Journal of Function Spaces
Volume 2022, Issue 1 2324774
Research Article
Open Access

Applications of the Bell Numbers on Univalent Functions Associated with Subordination

Sh. Najafzadeh, 

Sh. Najafzadeh

Department of Mathematics, Payame Noor University, Post Office Box: 19395–3697, Tehran, Iran pnu.ac.ir

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Mugur Acu, 

Corresponding Author

Mugur Acu

Lucian Blaga University of Sibiu, Faculty of Science, Department of Mathematics and Informatics, Street Dr. I. Ratiu 5–7, Sibiu 550012, Romania ulbsibiu.ro

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First published: 10 March 2022
https://doi.org/10.1155/2022/2324774
Academic Editor: Muhammad Arif

Abstract

The motivation of the present paper is to define a new subclass of univalent functions associated with the q-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.

1. Introduction

For a fixed nonnegative integer k, 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)
involving the Bell numbers was considered by Kumar et al. [1], see also [2, 3].
Let denote the class of all functions F which are analytic in the open unit disk,
(2)
and normalized by conditions:
(3)
Hence, has a Taylor–Maclaurin series representation:
(4)

Also, is the subclass of consisting of all well-known univalent functions in .

Furthermore, we denote by a subclass of consisting of functions with negative coefficients of the type:
(5)

Since (zF′(z)/F(z)) maps onto the right half-plane of ℂ, so Re(zF′(z)/F(z)) > 0, and it is a usual subclass of normalized univalent function class , which are star-like functions, see [4]. Thus, Q(z), given by (1), is star-like with respect to 1, and its coefficients are the Bell numbers.

The theory of q-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].

The q-analogue of the exponential function ez is given by
(6)
where q ∈ (0,1) and Γq(k + 1) is the q-gamma function defined by
(7)
and q-number [k]q is introduced by
(8)
see [13–15].
The Hadamard product (convolution) for function F(z), given by (5) and denoted by F∗G, is defined by
(9)
Let F and G be analytic in ; then, F is said to be subordinate to G, written F≺G, if there exists a function W analytic in , with W(0) = 0 and |W(z)| < 1, such that
(10)

If G is univalent, then F≺G if and only if F(0) = G(0) and , see [16].

Definition 1. A function H is said to belong to the class if

(11)
where 0 < β < 1, −1 ≤ γ ≤ 1, −1 ≤ α ≤ 1, 0 ≤ t ≤ 1, Ft(z) = (1 − t)z + tF(z), , and
(12)

Q(z), F(z), and are given by (1), (5), and (6), respectively.

2. Main Results

In this section, we obtain the coefficient bounds and extreme points of .

Theorem 1. Let H(z) be analytic in . Then, if and only if

(13)

Proof. The γ subordination relation (11) is equivalent to

(14)

Suppose that (13) holds true. We must show that (11) or equivalently (14) holds. However, we have

(15)

By (13) and letting |z| = 1, the above expression is less than or equal to zero, so (14) holds true.

To prove the converse, let ; thus,

(16)
for all . Since Re(z) ≤ |z|, we have
(17)

By letting z⟶1 through positive values and choosing the values of z such that (zH′(z)/Ft(z)) is real, we have

(18)
and this completes the proof.

Remark 1. We note that the function,

(19)
shows that inequality (13) is sharp.

Theorem 2. Let H1(z) = z and

(20)
where k = 2,3, …. Then, if and only if it can be expressed in the form:
(21)
where λk ≥ 0 and . In particular, the extreme points of are the functions Hk(z), where k = 1,2,3, ….

Proof. Let H be expressed by (21). This means that we can write

(22)

Since and

(23)
so, by Theorem 1, we conclude that .

Conversely, suppose that . Then, by (13), we have

(24)

By setting

(25)
and , we obtain the required result. So, the proof is complete.

3. Geometric Properties of

In this section, we show convexity of . Also, we obtain convolution preserving property.

Theorem 3. is a convex set.

Proof. We must show that if Hj(z), for j = 1,2, …, m, belong to , then the function,

(26)
is also in the same class, where 0 < σj < 1 and .

Since , we have

(27)

However,

(28)

It is enough to verify inequality (13) for H(z). However,

(29)

This inequality by (13) shows that , and the proof is complete.

Theorem 4. Let the functions Hj(z), j = 1,2, be in the class ; then, (H1∗H2)(z) belongs to , where β∗ ≤ 1 − X, and

(30)

Proof. Since , so

(31)

It is sufficient to show that

(32)

By using Cauchy–Schwarz inequality, from (13), we obtain

(33)

Hence, we find β∗ such that

(34)
or equivalently
(35)

This inequality holds if

(36)
or equivalently β∗ ≤ 1 − X, where X is given in (30). So, the proof is complete.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Data Availability

No data were used to support this study.

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