On Taylor and Logarithmic Coefficient Bounds for Functions Starlike with Respect to Symmetric Points Associated with the Lune

Abstract

     

This paper investigates Taylor and logarithmic coefficient bounds for a class of analytic functions that are starlike with respect to symmetric points and associated with the lune-shaped domain. The class is defined through a subordination condition involving the conformal function

????_????(????) = ???? + √1 + ????², which provides a geometric framework for studying the behavior of normalized analytic functions in the open unit disk. By employing the subordination principle and the coefficient expansion of an associated Schwarz function, we derive relations between the Taylor coefficients of the functions under consideration and the coefficients of the Schwarz function. These relations are then used to establish upper bounds for the initial Taylor coefficients. Furthermore, by expanding the logarithm of the normalized function, we obtain expressions for the initial logarithmic coefficients and investigate their corresponding bounds. The results contribute to the study of coefficient problems in geometric function theory and provide a framework for examining how the geometry of the lune-shaped domain influences Taylor and logarithmic coefficients. The estimates are presented with detailed derivations, while questions of sharpness are considered separately.