in this paper, we introduce and investigate a subclass of analytic and bi-univalent functions in the open unit disk. upper bounds for the second and third coefficients of functions in this subclass are founded. our results, which are presented in this paper, generalize and improve those in related works of several earlier authors.

The Koebe One Quarter Theorem states that the range of any Schlicht function contains centered disc radius 1/4 which is sharp due to value at −1. A natural question finding polynomials set sharpness for polynomials. In particular, it was asked in [7] whether Suffridge [15] are optimal. For degree 1 and 2 obviously true. It demonstrated [10] 3 not optimal a promising alternative family introduce...

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. ...

Abstract We study problems similar to the Koebe Quarter Theorem for close-to-convex polynomials with all zeros of derivative in $${\mathbb {T}}:=\{z\in {\mathbb {C}}:|z|=1\}$$ T : = { z ? C | ...

This paper utilizes the method of extremal length to study several diameter problems for functions conformal outside of a disc centered at the origin, with a standard normalization, which possess a quasiconformal extension to a ring subdomain of this disc. Known results on the diameter of a complementary component of the image domain of a univalent function are extended. Applications to the tra...

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...