New Developments in Geometric Function Theory

The book contains papers published in a Special Issue of Axioms, entitled "New Developments in Geometric Function Theory". An Editorial describes the 14 papers devoted to the study of complex-valued functions which present new outcomes related to special classes of univalent and bi-univalent functions, new operators and special functions associated with differential subordination and superordination theories, fractional calculus, and certain applications in geometric function theory

A generalization of starlike functions of order alpha

For every $q\in(0,1)$ and $0\le \alpha<1$ we define a class of analytic functions, the so-called $q$-starlike functions of order $\alpha$, on the open unit disk. We study this class of functions and explore some inclusion properties with the well-known class of starlike functions of order $\alpha$. The paper is also devoted to the discussion on the Herglotz representation formula for analytic functions $zf'(z)/f(z)$ when $f(z)$ is $q$-starlike of order $\alpha$. As an application we also discuss the Bieberbach conjecture problem for the $q$-starlike functions of order $\alpha$. Further application includes the study of the order of $q$-starlikeness of the well-known basic hypergeometric functions introduced by Heine.Comment: 13 pages, 4 figures, submitted to a journa

A new look at q-hypergeometric functions

For complex parameters ai, bj , q(i = 1, ..., r, j = 1, ..., s, bj ∈ C\{0, −1, −2, ...}, |q| < 1), define the q-hypergeometric function rΦs(a1, ..., ar; b1, ..., bs; q, z) by rΦs(ai; bj ; q, z) = ∑∞ n=0 (a1, q)n...(ar, q)n (q, q)n(b1, q)n...(bs, q)n z n (r = s + 1; r, s ∈ N0 = N ∪ {0}; z ∈ U) where N denote the set of positive integers and (a, q)n is the q-shifted factorial defined by (a, q)n = { 1, n = 0; (1 − a)(1 − aq)(1 − aq2)...(1 − aqn−1), n ∈ N. Recently, the authors [7] defined the linear operator M(ai, bj ; q)f. Using the operatör M(ai, bj ; q)f(z)f, Aldweby and Darus [13] gave a new integral operator. In this work we highlight a result related to the new integral operator.Publisher's Versio

On boundedness and compactness of a generalized Srivastava–Owa fractional derivative operator

AbstractThe purpose of this present effort is to define a new fractional differential operator Tzβ,τ,γ, involving Srivastava–Owa fractional derivative operator. Further, we investigate some geometric properties such as univalency, starlikeness, convexity for their normalization, we also study boundedness and compactness of analytic and univalent functions on weighted μ-Bloch space for this operator. The method in this study is based on the generalized hypergeometric function

New subclasses of bi-univalent functions of complex order associated with hypergeometric functions

In the present paper, new subclasses of bi-univalent functions of complex order associated with hypergeometric functions are introduced and coefficient estimates for functions in these classes are obtained. Several new (or known) consequences of the results are also pointed out.Publisher's Versio

Connections between normalized Wright functions with families of analytic functions with negative coefficients

In this article we present sufficient conditions that ensures that normalized Wright functions belong to certain subclasses of analytic univalent functions with negative coefficients in the unit disc U. We also provide some geometric properties of integral transforms involving normalized Wright functions