Steffensen Type Inequalities Over General Measure Spaces

Abstract

A recent comprehensive presentation of Steffensen's inequality is provided in [8, Chapter XI].In the present paper, we extend Steffensen's inequality from integrals over compact intervals of the real line to integrals over general measure spaces. We apply this generalization to establish integral inequalities for composed functions. Then we obtain stronger particular inequalities by means of the concept of Wrightconvexity. The key tools that we use here are the concepts of separating subsets for a measurable function, and continuous measure spaces. We establish convexity properties of integrals over separating subsets of a continuous measure space, which provide interpolations of our inequalities, and also new expressions for the monotone rearrangement of a function.

Some Steffensen-type dynamic inequalities on time scales

“…, n} with λ 2 ≤ n k=1 ψ(k) ≤ λ 1 . Inequality (1.2) is known as discrete Steffensen's inequality [15].…”

Steffensen’s inequality and 𝐿¹-𝐿^{∞} estimates of weighted integrals

“…The last step is to extend Corollary 2.3 to functions f on R N . The few existing generalizations of the Steffensen inequality to higher dimension or to arbitrary measure spaces ( [3], [4], [7], [15]) have no connection with the one given in Theorem 2.5 below. Recall that ω N is the volume of the unit ball of R N , so that Nω N is the area of the unit sphere S N −1 .…”

A simple proof of the discrete Steffensen's inequality

“…It was first given in [1] and then cited repeatedly in [2]- [5] the following discrete Steffensen's inequality:…”

“…It should be noted that the proof of this theorem in [1] states as an application of a generalized Steffensen's inequality over a general measure space to the discret case which seemed as if it is not quite clear. The purpose of this note is to give a new proof which is very simple and clear.…”

“…In fact, the Theorem can be restated in a more general version as follows: Let (y i ) n i=1 be a finite sequences of real numbers such that for every i, 0 ≤ y i ≤ 1. Then k 2 ≤ n i=1 y i ≤ k 1 is a necessary and sufficient condition for inequalities (1) to hold for all nonincreasing finite sequences (x i ) n i=1 of nonnegative real numbers, where k 1 , k 2 ∈ {1, . .…”