On a class of analytic functions with missing coefficients

摘要

Let Tn(A,B,α) denote the class of functions of the form:f(z)=z+∑k=n+1∞akzk(n∈N={1,2,3,…}),which are analytic in the open unit disk U and satisfy the following subordination condition:f′(z)+αzf″(z)≺1+Az1+Bz(z∈U;-1⩽B<1;B0).In this paper, we find the sharp lower bound on |z|=r<1 for the functional Re{f′(z)+αzf″(z)} over the class:Tn(A,B,0)=f(z)=z+∑k=n+1∞akzk:f′(z)≺1+Az1+Bz(n∈N={1,2,3,…};z∈U).By applying this result, several interesting consequences are given.