Functional inequalities for the incomplete gamma function

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