which are analytic in the open unit disc U {z ∈ C : |z| < 1} on the complex plane C. Let S, S∗ α , C α 0 ≤ α < 1 denote the subclasses of A consisting of functions that are univalent, starlike of order α, and convex of order α in U, respectively. In particular, the classes S∗ 0 S∗ and C 0 C are the familiar classes of starlike and convex functions inU, respectively. A function f ∈ C α if Re 1 z...

Ruscheweyh and Sheil-Small proved that convexity is preserved under the convolution of univalent analytic mappings in K. However, when we consider the convolution of univalent harmonic convex mappings in K H , this property does not hold. In fact, such convolutions may not be univalent. We establish some results concerning the convolution of univalent harmonic convex mappings provided that it i...

The object of this article is to explore a ?-pseudo-?-convex ?-fold symmetric bi-univalent function family satisfying subordinations condition generalizing certain previously examined families. We originate the initial Taylor–Maclaurin coefficient estimates functions in defined family. classical Fekete–Szegö inequalities for also estimated. Furthermore, we present some special cases main result...

it is known that the condition $mathfrak {re} left{zf&apos;(z)/f(z)right}>0$, $|z|

let $p$ be an analytic function defined on the open unit disc $mathbb{d}$ with $p(0)=1.$ the conditions on $alpha$ and $beta$ are derived for $p(z)$ to be subordinate to $1+4z/3+2z^{2}/3=:varphi_{c}(z)$ when $(1-alpha)p(z)+alpha p^{2}(z)+beta zp&apos;(z)/p(z)$ is subordinate to $e^{z}$. similar problems were investigated for $p(z)$ to lie in a region bounded by lemniscate of bernoulli $|w^{2}-1...

The main purpose of this paper is to introduce a generalization of modified Salagean operator for harmonic univalent functions. We define a new subclass of complex-valued harmonic univalent functions by using this operator ,and we investigate necessary and sufficient coefficient conditions, distortion bounds, extreme points and convex combination for the above class of harmonic univalent functi...

a function is said to be bi-univalent on the open unit disk d if both the function and its inverse are univalent in d. not much is known about the behavior of the classes of bi-univalent functions let alone about their coefficients. in this paper we use the faber polynomial expansions to find coefficient estimates for four well-known classes of bi-univalent functions which are defined by subord...

Different methods have been used in studying the univalence of the integral (1) Jα,β(f)(z) = ∫ z 0 ( f ′(t) )α(f(t) t )β dt, α, β ∈ R, where f belongs to one of the known families of holomorphic and univalent functions f(z) = z + a2z + · · · in the unit disk D = {z : |z| < 1} (see [5]). In this paper, we study a larger set than (1), namely the set of the minimal invariant family which contains ...