A monotonicity property for gamma distributions

Let X1,…,Xk be k(⩾2) independent identically distributed random variables having the gamma distribution with scale and shape parameters 1 and s(⩾1), respectively. Denote the density and cumulative distribution function of X1 by gs(·) and Gs(·), respectively. A quantity which naturally arises in certain ranking and selection procedures is P=P(k,b,s)=∫0∞[Gs(x/b)]k-1gs(x)dx, where 0