International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 1, Pages 47-54
doi:10.1155/S0161171288000092

Coefficient estimates for some classes of p-valent functions

Abstract

Let Ap, where p is a positive integer, denote the class of functions f(z)=zp+∑n=p+1anzn which are analytic in U={z:|z|<1}.For 0<λ≤1, |α|<π2, 0≤β<p, let Fλ(α,β,p) denote the class of functions f(z)∈Ap which satisfy the condition|H(f(z))−1H(f(z))+1|<λ   for   z∈U,where H(f(z))=eiαzf′(z)f(z)−βcosα−ipsinα(p−β)cosα.Also let Cλ(b,p), where p is a positive integer, 0<λ<1, and b≠0 is any complex number, denote the class of functions g(z)∈Ap which satisfy the condition|H(g(z))−1H(g(z))+1|<λ   for   z∈U,   whereH(g(z))=1+1pb(1+zg″(z)g′(z)−p).In this paper we obtain sharp coefficient estimates for the above mentioned classes.