On the Fekete-Szego problem for concave univalent functions

We consider the Fekete-Szego problem with real parameter lambda for the class Co(alpha) of concave univalent functions. (C) 2010 Elsevier Inc. All rights reserved.

On some results of A. E. Livingston and coefficient problems for concave univalent functions

We consider functions that map the open unit disc conformally onto the complement of a bounded convex set. We call these functions concave univalent functions. In 1994, Livingston presented a characterization for these functions. In this paper, we observe that there is a minor flaw with this characterization. We obtain certain sharp estimates and the exact set of variability involving Laurent and Taylor coefficients for concave functions. We also present the exact set of variability of the linear combination of certain successive Taylor coefficients of concave functions.

On concave univalent functions

We consider functions that map the open unit disc conformally onto the complement of an unbounded convex set with opening angle pa, a ? (1, 2], at infinity. In this paper, we show that every such function is close-to-convex of order (a - 1) and is included in the set of univalent functions of bounded boundary rotation. Many interesting consequences of this result are obtained. We also determine the extreme points of the set of concave functions with respect to the linear structure of the Hornich space.

Alternative Characterization of the Class k-UCV and Related Classes of Univalent Functions

In this paper an alternative characterization of the class of functions called k -uniformly convex is found. Various relations concerning connections with other classes of univalent functions are given. Moreover a new class of univalent functions, analogous to the ’Mocanu class’ of functions, is introduced. Some results concerning this class are derived.

On Two Saigo’s Fractional Integral Operators in the Class of Univalent Functions

2000 Mathematics Subject Classification: Primary 26A33, 30C45; Secondary 33A35

Some Notes about a Class of Univalent Functions with Negative Coefficients

MSC 2010: 30C45, 30C50

A Note on Univalent Functions with Finitely many Coefficients

MSC 2010: 30C45

The Koebe Domain for Concave Univalent Functions

2000 Mathematics Subject Classification: 30C25, 30C45.

On a Class of Univalent Functions with Negative Coefficients

Донка Пашкулева - Предмет на тази статия е получаването на точни оценки за коефициентите и ръста на функциите за някои класове еднолистни функции с отрицателни коефициенти.

On the Residuum of Concave Univalent Functions

2000 Mathematics Subject Classification: 30C25, 30C45.

Application of Salagean and Ruscheweyh Operators on Univalent Holomorphic Functions with Finitely many Coefficients

MSC 2010: 30C45, 30C50

Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [deBranges1985] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [AskeyGasper1976] about certain hypergeometric functions played a crucial role in de Branges' proof. In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [Weinstein1991] follows, and it is shown how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated.

Applications of the Owa-Srivastava Operator to the Class of K-Uniformly Convex Functions

2000 Mathematics Subject Classification: Primary 30C45, 26A33; Secondary 33C15

Subclasses of Starlike Functions of Complex Order Involving Generalized Hypergeometric Functions

MSC2010: 30C45, 33C45