Some properties of a subclass of analytic functions

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Some properties of a subclass of analytic functions. Presented by Dr. Wasim Ul-Haq Department of Mathematics College of Science in Al-Zulfi, Majmaah University KSA. Presentation Layout. Introduction Basic Conceps Preliminary Results Main Results. 3. Introduction.

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Some properties of a subclass of analytic functions

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Presentation Transcript

Some properties of a subclass of analytic functions

Presented by

Dr. Wasim Ul-Haq

Department of Mathematics

College of Science in Al-Zulfi, Majmaah University

KSA

Presentation Layout

Introduction

Basic Conceps

Preliminary Results

Main Results

3

Introduction

Geometric Function Theory is the branch of Complex Analysis

which deals with the geometric properties of analytic functions.

The famous Riemann mapping theorem about the replacement

of an arbitrary domain (of analytic function) with the open unit

disk is the founding stone of the geometric

function theory. Later, Koebe (1907) and Bieberbach (1916)

studied analytic univalent functions which map E onto the

domain with some nice geometric properties. Such functions and

their generalizations serve a key role in signal theory, constructing

Functions with bounded turning, that is, functions whose derivative has positive real part and their generalizations have very close connection to various classes of analytic univalent functions. These classes have been considered by many mathematicians such as Noshiro and Warchawski (1935), Chichra (1977), Goodman (1983) and Noor (2009).

In this seminar, we define and discuss a certain subclass of analytic functions related with the functions with bounded turning. An inclusion result, a radius problem, invariance under certain integral operators and some other interesting properties for this class will be discussed.

Basic Concepts

The class A (Goodman, vol.1)[2]

The class S of univalent functions[2]

Special classes of univalent functions [2]

Starlike functions(Nevanilinna, 1913)

Convex functions (Study, 1913)

Alexander relation (1915)

Lemma1 (Singh and Singh)[9]

Preliminary Results

Lemma 2(Lashin, 2005)[4]

Theorem 5

Proof.

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Theorem 6

Corollary 1

Miller and Mocanu [5] proved this result with a different technique.

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Conclusion

inclusion relations.

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References

[1] P.N. Chichra, New subclasses of the class of close-to-convex functions,

Proc. Amer. Math. Soc., 62(1977) 37-43.

[2] A.W. Goodman, Univalent functions, Vol. I, II, Mariner Publishing Company,

Tempa Florida, U.S.A 1983.

[3] J. Krzyz, A counter example concerning univalent functions, Folia Soc. Scient..

Lubliniensis 2(1962) 57-58.

[4] A.Y. Lashin, Applications of Nunokawa's theorem, J. Ineq. Pure Appl. Math., 5(2004),

1-5, Article 111.

[5] S. S. Miller and P. T. Mocanu, Differential subordination theory and applications,

Marcel Dekker Inc., New York, Basel, 2000.

[6] K.I. Noor , On a generalization of alpha convexity, J. Ineq. Pure Appl. Math., 8(2007),

1-4, Article 16.

[7] K.I. Noor and W. Ul-Haq, Some properties of a subclass of analytic functions, Nonlinear

Func. Anal. Appl, 13(2008)265-270.

[8] B. Pinchuk, Functions of bounded boundary rotations, Isr. J. Math., 10(1971),6-16.

[9] S. Singh and R. Singh, Convolution properties of a class of starlike functions, Proc.

Amer. Math. Soc., 106(1989), 145-152.

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