This article is concerned with the number B D (x) of integer points with relative prime coordinates in p x D, where x is a large real variable and D is a starlike set in the Euclidean plane. Assuming the truth of the Riemann Hypothesis, we establish an asymptotic formula for B D (x). Applications to certain special geometric and arithmetic problems are discussed .

We define a new class of multivalent meromorphic functions using the generalised hypergeometric function. We derived this class related to conic domain. It is also shown that this new class of functions, under certain conditions, becomes a class of starlike functions. Some results on inclusion and closure properties are also derived.

In this paper we introduce new subclasses of p-valent starlike, pvalent convex, p-valent close-to-convex and p-valent quasi-convex meromorphic functions and investigate some inclusion properties of these subclasses and investigate various inclusion properties and integral-preserving properties for the p-valent meromorphic function classes.

A number of interesting criteria were given by earlier workers for a normalized analytic function to be in the familiar class ∗ of starlike functions. The main object of the present paper is to extend and improve each of these earlier results. An application associated with an integral operator c(c >−1) is also considered.

We define and investigate a new class of Salagean-type k-starlike functions with negative coefficients. We obtain some properties of this subclass. Furthermore, we give Hadamard product of several functions and some distortion theorems for fractional calculus of this generalized class.

The purpose of the present paper is to establish some results concerning the partial sums of meromorphic starlike and meromorphic convex functions analogous to the results due to H. Silverman [J. Math. Anal. Appl. 209 (1997), 221-227]. Furthermore, we consider the partial sums of certain integral operator.