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

Iason Efraimidis

doi:10.1016/j.jmaa.2015.10.050

paper

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

arXiv:1506.07111 · doi:10.1016/j.jmaa.2015.10.050

Abstract

For functions $p (z) = 1 + \sum_{n = 1}^{\infty} p_{n} z^{n}$ holomorphic in the unit disk, satisfying $Re p (z) > 0$ , we generalize two inequalities proved by Livingston in 1969 and 1985, and simplify their proofs. One of our results states that $∣ p_{n} - w p_{k} p_{n - k} ∣ \leq 2 max {1, ∣1 - 2 w ∣}, w \in C$ . 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 J.E. Brown and various inequalities for the coefficients of holomorphic self-maps of the unit disk.

12 pages, LaTeX; An alternative proof of Theorem 2 and an alternative proof of a theorem of Brown are added in version 2. Available online in Journal of Mathematical Analysis and Applications

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Applications of Livingston-type inequalities to the generalized Zalcman functional