Acta Mathematica Sinica, English Series

, Volume 29, Issue 5, pp 833–840 | Cite as

A new class of generalized close-to-starlike functions defined by the Srivastava-Attiya operator

  • H. M. Srivastava
  • Dorina Răducanu
  • Grigore S. Sălăgean
Article

Abstract

In this paper, we consider a new class CSs,b* of generalized close-to-starlike functions, which is defined by means of the Srivastava-Attiya operator Js,b involving the Hurwitz-Lerch Zeta function Φ(z, s, a). Basic results such as inclusion relations, coefficient inequalities and other interesting properties of this class are investigated. Relevant connections of some of the results presented here with those that were obtained in earlier investigations are pointed out briefly.

Keywords

Analytic functions starlike functions convex functions close-to-starlike functions Srivastava-Attiya operator inclusion results coefficient bounds coefficient inequalities hadamard product (or convolution) 

MR(2010) Subject Classification

30C10 30C45 11M35 

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References

  1. [1]
    Reade, M. O.: On close-to-convex univalent functions. Michigan Math. J., 3, 59–62 (1955–56)MathSciNetCrossRefGoogle Scholar
  2. [2]
    Srivastava, H. M., Owa, S.: Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, Hong Kong, 1982Google Scholar
  3. [3]
    Owa, S., Nunokawa, M., Saitoh, H., Srivastava, H. M.: Close-to-convexity, starlikeness and convexity for certain analytic functions. Appl. Math. Lett., 15, 63–69 (2002)MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Blezu, D., Pascu, N. N.: Integral of univalent functions. Mathematica (Cluj), 46, 5–8 (1981)MathSciNetGoogle Scholar
  5. [5]
    Ruscheweyh, St., Sheil-Small, T.: Hadamard products of schlicht functions and the Pólya-Schoenberg conjecture. Comment. Math. Helv., 48, 119–135 (1973)MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Srivastava, H. M., Choi, J.: Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001MATHCrossRefGoogle Scholar
  7. [7]
    Srivastava, H. M., Choi, J.: Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London, New York, 2012Google Scholar
  8. [8]
    Choi, J., Srivastava, H. M.: Certain families of series associated with the Hurwitz-Lerch Zeta function. Appl. Math. Comput., 170, 399–409 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Ferreira, C., López, J. L.: Asymptotic expansions of the Hurwitz-Lerch Zeta function. J. Math. Anal. Appl., 298, 210–224 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Garg, M., Jain, K., Srivastava, H. M.: Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch Zeta functions. Integral Transforms Spec. Funct., 17, 803–815 (2006)MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Lin, S.-D., Srivastava, H. M.: Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations. Appl. Math. Comput., 154, 725–733 (2004)MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Lin, S.-D., Srivastava, H. M., Wang, P.-Y.: Some expansion formulas for a class of generalized Hurwitz-Lerch Zeta functions. Integral Transforms Spec. Funct., 17, 817–827 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Luo, Q.-M., Srivastava, H. M.: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl., 308, 290–302 (2005)MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Srivastava, H. M.: Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inform. Sci., 5, 390–444 (2011)Google Scholar
  15. [15]
    Srivastava, H. M., Attiya, A. A.: An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination. Integral Transforms Spec. Funct., 18, 207–216 (2007)MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Alexander, J. W.: Functions which map the interior of the unit circle upon simple regions. Ann. of Math. (Ser. 2), 17, 12–22 (1915)MATHCrossRefGoogle Scholar
  17. [17]
    Bernardi, S. D.: Convex and starlike functions. Trans. Amer. Math. Soc., 135, 429–446 (1969)MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Libera, R. J.: Some classes of regular univalent functions. Proc. Amer. Math. Soc., 16, 755–758 (1969)MathSciNetCrossRefGoogle Scholar
  19. [19]
    Jung, I. B., Kim, Y. C., Srivastava, H. M.: The Hardy space of analytic functions associated with certain one-parameter families of integral operators. J. Math. Anal. Appl., 176, 138–147 (1993)MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Sălăgean, G. S.: Subclasses of univalent functions. In: Complex Analysis, Fifth Romanian-Finnish Seminar, Part I (Bucharest, 1981), pp. 362–372, Lecture Notes in Mathematics, Vol. 1013, Springer-Verlag, Berlin, Heidelberg, New York, 1983CrossRefGoogle Scholar
  21. [21]
    Al-Oboudi, F. M.: On univalent functions defined by a generalized Sălăgean operator. Internat. J. Math. Math. Sci., 27, 1429–1436 (2004)MathSciNetCrossRefGoogle Scholar
  22. [22]
    Goodman, A. W.: Univalent Functions, Vols. 1 and 2, Mariner Publishing Company, Tampa, Florida, 1983Google Scholar
  23. [23]
    Răducanu, D., Srivastava, H. M.: A new class of analytic functions defined by means of a convolution operator involving the Hurwitz-Lerch Zeta function. Integral Transforms Spec. Funct., 18, 933–943 (2007)MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Ruscheweyh, St.: Convolutions in Geometric Function Theory, Les Presses de l’Université de Montréal, Montréal, 1982MATHGoogle Scholar
  25. [25]
    Sun, Y., Kuang, W.-P., Wang, Z.-G.: Properies of uniformly starlike and related functions under the Srivastava-Attiya operator. Appl. Math. Comput., 218, 3615–3623 (2011)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • H. M. Srivastava
    • 1
  • Dorina Răducanu
    • 2
  • Grigore S. Sălăgean
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Faculty of Mathematics and Computer ScienceTransilvania University of BraşovBraşovRomania
  3. 3.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania

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