Skip to main content
SpringerLink
Log in

Turán Type Inequalities for Classical and Generalized Mittag-Leffler Functions

  • Published:
Analysis Mathematica Aims and scope Submit manuscript

Abstract

In this paper some Turán type inequalities for classical and generalized Mittag-Leffler functions are considered. The method is based on proving monotonicity for the special ratio of sections for series of such functions. Some applications are considered to Lazarević type and Wilker type inequalities for classical and generalized Mittag-Leffler functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Á. Baricz, Turàn type inequalities for hypergeometric functions, Proc. Amer. Math. Soc., 136 (2008), 3223–3229.

    Article  MathSciNet  MATH  Google Scholar 

  2. Á. Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math., 26 (2008), 279–293.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. W. Barnard, M. B. Gordy and K. C. Richards, A note on Turàn type and mean inequalities for the Kummer function, J. Math. Anal. Appl., 349 (2009), 259–263.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Biernacki and J. Krzyz, On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. M. Curie-Sk-lodowska, 2 (1995), 134–145.

    Google Scholar 

  5. P. S. Bullen, D. S. Mitrinović and P. M. Vasić, Means and their Inequalities, D. Reidel Publ. (Dordrecht, 1988).

    Book  MATH  Google Scholar 

  6. K. Diethelm, The Analysis of Fractional Differential Equations: an Applicationoriented Exposition Using Differential Operators of Caputo Type, Lecture notes in math., Springer-Verlag (Heidelberg–New York, 2010).

    Book  MATH  Google Scholar 

  7. M. M. Dzherbashyan, Integral Transform Representations of Functions in the Complex Domain, Nauka (Moscow, 1966) (in Russian).

    Google Scholar 

  8. S. D. Eidelman, S. D. Ivasyshen and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-differential Equations of Parabolic Type, Birkhäuser (Basel, 2004).

    Book  MATH  Google Scholar 

  9. R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: Fractals and Fractional Calculus in Continuum Mechanics, Springer (Vienna, 1997), pp. 223–276.

    Chapter  Google Scholar 

  10. R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer (New York, 2014).

    Book  MATH  Google Scholar 

  11. S. I. Kalmykov and D. B. Karp, Log-concavity for series in reciprocal gamma functions, Integral Transforms Spec. Funct., 24 (2013), 859–872.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. I. Kalmykov and D. B. Karp, Log-convexity and log-concavity for series in gamma ratios and applications, J. Math. Anal. Appl., 406 (2013), 400–418.

    Article  MathSciNet  MATH  Google Scholar 

  13. D. B. Karp and S. M. Sitnik, Log-convexity and log-concavity of hypergeometric-like functions, J. Math. Anal. Appl., 364 (2010), 384–394.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. B. Karp and S. M. Sitnik, Inequalities and monotonicity of ratios for generalized hypergeometric function, J. Approx. Theory, 161 (2009), 337–352.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier (Amsterdam, 2006).

    MATH  Google Scholar 

  16. A. A. Kilbas, H-transforms, Theory and Applications, Chapman & Hall/CRC (Boca Raton, FL, 2004).

    Book  MATH  Google Scholar 

  17. V. Kiryakova, The multi-indexMittag-Leffler functions as an important class of special functions of fractional calculus, Comput. Math. Appl., 59 (2010), 1885–1895.

    Article  MathSciNet  MATH  Google Scholar 

  18. V. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math., 118 (2000), 241–259.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. N. Kochubei, A Cauchy problem for evolution equations of fractional order, Diff. Equ., 25 (1989), 967–974.

    MathSciNet  Google Scholar 

  20. A. N. Kochubei, Fractional-order diffusion, Diff. Equ., 26 (1990), 485–492.

    MATH  Google Scholar 

  21. I. Lazarević, Certain inequalities with hyperbolic functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 159–170 (1966), 41–48 (in Serbo-Croatian).

    MathSciNet  Google Scholar 

  22. R. J. McEliece, B. Reznick and J. B. Shearer, A Turàn inequality arising in information theory, SIAM J. Math. Anal., 12 (1981), 931–934.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. M. Mathai and R. Saxena, The H-function with Applications in Statistics and Other Disciplines, Halsted Press (John Wiley & Sons) (New York–London–Sidney, 1978).

    MATH  Google Scholar 

  24. A. M. Mathai, R. Saxena, R. Kishore and H. J. Haubold, The H-function, Springer–Verlag (Berlin–New York, 2010).

    Book  MATH  Google Scholar 

  25. K. Mehrez, M. Ben Said and J. El Kamel, Turàn type inequalities for Dunkl and q-Dunkl kernel, arXiv: 1503.04285.

  26. K. Mehrez and S. M. Sitnik, Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions, arXiv: 1410.6120v2 [math.CA] (2014).

