slide1 n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Some properties of a subclass of analytic functions PowerPoint Presentation
Download Presentation
Some properties of a subclass of analytic functions

Loading in 2 Seconds...

play fullscreen
1 / 33

Some properties of a subclass of analytic functions - PowerPoint PPT Presentation


  • 173 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Some properties of a subclass of analytic functions' - milica


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
some properties of a subclass of analytic functions

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
Presentation Layout

Introduction

Basic Conceps

Preliminary Results

Main Results

3

introduction
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

quadrature formulae and moment problems.

slide5

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
Basic Concepts

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

The class S of univalent functions[2]

slide11

Special classes of univalent functions [2]

Starlike functions(Nevanilinna, 1913)

Convex functions (Study, 1913)

Alexander relation (1915)

slide13

Convolution (or Hadamard Product)

Lemma1 (Singh and Singh)[9]

preliminary results
Preliminary Results

Lemma 2(Lashin, 2005)[4]

slide25
Theorem 5

Proof.

25

slide28
Radius problem (Inverse inclusion)

Theorem 6

Corollary 1

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

28

slide29
Conclusion

The arrow heads show the

inclusion relations.

29

slide30
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.

30