On the Raşa Inequality for Higher Order Convex Functions II

We give necessary and sufficient conditions for Borel measures μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} and ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} to satisfy the following (q-1)th\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q-1)\hbox {th}$$\end{document} convex ordering relation for qth\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\hbox {th}$$\end{document} convolution power of the difference of μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} and ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document}which was introduced by Komisarski and Rajba (Results Math 76(2):103–115, 2021), and we gave in Komisarski and Rajba (Results Math 76(2):103–115, 2021) a useful sufficient condition for its verification. We give also necessary and sufficient conditions for discrete probability distributions. This inequality is a generalization of the inequality given recently by Abel and Leviatan (Results Math 75(4):181–193, 2020) and the Raşa type inequalities given in Komisarski and Rajba (Math Anal Appl 478:182–194, 2019).

The proof given by Mrowiec et al. [11] makes use of probability theory. Let μ and ν be two finite Borel measures on R such that R ϕ(x)μ(dx) ≤ R ϕ(x)ν(dx) for all convex functions ϕ : R → R provided the integrals exist. Then μ is said to be smaller then ν in the convex order (denoted as μ ≤ cx ν ). In [11], the authors proved the following stochastic convex ordering relation for convolutions of binomial distributions B(n, x) and B(n, y) (n ∈ N, x, y ∈ [0, 1]): B(n, x) * B(n, y) ≤ cx 1 2 (B(n, x) * B(n, x) + B(n, y) * B(n, y)), (2) which is a probabilistic version of inequality (1) and is equivalent to (1). In [5], we gave a generalization of inequality (2). We introduced and studied the following convex ordering relation where μ and ν are two probability distributions on R. We note, that inequality (3) can be regarded as the Raşa type inequality. In [5], we proved Theorem 2.3 providing a very useful sufficient condition for verification that μ and ν satisfy (3). We applied Theorem 2.3 for μ and ν from various families of probability distributions. In particular, we obtained a new proof for binomial distributions, which is significantly simpler and shorter than that given in [11]. In [7], we gave also necessary and sufficient condition for verification that μ and ν satisfy (3).
for n ∈ N, x ∈ I and h, h 1 , . . . , h n , h n+1 ≥ 0 with all needed arguments belonging to I.
In [8], we give a probabilistic version of inequality (4) and consider some generalization of (4) in terms of higher order convex orders. Let us review some notations.
In the classical theory of convex functions their natural generalization are convex functions of higher-order.
The convexity of n-th order (or n-convexity) was defined in terms of divided differences by Popoviciu [12,13], however, we will not state it here. Instead we list some properties of n-th order convex functions (see [9]).

Proposition 2.
Let f ∈ C(I) and n ≥ 1. Then the following statements are equivalent: (a) f is n-convex on I.
Recall the definition of n-convex orders ( [17] provided the integrals exist. Then μ is said to be smaller then ν in the n-convex order (denoted as μ ≤ n-cx ν ).
In particular, μ ≤ 1-cx ν coincides with μ ≤ cx ν. In [8], we study the following generalization of (3) where μ, ν are probability measures. Inequality (5) can be regarded as the Raşa type inequality. Let μ be a probability distribution on R. For x ∈ R the delta symbol δ x denotes one-point probability distribution satisfying δ x ({x}) = 1. As usual, for all x ∈ R, then μ is said to be smaller than ν in the usual stochastic order (denoted by μ ≤ st ν).
In the following theorem, we give a very useful sufficient condition that will be used for verification of (5).

Theorem 3 ([8]). Let q ≥ 2. Let μ and ν be two probability distributions on
In particular, by Theorem 3, taking binomial distributions, we obtain which is equivalent to inequality (4). Consequently, we obtain a new proof of inequality (4) given by Abel and Leviatan [2], which is significantly simpler and shorter than that given in [2]. In [8], we apply Theorem 3 for μ and ν from various families of probability distributions. Using inequality (5), we can also obtain inequalities related to some approximation operators associated with μ and ν (such as Bernstein-Schnabl operators, Mirakyan-Szász operators, Baskakov operators and others). We proved also the following generalization of (6).
In this paper, we give necessary and sufficient conditions for verification of (7). We give also necessary and sufficient conditions for verification of (6) for discrete probability distributions.

Main Results
We will use the following characterization of n-convex orders for signed measures.
Proposition 3 (Corollary 2.1 [14]). Let γ be a signed measure on R, which is concentrated on the interval (a, b) (bounded or unbounded) and such that b it is necessary and sufficient that γ verify the following conditions: First we prove the following lemma on the moments of the convolutions of signed measures. Let γ be a signed measure on R such that ∞ −∞ |x| n |γ|(dx) < ∞. We denote by m n (γ) the n-th moment of γ, m n (γ) = ∞ −∞ x n γ(dx).
Let us consider one of the components of the sum above where n 1 + . . . + n m = k. Without loss of generality we may assume that n 1 ≥ n 2 ≥ . . . ≥ n m . Then there exists 1 ≤ j ≤ k such that n j > 0 and n j+1 = . . . = n m = 0. Taking into account that γ j+1 (R) = .
This completes the proof of (b).
We prove some generalization of Theorem 10.
Similarly, one can prove a generalization of Theorem 11. (1) For all (q − 1)-convex functions ϕ : Proof. Note that (13) is equivalent to the relation (ν −μ) * q ≥ (q−1)-cx 0. Indeed, let ϕ : R → R be a (q − 1)-convex function. Then, we have Consequently, the equivalence of (1) and (2) clearly follows from Corollary 8. It suffices to prove the equivalence of (2) and (3). In the following calculations, we use the existence and finiteness of the qth moments of the probability distributions μ and ν, which implies that all the following series are absolutely convergent for z ∈ [−1, 1] and we can change the summation order. By the equality ∞ k=0 a k = 1, we have for every z ∈ [−1, 1), where F is the tail distribution of μ. Note that ∞ i=0 F (i) = R x μ(dx) < ∞. Similarly, we obtain where G is the tail distribution of ν. Therefore, for every z ∈ [−1, 1) we have Here, * denotes the discrete convolution (the Euler product) of sequences. Condition (3) is equivalent to the non-negativity of all the coefficients in the above series. Note that (F − G) * q (i) = (F − G) * q (i), i = 0, . . . , ∞ (cf. the proof of Theorem 2.6 in [7]). Therefore, the non-negativity of all the terms (F − G) * q (i) is equivalent to (2). The theorem is proved.
Theorem 14 is a generalization of results [4,7] on the Raşa inequality for convex functions.