Differential Subordinations and Harmonic Means

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.