1. 1

2. 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 2

3. Presentation Layout Introduction Basic Conceps Preliminary Results Main Results 3

4. 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. 4

5. 5 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.

6. Basic Concepts The class A (Goodman, vol.1)[2] The class S of univalent functions[2] 6

7. The class P (Caratheodory functions) [2] The class 19() 6

8. Some related classes to the class P (Noor, 2007)[6] 5

9. Examples 6

11. Special classes of univalent functions [2] Starlike functions (Nevanilinna, 1913) Convex functions (Study, 1913) Alexander relation (1915) 11

12. Functions with bounded turning and related classes 12

13. Convolution (or Hadamard Product) Lemma1 (Singh and Singh)[9] 13

14. A class of analytic functions [Noor and Haq, 7] 14

15. Preliminary Results Lemma 2 (Lashin, 2005)[4] 15

16. Inclusion result Theorem 1 Main Results 16

17. Proof 17

18. Applications of Theorem 1 Theorem 2 18

19. Proof 19

20. Integral preserving property 20

21. Theorem 3 21

22. Convolution properties Theorem 4 22

23. Proof. 23

24. 24 Proof (Cont…..)

25. Theorem 5 Proof. 25

26. 26 Proof (Cont…..)

27. Applications of Theorem 5 27

28. Radius problem (Inverse inclusion) Theorem 6 Corollary 1 Miller and Mocanu [5] proved this result with a different technique. 28

29. Conclusion The arrow heads show the inclusion relations. 29

30. 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

33. THANK YOU 33,.