For α ≥ 0, λ > 0, we consider theM α, λ b of normalized analytic α−λ convex functions defined in the open unit disc U. In this paper, we investigate the classM α, λ b, that is, Re{ zf ′ b z /fb z 1− α α 1 − λ zf ′ b z /fb z αλ 1 zf ′′ b z /f ′ b z } > 0, with fb is Koebe type, that is, fb z : z/ 1−zn . The subordination result for the aforementioned class will be given. Further, bymaking use of...

In this research, we develop some differential subordination results involving harmonic means of $f_{b}(z),f_{b}(z)+zf_{b}^{\prime}(z)$ and $f_{b}(z)+\frac{zf_{b}^{\prime}(z)}{f_{b}(z)},$ where $f_{b}(z)=\frac{z}{\left(1-z^{n}\right) ^{b}},$ $b\geq0;n\in\mathbb{N}=1,2,3...$ is an $n$-fold symmetric Koebe type functions defined in the unit disk with $f_{b}(0)=0,f_{b}^{\prime}(z)\neq0.$ By using ...

Abstract In this article, it is shown that Koebe inner functions and squares of singular Nevanlinna functions, so called atomic density have minimal realizations in reproducing kernel Krein spaces.

A variant of the Circle Packing Theorem states that combinatorial class any convex polyhedron contains elements, called Koebe polyhedra, midscribed to unit sphere centered at origin, and these representatives are unique up Mobius transformations sphere. Motivated by this result, various papers investigate problem centering spherical configurations under transformations. In particular, Springbor...

Let (M, ∂M) be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric (but is not a solid torus). We consider the hyperbolic metrics on M such that ∂M looks locally like a hyperideal polyhedron, and we characterize the possible dihedral angles. We find as special cases the results of Bao and Bonahon [BB02] on hyperideal polyhedra, and those of Rousset [Rou02...

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...