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
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
Received 6 January 1985; Revised 7 May 1985
Copyright © 1988 M. K. Aouf. 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
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.