This paper aims to introduce a measure of the non-univalency harmonic mapping. By using it, we find best approximation an analytic function by univalent

In the present paper, we have established sufficient conditions for Gaus-sian hypergeometric functions to be in certain subclass of analytic univalent functions in the unit disc $mathcal{U}$. Furthermore, we investigate several mapping properties of Hohlov linear operator for this subclass and also examined an integral operator acting on hypergeometric functions.

The class SH consists of univalent, harmonic, and sense-preserving functions f in the unit disk, , such that f = h+g where h(z) = z+ P 1 2 akz , g(z) = P 1 1 bkz . S H will denote the subclass with b1 = 0. We present a collection of n-slit mappings (n 2) and prove that the 2-slit mappings are in SH while for n 3 the mappings are in S H . Finally we show that these mappings establish the sharpne...

The class S of functions g(z) = z + c 2 z 2 + c 3 z 3 + ... analytic and univalent in the unit disk Izr < 1 has been thoroughly studied, and its properties are well known. Our purpose is to investigate another class of functions which, by contrast, seems to have been rather neglected. This is the class S o of functions f ( z ) = 1 + a 1 z + a 2 z Z + . . , analytic, univalent, and nonvanishing ...

In the present work, the author determines some coefficient bounds for functions in a new class of analytic and bi-univalent functions, which are introduced by using of polylogarithmic functions. The presented results in this paper one the generalization of the recent works of Srivastava et al. [26], Frasin and Aouf [13] and Siregar and Darus [25].

In this paper, we introduce and investigate two new subclasses of the functions class $ Sigma $ of bi-univalent functions defined in the open unit disk, which are associated with the Aghalary-Ebadian-Wang operator. We  estimate the coefficients $|a_{2} |$ and  $|a_{3} |$ for functions in these new subclasses. Several  consequences of the result are also pointed out.