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.

In this paper we consider basic hypergeometric functions introduced by Heine. We study mapping properties of certain ratios of basic hypergeometric functions having shifted parameters and show that they map the domains of analyticity onto domains convex in the direction of the imaginary axis. In order to investigate these mapping properties, few useful identities are obtained in terms of basic hypergeometric functions. In addition, we find conditions under which the basic hypergeometric functions are in $q$-close-to-convex family.Comment: 18 pages, 5 figures, submitted to a journa

In this paper, for every q ∈ (0, 1), we obtain the Herglotz representation theorem and discuss the Bieberbach type problem for the class of q-convex functions of order α, 0 ≤ α < 1. In addition, we discuss the Fekete-szegö problem and the Hankel determinant problem for the class of q-starlike functions, leading to couple of conjectures for the class of q-starlike functions of order α, 0 ≤ α < 1.

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