Abstract
We prove that the inequality
holds for all \(x\in [0,1]\), \(\beta \ge {\beta ^{*}}\), with the best possible constant
where f is given by
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.
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References
M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1975)
H. Alzer, Inequalities for Euler’s gamma function. Forum Math. 20, 955–1004 (2008)
H. Alzer, Gamma function inequalities. Numer. Algorithms 49, 53–84 (2008)
J.J. Carmona, J.J. Donaire, Revisiting some properties of the gamma function. Math. Proc. R. Ir. Acad. 114A, 17–35 (2014)
P.J. Davis, Leonhard Euler’s integral: a historical profile of the gamma function. Am. Math. Mon. 66, 849–869 (1959)
B. Fuglede, A sharpening of Wielandt’s characterization of the gamma function. Am. Math. Mon. 115, 845–850 (2008)
L. Gordon, A stochastic approach to the gamma function. Am. Math. Mon. 101, 858–865 (1994)
P. Ivády, A note on a gamma function inequality. J. Math. Inequal. 3, 227–236 (2009)
G. Maksa, Z. Páles, On Hosszú’s functional inequality. Publ. Math. Debr. 36, 187–189 (1989)
W. Paulsen, Gamma triads. Ramanujan J. 50, 123–133 (2019)
R. Remmert, Wielandt’s theorem about the \(\Gamma \)-function. Am. Math. Mon. 103, 214–220 (1996)
J. Sándor, A bibliography on gamma functions: inequalities and applications, http://www.math.ubbcluj.ro/~jsandor/letolt/art42
G.K. Srinivasan, The gamma function: an eclectic tour. Am. Math. Mon. 114, 297–315 (2007)
B.L. van der Waerden, Algebra I (Springer, Berlin, 1971)
R. Vidūnas, Expressions for values of the gamma function. Kyushu J. Math. 59, 267–283 (2005)
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)
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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
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DOI: https://doi.org/10.1007/s10998-020-00356-9