Close-to-convex functions defined by fractional operator

Abstract

Let S denote the class of functions f (z) = z + a 2 z 2 + ... analytic and univalent in the open unit disc D = {z ∈ C||z| < 1}. Consider the subclass and S * of S, which are the classes of convex and starlike functions, respectively. In 1952, W. Kaplan introduced a class of analytic functions f (z), called close-to-convex functions, for which there exists φ(z) ∈ C, depending on f (z) with Re( f (z) φ (z) ) > 0 in , and prove that every close-to-convex function is univalent. The normalized class of close-to-convex functions denoted by K. These classes are related by the proper inclusions C ⊂ S * ⊂ K ⊂ S. In this paper, we generalize the close-to-convex functions and denote K(λ) the class of such functions. Various properties of this class of functions is alos studied.

Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric q-derivative operator

“…In fact, the theory of univalent functions can be described by using the theory of the q-calculus. Moreover, in recent years, such q-calculus operators as the fractional q-integral and fractional q-derivative operators were used to construct several subclasses of analytic functions (see, for example, [1], [6], [26], [27], [33], [35], [36], [37] and [38]). In particular, Purohit and Raina [36] investigated applications of fractional q-calculus operators to define several classes of functions which are analytic in the open unit disk U.…”

Faber polynomial coefficient estimates for certain classes of bi-univalent functions defined by using the Jackson (p,q)-derivative operator

“…In fact, the theory of univalent functions can be described by using the theory of the q-calculus. Moreover, in recent years, such q-calculus operators as the fractional q-integral and fractional q-derivative operators were used to construct several subclasses of analytic functions (see, for example, [5,8,23]). …”

Coefficient estimates and Fekete- Szegὅ inequality for a subclass of Bi-univalent functions defined by symmetric Q-derivative operator by using Faber polynomial techniques

“…For example, the theory of q-calculus is used to describe the extension of the theory of univalent functions. For basic definitions, applications, terminologies, geometric properties and approximation one can refer [ [5], [8], [9], [14], [17], [19], [20], [21]]. Let us suppose 01 q throughout this paper.…”