Logarithmic coefficients for a class of analytic functions defined by subordination

Abstract

In this paper we consider a class of functions Mα(φ) defined by subordination, consisting of functions f ∈ A satisfying the condition (1 −α) zf′(z)/ f(z) + α (1 + zf′′(z)/ f′(z) )≺ φ(z), z ∈ U. In the study of univalent functions, estimates on the Taylor coefficients are usually given. Another significant problem deals with the estimates of logarithmic coefficients. For the class S of univalent functions no sharp bounds for the modulus of the individual logarithmic coefficients are known if n ≥ 3. For different subclasses of S the results are not better and in most cases only th e first three initial coefficients of log f(z)/z are considered. For the class Mα(φ) we obtain upper bounds for the logarithmic coefficients γ_n, n ∈ {1, 2, 3} and also for Γ_n, n ∈ {1, 2, 3}, the logarithmic coefficients of the inverse of Mα(φ). Connections with previous known results are pointed out.