Koebe Function Research Articles

Sandwich Subordinations Imposed by New Generalized Koebe-Type Operator on Holomorphic Function Class

In the complex field, special functions are closely related to geometric holomorphic functions. Koebe function is a notable contribution to the study of the geometric function theory (GFT), which is a univalent function. This sequel introduces a new class that includes a more general Koebe function which is holomorphic in a complex domain. The purpose of this work is to present a new operator correlated with GFT. A new generalized Koebe operator is proposed in terms of the convolution principle. This Koebe operator refers to the generality of a prominent differential operator, namely the Ruscheweyh operator. Theoretical investigations in this effort lead to a number of implementations in the subordination function theory. The tight upper and lower bounds are discussed in the sense of subordinate structure. Consequently, the subordinate sandwich is acquired. Moreover, certain relevant specific cases are examined.

Some Transformation Properties with Omitted Value

The purpose of this paper is to show that for a holomorphic and univalent function in class S, an omitted –value transformation yields a class of starlike functions as a rotation transformation of the Koebe function, allowing both the image and rotation of the function to be connected. Furthermore, these functions have several properties that are not far from a convexity properties. We also show that Pre- Schwarzian derivative is not invariant since the convexity property of the function is so weak.

Coefficient Estimate on Second Hankel Determinant of the Logarithmic Coefficients for Close-To-Convex Function Subclass with Respect to the Koebe Function

Let S denote the subclass of the analytic function and univalent functions in D, where D is defined as the unit disk and having the Taylor representation form of S. In this paper, we will estimate the second Hankel determinant which the elements are the logarithmic coefficients of the class close-to-convex function with respect to the Koebe function in S

An Analytical Investigation of Uniformly Star Like Class of Functions via Generalized Koebe Function

Recently, several of the generalizations Koebe function are introduced and investigated. In this study, a linear complex operator is investigated in terms of the generalized Koebe function and Wright function. A new geometric class, namely a uniformly starlike class, is provided. Furthermore, certain properties are discussed such as coefficient bounds, growth, distortion results, radii of convexity, starlikeness and close-to-convexity.

An analytical investigation of uniformly star like class of functions via gener-alized koebe function

Recently, several of the generalizations Koebe function are introduced and investigated. In this study, a linear complex operator is investigated in terms of the generalized Koebe function and Wright function. A new geometric class, namely a uniformly starlike class, is provided. Furthermore, certain properties are discussed such as coefficient bounds, growth, distortion results, radii of convexity, starlikeness and close-to-convexity.

A Simple Proof of Bieberbach Conjecture

The famous Riemann mapping theorem, which was established in 1851, states that every proper subdomain of the complex plane that is simply connected (without “holes”) is the image of the unit disk under a univalent analytic mapping f(z). This is the most significant characteristic of univalent analytic functions. The domain D, the image point f(0) in D, and the restriction that f’(0) be a positive real integer each independently define the mapping function f(z). One naturally asks how geometric aspects of the image domain f(D) represent analytic properties of f when considering univalent analytic functions as “Riemann mappings”. This is the perspective of geometric function theory, whose primary issue was solved by de Branges’ proof. The Bieberbach conjecture (hereinafter referred to as “BC”) is essentially a claim on the extreme nature of the Koebe function. Therefore, a key subject in this article is to make an explanation of the Koebe function and intepret why it is the best contender to be “largest” in the sense of the conjecture. Also, a detailed but easy understanding noval proof is given for this conjecture.

On the Koebe Quarter Theorem for Polynomials

The Koebe One Quarter Theorem states that the range of any Schlicht function contains the centered disc of radius 1/4 which is sharp due to the value of the Koebe function at −1. A natural question is finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials. In particular, it was asked in [7] whether Suffridge polynomials [15] are optimal. For polynomials of degree 1 and 2 that is obviously true. It was demonstrated in [10] that Suffridge polynomials of degree 3 are not optimal and a promising alternative family of polynomials was introduced. These very polynomials were actually discovered earlier independently by M. Brandt [3] and D. Dimitrov [9]. In the current article we reintroduce these polynomials in a natural way and make a far-reaching conjecture that we verify for polynomials up to degree 6 and with computer aided proof up to degree 52. We then discuss the ensuing estimates for the value of the Koebe radius for polynomials of a specific degree.

