Function kernels and divisible groupoids

1.   Introduction

Let $\mathbb{A}$ stand for the collection of functions $\mathcal{G}$ of the type

that are holomorphic in the open unit disk $\Lambda : = \left\{ \xi \in \mathbb{C}:\left\vert \xi \right\vert < 1\right\}$ of the complex plane, and let $\mathfrak{S}$ indicate the subclass of functions of $\mathbb{A}$ which are univalent in $\Lambda$. For functions $\mathcal{G}\in \mathbb{A}$ given by (1.1) and $\mathcal{H}\in \mathbb{A}$ given by $\mathcal{H} (\zeta) = \zeta +\sum\limits_{j = 2}^{\infty }b_{j}\zeta ^{j},$ we define the convolution product (or Hadamard) of $\mathcal{G}$ and $\mathcal{H}$ by

Let $\mathcal{G}$ and $\mathcal{F}$ be two holomorphic functions in $\Lambda$. The function $\mathcal{G}$ is said to be subordinated to $\mathcal{F}$ if there are Schwarz function $w(\xi)$, that is, holomorphic in $\Lambda$ with $w(0) = 0$ and $\left\vert w(\xi)\right\vert < 1$, $\xi \in \Lambda$, such as $\mathcal{G}(\xi) = \mathcal{F}(w(\xi))$ for all $\xi \in \Lambda$. This subordination notion is indicated by

If the function $\mathcal{F}$ is univalent in $\Lambda$, then we have the inclusion equivalence

The subfamilies of $\mathfrak{S}$ which are the starlike and the convex function in $\Lambda$ defined by

and

respectively. Equivalently, we have

where

Janowski defined in ^[4] the extended function family $\mathfrak{S} ^{\ast }\left[\mathcal{A}, \mathcal{B}\right]$ of starlike functions called the Janwoski class of functions as follows: A function $\mathcal{ G}\in \mathbb{A}$ is in the family $\mathfrak{S}^{\ast }\left[\mathcal{A}, \mathcal{B}\right]$ if

The above subordination could be written as

where an analytical function with a real positive part in $\Lambda$ is denoted by $p(\xi)$.

The Janowski convex and Janowski starlike functions are obtained by reducing the above-described classes to the requirement $-1\leq \mathcal{B} < \mathcal{A }\leq 1$. For the special cases $\mathcal{A}: = 1-2\alpha$ and $\mathcal{B} : = -1$, where $0\leq \alpha < 1$, we obtain the families, namely the family of starlike and convex functions of order $\alpha$ $(0\leq \alpha < 1)$ previously defined by Robertson in ^[6], and considered respectively by

Babalola defined the operator $\mathcal{I}_{\upsilon }^{\rho }:\; \mathbb{A} \rightarrow \mathbb{A}\;$as

where

and $\rho _{\sigma, \upsilon }^{-1}$ is

For $\mathcal{G}\in \mathbb{A},$ then (1.7) is equivalent to

Making use the binomial series

for $\mathcal{G}\in \mathbb{A}, \ $El-Deeb ^[3] introduced the linear differential operator as follows:

where

and

From (1.8), we obtain that

In this article using the El-Deeb operator defined in (1.8), we define a new sub-family of $\mathbb{A}$:

where $-1\leq \mathcal{A} < \mathcal{B}\leq 1$; $\ \delta > 0;\ t, \sigma, \upsilon \in \mathbb{N}$ and $n, m\in \mathbb{N} _{0}$, that will lead us to the study of Fekete-Szegö problem. Further, coefficient estimates, characteristic properties and partial sums results will be established.

Specializing the values of $\mathcal{A}$ and $\mathcal{B}$ one can obtain the particular cases

and

2.   The Fekete-Szegö functional bounds for the class $\mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A }, \mathcal{B}\right)$

To solve the Fekete-Szegö type inequality for $\mathcal{G}\in \mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ we will use the next results (the first part is due to Carathéodory ^[1]):

Lemma 1. ^[1,5] If $P(\xi) = 1+p_{1}\xi +p_{2}\xi ^{2}+\dots \in \mathcal{P}$ where $\mathcal{P}$ the class of holomorphic functions with positive real part in $\Lambda$, with $P(0) = 1$, then

and for the complex number $\mu \in \mathbb{C}$ we have

If $\mu$ is a real parameter, then

When $\mu > 1$ or $\mu < 0$, equality (2.3) holds true if and only if $P_{1}(\xi) = \dfrac{1+\xi }{1-\xi }$ or one of its rotations. When $0 < \mu < 1$, the equality (2.3) holds if and only if $P_{2}(\xi) = \dfrac{ 1+\xi ^{2}}{1-\xi ^{2}}$ or one of its rotations. When $\mu = 0$, equality (2.3) holds if and only if

or one of its rotations. When $\mu = 1$, the equality (2.3) holds true if $P(\xi)$ is a reciprocal of one of the functions such that the equality holds true in the case when $\mu = 0$.

