Gamma function inequalities

摘要

We prove various new inequalities for Euler’s gamma function. One of our theorems states that the double-inequality $$alpha cdot Bigl(frac{1}{Gamma,(sqrt{x})}+frac{1}{Gamma,(sqrt{y})}Bigr) {kern-1pt}<{kern-1pt} frac{1}{Gamma,( sqrt{x+y-xy})}+ frac{1}{Gamma,( sqrt{xy})} {kern-1pt} <{kern-1pt} beta cdot Bigl( frac{1}{Gamma,(sqrt{x})}+frac{1}{Gamma,(sqrt{y})}Bigr)$$ is valid for all real numbers x,y ∈ (0,1) with the best possible constant factors $alpha=1/sqrt{2}=0.707...$ and β = 1.