarrow }\Psi_{e^{i\theta}, \xi}(z_{0}),$ where

Moreover, if $f(z_{0}) = \Psi_{e^{i\theta}, \xi}(z_{0})$ for $f\in \mathcal{P}_{\gamma}[\xi, A;B],$ then $f(z) = \Psi_{e^{i\theta}, \xi}(z)$.

Proof. Finally, suppose that the mapping $\theta\in \partial \mathbb{U}\rightarrow \Psi_{e^{i\theta}, \xi}(z_{0})$ is not injective. Then there exists $\theta _{1},$ $\theta_{2}\in \partial \mathbb{U}$ with $\theta_{1}\neq \theta_{2}$ such that $\Psi_{e^{i\theta_{1}}, \xi}(z_{0}) = \Psi_{e^{i\theta_{2}}, \xi}(z_{0}).$ Since $\Psi_{e^{i\theta_{1}}, \xi}, \Psi_{e^{i\theta_{2}}, \xi}\in \mathcal{P} _{\gamma}[\xi, A;B],$ we have $\Psi_{e^{i\theta_{1}}, \xi} = \Psi_{e^{i\theta_{2} }, \xi}$ from uniqueness. This contradicts $c_{1}\neq c_{2}.$ For proof of the theorem, we combine the results of Theorem 2.1 as well as Theorem 2.2 and it can be seen that a simple closed curve $\partial \mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ obviously comprises of $\theta \in \partial \mathbb{U\rightarrow }\Psi_{e^{i\theta}, \xi}.$ As any simple closed curve cannot surrounds such a curve other than itself. Therefore, $\partial \mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ is coincident with $\theta \in \left(-\pi, \pi \right] \rightarrow \Psi _{e^{i\theta}, \xi}.$ □

3.   Concluding remarks

The region of variability problems are more useful for the family $\mathcal{S}$, than the related classical theorems about this family $\mathcal{S}$. In this study, we discussed the region of variability $\mathcal{V}_{\gamma}[\psi, z_{0}, \xi, \beta, A;B]$ for $f(z_{0})$, where $f$ ranges over the family $\mathcal{B}_{\gamma}\left[\psi, \xi, \beta, A;B\right]$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through a Large Group research project under grant number RGP2/25/44.

Conflicts of interest

This research does not involve any conflicts of interest.