Functional inequalities for the incomplete gamma function

We present several inequalities forfa(x)=Γ(a,x)Γ(a,0)(a>0,x⩾0), where Γ(a,x)Γ(a,x) is the incomplete gamma function. One of our theorems states that the inequalitiesfa(Sp(x1,…,xn))⩽fa(x1)⋯fa(xn)⩽fa(Sq(x1,…,xn))(p,q>0) hold for all nonnegative real numbers x1,…,xnx1,…,xn(n⩾2)(n⩾2) if and only if p⩽min(a,1)p⩽min(a,1) and q⩾max(a,1)q⩾max(a,1). Here, St(x1,…,xn)St(x1,…,xn) denotes the power sum of order t. This extends and complements a result published by Ismail and Laforgia in 2006.