On a generalization of close-to-convex functions

Abstract

A motivation comes from M. Ismail and et al.: A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77-84 to study a generalization of close-to-convex functions by means of a q-analog of a difference operator acting on analytic functions in the unit disk D = {z ∈ C : |z| < 1}. We use the terminology q-close-to-convex functions for the q-analog of close-to-convex functions. The q-theory has wide applications in special functions and quantum physics which makes the study interesting and pertinent in this field. In this paper, we obtain some interesting results concerning conditions on the coefficients of power series of functions analytic in the unit disk which ensure that they generate functions in the q-close-to-convex family. As a result we find certain dilogarithm functions that are contained in this family. Secondly, we also study the famous Bieberbach conjecture problem on coefficients of analytic q-close-to-convex functions. This produces several power series of analytic functions convergent to basic hypergeometric functions.2010 Mathematics Subject Classification. 30C45; 30C50; 30C55; 30B10; 33B30; 33D15; 40A30; 47E05. Key words and phrases. Univalent and analytic functions; starlike and close-to-convex functions; Bieberbach-de Branges theorem; q-difference operator; q-starlike and q-close-to-convex functions; special functions.* The corresponding author.

Geometric properties of basic hypergeometric functions

“…The q-close-to-convex functions (see Definition 1.2), defined in Section 1, analytically characterizes by the fact that |g(z) + f (qz) − f (z)|/|g(z)| ≤ 1 for all z ∈ D (see [20,Lemma 3.1]). It shows that if the function g(z) vanishes at z then z has to be zero, else the quotient (g(z) + f (qz) − f (z))/g(z) would have a pole at z = 0.…”

“…For its proof we use the following result, a generalization of a result by MacGregor [11,Theorem 1], recently obtained in [20]. Lemma 3.5.…”

“…Lemma 3.5. [20] Let {A n } be a sequence of real numbers such that A 1 = 1 and for all n ≥ 1, define B n = A n (1 − q n )/(1 − q). Suppose that…”

“…In view of the above relationship between S * and K, with the help of the difference operator D q f , a similar q-analog of close-to-convex functions are studied in [16,20].…”

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

“…Recently, EsraÖzkan Uçar [13] studied the coefficient inequality for -closed-to-convex functions with respect to Janowski starlike functions. Here, many newsworthy results related to -calculus and subclasses of analytic functions theory are studied by various authors (see [14][15][16][17][18][19][20][21]). …”