Fekete–Szeg¨o Problems for Certain Classes of Meromorphic Functions Involving q -Al-Oboudi Differential Operator

In this paper, we introduce a new derivative operator involving q -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.


Introduction
For two analytic functions f and g in U, we say that f(z) is subordinate to g(z), written f≺g , if there is a Schwarz function w(z) with w(0) � 0, |w(z)| < 1(z ∈ U), such that f(z) � g(w(z)), (z ∈ U). If g is univalent, then f≺g if and only if f(0) � g(0) and f(U) ⊂ g(U), (see [8,24]). A function f is meromorphic if it is analytic throughout a domain D, except possibly for poles in D (see [40]).
Let φ(z) be an analytic function with a positive real part on U satisfying φ(0) � 1 and φ ′ (0) > 0 which maps U onto a region starlike with respect to 1 and symmetric with respect to the real axis.
Let * b (φ) be the class of functions f ∈ for which e class * b (φ) was introduced and studied by Mohammed and Darus [26] (see also Reddy and Sharma [30], with c � 1).
We note that for suitable choices of b and φ(z), we obtain the following subclasses: [4], with α � 1] and [33]); [29]); [17]); We note that 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 f ∈ given by (2), the δ-derivative of a function f(z) is defined by [3,11] (see also [14,15]) where As . Due to its use in numerous fields of mathematics and physics, the δ-derivative operator D * δ has fascinated and inspired many researchers. Jackson [14] was among the key contributors of all the scientists who introduced and developed the δ-calculus theory. In 1991, Ismail [13] was the first to demonstrate a crucial link between geometric function theory and the δ-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]).
Making use of D * n λ,δ , we define the following class * n λ,δ,α (b, φ) as follows: Noting that [12]); [12]); 2 2 | 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. is 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.

Fekete-Szeg Problems
We need the following lemmas, which will be used in our investigation.
Lemma 1 (see [22]). Let p(z) � 1 + c 1 z + c 2 z 2 + · · ·, be analytic in U and satisfy the following condition then for a complex number μ, we have The result is sharp for the functions given by Lemma 2 (see [22]). If h(z) � 1 + c 1 z + c 2 z 2 + · · · is a function with positive real part in U, then or one of its rotations. If ] � 1, equality holds if and only if or one of its rotations. Also, the above upper bound is sharp and it can be improved as follows when 0 < ] < 1: Unless otherwise mentioned, we assume throughout this paper that b ∈ C * , 0 ≤ α < δ/(δ + 1), λ ≥ 0, n ∈ N 0 , and 0 < δ < 1.

Theorem 1. Let f(z) be defined by (2) and φ(z)
The result is sharp.

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Taking α � 0 in eorem 1, we get Corollary 1. Let f(z) be defined by (2) and e result is sharp.
Taking n � 0 in eorem 1, we get Corollary 3. Let f(z) be defined by (2) and Journal of Mathematics e result is sharp.