A LINEAR INTEGRAL OPERATOR AND ITS APPLICATION

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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

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Presentation Layout

• Introduction
• Basic Conceps
• Preliminary Results
• Main Results
• 
  

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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.

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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.
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Basic Concepts

• The class A (Goodman, vol.1)2
• 
  
• The class S of univalent functions2

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• The class P (Caratheodory functions) 2
• The class 19()

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• Some related classes to the class P (Noor,
  2007)6

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• Examples

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• Special classes of univalent functions 2
• Starlike functions (Nevanilinna, 1913)
• Convex functions (Study, 1913)
• Alexander relation (1915)

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• Functions with bounded turning and related
  classes

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• Convolution (or Hadamard Product)
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• Lemma1 (Singh and Singh)9

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• A class of analytic functions Noor and Haq, 7

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Preliminary Results

• Lemma 2 (Lashin, 2005)4

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Main Results

• Inclusion result
• Theorem 1

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• Proof

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• Applications of Theorem 1
• Theorem 2

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• Proof

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• Integral preserving property

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• Theorem 3

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• Convolution properties
• Theorem 4

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• Proof.

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Proof (Cont..)
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• Theorem 5
• Proof.

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Proof (Cont..)
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• Applications of Theorem 5

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• Radius problem (Inverse inclusion)
• Theorem 6
• Corollary 1
• Miller and Mocanu 5 proved this result with a
  different technique.

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• Conclusion
• The arrow heads show the
• 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.

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• THANK YOU

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