Abstract and Applied Analysis

Convexity of Certain q -Integral Operators of p -Valent Functions

K. A. Selvakumaran, S. D. Purohit, Aydin Secer, and Mustafa Bayram

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


By applying the concept (and theory) of fractional q -calculus, we first define and introduce two new q -integral operators for certain analytic functions defined in the unit disc 𝒰 . Convexity properties of these q -integral operators on some classes of analytic functions defined by a linear multiplier fractional q -differintegral operator are studied. Special cases of the main results are also mentioned.

Article information

Abstr. Appl. Anal., Volume 2014 (2014), Article ID 925902, 7 pages.

First available in Project Euclid: 2 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Selvakumaran, K. A.; Purohit, S. D.; Secer, Aydin; Bayram, Mustafa. Convexity of Certain $q$ -Integral Operators of $p$ -Valent Functions. Abstr. Appl. Anal. 2014 (2014), Article ID 925902, 7 pages. doi:10.1155/2014/925902. https://projecteuclid.org/euclid.aaa/1412276989

Export citation


  • T. Ernst, A Comprehensive Treatment of q-Calculus, Springer, Basel, Switzerland, 2012.
  • G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 35 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, Mass, USA, 1990.
  • V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002.
  • M. H. Annaby and Z. S. Mansour, q-Fractional Calculus and Equations, vol. 2056 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2012.
  • S. D. Purohit and R. K. Raina, “Certain subclasses of analytic functions associated with fractional $q$-calculus operators,” Mathematica Scandinavica, vol. 109, no. 1, pp. 55–70, 2011.
  • S. D. Purohit and R. K. Raina, “Fractional $q$-calculus and certain subclass of univalent analytic čommentComment on ref. [20?]: Please update the information of this reference, if possible.functions,” Mathematica. In press.
  • S. D. Purohit and R. K. Raina, “On a subclass of $p$-valent analytic functions involving fractional $q$-calculus operators,” Kuwait Journal of Science, vol. 41, no. 3, 2014.
  • S. D. Purohit, “A new class of multivalently analytic functions associated with fractional $q$-calculus operators,” Fractional Differential Calculus, vol. 2, no. 2, pp. 129–138, 2012.
  • A. W. Goodman, “On uniformly starlike functions,” Journal of Mathematical Analysis and Applications, vol. 155, no. 2, pp. 364–370, 1991.
  • W. C. Ma and D. Minda, “Uniformly convex functions,” Annales Polonici Mathematici, vol. 57, no. 2, pp. 165–175, 1992.
  • F. Rønning, “Uniformly convex functions and a corresponding class of starlike functions,” Proceedings of the American Mathematical Society, vol. 118, no. 1, pp. 189–196, 1993.
  • R. Aghalary and J. M. Jahangiri, “Inclusion relations for $k$-uniformly starlike and related functions under certain integral operators,” Bulletin of the Korean Mathematical Society, vol. 42, no. 3, pp. 623–629, 2005.
  • F. M. Al-Oboudi, “On univalent functions defined by a generalized Sǎlǎgean operator,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 25-28, pp. 1429–1436, 2004.
  • G. Ş. Sǎlǎgean, “Subclasses of univalent functions,” in Complex Analysis–-Fifth Romanian-Finnish Seminar, vol. 1013 of Lecture Notes in Math, pp. 362–372, Springer, Berlin, Germany, 1983.
  • B. A. Frasin, “Convexity of integral operators of $p$-valent functions,” Mathematical and Computer Modelling, vol. 51, no. 5-6,pp. 601–605, 2010.
  • D. Breaz, H. Ö. Güney, and G. Ş. Sǎlǎgean, “A new general integral operator,” Tamsui Oxford Journal of Mathematical Sci-ences, vol. 25, no. 4, pp. 407–414, 2009.
  • D. Breaz and N. Breaz, “Two integral operators,” Studia Universitatis Babeş-Bolyai Mathematica, vol. 47, no. 3, pp. 13–19, 2002.
  • V. Pescar and S. Owa, “Sufficient conditions for univalence of certain integral operators,” Indian Journal of Mathematics, vol. 42, no. 3, pp. 347–351, 2000.
  • J. W. Alexander, “Functions which map the interior of the unit circle upon simple regions,” Annals of Mathematics Second Series, vol. 17, no. 1, pp. 12–22, 1915.
  • D. Breaz, S. Owa, and N. Breaz, “A new integral univalent operator,” Acta Universitatis Apulensis Mathematics Informatics, no.16, pp. 11–16, 2008.
  • J. A. Pfaltzgra, “Univalence of the integral of ${f}^{'}{(z)}^{\lambda }$,” The Bulletin London Mathematical Society, vol. 7, no. 3, pp. 254–256, 1975.
  • D. Breaz, “A convexity property for an integral operator on the class ${S}_{p}(\beta )$,” General Mathematics, vol. 15, no. 2, pp. 177–183, 2007.
  • D. Breaz and N. Breaz, “Some convexity properties for a general integral operator,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 5, article 177, 8 pages, 2006. \endinput