Abstract 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 ) = ′ - 1 ) such ...

In this paper we present a new method of determining Koebe domains. We apply this method by giving a new proof of the well-known theorem of A. W. Goodman concerning the Koebe domain for the class T of typically real functions. We applied also the method to determine Koebe sets for classes of the special type, i.e. for T = {f ∈ T : f(∆) ⊂ Mg(∆)}, g ∈ T ∩ S, M > 1, where ∆ = {z ∈ C : |z| < 1} and...

In 1986 A. Ancona showed, using the Koebe one-quarter Theorem, that for a simply-connected planar domain the constant in the Hardy inequality with the distance to the boundary is greater than or equal to 1/16. In this paper we consider classes of domains for which there is a stronger version of the Koebe Theorem. This implies better estimates for the constant appearing in the Hardy inequality. ...

We prove a technical lemma, the C-Denjoy-Koebe distortion lemma, estimating the distortion of a long composition of a C onedimensional mapping f : M 7→ M with finitely many, non-recurrent, power law critical points. The proof of this lemma combines the ideas of the distortion lemmas of Denjoy and Koebe.

We investigate the new class H b (α, β) of analytic functions with Koebe type. The subordination, superordination, best dominant result for that class was obtain. And by making use of Jack's Lemma as well as several differential and other inequalities, the authors derive sufficient conditions for starlikeness of the class H b (α, β) of n-fold symmetric analytic functions of Koebe type. Relevant...

Since the function z 7→ 1+z 1−z is univalent with image the right half plane, we see that z 7→ ( 1+z 1−z )2 is univalent, so k ∈ S, and the image of k is the entire complex plane except for real numbers ≤ −14 . In 1916, L. Bieberbach [Bi] conjectured that the Koebe function was maximal with respect to the absolute value of the coefficients of its power series. More precisely, he conjectured the...