In the present paper, we consider the coefficient estimates for functions in certain new subclasses of starlike and convex functions of complex order γ, which are introduced by means of a generalized differential operator and non-homogeneous Cauchy-Euler type differential equation. Several corollaries and consequences of the main results are also obtained.

In this paper, we investigate several inclusion relationships of certain subclasses of meromorphically p-valent functions which are defined here by means of a linear operator involving the generalized hypergeometric function . We introduce and investigate several new subclasses of p-valent starlike, p-valent convex, p-valent close-to-convex and p-valent quasiconvex meromorphic functions.

We construct polynomial norming meshes with optimal cardinality growth, on planar compact starlike domains that satisfy a uniform interior ball condition. Moreover, we provide an algorithm that computes such meshes on planar C convex domains by Blaschke’s rolling ball theorem. 2000 AMS subject classification: 41A10, 41A63, 65D05.

We consider local convexity properties of balls in the Apollonian and Seittenranta’s metrics. Balls in the Apollonian metric are considered in the twice punctured space and starlike domains. Balls in Seittenranta’s metric are considered in the twice punctured space and in the punctured ball.

A new subclass R(u), 0 < u < I, of the class St(I/2) the class of starlike functions of order I/2 is introduced and it is shown that R(u) is closed with respect to the Hadamard product of analytic functions. Some sufficient conditions for the normalized regular functions to be univalent in the unit disk E are given.

The criteria that embed a normalized analytic function in the class of functions that are starlike with respect to N-symmetric points are presented. The criteria are based on the quotient of analytical representations of starlikeness and convexity with respect to N -symmetric points. AMS Mathematics Subject Classification (2000): 30C45