in the present paper, we prove subordination, superordination and sandwich-type properties of a certain integral operators for univalent functions on open unit disc, moreover the special behavior of this class is investigated.

For $0\leq\alpha\leq 1,$ let $H_{\alpha}(x,y)$ be the convex weighted harmonic mean of $x$ and $y.$ We establish differential subordination implications form $$H_{\alpha}(p(z),p(z)\Theta(z)+zp'(z)\Phi(z))\prec h(z)\Rightarrow p(z)\prec h(z),$$ where $\Phi$, $\Theta$ are analytic functions $h$ is a univalent function satisfying some special properties. Further, we prove involving combination thr...

In this work, we obtain the Fekete-Szegö inequalities for the class $P_{Sigma }left( lambda ,phi right) $ of bi-univalent functions. The results presented in this paper improve the recent work of Prema and Keerthi [11].

The valence of a function f at a point w is the number of distinct, finite solutions to f(z) = w. Let f be a complex-valued harmonic function in an open set R ⊆ C. Let S denote the critical set of f and C(f) the global cluster set of f . We show that f(S)∪C(f) partitions the complex plane into regions of constant valence. We give some conditions such that f(S) ∪ C(f) has empty interior. We also...