# Applications of the Bell Numbers on Univalent Functions Associated with Subordination

Authors: Sh. Najafzadeh; Mugur Acu
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The motivation of the present paper is to define a new subclass of univalent functions associated with theq-analogue of the exponential function and the well-known Bell numbers based on subordination structure. Furthermore, we estimate the coefficient bound and extreme points. Also, geometric properties such as convexity and convolution preserving concept are investigated.


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## Body

## 1. Introduction

For a fixed nonnegative integerk, the Bell numbers Bk is the number of equivalent relations on a set with k elements or equivalently the number of possible disjoint partitions of a set with k elements into nonempty subsets. The function(1)Qz=eez−1=∑k=0∞Bkzkk!,involving the Bell numbers was considered by Kumar et al. [1], see also [2, 3].LetA denote the class of all functions F which are analytic in the open unit disk,(2)D=z∈ℂ:z<1,and normalized by conditions:(3)F0=F′0−1=0.Hence,F∈A has a Taylor–Maclaurin series representation:(4)Fz=z+∑k=2∞akzk,z∈D.Also,S is the subclass of A consisting of all well-known univalent functions in D.Furthermore, we denote byN a subclass of A consisting of functions with negative coefficients of the type:(5)Fz=z−∑k=2∞akzk,z∈D,ak≥0.SincezF′z/Fz maps D onto the right half-plane of ℂ, so RezF′z/Fz>0, and it is a usual subclass of normalized univalent function class S, which are star-like functions, see [4]. Thus, Qz, given by (1), is star-like with respect to 1, and its coefficients are the Bell numbers.The theory ofq-calculus (or quantum calculus) operators is used in various areas of science and also in the geometric function theory. Also, the theory of q-derivative operators has played an important role in differential equations, physics, mechanics, and so on. The application of q-calculus was initiated by Jackson [5, 6]. He was the first to develop q-integral and q-derivative in a systematic way. Q-calculus is equivalent to classical calculus without the notion of limits. A comprehensive study on applications of q-calculus and q-analogue of well-known operators in theory of univalent functions may be found in [7–12].Theq-analogue of the exponential function ez is given by(6)eqz=∑k=0∞zkΓqk+1,where q∈0,1 and Γqk+1 is the q-gamma function defined by(7)Γqk+1=kqΓqk,Γq1=1,and q-number kq is introduced by(8)kq=1−qk1−q,k∈ℂ,∑n=0k−1qn=1+q+q2+⋯+qk−1,k∈ℕ,see [13–15].The Hadamard product (convolution) for functionFz, given by (5) and Gz=z−∑k=2∞bkzk denoted by F∗G, is defined by(9)F∗Gz=z−∑k=2∞akbkzk=G∗Fz,z∈D.LetF and G be analytic in D; then, F is said to be subordinate to G, written F≺G, if there exists a function W analytic in D, with W0=0 and Wz<1, such that(10)Fz=GWz.IfG is univalent, then F≺G if and only if F0=G0 and FD⊂GD, see [16].Definition 1.
A functionH is said to belong to the class Wtα,β,γ if(11)zH′zFtz≺1+γ+α1−βz1+γz,where 0<β<1, −1≤γ≤1, −1≤α≤1, 0≤t≤1, Ftz=1−tz+tFz, Fz∈N, and(12)Hz=1+2z−eqz∗2z+1−Qz∗Fz.
Qz, Fz, and eqz are given by (1), (5), and (6), respectively.


