On neighbourhoods of univalent starlike functions

Neighbourhoods of Univalent Functions

The main result shows that a small perturbation of a univalent function is again a univalent function, hence a univalent function has a neighbourhood consisting entirely of univalent functions. For the particular choice of a linear function in the hypothesis of the main theorem, we obtain a corollary which is equivalent to the classical Noshiro–Warschawski–Wolff univalence criterion. We also pr...

Convex and Starlike Univalent Functions

Generating Starlike and Convex Univalent Functions

Alexander [1] was the first to introduce certain subclasses of univalent functions examining the geometric properties of the image f(D) of D under f . The convex functions are those that map D onto a convex set. A function w = f(z) is said to be starlike if, together with any of its points w, the image f(D) contains the entire segment {tw : 0 ≤ t ≤ 1}. Thus we introduce the denotations S = {f ∈...

Horadam Polynomials Estimates for $lambda$-Pseudo-Starlike Bi-Univalent Functions

In the present investigation, we use the Horadam Polynomials to establish upper bounds for the second and third coefficients of functions belongs to a new subclass of analytic and $lambda$-pseudo-starlike bi-univalent functions defined in the open unit disk $U$. Also, we discuss Fekete-Szeg$ddot{o}$ problem for functions belongs to this subclass.

Differential Inequalities and Criteria for Starlike and Univalent Functions

The main aim of this paper is to use the method of differential subordination to obtain a number of sufficient conditions for a normalized analytic function to be univalent or starlike in the unit disc. In particular, we find a condition on β so that each normalized analytic function f satisfying the condition ∣∣∣1 + zf ′′(z) 2f ′(z) − zf ′(z) f(z) ∣∣∣ < β, z ∈ Δ implies that f is univalent or ...