On the Raşa Inequality for Higher Order Convex Functions

Abstract

We study the following $$(q-1)$$ ( q - 1 ) th convex ordering relation for qth convolution power of the difference of probability distributions $$\mu $$ μ and $$\nu $$ ν $$\begin{aligned} (\nu -\mu )^{*q}\ge _{(q-1)cx} 0 , \quad q\ge 2, \end{aligned}$$ ( ν - μ ) ∗ q ≥ ( q - 1 ) c x 0 , q ≥ 2 , and we obtain the theorem providing a useful sufficient condition for its verification. We apply this theorem for various families of probability distributions and we obtain several inequalities related to the classical interpolation operators. In particular, taking binomial distributions, we obtain a new, very short proof of the inequality given recently by Abel and Leviatan (2020).