Skip to main content
Log in

On Ivády’s bounds for the gamma function and related results

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We prove that the inequality

$$\begin{aligned} \Gamma (x+1)\le \frac{x^2+\beta }{x+\beta } \end{aligned}$$

holds for all \(x\in [0,1]\), \(\beta \ge {\beta ^{*}}\), with the best possible constant

$$\begin{aligned} \beta ^*=\max _{0.1\le x\le 0.3} f(x)=1.75527\ldots , \end{aligned}$$

where f is given by

$$\begin{aligned} f(x)=\frac{x\Gamma (x+1)-x^2}{1-\Gamma (x+1)}. \end{aligned}$$

This refines bounds given by Ivády (J Math Inequal 3:227–236, 2009) and Yang et al. (J Inequal Appl 2017(1):210, 2017). Moreover, we show that f is strictly concave on [0, 1] and we apply this result to obtain some functional inequalities for the gamma function.

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

Access this article

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. M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1975)

    MATH  Google Scholar 

  2. H. Alzer, Inequalities for Euler’s gamma function. Forum Math. 20, 955–1004 (2008)

    Article  MathSciNet  Google Scholar 

  3. H. Alzer, Gamma function inequalities. Numer. Algorithms 49, 53–84 (2008)

    Article  MathSciNet  Google Scholar 

  4. J.J. Carmona, J.J. Donaire, Revisiting some properties of the gamma function. Math. Proc. R. Ir. Acad. 114A, 17–35 (2014)

    Article  MathSciNet  Google Scholar 

  5. P.J. Davis, Leonhard Euler’s integral: a historical profile of the gamma function. Am. Math. Mon. 66, 849–869 (1959)

    MathSciNet  MATH  Google Scholar 

  6. B. Fuglede, A sharpening of Wielandt’s characterization of the gamma function. Am. Math. Mon. 115, 845–850 (2008)

    Article  MathSciNet  Google Scholar 

  7. L. Gordon, A stochastic approach to the gamma function. Am. Math. Mon. 101, 858–865 (1994)

    Article  MathSciNet  Google Scholar 

  8. P. Ivády, A note on a gamma function inequality. J. Math. Inequal. 3, 227–236 (2009)

    Article  MathSciNet  Google Scholar 

  9. G. Maksa, Z. Páles, On Hosszú’s functional inequality. Publ. Math. Debr. 36, 187–189 (1989)

    MATH  Google Scholar 

  10. W. Paulsen, Gamma triads. Ramanujan J. 50, 123–133 (2019)

    Article  MathSciNet  Google Scholar 

  11. R. Remmert, Wielandt’s theorem about the \(\Gamma \)-function. Am. Math. Mon. 103, 214–220 (1996)

    MathSciNet  MATH  Google Scholar 

  12. J. Sándor, A bibliography on gamma functions: inequalities and applications, http://www.math.ubbcluj.ro/~jsandor/letolt/art42

  13. G.K. Srinivasan, The gamma function: an eclectic tour. Am. Math. Mon. 114, 297–315 (2007)

    Article  MathSciNet  Google Scholar 

  14. B.L. van der Waerden, Algebra I (Springer, Berlin, 1971)

    Book  Google Scholar 

  15. R. Vidūnas, Expressions for values of the gamma function. Kyushu J. Math. 59, 267–283 (2005)

    Article  MathSciNet  Google Scholar 

  16. Z.-H. Yang, W.-M. Qian, Y.-M. Chu, W. Zhang, On rational bounds for the gamma function. J. Inequal. Appl. 2017(1), 210 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Horst Alzer.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alzer, H., Kwong, M.K. On Ivády’s bounds for the gamma function and related results. Period Math Hung 82, 115–124 (2021). https://doi.org/10.1007/s10998-020-00356-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-020-00356-9

Keywords

Mathematics Subject Classification

Navigation