On Ivády's bounds for the gamma function and related results (Q2043710)

Improving a bound given by \textit{P. Ivády} [J. Math. Inequal. 3, No. 2, Article ID 23, 227--236 (2009; Zbl 1179.26052)] and answering a question of \textit{Z.-H. Yang} et al. [J. Inequal. Appl. 2017, Paper No. 210, 17 p. (2017; Zbl 1370.41056)], the authors prove that the inequality \[ \Gamma(x+1)\leq \frac{x^2+\beta}{x+\beta} \] holds for all \(x\in [0,1]\), \(\beta\geq \beta^*\), with the best possible constant \[\beta^{*}=\max_{0.1\leq x\leq 0.3}f(x)= 1.75527\ldots,\] where \(f\) is given by \[ f(x)=\frac{x\Gamma(x+1)-x^2}{1-\Gamma(x+1)}. \] Moreover, they show that \(f\) is strictly concave on \([0, 1]\) and apply this result to obtain some interesting functional inequalities for the gamma function.