Some Properties of Certain New Subclasses of Analytic Functions

Some Properties of Certain New Subclasses of Analytic Functions

Authors

  • Rabha M. El-Ashwah Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt
  • Alaa H. Hassan Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

Keywords:

Analytic function, starlike function, convex function, close-to-convex, quasi-convex, 2010 MSC: 30C45

Abstract

In this paper, we introduce some new subclasses of analytic functions related to starlike, convex, close-to-convex and quasi-convex functions defined by using a generalized operator. Inclusion relationships for these subclasses are established. Also, we introduce some integral-preserving properties. Moreover, connections of the results presented here with those obtained in earlier works are pointed out.

References

Aouf, M.K. Some inclusion relationships associated with the Komatu integral operator. Mathematical and Computer Modelling 50: 1360-1366 (2009).

Aouf, M.K., A. Shamandy, A.O. Mostafa & S.M. Madian. Inclusion properties of certain subclasses of analytic functions defined by Salagean operator. Annales Universitatis Mariae Curie-Sklodowska Sectio A Mathematica 64(no. 1): 17-26 (2010).

Aouf, M.K. & R.M. El-Ashwah. Inclusion properties of certain subclass of analytic functions defined by multiplier transformations. Annales Universitatis Mariae Curie-Sklodowska Sectio A Mathematica 63: 29-38 (2009).

Aouf, M.K., R.M. El-Ashwah & A.M. Abd-Eltawab. Some inclusion relationships of certain subclasses of pvalent functions associated with a family of integral operators. ISRN Mathematical Analysis Art. ID 384170, 8 pages (2013).

Bernardi, S.D. Convex and starlike univalent functions. Transactions of the American Mathematical Society 135: 429-446 (1969).

Carlson, B.C. & D.B. Shaffer. Starlike and prestarlike hypergeometric functions. SIAM Journal on Mathematical Analysis 15: 737-745 (1984).

Catas, A. A note on a certain subclass of analytic functions defined by multiplier transformations, in Proc. Book of the International Symposium on Geometric Function Theory and Applications, Istanbul, Turkey 241-250 (2007).

Cho, N.E. & J.A. Kim. Inclusion properties of certain subclasses of analytic functions defined by a multiplier transformation. Computer and Mathematics with Applications 52: 323-330 (2006).

Choi, J.H., M. Saigo & H.M. Srivastava. Some inclusion properties of a certain family of integral operators. Journal of Mathematical Analysis and Applications 276: 432-445 (2002).

El-Ashwah, R.M. & M.K. Aouf. Some properties of new integral operator. Acta Universitatis Apulensis 24: 51-61 (2010).

El-Ashwah, R.M. & M.K. Aouf. Differential subordination and superordination for certain subclasses of analytic functions involving an extended integral operator. Acta Universitatis Apulensis 28: 341-350 (2011).

Flett, T.M. The dual of an inequality of Hardy and Littlewood and some related inequalities. Journal of Mathematical Analysis and Applications 38: 746-765 (1972).

Gao, C.Y., S.M. Yuan & H.M. Srivastava. Some functional inequalities and inclusion relationships associated with certain families of integral operators. Computer and Mathematics with Applications 49: 1787-1795 (2005).

Jack, I.S. Functions starlike and convex of order α. Journal of London Mathematical Society 3 (no. 2): 469-474 (1971).

Jung, I.B., Y.C. Kim & H.M. Srivastava. The Hardy space of analytic functions associated with certain oneparameter families of integral operators. Journal of Mathematical Analysis and Applications 176: 138-147 (1993).

Komatu, Y. On analytical prolongation of a family of operators. Mathematica (Cluj) 32 (no. 2): 141-145 (1990).

Li, J.L. Some properties of two integral operators. Soochow Journal of Mathematics, 25 (no. 1): 91-96 (1999).

Libera, R.J. Some classes of regular univalent functions. Proceedings of the American Mathematical Society 16 (no. 4): 755-758 (1965).

Liu, J.-L. Some applications of certain integral operator. Kyungpook Mathematical Journal, 43: 211-219 (2003).

Liu, J.-L. Notes on Jung-Kim-Srivastava integral operator. Journal of Mathematical Analysis and Applications 294: 96-103 (2004).

Liu, J.-L. & S. Owa. Properties of certain integral operator. International Journal of Mathematical Sciences 3 (no. 1): 45-51 (2004).

Livingston, A. E. On the radius of univalence of cerain analytic functions. Proceedings of the American Mathematical Society 17: 352-357 (1966).

MacGregor, T.H. The radius of convexity for starlike function of order α. Proceedings of the American Mathematical Society 14: 71-76 (1963).

Miller, S.S. & P.T. Mocanu. Second order differential inequalities in the complex plane. Journal of Mathematical Analysis and Applications 65: 289-305 (1978).

Noor, K.I. On quasi-convex functions and related topics. International Journal of Mathematics and Mathematical Sciences 10: 241-258 (1987).

Noor, K.I. On new classes of integral operators. Journal of Natural Geometry 16: 71-80 (1999).

Noor, K.I. & H.A. Alkhorasani. Properties of close-to-convexity preserved by some integral operators. Journal of Mathematical Analysis and Applications 112: 509-516 (1985).

Noor, K.I. & M.A. Noor. On integral operators. Journal of Mathematical Analysis and Applications 238: 341-352 (1999).

Owa, S. & H.M. Srivastava. Some applications of the generalized Libera integral operator. Proceedings of the Japan Academy, Series A, Mathematical Sciences 62: 125-128 (1986).

Patel, J. Inclusion relations and convolution properties of certain subclasses of analytic functions defined by a generalized Salagean operator. Bulletin of the Belgian Mathematical Society-Simon Stevin 15: 33-47 (2008).

Pinchuk, B. On the starlike and convex functions of order α. Duke Mathematical Journal 35: 721-734 (1968).

Raina, R.K. & P. Sharma. Subordination preserving properties associated with a class of operators. Le Matematiche 68 (no. 1): 217-228 (2013).

Robertson, M.S. On the theory of univalent functions. Annals of Mathematics 37: 374-408 (1936).

Ruscheweyh, S. New criteria for univalent functions. Proceedings of the American Mathematical Society 49:109-115 (1975).

Salagean, G.S. Subclasses of univalent functions. Lecture Notes in Math. (Springer-Verlag) 1013: 362-372 (1983).

Schild, A. On starlike function of order α. American Journal of Mathematics 87: 65-70 (1965).

Silverman, H. On a class of close-to-convex schlicht functions. Proceedings of the American Mathematical Society 36: 477-484 (1972).

Srivastava, H.M. & S. Owa. Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, Horwood Limited, Chichester and John Wiley and Sons, New York, Chichester, Brisbane and Toronto (1989).

Uralegaddi, B.A. & C. Somanatha. Certain classes of univalent functions, In: Current Topics in Analytic Function Theory, H.M. Srivastava and S. Owa (Ed.). World Scientific Publishing Company, Singapore p. 371-374 (1992).

Published

2021-06-17

How to Cite

El-Ashwah, R. M., & Hassan, A. H. . (2021). Some Properties of Certain New Subclasses of Analytic Functions: Some Properties of Certain New Subclasses of Analytic Functions. Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences, 53(1), 1–16. Retrieved from https://ppaspk.org/index.php/PPAS-A/article/view/353

Issue

Section

Articles