• Received : 2014.02.17
  • Published : 2015.11.30


In this article, we first consider a linear multiplier fractional q-differintegral operator and then use it to define new subclasses of p-valent analytic functions in the open unit disk U. An attempt has also been made to obtain two new q-integral operators and study their sufficient conditions on some classes of analytic functions. We also point out that the operators and classes presented here, being of general character, are easily reducible to yield many diverse new and known operators and function classes.


  1. J.W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. of Math. (2) 17 (1915), no. 1, 12-22.
  2. F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci. 2004 (2004), no. 25-28, 1429-1436.
  3. D. Breaz, M. K. Aouf, and N. Breaz, Some properties for integral operators on some analytic functions with complex order, Acta Math. Acad. Paedagog. Nyhazi. (N.S.) 25 (2009), no. 1, 39-43.
  4. D. Breaz and N. Breaz, Two integral operators, Studia Univ. Babes-Bolyai Math. 47 (2002), no. 3, 13-19.
  5. D. Breaz and H. O. Guney, The integral operator on the classes $S_{\alpha}^{*}(b)$ and $C_{\alpha}(b)$, J. Math. Inequal. 2 (2008), no. 1, 97-100.
  6. D. Breaz, H. O. Guney, and G. S. Salagean, A new general integral operator, Tamsui Oxf. J. Math. Sci. 25 (2009), no. 4, 407-414.
  7. D. Breaz, S. Owa, and N. Breaz, A new integral univalent operator, Acta Univ. Apulensis Math. Inform. (2008), no. 16, 11-16.
  8. S. Bulut, A note on the paper of Breaz and Guney, J. Math. Inequal. 2 (2008), no. 4, 549-553.
  9. P. N. Chichra, Regular functions f(z) for which zf′(z) is $\alpha$-spiral-like, Proc. Amer. Math. Soc. 49 (1975), 151-160.
  10. B. A. Frasin, Family of analytic functions of complex order, Acta Math. Acad. Paedagog. Nyhazi. (N.S.) 22 (2006), no. 2, 179-191
  11. B. A. Frasin, Convexity of integral operators of p-valent functions, Math. Comput. Modelling 51 (2010), no. 5-6, 601-605.
  12. G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 35, Cambridge Univ. Press, Cambridge, 1990.
  13. R. J. Libera, Univalent $\alpha$-spiral functions, Canad. J. Math. 19 (1967), 449-456.
  14. M. A. Nasr and M. K. Aouf, Starlike function of complex order, J. Natur. Sci. Math. 25 (1985), no. 1, 1-12.
  15. V. Pescar and S. Owa, Sufficient conditions for univalence of certain integral operators, Indian J. Math. 42 (2000), no. 3, 347-351.
  16. J. A. Pfaltzgraff, Univalence of the integral of f′(z), Bull. London Math. Soc. 7 (1975), no. 3, 254-256.
  17. S. D. Purohit, A new class of multivalently analytic functions associated with fractional q-calculus operators, Fract. Differ. Calc. 2 (2012), no. 2, 129-138.
  18. S. D. Purohit and R. K. Raina, Certain subclasses of analytic functions associated with fractional q-calculus operators, Math. Scand. 109 (2011), no. 1, 55-70.
  19. S. D. Purohit and R. K. Raina, Fractional q-calculus and certain subclass of univalent analytic functions, Mathematica 55(78) (2013), no. 1, 62-74.
  20. S. D. Purohit and R. K. Raina, Some classes of analytic and multivalent functions associated with q-derivative operators, Acta Univ. Sapientiae Math. 6 (2014), no. 1, 5-23.
  21. S. D. Purohit and R. K. Raina, On a subclass of p-valent analytic functions involving fractional q-calculus operators, Kuwait J. Sci. 42 (2015), no. 1, 1-15.
  22. G. S. Salagean, Subclasses of univalent functions, in Complex analysisfifth Romanian- Finnish seminar, Part 1 (Bucharest, 1981), 362-372, Lecture Notes in Math., 1013, Springer, Berlin, 1983.
  23. K. A. Selvakumaran, S. D. Purohit, A. Secer, and M. Bayram, Convexity of certain q-integral operators of p-valent functions, Abstr. Appl. Anal. 2014 (2014), Article ID 925902, 7 pp.
  24. P. Wiatrowski, The coefficients of a certain family of holomorphic functions, Zeszyty Nauk. Uniw. Lodz. Nauki Mat. Przyrod. Ser. II No. 39 Mat. (1971), 75-85.