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

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.