Maximal coefficient functionals on univalent functions

Recently the author presented a new approach to solving the coefficient problems for holomorphic functions based on the features of Bers’ fiber spaces for punctured Riemann surfaces. The holomorphy of functionals causes strong rigid constrains.This paper extends the previous results to a broad class of plurisubharmonic coefficient functionals and provides new extremal features of the Koebe function.

Maximal coefficient functionals on univalent functions

Recently the author presented a new approach to solving the coefficient problems for holomorphic functions based on the features of Bers' fiber spaces for punctured Riemann surfaces. The holomorphy of functionals causes strong rigid constrains. This paper extends the previous results to a broad class of plurisubharmonic coefficient functionals and provides new extremal features of the Koebe function.

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) such that satisfy the following subordination relation zf′(z)f(z)-1≺z(1-pz)(1-qz),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( \\frac{zf'(z)}{f(z)}-1\\right) \\prec \\frac{z}{(1-pz)(1-qz)}, \\end{aligned}$$\\end{document}where (p,q) in [-1,1] times [-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), image of fleft( {z:|z|<r}right) where rin (0,1), radius of convexity, estimation of initial and logarithmic coefficients, and Fekete–Szegö problem.

Teichmüller spaces and coefficient problems for univalent holomorphic functions

We show that the features of Teichmuller spaces give rise to a completely new powerful tool for solving the classical coefficient problems in geometric function theory. Here we concern the univalent functions. Using the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces and some of their other complex geometric features, we prove a general theorem on maximization of homogeneous polynomial (in fact, more general holomorphic) coefficient functionals $$J(f) = J(a_{m_1}, a_{m_2},\dots , a_{m_n}) $$ on some classes of univalent functions in the unit disk naturally connected with the canonical class S. The given functional J is lifted to the Teichmuller space $$\mathbf{T}_1$$ of the punctured disk $$\mathbb {D}_{*} = \{0< |z| < 1\}$$ which is biholomorphically equivalent to the Bers fiber space over the universal Teichmuller space. This generates a positive subharmonic function on the disk $$\{|t| < 4\}$$ with $$\sup _{|t|<4} u(t) = \max _{{\mathbf {T}}_1} |J|$$ attaining this maximal value only on the boundary circle, which correspond to rotations of the Koebe function. This theorem implies new sharp distortion estimates for univalent functions giving explicitly the extremal functions, and creates a new bridge between Teichmuller space theory and geometric complex analysis. In particular, it provides an alternate and direct proof of the Bieberbach conjecture.

Geometric properties of partial sums of generalized Koebe function

The aim of the present paper is to investigate the starlikeness, convexity, and close-to-convexity of some partial sums of the generalized Koebe function. Furthermore, we give some special results related with special cases of c constant. The results, which are presented in this paper, would generalize those in related works of several earlier authors.

Qualitative Results in the Bombieri Problem for Conformal Mappings

Bombieri’s numbers σmn characterize a behavior of the coefficient body for the class S of all holomorphic and univalent functions f in the unit disk normalized by f(z) = z + a2z2 + …. The number σmn is the limit of ratio for Re(n − an) and Re (m − am) as f tends to the Koebe function K(z) = z(1 − z)−2. It is showed in the paper that Bombieri’s conjecture about explicit values of σmn implies a sliding regime in an associated control theory problem generated by the Loewner differential equation. We develop also an asymptotical approach in verification of necessary criteria for Bombieri’s conjecture.

The Bombieri Problem for Bounded Univalent Functions

Bombieri proposed to describe the structure of the sets of values of the initial coefficients of normalized conformal mappings of the disk in a neighborhood of the corner point corresponding to the Koebe function. The Bombieri numbers characterize the limit position of the support hyperplane passing through a critical corner point. In this paper, the Bombieri problem is studied for the class of bounded normalized conformal mappings of the disk, where the role of the Koebe function is played by the Pick function. The Bombieri numbers for a pair of two nontrivial initial coefficients are calculated.