Theorem 1. If $\mathcal{G}\in \mathbb{A}$ defined as (1.1), belongs to $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B} \right)$, then

and for a complex number $\tau$, we have

where

and $\psi _{j}^{n}$ is given by (1.9).

Proof. We will show that the relations $\left(2.4\right)$–$\left(2.6\right)$ and $\left(2.16\right)$ hold true for $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$. If $\mathcal{G}\in \mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, then

which yields

Since we can write $w\left(\xi \right)$ as

where $h\left(\xi \right) \in \mathcal{P}$ and have the form $h\left(\xi \right) = 1+p_{1}\xi +p_{2}\xi ^{2}+p_{3}\xi ^{3}+\dots,$ so

and therefore

If we compare the first coefficients of $\left(2.9\right)$ and $\left(2.10\right),$ we get

and by using $\left(2.1\right)$ in $\left(2.11\right)$ and $\left(2.2\right)$ in $\left(2.12\right)$, we get

For a complex nubmer $\tau$, and from $\left(2.11\right)$ together with $\left(2.12\right)$, we have

where $\Theta \left(\tau, \sigma, \upsilon, \mathcal{A}, \mathcal{B}\right)$ is denoted by (2.7). Now, we apply Lemma 1 to $\left(2.15 \right)$ and obtain the required results. □

Theorem 2. If the function $\mathcal{G}\in \mathbb{A}$ defined as (1.1) belongs to $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, then for any real parameter $\tau$ we obtain

where $\Theta \left(\tau, \sigma, \upsilon, \mathcal{A}, \mathcal{B}\right)$ is given by (2.7),

and

Proof. The proof can be produced directly by making use of Lemma 1 in $\left(2.15\right),$ so we choose to omit it. □

3.   The coefficient inequalities for $\mathcal{G}^{-1}\in \mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{D }, \mathcal{E}\right)$

The "Koebe one quarter theorem" ^[2] ensures that the image of $\Lambda$ under each univalent function $\mathcal{G}\in \mathbb{A}$ consists a disk of radius $\dfrac{1}{4}$. Thus each univalent function $\mathcal{G}$ has an inverse $\mathcal{G}^{-1}$ satisfying

A function $\mathcal{G}\in \mathbb{A}$ is called bi-univalent in $\Lambda$ if both $\mathcal{G}$ and $\mathcal{G}^{-1}$ are univalent in $\Lambda$. We mention that the collection of bi-univalent functions defined in the unit disk $\Lambda$ is not empty. For example, the functions $\xi$, $\dfrac{\xi }{1-\xi }$, $-\log (1-\xi)$ and $\dfrac{1}{2}\log \dfrac{1+\xi }{1-\xi }$ are members of bi-univalent function family, however the Koebe function is not a member.

Theorem 3. If $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ and the inverse function of $\mathcal{G}$ is $\mathcal{G}^{-1}(w) = w+\sum\limits_{j = 2}^{ \infty }d_{j}w^{j}$, then

and for any $\mu \in \mathbb{C}$, we have

where $\Theta \left(2, \sigma, \upsilon, \mathcal{A}, \mathcal{B}\right)$ given by (2.7).

Proof. Since

is the inverse of the function $\mathcal{G}$, it can be seen that

From (1.1) and (3.4), we obtain that

Therefore from (3.4) and (3.5) we get

Equating the corresponding coefficients of the relation (3.6), we conclude that

First, from the relations (2.11) and (3.7) we have

To find $|d_{3}|$, from (3.8) we have

Hence, by using (2.15) for real $\tau = 2$ we deduce that

where $\Theta \left(2, \sigma, \upsilon, \mathcal{A}, \mathcal{B}\right)$ given by (2.7). For any complex number $\mu$, a simple computation gives us that

By taking modulus on both sides of (3.11) and applying Lemma 1 and (2.1), we find that

and this completes our proof. □

4.   Characterization properties

By applying the techniques introduced by Silverman in ^[7], we will introduce some characteristic properties of the functions $\mathcal{G} \in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ such as partial sums results, necessary and sufficient conditions, radii of close-to-convexity, distortion bounds, radii of starlikeness and convexity.

Theorem 4. If $\mathcal{G}\in \mathbb{A}$ and be defined as (1.1) belongs to $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, then

where $\psi _{j}^{n}$ given by (1.9).

