(Sharp) inequality for Beta function
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I am trying to prove the following inequality concerning the Beta Function: $$ alpha x^alpha B(alpha, xalpha) geq 1 quad forall 0 < alpha leq 1, x > 0, $$ where as usual $B(a,b) = int_0^1 t^{a-1}(1-t)^{b-1}dt$ . In fact, I only need this inequality when $x$ is large enough, but it empirically seems to be true for all $x$ . The main reason why I'm confident that the result is true is that it is very easy to plot, and I've experimentally checked it for reasonable values of $x$ (say between 0 and $10^{10}$ ). For example, for $x=100$ , the plot is: Varying $x$ , it seems that the inequality is rather sharp, namely I was not able to find a point where that product is larger than around $1.5$ (but I do not need any such reverse inequality). I know very little about Beta functions, therefore