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Treatise on Generalized Krätzel Function

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Abstract

The paper deals with complete monotonicity and log-convexity of the type-1 and type-2 generalized Krätzel functions. The Laguerre type and Turán type inequalities for these functions are also substantiated. The generalized Krätzel functions of both types were brought in closed form via Fox’s H-function and existence conditions are given. The asymptotic behavior of these functions at zero and infinity were deduced and methods for the evaluation of some integrals involving the generalized Krätzel function are indicated.

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References

  1. Alzer, H., Felder, G.: A Turán-type inequality for the gamma function. J. Math. Anal. Appl. 350, 276–282 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baricz, Á.: Turán type inequalities for the generalized complete elliptic integrals. Math. Z. 256, 895–911 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baricz, Á.: Turán type inequalities for hypergeometric functions. Proc. Am. Math. Soc. 136, 3223–3229 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baricz, Á.: Turán type inequalities for some probability density functions. Studia Sci. Math. Hung. 47, 175–189 (2010)

    MATH  MathSciNet  Google Scholar 

  5. Baricz, Á.: Turán type inequalities for modified Bessel functions. Bull. Aust. Math. Soc. 82, 254–264 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Baricz, Á., Jankov, D., Pogány, T.K.: Turán type inequalities for Krätzel functions. J. Math. Anal. Appl. 388, 716–724 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bonilla, B., Rivero, M., Rodríguez, J., Trujillo, J.J., Kilbas, A.A.: Bessel-type functions and Bessel-type integral transforms on spaces \(\mathcal{F}_{p,\mu}\) and \(\mathcal{F}_{p,\mu}^{\prime}\). Integral Transforms Spec. Funct. 8, 13–30 (1999)

    Article  MATH  Google Scholar 

  8. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, Vol. II. McGraw-Hill, New York (1953)

    MATH  Google Scholar 

  9. Fox, C.: The G and H functions as symmetrical Fourier kernels. Trans. Am. Math. Soc. 98, 395–429 (1961)

    MATH  Google Scholar 

  10. Kilbas, A.A., Kumar, D.: On generalized Krätzel function. Integral Transforms Spec. Funct. 20, 835–846 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kilbas, A.A., Saigo, M.: H-Transforms: Theory and Applications. Chapman & Hall/CRC Press, Boca Raton (2004)

    Book  Google Scholar 

  12. Kilbas, A.A., Saxena, R.K., Trujillo, J.J.: Krätzel function as a function of hypergeometric type. Fract. Calc. Appl. Anal. 9, 109–131 (2006)

    MATH  MathSciNet  Google Scholar 

  13. Kilbas, A.A., Rodríguez-Germá, L., Saigo, M., Saxena, R.K., Trujillo, J.J.: The Krätzel function and evaluation of integrals. Comput. Math. Appl. 59, 1790–1800 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Krätzel, E.: Integral transformations of Bessel type. In: Generalized Functions and Operational Calculus, Proc. Conf., Varna, 1975. pp. 148–155. Bulgarian Academy of Sciences, Sofia (1979)

    Google Scholar 

  15. Kumar, D.: Some connections among generalized Krätzel function, \(\mathcal{P}\)-transform and their applications. In: Proc. Natl. Workshop Fract. Cal. Stat. Distrib., vol. 41, pp. 47–60 (2009)

    Google Scholar 

  16. Kumar, D.: Type-1 \(\mathcal{P}\)-transform. In: Proc. Int. Conf. Math. Sci. Honour of Prof. A.M. Mathai, pp. 191–204 (2010)

    Google Scholar 

  17. Kumar, D.: \(\mathcal{P}\)-transform. Integral Transforms Spec. Funct. 22, 603–616 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kumar, D., Kilbas, A.A.: Fractional calculus of \(\mathcal {P}\)-transforms. Fract. Calc. Appl. Anal. 13, 309–328 (2010)

    MATH  MathSciNet  Google Scholar 

  19. Mathai, A.M.: A pathway to matrix-variate gamma and normal densities. Linear Algebra Appl. 396, 317–328 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mathai, A.M., Haubold, H.J.: Pathway model, superstatistics, Tsallis statistics, and a generalized measure of entropy. Physica A 375, 110–122 (2007)

    Article  MathSciNet  Google Scholar 

  21. Mathai, A.M., Saxena, R.K., Haubold, H.J.: The H-Function Theory and Applications. Springer, New York (2010)

    MATH  Google Scholar 

  22. Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)

    Google Scholar 

  23. Rodríguez, J., Trujillo, J.J., Rivero, M.: Operational fractional calculus of Krätzel transformation. In: Lect. Notes Pure Appl. Math., vol. 118, pp. 613–620. Denver, New York (1989)

    Google Scholar 

  24. Szegő, G.: On an inequality of P. Turán concerning Legendre polynomials. Bull. Am. Math. Soc. 54, 401–405 (1948)

    Article  Google Scholar 

  25. Turán, P.: On the zeros of the polynomial of Legendre. Čas. Pěst. Math. Fys. 75, 113–122 (1950)

    MATH  Google Scholar 

  26. Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1941)

    Google Scholar 

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Acknowledgements

The authors would like to acknowledge the unknown referees for their valuable comments and suggestions which enhanced the paper in the present form. The authors would like to thank the Department of Science and Technology, Government of India, New Delhi, for the financial assistance for this work under project No. SR/S4/MS:287/05, and the Centre for Mathematical Sciences for providing all facilities.

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  1. Centre for Mathematical Sciences, Arunapuram P.O., Palai, Kerala, 686 574, India

    Deepa H. Nair & Dilip Kumar

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  1. Deepa H. Nair
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  2. Dilip Kumar
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Correspondence to Dilip Kumar.

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Nair, D.H., Kumar, D. Treatise on Generalized Krätzel Function. Vietnam J. Math. 43, 23–36 (2015). https://doi.org/10.1007/s10013-014-0058-2

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  • DOI: https://doi.org/10.1007/s10013-014-0058-2

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