scholarly journals Note on Schiffer’s Variation in the Class of Univalent Functions in the Unit Disc

1968 ◽  
Vol 32 ◽  
pp. 273-276
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Univalent Functions ◽  
The Real ◽  
Maximum Value ◽  
The Extremal Function

Let S denote the class of univalent functions f(z) in the unit disc D: | z | < 1 with the following expansion: (1) f(z) = z + a2z2 + a3z3 + · · · · anzn + · ··.We denote by fn(z) the extremal function in S which gives the maximum value of the real part of an and by Dn the image of D under w = fn(z).

scholarly journals A majorant problem

1992 ◽  
Vol 15 (3) ◽  
pp. 441-447
Author(s):  
Ronen Peretz
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Fourier Coefficients ◽  
Positive Real Part ◽  
Complex Vector ◽  
Positive Real ◽  
Zero Sets ◽  
The Extremal Function

Letf(z)=∑k=0∞akzk,a0≠0be analytic in the unit disc. Any infinite complex vectorθ=(θ0,θ1,θ2,…)such that|θk|=1,k=0,1,2,…, induces a functionfθ(z)=∑k=0∞akθkzkwhich is still analytic in the unit disc.In this paper we study the problem of maximizing thep-means:∫02π|fθ(reiϕ)|pdϕover all possible vectorsθand for values ofrclose to0and for allp<2.It is proved that a maximizing function isf1(z)=−|a0|+∑k=1∞|ak|zkand thatrcould be taken to be any positive number which is smaller than the radius of the largest disc centered at the origin which can be inscribed in the zero sets off1. This problem is originated by a well known majorant problem for Fourier coefficients that was studied by Hardy and Littlewood.One consequence of our paper is that forp<2the extremal function for the Hardy-Littlewood problem should be−|a0|+∑k=1∞|ak|zk.We also give some applications to derive some sharp inequalities for the classes of Schlicht functions and of functions of positive real part.

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

2014 ◽  
Vol 25 (07) ◽  
pp. 1450064 ◽  
Author(s):  
Samaneh G. Hamidi ◽  
Jay M. Jahangiri
Keyword(s):  
Extremal Function ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Square Root ◽  
Koebe Function ◽  
The Extremal Function

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.

scholarly journals Univalent functions maximizingRe[f(ζ1)+f(ζ2)]

1996 ◽  
Vol 19 (4) ◽  
pp. 789-795
Author(s):  
Intisar Qumsiyeh Hibschweiler
Keyword(s):  
Extremal Function ◽  
Existence Of Solutions ◽  
Univalent Functions ◽  
Real Numbers ◽  
Koebe Function ◽  
Straight Line ◽  
The Extremal Function ◽  
Extremal Value

We study the problemmaxh∈Sℜ[h(z1)+h(z2)]withz1,z2inΔ. We show that no rotation of the Koebe function is a solution for this problem except possibly its real rotation, and only whenz1=z¯2orz1,z2are both real, and are in a neighborhood of thex-axis. We prove that if the omitted set of the extremal functionfis part of a straight line that passes throughf(z1)orf(z2)thenfis the Koebe function or its real rotation. We also show the existence of solutions that are not unique and are different from the Koebe function or its real rotation. The situation where the extremal value is equal to zero can occur and it is proved, in this case, that the Koebe function is a solution if and only ifz1andz2are both real numbers andz1z2<0.

Starlike Univalent Functions Bounded on the Real Axis

1989 ◽  
Vol 41 (4) ◽  
pp. 642-658
Author(s):  
Richard Fournier
Keyword(s):  
Real Axis ◽  
Analytic Functions ◽  
Unit Disc ◽  
Univalent Functions ◽  
Uniform Norm ◽  
Open Unit ◽  
Open Unit Disc ◽  
The Real ◽  
Typically Real ◽  
Image Position

We denote by E the open unit disc in C and by H(E) the class of all analytic functions f on E with f(0) = 0. Let (see [3] for more complete definitions)S = {ƒ ∈ H(E)|ƒ is univalent on E}S0 = {ƒ ∈ H(E)|ƒ is starlike univalent on E}TR = {ƒ ∈ H(E)|ƒ is typically real on E}.The uniform norm on (— 1, 1) of a function ƒ ∈ H(E) is defined by

scholarly journals ON EXTREME POINTS OF A CERTAIN LINEAR SPACE OF LOCALLY UNIVALENT FUNCTIONS

1992 ◽  
Vol 23 (4) ◽  
pp. 321-325
Author(s):  
KUALIDA INAYAT NOOR
Keyword(s):  
Linear Space ◽  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Space Structure ◽  
Real Linear Space ◽  
The Real ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Real Linear

Let $H = (H, \oplus, \odot)$ denote the real linear space of locally univalent normalized functions in the unit disc as defined by Hornich. For $-1\le B <A\le 1$, $k>2$, the classes $V_k[A,B]$ of functions with bounded boundary rotation are introduced and this linear space structure is used to determine the extreme points of the classes $V_k[A,B]$.

Hankel determinants of second and third order for the class 𝓢 of univalent functions

2021 ◽  
Vol 71 (3) ◽  
pp. 649-654
Author(s):  
Milutin Obradović ◽  
Nikola Tuneski
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Third Order ◽  
Hankel Determinants

Abstract In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class 𝓢 of univalent functions in the unit disc.

scholarly journals Harmonic univalent functions defined by post quantum calculus operators

2019 ◽  
Vol 11 (1) ◽  
pp. 5-17 ◽  
Author(s):  
Om P. Ahuja ◽  
Asena Çetinkaya ◽  
V. Ravichandran
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Quantum Calculus ◽  
Special Cases ◽  
The Family ◽  
Covering Theorems

Abstract We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion and covering theorems were also obtained. The extreme points of the family and a radius result were also obtained. The results obtained include several known results as special cases.

Formulas for the Nehari Coefficients of Bounded Univalent Functions

1977 ◽  
Vol 29 (3) ◽  
pp. 587-605
Author(s):  
Duane W. De Temple ◽  
David B. Oulton
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Bieberbach Conjecture ◽  
Grunsky Inequalities

The Grunsky inequalities [6] and their generalizations (e.g., [5; 14; 17]) have become an increasingly important tool for the study of the coefficients of normalized univalent functions defined on the unit disc. In particular, proofs based upon the Grunsky inequalities have now settled the Bieberbach conjecture for the fifth [15] and sixth [13] coefficients. For bounded univalent functions the situation is similar, although the Grunsky inequalities go over to those of Nehari [11].

scholarly journals On Certain Subclass of Harmonic Starlike Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
A. Y. Lashin
Keyword(s):  
Integral Operator ◽  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disc ◽  
New Class ◽  
Distortion Bounds ◽  
Harmonic Starlike

Coefficient conditions, distortion bounds, extreme points, convolution, convex combinations, and neighborhoods for a new class of harmonic univalent functions in the open unit disc are investigated. Further, a class preserving integral operator and connections with various previously known results are briefly discussed.

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