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

Abstract

Abstract. Let Φ : [0, ∞) → R be a continuous convex function with Φ(0) = 0. We prove that Φwhere ω N is the measure of the unit ball of R N . This can be used to obtain lower or upper bounds for weighted integrals R N |f (x)|η(|x|)dx in terms of the L 1 and L ∞ norms of f, which are often much sharper than crude estimates that may be obtained, if at all, by a visual inspection of the integrand. The basic inequality is essentially independent of Jensen's inequality, but it is closely related to Steffensen's inequality.

Generalized Steffensen’s inequality by Montgomery identity

“…Integral inequalities such as Hardy's inequality, Steffensen's inequality, and Ostrowski's inequality have been topics of interest of many mathematicians since their pronouncement. Several generalizations of these inequalities have been proved for convex functions, beta m-convex functions, n-convex functions, and other classes of functions, for example, see [4,7,12,16], and [17]. Steffensen's inequality was proved in [18]: if ψ, f : [c, d] → R, with ψ being a decreasing function and function f having range in [0, 1], then…”

Generalized Steffensen’s Inequality by Fink’s Identity

“…Integral inequalities such as Hardy's inequality, Steffensen's inequality, and Ostrowski's inequality are topics of interest of many Mathematicians since their pronouncement. Several generalizations of these inequalities have been proved for different classes of functions, such as convex functions, n-convex functions, and other types of functions, for example see [1][2][3][4]. Moreover, integral inequalities have been proved for different integrals, such as Jensen-steffensen inequality for diamond integral and bounds of related identities have been obtained in [5].…”

“…A massive literature dealing with several variants and improvements of Steffensen's inequality can be seen in [8,9] and references therein. A well known generalization of Steffensen's inequality has been presented in [4]. Several results of [4] have been recently generalized by using non-bounded Montgomery's identity in [10].…”

“…A well known generalization of Steffensen's inequality has been presented in [4]. Several results of [4] have been recently generalized by using non-bounded Montgomery's identity in [10]. To proceed further, we recall a nice generalization of Steffensen's inequality proved by Pečarić, see [11].…”

Generalized Steffensen's inequality

“…We give a generalization of Steffensen's inequality by extending the results of Pečarić [4] and Rabier [5]. We make use of the n -order Taylor expansion of a composition of functions and Faà di Bruno's formula for higher order derivatives of the composition.…”