Some Results for Analytic Functions Related to Bernoulli Numbers

Abstract

This paper explores the interesting relationship between Bernoulli numbers and the Schwarz lemma, two seemingly disparate concepts from number theory and complex analysis. Bernoulli numbers, a sequence of rational numbers, play a crucial role in various mathematical contexts, including power series expansions, the Riemann zeta function, and the Euler–Maclaurin formula. The Schwarz lemma, a fundamental result in complex analysis, provides constraints on analytic functions mapping the unit disc to itself. We investigate how these two areas intersect in the study of analytic functions and power series, highlighting how the properties of Bernoulli numbers influence the coefficients and behaviour of functions subject to the Schwarz lemma. By examining specific examples and deriving relevant identities, we uncover the subtle yet significant connections between these mathematical constructs, offering new insights into their combined application in modern mathematical research.