A characterization of Euler's constant (Q392150)

For Euler's gamma function \(\Gamma(x)\) the author proves some new interesting inequalities. The main result says that if \(a\) and \(b\) are real numbers then the inequality NEWLINE\[NEWLINE\Gamma\left(x^a +y^b\right) \leq \Gamma\left(\Gamma(x)+\Gamma(y)\right)NEWLINE\]NEWLINE is true for all \(x, y>0\) if and only if \(a=b=-\gamma\), where \(\gamma=0.57721\ldots\) is the Euler constant. This result gives ``a new characterization of the famous Euler constant \(\gamma\)''.NEWLINENEWLINETo obtain the main theorem the author applies a number of very subtle inequalities for the gamma function which are interesting in their own right. The one particularly worth noting is: NEWLINE\[NEWLINE\Gamma(x)\geq x^{-\gamma}NEWLINE\]NEWLINE for all positive \(x\), where the equality holds only for \(x=1\).