Abstract
We introduce and study a new class of analytic functions defined in the unit disc using a certain multiplier transformation. Some inclusion results and other interesting properties of this class are investigated.
Research Article | Open Access
On Some Analytic Functions Defined by a Multiplier Transformation
Academic Editor: Heinrich Begehr
Received26 Jul 2007
Accepted19 Nov 2007
Published22 Jan 2008
References
B. Pinchuk, “Functions of bounded boundary rotation,” Israel Journal of Mathematics, vol. 10, pp. 6–16, 1971.
View at: Google Scholar | Zentralblatt MATH | MathSciNetK. I. Noor, “On subclasses of close-to-convex functions of higher order,” International Journal of Mathematics and Mathematical Sciences, vol. 15, no. 2, pp. 279–289, 1992.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetW. Kaplan, “Close-to-convex schlicht functions,” The Michigan Mathematical Journal, vol. 1, p. 169–185 (1953), 1952.
View at: Google Scholar | MathSciNetB. A. Uralegaddi and C. Somanatha, “Certain classes of univalent functions,” in Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa, Eds., pp. 371–374, World Scientific, River Edge, NJ, USA, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNetS. Owa and H. M. Srivastava, “Some applications of the generalized Libera integral operator,” Proceedings of the Japan Academy, Series A, Mathematical Sciences, vol. 62, no. 4, pp. 125–128, 1986.
View at: Google Scholar | Zentralblatt MATH | MathSciNetI. B. Jung, Y. C. Kim, and H. M. Srivastava, “The Hardy space of analytic functions associated with certain one-parameter families of integral operators,” Journal of Mathematical Analysis and Applications, vol. 176, no. 1, pp. 138–147, 1993.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. D. Bernardi, “Convex and starlike univalent functions,” Transactions of the American Mathematical Society, vol. 135, pp. 429–446, 1969.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetG. S. Sălăgean, “Subclasses of univalent functions,” in Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Math, pp. 362–372, Springer, Berlin, Germany, 1983.
View at: Google Scholar | Zentralblatt MATH | MathSciNetT. M. Flett, “The dual of an inequality of Hardy and Littlewood and some related inequalities,” Journal of Mathematical Analysis and Applications, vol. 38, pp. 746–765, 1972.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetJ. H. Choi, M. Saigo, and H. M. Srivastava, “Some inclusion properties of a certain family of integral operators,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 432–445, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetJ. L. Liu, “The Noor integral and strongly starlike functions,” Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 441–447, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetJ. L. Liu and K. I. Noor, “Some properties of Noor integral operator,” Journal of Natural Geometry, vol. 21, no. 1-2, pp. 81–90, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNetK. I. Noor, “On new classes of integral operators,” Journal of Natural Geometry, vol. 16, no. 1-2, pp. 71–80, 1999.
View at: Google Scholar | Zentralblatt MATH | MathSciNetK. I. Noor and M. A. Noor, “On integral operators,” Journal of Mathematical Analysis and Applications, vol. 238, no. 2, pp. 341–352, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetN. E. Cho and J. A. Kim, “Inclusion properties of certain subclasses of analytic functions defined by a multiplier transformation,” Computers & Mathematics with Applications, vol. 52, no. 3-4, pp. 323–330, 2006.
View at: Publisher Site | Google Scholar | MathSciNetS. Ponnusamy, “Differential subordination and Bazilevič functions,” Indian Academy of Sciences. Proceedings. Mathematical Sciences, vol. 105, no. 2, pp. 169–186, 1995.
View at: Google Scholar | Zentralblatt MATH | MathSciNetR. Singh and S. Singh, “Convolution properties of a class of starlike functions,” Proceedings of the American Mathematical Society, vol. 106, no. 1, pp. 145–152, 1989.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. Ponnusamy and V. Singh, “Convolution properties of some classes of analytic functions,” 1990, Internal Report, SPIC Science Foundation.
View at: Google ScholarY. Ling and S. Ding, “A new criterion for starlike functions,” International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 3, pp. 613–614, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetR. J. Libera, “Some classes of regular univalent functions,” Proceedings of the American Mathematical Society, vol. 16, pp. 755–758, 1965.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetR. W. Barnard and C. Kellogg, “Applications of convolution operators to problems in univalent function theory,” The Michigan Mathematical Journal, vol. 27, no. 1, pp. 81–94, 1980.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetSt. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical Society, vol. 49, pp. 109–115, 1975.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetR. V. Nikolaeva and L. G. Repnīna, “A certain generalization of theorems due to Livingston,” Ukrainskiĭ Matematicheskiĭ Zhurnal, vol. 24, pp. 268–273, 1972.
View at: Google Scholar | MathSciNet
Copyright
Copyright © 2007 Khalida Inayat Noor. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.