Some results for Analytic functions associated with Gregory coefficients
DOI:
https://doi.org/10.5644/SJM.22.01.05Keywords:
Gregory coefficients, Schwarz lemma, Angular derivative, Principle of SubordinationAbstract
This paper investigates the intricate relationship between Gregory
coefficients and the Schwarz Lemma, two concepts arising from different
branches of mathematics---numerical analysis and complex analysis,
respectively. By examining the analytic function defined by $f^{\prime }(z)=%
\frac{\varphi (z)}{\ln \left( 1+\varphi (z)\right) }$ , we derive upper
bounds for Taylor coefficients and analyze the modulus of the second
derivative, utilizing the Schwarz Lemma. Through this exploration, we
highlight the sharp inequalities and demonstrate their significance in the
geometric theory of functions. The results obtained extend the classical
Schwarz Lemma to boundary behaviors and offer valuable contributions to both
numerical and complex analysis.
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