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Logarithmically completely monotonic functions related to the q-gamma function and its applications

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Abstract

Our main goal in this paper is to introduce new classes of logarithmically completely monotonic functions involving q-gamma function. As applications, new classes of Bernstein functions related to the q-gamma function and dilogarithm are established with its integral representation. Moreover, various new sharp bounds for the q-digamma and q-trigamma functions are derived. In particular, sharp bounds for the q-analogue of harmonic numbers are established as a consequence. The results obtained in this work are new. The limiting case \(q\rightarrow 1\), in the results obtained in this paper leads to the results for a class of Bernstein functions and logarithmically completely monotonic function involving Euler’s gamma function and dilogarithm, which are also new in the literature.

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References

  1. Gasper, G., Rahman, M.: Basic hypergeometric series, second edition. Encyclopedia of mathematics and its applications, 96. Cambridge University Press, Cambridge (2004)

  2. Floreanini, R., Vinet, L.: \(q\)-gamma and \(q\)-beta functions in quantum algebra representation theory. J. Comput. Appl. Math. 68(1–2), 57–68 (1996)

    Article  MathSciNet  Google Scholar 

  3. Aral, A., Gupta, V., Agarwal, R.P.: Applications of q-calculus in operator theory. Springer, New York (2013)

    Book  Google Scholar 

  4. Jackson, F.H.: On \(q\)-functions and a certain difference operator. Trans. R. Soc. Lond. 46, 253–281 (1908)

    Google Scholar 

  5. Jackson, F.H.: On a q-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)

    MATH  Google Scholar 

  6. Askey, R.: The \(q\)-gamma and \(q\)-beta functions. Appl. Anal. 8(2), 125–141 (1978)

    Article  MathSciNet  Google Scholar 

  7. Alzer, H.: Sharp inequalities for the digamma and polygamma functions. Forum Math. 16(2), 181–221 (2004)

    Article  MathSciNet  Google Scholar 

  8. Batir, N.: \(q\)-extensions of some estimates associated with the digamma function. J. Approx. Theory 174, 54–64 (2013)

    Article  MathSciNet  Google Scholar 

  9. Batir, N.: Monotonicity properties of \(q\)-digamma and \(q\)-trigamma functions. J. Approx. Theory 192, 336–346 (2015)

    Article  MathSciNet  Google Scholar 

  10. Batir, N.: On some properties of digamma and polygamma functions. J. Math. Anal. Appl. 328(1), 452–465 (2007)

    Article  MathSciNet  Google Scholar 

  11. Mehrez, K.: A class of logarithmically completely monotonic functions related to the \(q\)-gamma function and applications. Positivity 21(1), 495–507 (2017)

    Article  MathSciNet  Google Scholar 

  12. Mortici, C.: Very accurate estimates of the polygamma functions. Asymptot. Anal. 68(3), 125–134 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Salem, A.: A completely monotonic function involving \(q\)-gamma and \(q\)-digamma functions. J. Approx. Theory 164(7), 971–980 (2012)

    Article  MathSciNet  Google Scholar 

  14. Salem, A.: Complete monotonicity properties of functions involving \(q\)-gamma and \(q\)-digamma functions. Math. Inequal. Appl. 17(3), 801–811 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Qi, F.: A completely monotonic function related to the \(q\)-trigamma function. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys 76(1), 107–114 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Qi, F.: Complete monotonicity of functions involving the \(q\)-trigamma and \(q\)-tetragamma functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109(2), 419–429 (2015)

    Article  MathSciNet  Google Scholar 

  17. Guo, B.N., Qi, F.: Some properties of the psi and polygamma functions. Hacet. J. Math. Stat. 39(2), 219–231 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Qi, F., Cerone, P., Dragomir, S.S.: Complete monotonicity of a function involving the divided difference of PSI functions. Bull. Aust. Math. Soc. 88(2), 309–319 (2013)

    Article  MathSciNet  Google Scholar 

  19. Qi, F., Liu, F.F., Shi, X.T.: Comments on two completely monotonic functions involving the \(q\)-trigamma function. J. Inequal. Spec. Funct. 7(4), 211–217 (2016)

    MathSciNet  Google Scholar 

  20. Bernstein, S.: Sur les fonctions absolument monotones. Acta Math. 52(1), 1–66 (1929)

    Article  MathSciNet  Google Scholar 

  21. Schilling, R.L., Song, R., Vondraček, Z.: Bernstein functions, second edition. In De Gruyter Studies in Mathematics, vol. 37. Walter de Gruyter & Co., Berlin (2012)

  22. Horn, R.A.: On infinitely divisible matrices, kernels and functions. Z. Wahrscheinlichkeitstheorie. verw. Geb. 8, 219–230 (1967)

    Article  MathSciNet  Google Scholar 

  23. Berg, C.: Integral representation of some functions related to the gamma function. Mediterr. J. Math. 1, 433–439 (2004)

    Article  MathSciNet  Google Scholar 

  24. Berg, C.: Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity Chapter 2. In: Jorge, M., Emilio, P. (eds.) Positive denite functions: from Schoenberg to space-time challenges. University Jaume I, Castellon (2008)

    Google Scholar 

  25. Qi, F., Guo, B.N.: A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72(2), 21–30 (2010)

    MathSciNet  MATH  Google Scholar 

  26. El Kamel, J., Mehrez, K.: A function class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions. Positivity 22(5), 1403–1417 (2018)

    Article  MathSciNet  Google Scholar 

  27. Moak, D.S.: The \(q\)-analogue of Stirling’s formula. Rocky Mt. J. Math. 14(2), 403–413 (1984)

    Article  MathSciNet  Google Scholar 

  28. Salem, A.: A certain class of approximations for the \(q\)-digamma function. Rocky Mt. J. Math. 46(5), 1665–1677 (2016)

    Article  MathSciNet  Google Scholar 

  29. Wei, C., Gu, Q.: \(q\)-generalizations of a family of harmonic number identities. Adv. Appl. Math. 45(1), 24–27 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very grateful to the anonymous referee for the careful reading of the manuscript and for constructive comments, observations, and suggestions that has helped in significantly improving the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Kairouan Preparatory Institute for Engineering Studies, University of Kairouan, 3100, Kairouan, Tunisia

    Khaled Mehrez

  2. Department of Mathematics, National Institute of Technology Jamshedpur, Jamshedpur, Jharkhand, 831014, India

    Sourav Das

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  1. Khaled Mehrez
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  2. Sourav Das
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Mehrez, K., Das, S. Logarithmically completely monotonic functions related to the q-gamma function and its applications. Anal.Math.Phys. 12, 65 (2022). https://doi.org/10.1007/s13324-022-00678-6

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  • DOI: https://doi.org/10.1007/s13324-022-00678-6

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