New Results Related to the Convexity and Starlikeness of the Bernardi Integral Operator

Yıl 2009, Cilt: 38 Sayı: 2, 137 - 143, 01.02.2009

G. İ. Oros Georgia İrina Oros

Öz

New Results Related to the Convexity and Starlikeness of the Bernardi Integral Operator

Öz

In (Lewandowski, Z., Miller, S. S. and Zlotkiewicz, E. Generating functions for some classes of univalent functions, Proc. Amer. Math. Soc. 56, 111–117, 1976) and (Pascu, N. N. Alpha-close-to-convex functions, Romanian–Finish Seminar on Complex Analysis, Springer Berlin, 331– 335, 1979) it has been proved that the integral operator defined by S. D. Bernardi (Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135, 429–446, 1969) and given by (1) Lγ(f)(z) = F(z) = γ + 1
z γ Z z 0 f(t)t γ−1 dt, z ∈ U preserves certain classes of univalent functions, such as the class of starlike functions, the class of convex functions and the class of closeto-convex functions. In this paper we determine conditions that a function f ∈ A needs to satisfy in order that the function F given by (1) be convex. We alsoprove two duality theorems between the classes K−12γand S∗, andbetween K−12γand S∗−12γ, respectively.

Anahtar Kelimeler

Analytic function, Univalent function, Integral operator, Convex function, Starlike function, Close-to-convex function

Kaynakça

• Bernardi, S. D. Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135, 429–446, 1969.
• Lewandowski, Z., Miller, S. S. and Zlotkiewicz, E. Generating functions for some classes of univalent functions, Proc. Amer. Math. Soc. 56, 111–117, 1976.
• Miller, S. S. and Mocanu, P. T. Differential subordinations and univalent functions, Michigan Math. J. 28, 157–171, 1981.
• Miller, S. S. and Mocanu, P. T. Differential subordinations and inequalities in the complex plane, J. Diff. Eqns. 67 (2), 199–211, 1987.
• Miller, S. S. and Mocanu, P. T. Classes of univalent integral operators, J. Math. Anal. Appl. 157(1), 147–165, 1991.
• Miller, S. S. and Mocanu, P. T. Differential Subordinations. Theory and Applications (Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 2000).
• Mocanu, P. T., Bulboac˘a, T. and S˘al˘agean, S¸t. G. Teoria geometric˘a a funct¸iilor univalente (Casa C˘art¸ii de S¸tiint¸˘a, Cluj-Napoca, 1999).
• Pascu, N. N. Alpha-close-to-convex functions, Romanian-Finish Seminar on Complex Anal- ysis, Springer Berlin, 331–335, 1979.