Note on the second derivative of bounded analytic functions

2024/03/15 by Gangqiang Chen, Chen, Gangqiang
Mathematics · #30C80 #30F45 #Advanced Banach Space Theory #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Functional Equations Stability Results

paper · pdf · doi:10.48550/arxiv.2403.10213

openalex publication_date 2024/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Assume z₀ lies in the open unit disk \mathbbD and g is an analytic self-map of \mathbbD. We will determine the region of values of g''(z₀) in terms of z₀, g(z₀) and the hyperbolic derivative of g at z₀, and give the form of all the extremal functions. In particular, we obtain a smaller sharp upper bound for |g''(z₀)| than Ruscheweyh's inequality for the case of the second derivative. Moreover, we use a different method to obtain Szász's inequality, which provides a sharp upper bound for |g''(z₀)| depending only on |z₀|.

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