Starlike functions associated with the generalized Koebe function

In this paper we study some properties of functions f which are analytic and normalized (i.e. $$f(0)=0=f'(0)-1$$ f ( 0 ) = 0 = f ′ ( 0 ) - 1 ) such that satisfy the following subordination relation $$\begin{aligned} \left( \frac{zf'(z)}{f(z)}-1\right) \prec \frac{z}{(1-pz)(1-qz)}, \end{aligned}$$ z f ′ ( z ) f ( z ) - 1 ≺ z ( 1 - p z ) ( 1 - q z ) , where $$(p,q) \in [-1,1] \times [-1,1]$$ ( p , q ) ∈ [ - 1 , 1 ] × [ - 1 , 1 ] . These types of functions are starlike related to the generalized Koebe function. Some of the features are: radius of starlikeness of order $$\gamma \in [0,1)$$ γ ∈ [ 0 , 1 ) , image of $$f\left( \{z:|z|<r\}\right) $$ f { z : | z | < r } where $$r\in (0,1)$$ r ∈ ( 0 , 1 ) , radius of convexity, estimation of initial and logarithmic coefficients, and Fekete–Szegö problem.

COEFFICIENT BOUNDS FOR A CERTAIN SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS ASSOCIATED WITH THE KOEBE FUNCTION

We consider here the functions which are analytic and univalent in the open unit disc normalized by and . By , we denote a new subclass of close-to-convex function such that for which and . In this paper, we give the representation theorem and obtain the coefficient bounds for functions in

On Use of Meijer's G-functions In The Theory of Univalent Functions

Using some properties of Meijer's G-functions and univalent functions, in this paper some definitions, transformations and theorems in univalent function theory are discussed and then reformulated in the language of Meijer's G-functions. The starting point is to consider the Koebe function as a Meijer’s G-function.

Unpredictability of the coefficients of m-fold symmetric bi-starlike functions

In 1984, Libera and Zlotkiewicz proved that the inverse of the square-root transform of the Koebe function is the extremal function for the inverses of odd univalent functions. The purpose of this paper is to point out that this is not the case for the m-fold symmetric bi-starlike functions by demonstrating the unpredictability of the coefficients of such functions.

Proof of the Zalcman conjecture for initial coefficients

The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions on the unit disk satisfy the inequality for all n > 2, with the equality only for the Koebe function. This conjecture remained open for n > 3. We provide here the proof of this inequality for n = 4, 5, 6. It relies on the holomorphic homotopy of univalent functions and comparison of generated singular conformal metrics in the disk. The extremality of Koebe's function follows from hyperbolic properties.