A generalization of Livingston's coefficient inequalities for functions with positive real part

Abstract

For functions p(z)=1+∑n=1∞pnzn holomorphic in the unit disk, satisfying Rep(z)>0, we generalize two inequalities proved by Livingston [10,11] and simplify their proofs. One of our results states that |pn−wpkpn−k|≤2max⁡{1,|1−2w|}, w∈C. Another result involves certain determinants whose entries are the coefficients pn. Both results are sharp. As applications we provide a simple proof of a theorem of Brown [2] and various inequalities for the coefficients of holomorphic self-maps of the unit disk.