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Majorization of starlike and convex functions of complex order involving linear operators

  • G. Murugusundaramoorthy EMAIL logo and K. Thilagavathi
From the journal Mathematica Slovaca

Abstract

The main object of this present paper is to investigate the problem of majorization of certain class of analytic functions of complex order defined by the Dziok-Raina linear operator. Moreover we point out some new or known consequences of our main result.


Dedicated to the Mathematician Srinivasan Ramanujan on his 125th Birth Anniversary


  1. This paper has been communicated by Stanisława Kanas

Acknowledgement

The authors thank the referee for his insightful suggestions to improve this paper in the present form.

References

[1] Altintaş, O.—Özkan, Ö.—Srivastava, H. M.: Majorization by starlike functions of complex order, Complex Variables Theory Appl. 46 (2001), 207–218.10.1080/17476930108815409Search in Google Scholar

[2] Bernardi, S. D.: Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429–446.10.1090/S0002-9947-1969-0232920-2Search in Google Scholar

[3] Carlson, B. C.—Shaffer, D. B.: Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984), 737–745.10.1137/0515057Search in Google Scholar

[4] Dziok, J.—Srivastava, H. M.: Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec. Funct. 14 (2003), 7–18.10.1080/10652460304543Search in Google Scholar

[5] Dziok, J.—Raina, R. K.: Families of analytic functions associated with the Wright generalized hypergeometric function, Demonstr. Math. 37 (2004), 533–542.10.1515/dema-2004-0305Search in Google Scholar

[6] Goyal, S. P.—Goswami, P.: Majorization for certain classes of analytic functions defined by fractional derivatives, Appl. Math. Lett. 22 (2009), 1855–1858.10.1016/j.aml.2009.07.009Search in Google Scholar

[7] Hohlov, Y. E.: Hadamard convolution, hypergeometric functions and linear operators in the class of univalent functions, Dokl. Akad. Nauk Ukr. SSR Ser. A 7 (1984), 25–27.Search in Google Scholar

[8] Libera, R. J.: Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755–758.10.1090/S0002-9939-1965-0178131-2Search in Google Scholar

[9] Libera, R. J.: Univalent α spiral-like functions, Canad. J. Math. Soc. 19 (1967), 449–456.10.4153/CJM-1967-038-0Search in Google Scholar

[10] Livingston, A. E.: On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 17 (1966), 352–357.10.1090/S0002-9939-1966-0188423-XSearch in Google Scholar

[11] MacGregor, T. H.: Majorization by univalent functions, Duke Math. J. 34 (1967), 95–102.10.1215/S0012-7094-67-03411-4Search in Google Scholar

[12] Miller, S. S.—Mocanu, P. T.: Differential Subordinations. Monographs and Textbooks in Pure and Applied Mathematics 225, Dekker, New York, 2000.10.1201/9781482289817Search in Google Scholar

[13] Murugusundaramoorthy, G.—Magesh, N.: Starlike and convex functions of complex order involving the Dziok-Srivastava operator, Integral Transforms Spec. Funct. 18 (2007), 419–425.10.1080/10652460701318111Search in Google Scholar

[14] Nahari, Z.: Conformal Mapping, MacGra-Hill Book Company, New York-Toronto-London, 1952.Search in Google Scholar

[15] Owa, S.: On the distortion theorems. I, Kyungpook Math. J. 18 (1978), 53–59.Search in Google Scholar

[16] Ruscheweyh, S.: New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–115.10.1090/S0002-9939-1975-0367176-1Search in Google Scholar

[17] Špaček, L.: A contributin to the theory of simple functions, Časopis Pěst. Mat. 63 (1933), 12–19 (Czech).10.21136/CPMF.1933.121951Search in Google Scholar

[18] Srivastava, H. M.—Owa, S.: Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions, Nagoya Math. J. 106 (1987), 1–28.10.1017/S0027763000000854Search in Google Scholar

[19] Srivastava, H. M.—Owa, S.: A note on certain class of spiral-like functions, Rend. Semin. Mat. Univ. Padova 80 (1988), 17–24.Search in Google Scholar

[20] Wright, E. M.: The asymptotic expansion of the generalized hypergeometric function, Proc. Lond. Math. Soc. (3) 46 (1940), 389–408.10.1112/plms/s2-46.1.389Search in Google Scholar

Received: 2012-1-9
Accepted: 2013-8-5
Published Online: 2016-4-29
Published in Print: 2016-2-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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