International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 3, Pages 517-522
doi:10.1155/S016117129200067X

A class of univalent functions with varying arguments

K. S. Padmanabhan1,2 and M. Jayamala1,2

1The Ramanujan Institute, University of Madras, Madras 600 005, India
2Department of Mathematics, Queen Mary's College, Madras 600 005, India

Received 24 April 1990; Revised 7 August 1991

Copyright © 1992 K. S. Padmanabhan and M. Jayamala. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

f(z)=z+∑m=2∞amzm is said to be in V(θn) if the analytic and univalent function f in the unit disc E is nozmalised by f(0)=0, f′(0)=1 and arg an=θn for all n. If further there exists a real number β such that θn+(n−1)β≡π(mod2π) then f is said to be in V(θn,β). The union of V(θn,β) taken over all possible sequence {θn} and all possible real number β is denoted by V. Vn(A,B) consists of functions f∈V such thatDn+1f(z)Dnf(z)=1+Aw(z)1+Bw(z),−1≤A<B≤1, where n∈NU{0} and w(z) is analytic, w(0)=0 and |w(z)|<1, z∈E. In this paper we find the coefficient inequalities, and prove distortion theorems.