Coefficient estimates for some classes ofp-valent functions

Abstract

LetAp, wherepis a positive integer, denote the class of functionsf(z)=zp+∑n=p+1anznwhich are analytic inU={z:|z|<1}.For0<λ≤1,|α|<π2,0≤β<p, letFλ(α,β,p)denote the class of functionsf(z)∈Apwhich satisfy the condition|H(f(z))−1H(f(z))+1|<λ   for   z∈U,whereH(f(z))=eiαzf′(z)f(z)−βcosα−ipsinα(p−β)cosα.Also letCλ(b,p), wherepis a positive integer,0<λ<1, andb≠0is any complex number, denote the class of functionsg(z)∈Apwhich 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.