Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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A NEW SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY CONVOLUTION

  • Lee, S.K. (Department of Mathematics and The Research Institute of Natural Science Gyeongsang National University) ;
  • Khairnar, S.M. (Department of Mathematics Maharashtra Academy of Engineering Alandi)
  • Received : 2011.08.05
  • Accepted : 2011.12.10
  • Published : 2011.12.30
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Abstract

In the present paper we introduce a new subclass of analytic functions in the unit disc defined by convolution $(f_{\mu})^{(-1)}*f(z)$; where $$f_{\mu}=(1-{\mu})z_2F_1(a,b;c;z)+{\mu}z(z_2F_1(a,b;c;z))^{\prime}$$. Several interesting properties of the class and integral preserving properties of the subclasses are also considered.

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References

  1. R.W. Barnard and Ch. Kellogg, Applications of convolution operators to problems in univalent function theory, Michigan Math. J. 27 (1980), no. 1, 81-94. https://doi.org/10.1307/mmj/1029002312
  2. B.C. Carlson and D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984), no. 4, 737-745. https://doi.org/10.1137/0515057
  3. N.E. Cho, The Noor integral operator and strongly close-to-convex functions, J. Math. Anal. Appl. 283 (2003), no. 1, 202-212. https://doi.org/10.1016/S0022-247X(03)00270-1
  4. N.E. Cho, Inclusion properties for certain subclasses of analytic functions defined by a linear operator, Abstr. Appl. Anal. Article id. 246876 (2008), 8 pages.
  5. N.E. Cho and T.H. Kim, Multiplier transformations and strongly close-toconvex functions, Bull. Korean Math. Soc. 40 (2003), no. 3, 399-410. https://doi.org/10.4134/BKMS.2003.40.3.399
  6. J.H. Choi, M. Saigo and H.M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl. 276 (2002), no. 1, 432-445. https://doi.org/10.1016/S0022-247X(02)00500-0
  7. P. Eenigenburg, S.S. Miller, P.T. Mocanu and M.O. Reade, On a Briot-Bouquet differential subordination in general inequalities, Internat. Schriftenreihe Numer. Math. 64 (1983), 339-348 Birkhauser, Basel, Switzerland. https://doi.org/10.1007/978-3-0348-6290-5_26
  8. R.M. Goel and B.S. Mehrok, On the coefficients of a subclass of starlike functions, Indian J. Pure Appl. Math. 12 (1981), no. 5, 634-647.
  9. J.E. Hohlov, Operators and operations on the class of univalent fucntions, Izv. Vyssh. Uchebn. Zaved.i Mat. 197 (1978), no. 10, 83-89.
  10. W. Janowski,, Some external problems for certain families of analytic functions, Bull. Acad. Pol. Sci., Ser. Sci. Phys. Astron 21 (1973), 17-25.
  11. J.A. Kim and K.H. Shon, Mapping properties for convolutions involving hypergeometric functions, Int. J. Math. Math. Sci. 17 (2003), 1083-1091.
  12. J. Liu, The Noor integral operator and strongly close-to-convex functions, J. Math. Anal. Appl. 261 (2001), no. 2, 441-447. https://doi.org/10.1006/jmaa.2001.7489
  13. W. Ma and D. Minda, An internal geometric characterization of strongly starlike functions, Ann. Univ. Mariae Curie Sklodowska. Sect. A 45 (1991), 89-97.
  14. S.S. Miller and P.T. Mocanu, Differential subordinations and inequalities in the complex plane, J. Differential Equations 67 (1987), no. 2, 199-211. https://doi.org/10.1016/0022-0396(87)90146-X
  15. K.I. Noor, Integral operators defined by convolution with hypergeometric functions, Appl. Math. Comput. 182 (2006), no. 2, 1872-1881. https://doi.org/10.1016/j.amc.2006.06.023
  16. K.I. Noor, On new classes of integral operators, J. Natur. Geom. 283 (1999), no. 2, 71-80.
  17. K.I. Noor and M.A. Noor, On integral operators, J. Math. Anal. Appl. 283 (1999), no. 2, 341-352.
  18. S. Owa and H.M. Srivastava, Some applications of the generalized Libera integral operator, Proc. Japan Acad. Ser. A 62 (1986), no. 4, 125-128.
  19. K.S. Padmanabhan and R. Parvatham, Some applications of differential subordination, Bull. Aust. Math. Soc. 32 (1985), no. 3, 321-330. https://doi.org/10.1017/S0004972700002410
  20. St. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Polya-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), no. 1, 119-135. https://doi.org/10.1007/BF02566116
  21. St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109-115. https://doi.org/10.1090/S0002-9939-1975-0367176-1
  22. N. Shukla and P. Shukla, Mapping properties of analytic function defined by hypergeometric function II, Soochow J. Math. 25 (1999), no. 1, 29-36.

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