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FUNCTIONAL INEQUALITIES FOR THE q-DIGAMMA FUNCTION

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Abstract

We present functional inequalities for the q-digamma function \(\psi_q=\Gamma{'}_q/{\Gamma}_{q}\), where \(\Gamma_q\) denotes the q-gamma function. Among others, we prove that for all positive real numbers a, b with \(a\neq b\) we have

$$\frac{\psi_q(a)+\psi_q(b)}{2}<\psi_q\bigl( \sqrt{ab} \bigr), \quad 0<q<1,$$

and

$$a\psi_q \bigl( e^{1/\sqrt{a}} \bigr)+b\psi_q \bigl( e^{1/\sqrt{b}} \bigr)<(a+b)\psi_q \bigl( e^{\sqrt{2/(a+b)} } \bigr), \quad 0<q<1.$$

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References

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

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

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

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

  5. M. El Bachraoui and J. Griffin, Reciprocity theorems involving the q-gamma function,Ramanujan J., 45 (2018), 683–694.

  6. M. E.H. Ismail and M. E. Muldoon, Inequalities and monotonicity properties for gamma and q-gamma functions, in: Approximation and Computation, R.V.M.Zahar, Ed., Birkhäuser (Boston, 1994), pp. 309–322.

  7. H.-H. Kairies, Charakterisierungen und Ungleichungen für die q-Factorial-Funktionen, in: General Inequalities 4, W. Walter, Birkhäuser (Basel, 1984), pp. 257–267.

  8. K. Knopp, Theorie und Anwendung der unendlichen Reihen, Springer (Berlin, 1964).

  9. Gy. Maksa and Zs. Páles, On Hosszú’s functional inequality, Publ. Math. Debrecen,36 (1989), 187–189.

  10. T. Mansour and A. Sh. Shabani, Some inequalities for the q-digamma function, J. Inequal.Pure Appl. Math., 10 (2009), Article 12, 8 pp.

  11. D.S. Mitrinović, Analytic Inequalities, Springer (New York, 1970).

  12. D.S. Moak, The q-gamma function for q > 1, Aequat. Math., 20 (1980), 278–285.

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

  14. A. Salem, An infinite class of completely monotonic functions involving the q-gamma function, J. Math. Anal. Appl., 406 (2013), 392–399.

  15. A. Salem, Two classes of bounds for the q-gamma and the q-digamma functions in terms of the q-zeta functions, Banach J. Math. Anal., 8 (2014), 109–117.

  16. A. Salem, Monotonic functions related to the q-gamma function, Monatsh. Math., 179 (2016), 281–292.

  17. A. Salem, Some classes of completely monotonic functions related to q-gamma and q-digamma functions, Math. Inequal. Appl., 19 (2016), 853–862.

  18. A. Salem, Generalized the q-digamma and the q-polygamma functions via neutrices,Filomat, 31 (2017), 1475–1481.

  19. A. Salem, Sharp bounds for the q-gamma function in terms of the Lambert W function,Ramanujan J., 49 (2018), 321–339.

  20. A. Salem, Sharp lower and upper bounds for the q-gamma function, Math. Inequal.Appl., 23 (2020), 855–872.

  21. A. Salem and F. Alzahrani, Improvements of bounds for the q-gamma and the q-polygamma functions, J. Math. Inequal., 11 (2017), 873–883.

  22. E.Ṁ . Wright, A generalisation of Schur’s inequality, Math. Gaz., 40 (1956), 217.

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We thank the referee for the careful reading of the manuscript.

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ALZER, H., SALEM, A. FUNCTIONAL INEQUALITIES FOR THE q-DIGAMMA FUNCTION. Acta Math. Hungar. 167, 561–575 (2022). https://doi.org/10.1007/s10474-022-01247-w

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