Abstract
In this paper, we introduce a new derivative operator involving -Al-Oboudi differential operator for meromorphic functions. By using this new operator, we define a new subclass of meromorphic functions and obtain the Fekete–Szegő inequalities.
1. Introduction
For two analytic functions and in , we say that is subordinate to , written , if there is a Schwarz function with , , such that . If is univalent, then
(see [8, 24]). A function is meromorphic if it is analytic throughout a domain , except possibly for poles in (see [40]).
Let denote the class of meromorphic functions of the formwhich are analytic in the open punctured unit disc A function is said to be a meromorphic starlike function of order , denoted by , if
The class was studied by Pommerenke [29], Miller [23], and many others (see [9, 25]).
Let be an analytic function with a positive real part on satisfying and which maps onto a region starlike with respect to 1 and symmetric with respect to the real axis.
Let be the class of functions for which
The class was introduced and studied by Mohammed and Darus [26] (see also Reddy and Sharma [30], with ).
We note that for suitable choices of and , we obtain the following subclasses:(1) (see [4], with ] and [33]);(2) (see [6]);(3) (see [29]);(4) (see [17]);(5) (see [16, 31]).
Let be the class of functions for which
We note that(1). (see Aouf [6]);(2).
In geometric function theory, operators play an important role. Many authors present differential and integral operators, for example ([1, 20, 32, 37]). For a function given by (2), the -derivative of a function is defined by [3, 11] (see also [14, 15])where
As , we have and .
Due to its use in numerous fields of mathematics and physics, the -derivative operator has fascinated and inspired many researchers. Jackson [14] was among the key contributors of all the scientists who introduced and developed the -calculus theory. In 1991, Ismail [13] was the first to demonstrate a crucial link between geometric function theory and the -derivative operator, but a solid and comprehensive foundation was provided in 1989 in a book chapter by Srivastava [34]. Several recent works on this operator can be found in ([7, 18, 19, 35, 36]).
For , and , we define the following operator as follows:
From (9), it is easy to see that, for ,
We note that(i) = (see [5], with ).(ii) = (see [21, 38], with ).
Making use of , we define the following class as follows:
Definition 1. For,,,, and, we say that a functionis in the classif and only if
Noting that(1);(2);(3);(4) (see [12]);(5)(6)(7)(8) (see [12]);(9)(10);(11).
A Fekete–Szegő inequality is one of the inequalities for coefficients of univalent analytic functions found by Fekete and Szegő (1933) (see [10]), which is related to the Bieberbach conjecture. Providing similar estimates for other classes of functions is called the Fekete–Szegő problem. This problem has been considered by many authors for typical classes of univalent functions (see [2, 39]). In this paper, several properties, such as coefficient inequalities and Fekete–Szegő functionals, for the currently established families are derived.
2. Fekete–Szeg Problems
We need the following lemmas, which will be used in our investigation.
Lemma 1 (see [22]). Let, be analytic inand satisfy the following conditionthen for a complex number , we haveThe result is sharp for the functions given by
Lemma 2 (see [22]). Ifis a function with positive real part in, thenWhen or , the equality holds if and only if is or one of its rotations. If , then the equality holds if and only if is or one of its rotations. If , the equality holds if and only ifor one of its rotations. If , equality holds if and only ifor one of its rotations. Also, the above upper bound is sharp and it can be improved as follows when :
Unless otherwise mentioned, we assume throughout this paper that , , , and .
Theorem 1. Letbe defined by (2) and. Ifand, thenThe result is sharp.
Proof. If , then there is a Schwarz function in with and in , such thatIf we setSince is a Schwarz function, then and . Let us defineUsing (21)–(23), we getSinceThen,From (24) and (26), we haveThen, from (2) and (23), we see that and , or, equivalently, we obtainThereforeApplying Lemma 1, we obtain the result (19). Also, if , thenSince , then (see [28]), hencethis proves (20). The result is sharp for the functionswhich completes the proof of Theorem 1.
Taking in Theorem 1, we get
Corollary 1. Let be defined by (2) and . If and , thenThe result is sharp.
Taking in Theorem 1, we get
Corollary 2. Let be defined by (2) and . If and , thenThe result is sharp.
Remark 1. (1)For in Corollary 2, we get the result obtained by [12], [Theorem 2.3].(2)For in Corollary 2, we get the results obtained by [26, 30].(3)For and in Corollary 2, we get the results obtained by [33] and [4], [Theorem 5.2].
Taking in Theorem 1, we get
Corollary 3. Let be defined by (2) and . If and , thenThe result is sharp.
Remark 2. (1)Taking in Corollary 3, we get the result obtained by [12], [Theorem 2.8].(2)Letting in Corollary 3, we get the result obtained by [30].
Taking in Theorem 1, we get
Corollary 4. Let be defined by (2) and . If and , thenThe result is sharp.
Taking and in Theorem 1, we get
Corollary 5. Let be defined by (2) and . If and , thenThe result is sharp.
Remark 3. Letting and taking and in Corollary 5, we get the result obtained by [27], [Example 1.1].
By using Lemma 2, we can obtain the following theorem.
Theorem 2. For real , let . If given by (2) belongs to the class , thenThe result is sharp. Further, let .(i)If , then(ii)If , then
Proof. First, let . ThenLet, . Then, we obtainFinally, if . ThenThe sharpness is an immediate consequence of Lemma 2. This completes the proof of Theorem 2.
Remark 4. Taking in Theorem 2, we get the result obtained by [12], [Theorem 2.10].
Taking in Theorem 2, we get
Corollary 6. For real , let . If given by (2) belongs to the class , then
The result is sharp. Further, let .(i)If , then(ii)If , then
Remark 5. (1)Taking in Corollary 6, we get the result obtained by [12].(2)Letting and taking in Corollary 6, we get the result obtained by [4].
3. Conclusion
In the fields of combinatorics and quantum calculus, the -derivative introduced by Frank Hilton Jackson [14] plays an important role in the theory of functions of a complex variable and other fields of mathematics. In this paper, we define a new differential operator for meromorphic functions. By using this new operator, we define and study a new family of meromorphic functions. Several properties of the abovementioned family of functions are investigated, including coefficient inequalities and Fekete–Szegő functionals.
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interest.