# Logarithmic coefficients for a class of analytic functions defined by subordination

## DOI:

https://doi.org/10.31926/but.mif.2023.3.65.2.11## Keywords:

Analytic functions, differential subordination, logarithmic coefficients## 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.