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.

Let S be the family of analytic and univalent functions f in the unit disk D with the normalization f (0) = f ′ (0) − 1 = 0, and let γn(f ) = γn denote the logarithmic coefficients of f ∈ S. In this paper, we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families F(c) and G(δ) of functions f ∈ S defined by c XXXX Australian Mathematical Society 0263-6115/XX $A2.00 + 0.00

I. M. Milin proposed, in his 1971 paper, a system of inequalities for the logarithmic coefficients of normalized univalent functions on the unit disk of the complex plane. This is known as the Lebedev-Milin conjecture and implies the Robertson conjecture which in turn implies the Bieberbach conjecture. In 1984, Louis de Branges settled the long-standing Bieberbach conjecture by showing the Lebedev-Milin conjecture. Recently, O. Roth proved an interesting sharp inequality for the logarithmic coefficients based on the proof by de Branges. In this paper, following Roth's ideas, we will show more general sharp inequalities with convex sequences as weight functions and then establish several consequences of them. We also consider the inequality with the help of de Branges system of linear ODE for non-convex sequences where the proof is partly assisted by computer. Also, we apply some of those inequalities to improve previously known results.2010 Mathematics Subject Classification. Primary 30C50; Secondary 30C75.

For an analytic function f defined on the unit disk |z| < 1, let ∆(r, f ) denote the area of the image of the subdisk |z| < r under f , where 0 < r ≤ 1. In 1990, Yamashita conjectured that ∆(r, z/f ) ≤ πr 2 for convex functions f and it was finally settled in 2013 by Obradović and et. al.. In this paper, we consider a class of analytic functions in the unit disk satisfying the subordination relation zf ′ (z)/f (z) ≺ (1 + (1 − 2β)αz)/(1 − αz) for 0 ≤ β < 1 and 0 < α ≤ 1. We prove Yamashita's conjecture problem for functions in this class, which solves a partial solution to an open problem posed by Ponnusamy and Wirths.

No abstract