Necessary criteria for the Bombieri conjecture

We consider functionals L which are the real parts of linear combinations of two Taylor coefficients on the class S of holomorphic univalent functions in the unit disk. The Bombieri conjecture can be interpreted in the form that L are locally maximized by the Koebe function simultaneously on S and on its subclass $$S_R$$ consisting of typically real functions. We derive necessary criteria for the Bombieri conjecture in terms of inequalities for solutions to systems of differential equations in variations for the Loewner ODE.

On logarithmic coefficients of some close-to-convex functions

The logarithmic coefficients γ n \gamma _n of an analytic and univalent function f f in the unit disk D = { z ∈ C : | z | &gt; 1 } \mathbb {D}=\{z\in \mathbb {C}:|z|&gt;1\} with the normalization f ( 0 ) = 0 = f ′ ( 0 ) − 1 f(0)=0=f’(0)-1 are defined by log ⁡ f ( z ) z = 2 ∑ n = 1 ∞ γ n z n \log \frac {f(z)}{z}= 2\sum _{n=1}^{\infty } \gamma _n z^n . Recently, D. K. Thomas [Proc. Amer. Math. Soc. 144 (2016), 1681–1687] proved that | γ 3 | ≤ 7 12 |\gamma _3|\le \frac {7}{12} for functions in a subclass of close-to-convex functions (with argument 0 0 ) and claimed that the estimate is sharp by providing a form of an extremal function. In the present paper, we point out that such extremal functions do not exist and the estimate is not sharp by providing a much more improved bound for the whole class of close-to-convex functions (with argument 0 0 ). We also determine a sharp upper bound of | γ 3 | |\gamma _3| for close-to-convex functions (with argument 0 0 ) with respect to the Koebe function.

A Note on the Kirwan’s Conjecture

We continue our investigation on a second variation formula of the Koebe function in the class $$\Sigma $$ of functions analytic and univalent in the exterior of the unit disk. Our aim is to give some supporting evidence of a conjecture raised by William Kirwan on the coefficients of functions in this class.

Analogy of Bombieri’s number for bounded univalent functions

Bombieri’s numbers σ mn characterize the behavior of the coefficient body for the class S of all holomorphic and univalent functions f in the unit disk normalized by f(z) = z + a 2 z 2 +.... The number σ mn is the limit of ratio for Re(n−a n) and Re(m−am) as f tends to the Koebe function K(z) = z(1 − z)−2. In particular, σ 23=0. We define analogous numbers σ mn(M) for the class S(M) ⊂ S of bounded functions |f(z)| 1, with the limit of ratio for Re(p n(M) − a n) and Re(p m(M) − a m) as f tends to the Pick function P M(z) = MK −1(K(z)/M) = z + Σ n=2 ∞ p n(M)z n. We prove that σ 23(M) = −4/M, M > 1.

On the Bombieri numbers for the class S

We derive a second variation formula of the Koebe function developed by Bombieri by using a linear version of the Loewner differential equation. We then express the formula in terms of classical orthogonal polynomials to consider the Bombieri numbers obtained by Greiner and Roth (Proc Am Math Soc 129(12):3657–3664, 2001) and by Prokhorov and Vasil’ev (Georgian Math J 12(4):743–761, 2005). We show a question raised by Bombieri on these numbers has a negative answer in the low order cases.

Advances on the coefficients of bi-prestarlike functions

Since 1923, when Löwner proved that the inverse of the Koebe function provides the best upper bound for the coefficients of the inverses of univalent functions, finding sharp bounds for the coefficients of the inverses of subclasses of univalent functions turned out to be a challenge. Coefficient estimates for the inverses of such functions proved to be even more involved under the bi-univalency requirement. In this paper, we use the Faber polynomial expansions to find upper bounds for the coefficients of bi-prestarlike functions and consequently advance some of the previously known estimates.