SOME CONVEXITY PROPERTIES FOR TWO NEW P -VALENT INTEGRAL OPERATORS

Abstract

In this paper, we define two new general p-valent integral operators in the unit disc U, and obtain the convexity properties of these integral operators of p-valent functions on some classes of β-uniformly p-valent starlike and β-uniformly p-valent convex functions of complex order. As special cases, the convexity properties of the operators R z 0 f(t) t µ dt and R z 0 (g ′ (t))µ dt are given.

Keywords

• Analytic functions,Integral operators,β-uniformly p-valent starlike and β-uniformly p-valent convex functions,Complex order,2000 AMS Classification: Primary 30 C 80. Secondary 30 C 45

SOME CONVEXITY PROPERTIES FOR TWO NEW P -VALENT INTEGRAL OPERATORS

Abstract

References

1. Alexander, J. W. Functions which map the interior of the unit circle upon simple regions, Annals of Mathematics 17 (1), 12–22, 1915.
2. Bharti, R., Parvatham, R. and Swaminathan, A. On subclasses of uniformly convex func- tions and corresponding class of starlike functions, Tamkang J. Math. 28 (1), 17–32, 1997.
3. Bulut, S. A note on the paper of Breaz and G¨uney, J. Math. Ineq. 2 (4), 549–553, 2008.
4. Breaz, D. A convexity properties for an integral operator on the classes Sp(α), Gen. Math. (2-3), 177–183, 2007.
5. Breaz, D., Aouf, M. K. and Breaz, N. Some properties for integral operators on some analytic functions with complex order, Acta. Math. Acad. Paedagog. Nyhazi. 25, 39–43, 2009.
6. Breaz, D. and Breaz, N. Two integral operators, Stud. Univ. Babes-Bolyai Math. 47 (3), –19, 2002.
7. Breaz, D. and Breaz, N. Some convexity properties for a general integral operator, J. Ineq. Pure Appl. Math. 7 (5), Art. 177, 2006.
8. Breaz, D., Owa, S. and Breaz, N. A new integral univalent operator, Acta Univ. Apulensis Math. Inform. 16, 11–16, 2008.

9. Breaz, D. and G¨uney, H. ¨O. The integral operator on the classes S*α(b) and Cα(b), J. Math. Ineq. 2 (1), 97–100, 2008.
10. Deniz, E., Orhan, H. and Sokol, J. Classes of analytic functions defined by a differential operator related to conic domains, submitted. Frasin, B. A. Convexity of integral operators of p-valent functions, Math. Comput. Model. , 601–605, 2010.
11. Frasin, B. A. Family of analytic functions of complex order, Acta Math. Acad. Paed. Ny. , 179–191, 2006.
12. Goel, R. M. and Sohi, N. S. A new criterion for p-valent functions, Proc. Amer. Math. Soc. , 353–357, 1980.
13. Goodman, A. W. On uniformly convex functions, Ann. Polon. Math. 56, 87–92, 1991.
14. Kanas, S. and Wisniowska, A. Conic regions and k-uniform convexity, Comput. Appl. Math. , 327–336, 1999.
15. Kim, Y. J. and Merkes, E. P. On an integral of powers of a spirallike function, Kyungpook Math. J. 12, 249–252, 1972.
16. Miller, S. S., Mocanu, P. T. and Reade, M. O. Starlike integral operators, Pacific J. Math. (1), 157–168, 1978.
17. Nasr, M. A. and Aouf, M. K. Starlike function of complex order, J. Natur. Sci. Math. 25 (1), –12, 1985.
18. Orhan, H., Deniz, E. and R˘aducanu, D. The Fekete–Szeg¨o problem for subclasses of analytic functions defined by a differential operator related to conic domains, Comput. Math. Appl. (1), 283–295, 2010.
19. Oros, G. I. New results related to the convexity and starlikeness of the Bernardi integral operator, Hacet. J. Math. Stat. 38 (2), 137–143, 2009.
20. Oros, G. I. and Oros, G. A convexity property for an integral operator Fm, Stud. Univ. Babes-Bolyai Math. 55 (3), 169–177, 2010.
21. Pescar, V. and Owa, S. Sufficient conditions for univalence of certain integral operators, Indian J. Math. 42 (3), 347–35, 2000.
22. Pfaltzgraff, J. A. Univalence of the integral of (f′(z))λ, Bull. London Math. Soc. 7 (3), 254– , 1975.
23. Rİnning, F. On starlike functions associated with parabolic regions, Ann. Univ. Mariae Curie-Sk ˜Alodowska Sect. A. 45 (14), 117–122, 1991.
24. Wiatrowski, P. The coefficients of a certain family of holomorphic functions, Zeszyty Nauk. Uniw.Lodz. Nauki Mat. Pryrod. Ser. 3, 75–85, 1971.