    Google Scholar 

  27. K. Mehrez and S. M. Sitnik, Inequalities for sections of exponential function series and proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions, RGMIA Res. Rep. Collect., 17 (2014), Article ID 132.

  28. K. Mehrez and S. M. Sitnik, Proofs of some conjectures on monotonicity of ratios of Kummer and Gauss hypergeometric functions and related Turàn-type inequalities, Analysis (2016) (accepted for publication).

    Google Scholar 

  29. K. Mehrez and S. M. Sitnik, On monotonicity of ratios of some hypergeometric functions, Sib. Electron. Mat. Izv., 13 (2016), 260–268 (in Russian).

    MathSciNet  MATH  Google Scholar 

  30. K. Mehrez and S. M. Sitnik, On monotonicity of ratios of q-Kummer confluent hypergeometric and q-hypergeometric functions and associated Turàn types inequalities, RGMIA Res. Rep. Collect., 17 (2014), Article ID 150.

  31. K. Mehrez and S. M. Sitnik, On monotonicity of ratios of some q-hypergeometric functions, Mat. Vesnik, 68 (2016), 225–231.

    MathSciNet  MATH  Google Scholar 

  32. D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer (Dordrecht, 1993).

    Book  MATH  Google Scholar 

  33. I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of their Solution and Some of their Applications, Mathematics in Science and Engineering, vol. 198, Academic Press (San Diego, CA, 1998).

    MATH  Google Scholar 

  34. S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika, 44 (1997), 278–301.

    Article  MathSciNet  MATH  Google Scholar 

  35. T. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7–15.

    MathSciNet  MATH  Google Scholar 

  36. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach (Yverdon, 1993).

    MATH  Google Scholar 

  37. S. M. Sitnik, Inequalities for the exponential remainder (preprint), Institute of Automation and Control Process, Far Eastern Branch of the Russian Academy of Sciences, Vladivostok, 1993 (in Russian).

    Google Scholar 

  38. S. M. Sitnik, A conjecture on monotonicity of a ratio of Kummer hypergeometric functions, arXiv: 1207.0936 (2012, 2014).

    Google Scholar 

  39. S. M. Sitnik, Conjectures on monotonicity of ratios of Kummer and Gauss hypergeometric functions, RGMIA Res. Rep. Collect., 17 (2014), Article Id. 107.

  40. S. M. Sitnik, Generalized Young and Cauchy–Bunyakowsky inequalities with applications: a survey, arXiv: 1012.3864 (2012).

    Google Scholar 

  41. H. M. Srivastava and H. A. Manocha, A treatise on generating functions, Bull. Amer. Math. Soc., 19 (1988), 346–348.

    Article  MathSciNet  Google Scholar 

  42. H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-functions of One and Two Variables, South Asian Publishers Pvt. Ltd. (New Delhi, 1988).

    Google Scholar 

  43. A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Lefer function and its properties, J. Math. Anal. Appl., 336 (2007), 797–811.

    Article  MathSciNet  MATH  Google Scholar 

  44. P. Turán, On the zeros of the polynomials of Legendre, Casopis Pest. Mat. Fys., 75 (1950), 113–122.

    MathSciNet  MATH  Google Scholar 

  45. Y. Sun and Á. Baricz, Inequalities for the generalized Marcum Q-function, Appl. Math. Comput., 203 (2008), 134–141.

    MathSciNet  MATH  Google Scholar 

  46. G. Szegő, Orthogonal Polynomials, AMS (Providence, RI, 1939).

    MATH  Google Scholar 

  47. F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press (Cambridge, 2010).

    MATH  Google Scholar 

  48. J. B. Wilker, Problem E 3306, Amer. Math. Monthly, 96 (1989), 55.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques ISSAT Kasserine, Université de Kairouan, Kairouan, Tunisia

    K. Mehrez

  2. Faculty of Sciences of Tunis, University of Tunis El Manar, 1060, Tunis, Tunisia

    K. Mehrez

  3. Voronezh Institute of the Ministry of Internal Affairs of Russia, Patriotov pr. 53, Voronezh, 394065, Russia

    S. M. Sitnik

Authors
  1. K. Mehrez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. M. Sitnik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. M. Sitnik.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mehrez, K., Sitnik, S.M. Turán Type Inequalities for Classical and Generalized Mittag-Leffler Functions. Anal Math 44, 521–541 (2018). https://doi.org/10.1007/s10476-018-0404-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10476-018-0404-9

Key words and phrases

Mathematics Subject Classification

Search

Navigation