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.
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.
Dr. Wasim Ul-Haq
Department of Mathematics
College of Science in Al-Zulfi, Majmaah University
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.
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.
The class A (Goodman, vol.1)
The class S of univalent functions
The class 19()
Starlike functions(Nevanilinna, 1913)
Convex functions (Study, 1913)
Alexander relation (1915)
Lemma1 (Singh and Singh)
Lemma 2(Lashin, 2005)
Miller and Mocanu  proved this result with a different technique.
The arrow heads show the
 P.N. Chichra, New subclasses of the class of close-to-convex functions,
Proc. Amer. Math. Soc., 62(1977) 37-43.
 A.W. Goodman, Univalent functions, Vol. I, II, Mariner Publishing Company,
Tempa Florida, U.S.A 1983.
 J. Krzyz, A counter example concerning univalent functions, Folia Soc. Scient..
Lubliniensis 2(1962) 57-58.
 A.Y. Lashin, Applications of Nunokawa's theorem, J. Ineq. Pure Appl. Math., 5(2004),
1-5, Article 111.
 S. S. Miller and P. T. Mocanu, Differential subordination theory and applications,
Marcel Dekker Inc., New York, Basel, 2000.
 K.I. Noor , On a generalization of alpha convexity, J. Ineq. Pure Appl. Math., 8(2007),
1-4, Article 16.
 K.I. Noor and W. Ul-Haq, Some properties of a subclass of analytic functions, Nonlinear
Func. Anal. Appl, 13(2008)265-270.
 B. Pinchuk, Functions of bounded boundary rotations, Isr. J. Math., 10(1971),6-16.
 S. Singh and R. Singh, Convolution properties of a class of starlike functions, Proc.
Amer. Math. Soc., 106(1989), 145-152.