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.

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In this paper, we mainly study the order of q-starlikeness of the well-known basic hypergeometric function. In addition, we obtain the Bieberbach-type problem for a generalized class of starlike functions. We also discuss the Fekete-szegö and the Hankel determinant problems for the same class of functions.

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Abstract. The Cassinian metric and its inner metric have been studied for subdomains of the n-dimensional Euclidean space R n (n ≥ 2) by the first named author. In this paper we obtain various inequalities between the Cassinian metric and other related metrics in some specific subdomains of R n . Also, a sharp distortion property of the Cassinian metric under Möbius transformations of the unit ball is obtained.2010 Mathematics Subject Classification. 30C35, 30C20, 30F45, 51M10. Key words and phrases. Möbius transformation, the hyperbolic metric, the Cassinian metric, the distance ratio metric, the visual angle metric, the triangular ratio metric, inner metric.