The aim of a study of the presented paper is the differential subordination involving harmonic means of the expressions \(p(z)\) , \(p(z) + zp'(z)\) , and \(p(z) + \frac{zp'(z)}{p(z)}\) when \(p\) is an analytic function in the unit disk, such that \(p(0)=1, p(z)\not \equiv 1\) . Several applications in the geometric functions theory are given.

In the present paper, we consider a generalised subclass of analytic functions involving arithmetic, geometric and harmonic means. For this function class we obtain an inclusion result, Fekete-Szegö inequality and coefficient bounds for bi-univalent functions.

In this paper we introduce a new class of analytic functions involving the integral operator I n defined by Sălăgean. For this class we provide inclusion relation, extreme points and coefficient estimates.

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.