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

By I. Efraimidis. 2015

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Abstract For functions p ( z ) = 1 + ∑ n = 1 ∞ p n z n holomorphic in the unit disk, satisfying Re p ( z ) > 0 , we generalize two inequalities proved by Livingston [10] , [11] and simplify their proofs. One of our results states that | p n − w p k p n − k | ≤ 2 max ⁡ { 1 , | 1 − 2 w | } , w ∈ C . Another result involves certain determinants whose entries are the coefficients p n . 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/db3a1f5009518c1c61cd59d01b29079b74d70779