Certain Properties of Some Families of Generalized Starlike Functions with respect toq-Calculus

Abstract

By making use of the concept of -calculus, various types of generalized starlike functions of order were introduced and studied from different viewpoints. In this paper, we investigate the relation between various former types of -starlike functions of order . We also introduce and study a new subclass of -starlike functions of order . Moreover, we give some properties of those -starlike functions with negative coefficient including the radius of univalency and starlikeness. Some illustrative examples are provided to verify the theoretical results in case of negative coefficient functions class.

Some General Classes of q-Starlike Functions Associated with the Janowski Functions

“…in Definition 8, we are led to the class S * (q,2) (α), which was introduced and studied by Wongsaijai and Sukantamala (see [17], Definition 2).…”

“…The coefficient inequality problems for q-closed-to-convex functions with respect to Janowski starlike functions were studied recently (see, for example, [16]). In the year 2016, Wongsaijai and Sukantamala [17] published a paper, in which they generalized certain subclasses of starlike functions in a systematic way. In fact, they made a very significant usage of the q-calculus basically in the context of Geometric Function Theory.…”

“…Motivated by the works of Wongsaijai and Sukantamala [17] and other related works cited above in this paper, we shall consider three new subfamilies of q-starlike functions with respect to Janowski functions. Several properties and characteristics, for example, sufficient conditions, inclusion results, distortion theorems, and radius problems, shall be discussed in this investigation.…”

Properties of a generalized class of analytic functions with coefficient inequality

“…For example, Purohit and Raina [23] introduced a subclass of analytic functions defined by the q -analogue operator of fractional calculus and derived some results including the coefficient inequality. In [35], the authors obtained some relations between various types of q -starlike functions of order α by using the coefficient inequality. Recently, Mahmood and Sokol [19] defined a new class of analytic functions by using the Ruscheweyh q -differential operator, while Govindaraj and Sivasubramanian [14] applied the concept of q -calculus to define the new class of analytic functions that are closely related to the domains bounded by conic sections.…”

A certain class of q-close-to-convex functions of order α

“…Up to date, there are many applications of q-calculus on subclasses of analytic functions, especially generalization of subclasses of univalent functions (see [19][20][21][22][23][24][25][26][27][28][29][30]). In the context of geometric function theory, the usage of q-calculus was firstly applied in a book chapter by Srivastava [19], in which the basis q-hypergeometric functions was also provided.…”