On certain subclass of $p$-valent analytic functions associated with differintegral operator

Chellian Selvaraj; Ganapathi Thirupathi

doi:10.26637/mjm304/009

Abstract

In this paper, by making use of the fractional differintegral operator, we introduce a certain subclass of multivalent analytic functions. We study some properties such as inclusion relationship, integral preserving, convolution and some interesting results for multivalent starlikeness are proved.

Keywords:

Multivalent function, subordination, superordination, hadamard product, differintegral operator, starlike function

Mathematics Subject Classification:

30C45

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Chellian Selvaraj Department of Mathematics, Presidency College, Chennai-600 005, Tamil Nadu, India.
Ganapathi Thirupathi Department of Mathematics, R.M.K.Engineering College, R.S.M.Nagar, Kavaraipettai-601 206, Tamil Nadu, India. https://orcid.org/0000-0001-9572-5837

Pages: 511-522
Date Published: 01-10-2015
Vol. 3 No. 04 (2015): Malaya Journal of Matematik (MJM)

Ahmed S. Galiz, On a Certain Subclass of Multivalent Analytic Functions Associated with a Generalized Fractional Differintegral Operator, International Journal of Mathematical Analysis, Vol. 9, 2015, no. 8, 365 $-383$. DOI: https://doi.org/10.12988/ijma.2015.412402

M. K. Aouf, A. O. Mostafa, A. M. Shahin, S. M. Madian, Applications of differential subordinations for certain classes of $p$-valent functions associated with generalized Srivastava-Attiya operator, J. Inequal. Appl, 2012, 2012:153, $15 mathrm{pp}$. DOI: https://doi.org/10.1186/1029-242X-2012-153

M.K. Aouf, T.M. Seoudy, Some properties of a certain subclass of multivalent analytic functions involving the Liu-Owa operator, Comput. Math. Appl., 60 (2010) 1525?535. DOI: https://doi.org/10.1016/j.camwa.2010.06.036

R. M. El-Ashwah, M. K. Aouf, Differential subordination and superordination for certain subclasses of p-valent functions, Math. Comput. Modelling, 51 (2010), no. 5-6, 349-360. DOI: https://doi.org/10.1016/j.mcm.2009.12.027

S. D. Bernardi, 1969, "Convex and starlike univalent functions", Trans. Amer. Math. Soc., 135, 429-446. DOI: https://doi.org/10.1090/S0002-9947-1969-0232920-2

J. Dziok, H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput., 103 (1999), no. 1, 1-13. DOI: https://doi.org/10.1016/S0096-3003(98)10042-5

S. P. Goyal, J. K. Prajapat, A new class of analytic p-valent functions with negative coefficients and fractional calculus operators, Tamsui Oxf. J. Math. Sci., 20 (2004), no. 2, 175-186.

Jung, Il Bong; Y. C. Kim, H. M. Srivastava, 1993, "The Hardy space of analytic functions associated with certain one-parameter families of integral operators", J. Math. Anal. Appl., 176 , no. 1, 138-147. DOI: https://doi.org/10.1006/jmaa.1993.1204

TH. MacGregor, Radius of univalence of certain analytic functions, Proc. Am. Math. Soc., (1963), 14, 514520. DOI: https://doi.org/10.1090/S0002-9939-1963-0148891-3

Sanford S. Miller, P. T. Mocanu, Differential subordinations, Theory and applications, Monographs and Textbooks in Pure and Applied Mathematics, 225. Marcel Dekker, Inc., New York, 2000. xii+459 pp. ISBN: 0-8247-0029-5.

Liu, Jin-Lin; Owa, Shigeyoshi. 2004, "Properties of certain integral operator", Int. J. Math. Sci., 3 , no. 1, 69-75.

S.S. Miller, P.T. Mocanu, On some classes of first order differential subordinations, Michigan Math. J., 32 (1985) $185 ? 95$. DOI: https://doi.org/10.1307/mmj/1029003185

A. O. Mostafa, M. K. Aouf, Some subordination results for $p$ - valent functions associated with differintegral operator, Journal of Fractional Calculus and Applications, Vol. 5(1) Jan. 2014, pp. 11-25.

PT. Nehari, Conformal Mapping, McGraw-Hill, New York (1952).

M. Obradovic, S. Owa, On certain properties for some classes of starlike functions, J. Math. Anal. Appl., (1990) 145, 357-364 DOI: https://doi.org/10.1016/0022-247X(90)90405-5

DZ. Pashkouleva, The starlikeness and spiral-convexity of certain subclasses of analytic functions. In: Srivastava, HM, Owa, S (eds.) Current Topics in Analytic Function Theory, pp. 266-273. World Scientific, Singapore (1992) DOI: https://doi.org/10.1142/9789814355896_0022

J. Patel, A.K. Mishra, On certain subclasses of multivalent functions associated with an extended fractional differintegral operator, J. Math. Anal. Appl., 332 (2007) 109?22. DOI: https://doi.org/10.1016/j.jmaa.2006.09.067

J.K. Prajapat, R.K. Raina, New sufficient conditions for starlikeness of analytic functions involving a fractional differintegral operator, Demonstratio Math., 43 (1) (2010) 805?13. DOI: https://doi.org/10.1515/dema-2010-0408

Salagean, Grigore Stefan, Subclasses of univalent functions, Complex analysis?ifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), 362-372, Lecture Notes in Math., 1013, Springer, Berlin, 1983. http://dx.doi.org/10.1007/bfb0066543. DOI: https://doi.org/10.1007/BFb0066543

R. Singh, S. Singh, Convolution properties of a class of starlike functions, Proc. Am. Math. Soc.,(1989) 106, $145-152$. DOI: https://doi.org/10.1090/S0002-9939-1989-0994388-6

C. Selvaraj, K. R. Karthikeyan, Differential subordination and superordination for certain subclasses of analytic functions, Far East J. Math. Sci., (FJMS), 29 (2008), no. 2, 419-430.

C. Selvaraj, O. S. Babu and G. Murugusundaramoorthy, Some Subclasses of p-Valent Functions Defined by Generalized Fractional Differintegral Operator -II, J. Ana. Num. Theor., (2015) 3, No. 1, 39-45 DOI: https://doi.org/10.18576/msl/050107

H. M. Srivastava, A. A. Attiya, An integral operator associated with the Hurwitz-Lerch zeta function and differential subordination, Integral Transforms Spec. Funct., 18 (2007), no. 3-4, 207-216. DOI: https://doi.org/10.1080/10652460701208577

Huo Tang, Guan-Tie Deng, Shu-Hai Li, Certain subclasses of $p$-valently analytic functions involving a generalized fractional differintegral operator, J. Egyptian Math. Soc., 22 (2014), no. 1,36-44. DOI: https://doi.org/10.1016/j.joems.2013.06.009

ET. Whittaker, GN. Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions, 4th edn. Cambridge University Press, Cambridge (1927)

DR. Wilken, J. Feng, A remark on convex and starlike functions, J. Lond. Math. Soc., Ser., (1980) 2, 21, 287-290. DOI: https://doi.org/10.1112/jlms/s2-21.2.287