Proof. Letting $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, by (1.11) we deduce that

where $w\left(\xi \right)$ is a Schwarz function, or equivalently

Thus, the above relation leads us to

and taking $\left\vert \xi \right\vert = r\rightarrow 1^{-}$ simple computation yields $\left(4.1\right)$. □

Example 1. For

such that $\sum\limits_{j = 2}^{\infty }\ell _{j} = 1$, we get

Then $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, and we note that the inequality $\left(4.1\right)$ is sharp.

Corollary 1. Let $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ given by (1.1). Then

where $\psi _{j}^{n}$ is defined by (1.9). The approximation is sharp for the function

Theorem 5. If $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, then

For the function defined by

the approximation is sharp.

Proof. For $\left\vert \xi \right\vert = r < 1$ we have

Moreover, since for $\left\vert \xi \right\vert = r < 1$ we get $r^{j} < r^{2}$ for all $j\geq 2$, the above relation implies that

Similarly, we get

From the relation $\left(4.1\right)$ we have

but

Therefore,

and by using (4.3) in $\left(4.7\right)$ and $\left(4.8 \right)$ we get the desired result. □

The next distortion theorem for the family $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ could be similarly obtained:

Theorem 6. If $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, then

The equality holds if the function is $\widehat{\mathcal{G}}$ given by $(4.6)$.

Proof. Since the proof is quite analogous with those of Theorem 5, so it will be omitted. □

The next result deals with the fact that a convex combination of functions of the class $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{ A}, \mathcal{B}\right)$ belongs to the same class, as follows:

Theorem 7. Let $\mathcal{G}_{i}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ given by

Then $H\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, where

Proof. By Theorem 4 we have

and,

Therefore

thus $H\left(\xi \right) \in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$. □

Regarding the arithmetic means of the functions of the family $\mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ the next result holds:

Theorem 8. If $\mathcal{G}_{i}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ are given by (4.10), then

Where $\mathcal{G}$ is the arithmetic mean of $\mathcal{G}_{i}$, $i = 1, 2, 3, \dots, k$.

Proof. From the definition relation (4.12) we get

and to prove that $\mathcal{G}\left(\xi \right) \in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, according to the Theorem 4.1 it is sufficient to prove that

A simple computation shows that

Therefore $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$. □

Theorem 9. If $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, then $\mathcal{G}$ is a starlike functions of order $\vartheta$ $\left(0\leq \vartheta < 1\right),$ $\left\vert \xi \right\vert < r_{1}^{\ast },$

The equality holds for $\mathcal{G}$ given in $\left(4.4\right).$

Proof. Let $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$. We see that $\mathcal{G}$ is a starlike functions of order $\vartheta$, if

By simple calculation, we deduce

Since $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, from $\left(4.1\right)$ we get

The relation $\left(4.13\right)$ will holds true if

which implies that

or, equivalently

which yields the starlikeness of the family. □

Theorem 10. If $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, then $\mathcal{G}$ is a close-to-convex function of order $\vartheta$ $\left(0\leq \vartheta < 1\right),$ $\left\vert \xi \right\vert < r_{2}^{\ast },$

Proof. Let $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$. If $\mathcal{G}$ is close-to-convex function of order $\vartheta$, then we find that

that is

Since $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$, by $\left(4.1\right)$ we have

The relation $\left(4.13\right)$ will holds true if

which implies that

or, equivalently

which yields the desired result. □

5.   Conclusions

In this paper, we introduced a new class $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ of holomorphic functions defined in the open unit disk, which is connected to the combination of the Binomial series and the Babalola operator. We employed differential subordination involving Janowski-type functions to investigate these properties. Utilizing well-established results, such as Carathéodory's inequality for functions with real positive parts, as well as the Keogh-Merkes and Ma-Minda inequalities, we established upper bounds for the first two initial coefficients of the Taylor-Maclaurin power series expansion. Additionally, we derived an upper bound for the Fekete-Szegő functional for functions within this family.

We also extended our findings to include similar results for the first two coefficients and for the Fekete-Szegő inequality for functions $\mathcal{G}^{-1}$ when $\mathcal{G}\in \mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$. Furthermore, we determined coefficient estimates, distortion bounds, radius problems, and the radius of starlikeness and close-to-convexity for these newly defined functions.

Author contributions

Kholood M. Alsager: Conceptualization, validation, formal analysis, investigation, supervision; Sheza M. El-Deeb: Methodology, formal analysis, investigation; Ala Amourah: Methodology, validation, writing-original draft; Jongsuk Ro: Writing-original draft, writing-review. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (No. NRF-2022R1A2C2004874). This work was supported by the Korea Institute of Energy Technology Evaluation and Planning(KETEP) and the Ministry of Trade, Industry & Energy(MOTIE) of the Republic of Korea (No. 20214000000280).

Conflict of interest

The authors declare no conflict of interest.