Some Coefficient of Inequility for Subclass Of holomorphic Functions

Abstract Since the early twentieth century, the investigation of bound fordeterminant of Hankel matrices in analytic univalent functions has been a focalpoint for mathematical research, providing a rich avenue for the exploration ofgeometric properties. Over the years, numerous researchers have striven to establishupper bounds for determinant of the third order Hankel matrix withinvarious subclasses of analytic univalent functions. While these efforts haveyielded valuable insights, the attainment of sharp bounds remained a challengeuntil Know et al. (2018) achieved an exact estimation of the fourth coefficientin the Carath´eodory class. In more recent developments, investigators haveleveraged this precise estimation of the fourth coefficient, alongside the welldefinedvalues of the second and third coefficients within the Carath´eodoryclass. This breakthrough has empowered them to delineate sharp bounds forthe third Hankel determinant in diverse subclasses of analytic univalent functions.This present study embarks on an exploration of upper bounds for thesecond-order Hankel determinant within the realm of analytic functions, subjectto the normalized conditions of f(0) = 0 and f′(0) = 1. We further extendour investigation to specific sub-classes of holomorphic functions, culminatingin the derivation of sharp bounds for a parameter of particular interest. 2020 Mathematics Subject Classification. 30C45, 30C50.