ON A SUBCLASS OF UNIFORMLY QUASI CONVEX FUNCTIONS OF ORDER $\alpha$

Year 2015, Volume: 3 Issue: 2, 33 - 41, 01.10.2015

D. Vamshee Krıshna B. Venkateswarlu , T. Ramreddy

Abstract

In this paper, we introduce two new classes of analytic functions namely uniformly quasi convex functions of order and quasi uniformly con- vex functions of order denoted by UQCV ($\alpha$) and QUCV ($\alpha$ ) ($0\qeq $\alpha$ < 1$) respectively and study certain properties of functions belonging to these two classes. Further, we obtain a necessary and sucient condition for the function f(z) to be in the class UQCV ($\alpha$ ): These results are generalized recent results of Rajalakshmi Rajagopal and Selvaraj [7]:

Keywords

Analytic function, uniformly quasi convex function, quasi uniformly convex function, positive real function

References

• [1] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Annals. Math., 17(1915), 12 -22.
• [2] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math., 56( 1991), 87 - 92. [3] A. W. Goodman, Univalent functions vol. I and vol. II, Mariner Publishing Comp. Inc., Tampa, Florida, 1983.
• [4] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc., 16(1965), 755 -758.
• [5] K. I. Noor, On quasi convex functions and related topics, Int. J. Math. Sci., 10(2)(1987), 241 - 258.
• [6] K. S. Padmanabhan, On certain subclasses of Bazilevic functions, Ind. J. Maths., 39(3)(1997).
• [7] Rajalakshmi Rajagopal and C. Selvaraj, On a class of uniformly quasi-convex functions, Bull. Calcutta Math. Soc., 95(2003), 199 -206.
• [8] F. Ronning, A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie - Sklodowska Sect. A, 47(1993), 123 - 134.
• [9] F. Ronning, Uniformly convex functions and corresponding class of star like functions, Proc. Amer. Math. Soc., 118(1) (1993), 189 - 196.