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Complete monotonicity of functions involving the \(q\)-trigamma and \({q}\)-tetragamma functions

Original Paper

Abstract

Let \(\psi _q(x)\), \(\psi _q'(x)\), and \(\psi _q''(x)\) for \(q>0\) stand respectively for the \(q\)-digamma, \(q\)-trigamma, and \(q\)-tetragamma functions. In the paper, the author proves along two different approaches that the functions \([\psi '_q(x)]^2+\psi ''_q(x)\) for \(q>1\) and \([\psi _{q}'(x)-\ln q]^2 +\psi ''_{q}(x)\) for \(0<q<1\) are completely monotonic on \((0,\infty )\). Applying these results, the author derives monotonic properties of four functions involving the \(q\)-digamma function \(\psi _q(x)\) and two double inequalities for bounding the \(q\)-digamma function \(\psi _q(x)\).

Keywords

Completely monotonic function Monotonicity Inequality \(q\)-Digamma function \(q\)-Trigamma function \(q\)-Tetragamma function 

Mathematics Subject Classification

Primary 33D05 Secondary 26A12 26A48 26D07 33B15 

Notes

Acknowledgments

The author appreciates anonymous referees for their comments on and helpful suggestions to the original version of this paper.

References

  1. 1.
    Alzer, H., Grinshpan, A.Z.: Grinshpan, Inequalities for the gamma and \(q\)-gamma functions. J. Approx. Theory 144(1), 67–83 (2007). doi:  10.1016/j.jat.2006.04.008 zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E., Askey, R., Roy, R.: Special functions. In: Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  3. 3.
    Batir, N.: An interesting double inequality for Euler’s gamma function. J. Inequal. Pure Appl. Math. 5(4). Article 97 (2004). http://www.emis.de/journals/JIPAM/article452.html
  4. 4.
    Batir, N.: On some properties of digamma and polygamma functions. J. Math. Anal. Appl. 328(1), 452–465 (2007). doi: 10.1016/j.jmaa.2006.05.065 zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Batir, N.: Some new inequalities for gamma and polygamma functions. J. Inequal. Pure Appl. Math. 6(4). Article 103 (2005). http://www.emis.de/journals/JIPAM/article577.html
  6. 6.
    Elezović, N., Giordano, C., Pečarić, J.: The best bounds in Gautschi’s inequality. Math. Inequal. Appl. 3(2), 239–252 (2000). doi: 10.7153/mia-03-26 zbMATHMathSciNetGoogle Scholar
  7. 7.
    Gasper, G., Rahman, M.: Encyclopedia of Mathematics and its Applications. Basic hypergeometric series, 2nd edn. Cambridge University Press, Cambridge (2004)Google Scholar
  8. 8.
    Guo, B.N., Qi, F.: A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications. J. Korean Math. Soc. 48(3), 655–667 (2011). doi: 10.4134/JKMS.2011.48.3.655 zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Guo, B.N., Qi, F.: Properties and applications of a function involving exponential functions. Commun. Pure Appl. Anal. 8(4), 1231–1249 (2009). doi: 10.3934/cpaa.2009.8.1231 zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Guo, B.N., Qi, F.: Some properties of the psi and polygamma functions. Hacet. J. Math. Stat. 39(2), 219–231 (2010)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Ismail, M.E.H., Muldoon, M.E.: Inequalities and monotonicity properties for gamma and \(q\)-gamma functions. In: Zahar, R.V.M. (ed.) Approximation and Computation: A Festschrift in Honour of Walter Gautschi, ISNM, vol. 119. pp. 309–323. Birkhauser, Basel (1994). doi: 10.1007/978-1-4684-7415-2_19
  12. 12.
    Ismail, M.E.H., Muldoon, M.E.: Inequalities and monotonicity properties for gamma and \(q\)-gamma functions. http://arxiv.org/abs/1301.1749
  13. 13.
    Li, W.H., Qi, F., Guo, B.N.: On proofs for monotonicity of a function involving the psi and exponential functions. Analysis (Munich) 33(1), 45–50 (2013). doi: 10.1524/anly.2013.1175 zbMATHMathSciNetGoogle Scholar
  14. 14.
    Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer, Dordrecht (1993)zbMATHCrossRefGoogle Scholar
  15. 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)MathSciNetGoogle Scholar
  16. 16.
    Qi, F.: Bounds for the ratio of two gamma functions. J. Inequal. Appl. 2010, 1–84 (2010). Article Id 493058. doi: 10.1155/2010/493058
  17. 17.
    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). doi: 10.1017/S0004972712001025 zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Qi, F., Guo, B.N.: Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications. Commun. Pure Appl. Anal. 8(6), 1975–1989 (2009). doi: 10.3934/cpaa.2009.8.1975 zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Qi, F., Guo, B.N.: Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic. Adv. Appl. Math. 44(1), 71–83 (2010). doi: 10.1016/j.aam.2009.03.003 zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Qi, F., Luo, Q.M., Guo, B.N.: Complete monotonicity of a function involving the divided difference of digamma functions. Sci. China Math. 56(11), 2315–2325 (2013). doi: 10.1007/s11425-012-4562-0 zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Qi, F., Luo, Q.M.: Bounds for the ratio of two gamma functions—from Wendel’s and related inequalities to logarithmically completely monotonic functions. Banach J. Math. Anal. 6(2), 132–158 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1946)Google Scholar

Copyright information

© Springer-Verlag Italia 2014

Authors and Affiliations

  1. 1.Institute of MathematicsHenan Polytechnic UniversityJiaozuoChina

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