## 2. Main Results

In this section, we obtain the coefficient bounds and extreme points ofWtα,β,γ.Theorem 1.
LetHz be analytic in D. Then, H∈Wtα,β,γ if and only if(13)∑k=2∞Bkk−1!Γqk+1−t1−γ+tα1−βak≤α1−β.Proof.
Theγ subordination relation (11) is equivalent to(14)zH′z/Ftz−1γ+α1−β−γzH′z/Ftz<1.
Suppose that (13) holds true. We must show that (11) or equivalently (14) holds. However, we have(15)zH′z−Ftz−γ+α1−βFtz−γzH′z=z−∑k=2∞Bkk−1!Γqk+1akzk−z+∑k=2∞takzk−α1−βz−∑k=2∞tγ+α1−β−γBkk−1!Γqk+1akzk≤∑k=2∞Bkk−1!Γqk+1−t1−γ+tα1−βak−α1−β.
By (13) and letting z=1, the above expression is less than or equal to zero, so (14) holds true.
To prove the converse, letHz∈Wtα,β,γ; thus,(16)zH′z/Ftz−1γ+α1−β−γzH′z/Ftz=z1−∑k=2∞Bk/k−1!Γqk+1akzk−1−1−tz−tz−∑k=2∞akzkγ+α1−β1−tz+tz−∑k=2∞akzk−γz1−∑k=2∞Bk/k−1!Γqk+1akzk−1<1,for all z∈D. Since Rez≤z, we have(17)Re∑k=2∞Bk/k−1!Γqk+1−takzkα1−βz−∑k=2∞tγ+α1−β−γBk/k−1!Γqk+1akzk<1.
By lettingz⟶1 through positive values and choosing the values of z such that zH′z/Ftz is real, we have(18)∑k=2∞Bkk−1!Γqk+1−t1−γ+tα1−βak≤α1−β,and this completes the proof.Remark 1.
We note that the function,(19)Vz=z−α1−βB2/Γq3−t1−γ+tα1−βz2,shows that inequality (13) is sharp.Theorem 2.
LetH1z=z and(20)Hkz=z−α1−βBk/k−1!Γqk+1−t1−γ+tα1−βzk,where k=2,3,…. Then, H∈Wtα,β,γ if and only if it can be expressed in the form:(21)Hz=∑k=1∞λkHkz,where λk≥0 and ∑k=1∞λk=1. In particular, the extreme points of Wtα,β,γ are the functions Hkz, where k=1,2,3,….Proof.
LetH be expressed by (21). This means that we can write(22)Hz=∑k=1∞λkHkz=λ1H1z+∑k=2∞λkHkz=∑k=1∞λkz−∑k=2∞α1−βλkBk/k−1!Γqk+1−t1−γ+tα1−βzk.
Since∑k=1∞λk=1 and(23)∑k=2∞Bkk−1!Γqk+1−t1−γ+tα1−βα1−βλkBk/k−1!Γqk+1−t1−γ+tα1−β=∑k=2∞α1−βλk=α1−β∑k=2∞λk=α1−β1−λ1<α1−β,so, by Theorem 1, we conclude that H∈Wtα,β,γ.
Conversely, suppose thatH∈Wtα,β,γ. Then, by (13), we have(24)ak<α1−βBk/k−1!Γqk+1−t1−γ+tα1−β,k=2,3,….
By setting(25)λk=Bk/k−1!Γqk+1−t1−γ+tα1−βα1−βak,k≥2,and λ1=1−∑k=2∞λk, we obtain the required result. So, the proof is complete.


## 3. Geometric Properties ofWtα,β,γ

In this section, we show convexity ofWtα,β,γ. Also, we obtain convolution preserving property.Theorem 3.
Wtα,β,γ is a convex set.Proof.
We must show that ifHjz, for j=1,2,…,m, belong to Wtα,β,γ, then the function,(26)Hz=∑j=1mσjHjz,is also in the same class, where 0<σj<1 and ∑j=1mσj=1.
SinceHjz∈Wtα,β,γ, we have(27)∑k=2∞Bkk−1!Γqk+1−t1−γ+tα1−βak,j≤α1−β,j=1,2,…,m.
However,(28)Hz=∑j=1mσjHjz=∑j=1mσjz−∑k=2∞BkΓqk+1k!ak,jzk=z−∑k=2∞∑j=1mσjak,jBkΓqk+1k!zk.
It is enough to verify inequality (13) for Hz. However,(29)∑k=2∞Bkk−1!Γqk+1−t1−γ+tα1−β∑j=1mσjak,j=∑j=1mσj∑k=2∞Bkk−1!Γqk+1−t1−γ+tα1−β<∑j=1mσjα1−β=α1−β.
This inequality by (13) shows that H∈Wtα,β,γ, and the proof is complete.Theorem 4.
Let the functionsHjz, j=1,2, be in the class Wtα,β,γ; then, H1∗H2z belongs to Wtα,β∗,γ, where β∗≤1−X, and(30)X=αBk/k−1!Γqk+1−t1−γBk/k−1!Γqk+1−t1−γ+tα1−β/1−β−tα2.Proof.
SinceHjz∈Wtα,β,γ, so(31)Hjz=z−∑k=2∞BkΓqk+1k!ak,jzk.
It is sufficient to show that(32)∑k=2∞Bkk−1!Γqk+1−t1−γα1−β+tak,1ak,2≤1.
By using Cauchy–Schwarz inequality, from (13), we obtain(33)∑k=2∞Bkk−1!Γqk+1−t1−γα1−β+tak,1ak,2≤1.
Hence, we findβ∗ such that(34)∑k=2∞Bkk−1!Γqk+1−t1−γα1−β∗+tak,1ak,2≤∑k=2∞Bkk−1!Γqk+1−t1−γα1−β+tak,1ak,2≤1,or equivalently(35)ak,1ak,2≤Bk/k−1!Γqk+1−t1−γ/α1−β+tBk/k−1!Γqk+1−t1−γ/α1−β∗+t.
This inequality holds if(36)α1−βBk/k−1!Γqk+1−t1−γ+tα1−β≤Bk/k−1!Γqk+1−t1−γ/α1−β+tBk/k−1!Γqk+1−t1−γ/α1−β∗+t,or equivalently β∗≤1−X, where X is given in (30). So, the proof is complete.



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*Source: 2324774-2022-03-10.xml*