Assume that $mathbb{D}$ is the open unit disk. Applying Ozaki's conditions, we consider two classes of locally univalent, which denote by $mathcal{G}(alpha)$ and $mathcal{F}(mu)$ as follows begin{equation*}  mathcal{G}(alpha):=left{fin mathcal{A}:mathfrak{Re}left( 1+frac{zf^{prime prime }(z)}{f^{prime }(z)}right) <1+frac{alpha }{2},quad 0<alphaleq1right}, end{equation*} and begin{equation*}  ma...

In this paper, we investigate the properties of neighbourhoods of functions for the classes U CV (α) and Sp(α). First we established an inclusion relationship between them and proved a necessary and sufficient condition interms of convolutions for a function f to be in Sp(α). Next we show that the class Sp(α) is closed under convolution with functions f (z) which are convex univalent. The resul...

In this paper, we introduce a new class of analytic functions defined by a new convolution operator La (α, β). The new class of analytic functions Σ a,t α,β (ρ; h) in U ∗ = {z : 0 < |z| < 1} is defined by means of a hypergeometric function with an integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination. The authors also introduces and investigates variou...

One of the most important problems in study geometric function theory is knowing how to obtain sharp bounds coefficients that appear Taylor–Maclaurin series univalent functions. In present investigation, our aim calculate some estimates involving for family convex functions with respect symmetric points and associated a hyperbolic tangent function. These include first four initial coefficients,...

In this paper we consider the classes of starlike functions, starlike functions of order α, convex functions, convex functions of order α and the classes of the univalent functions denoted by SH (β), SP and SP (α, β). On these classes we study the convexity and αorder convexity for a general integral operator.