Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions

Abstract

It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If S denotes the class of functions fz=z+∑n=2∞anzn analytic and univalent in the open unit disk U, then the logarithmic coefficients γnf of the function f∈S are defined by logfz/z=2∑n=1∞γnfzn. In the current paper, the bounds for the logarithmic coefficients γn for some well-known classes like C1+αz for α∈0,1 and CVhpl1/2 were estimated. Further, conjectures for the logarithmic coefficients γn for functions f belonging to these classes are stated. For example, it is forecasted that if the function f∈C1+αz, then the logarithmic coefficients of f satisfy the inequalities γn≤α/2nn+1,n∈ℕ. Equality is attained for the function Lα,n, that is, logLα,nz/z=2∑n=1∞γnLα,nzn=α/nn+1zn+⋯,z∈U.