Generalizations and applications of Young’s integral inequality by higher order derivatives

Abstract

In the paper, the authors 1. generalize Young's integral inequality via Taylor's theorems in terms of higher order derivatives and their norms, and 2. apply newly-established integral inequalities to estimate several concrete definite integrals, including a definite integral of a function which plays an indispensable role in differential geometry and has a connection with the Lah numbers in combinatorics, the exponential integral, and the logarithmic integral.

A Themed Issue on Mathematical Inequalities, Analytic Combinatorics and Related Topics in Honor of Professor Feng Qi

“…Concerning the integral inequalities, it is also worth noting that F. Qi and his coauthors have generalized, in [144][145][146][147], the Young integral inequality using the Taylor theorems in terms of higher order derivatives and their norms; the authors have applied their results for the estimation of several concrete definite integrals.…”

Geometric interpretations and reversed versions of Young's integral inequality

“…In the papers [8,10,11,21,24], Young's integral inequality (1.1) was rened by estimating the area C and bounding…”

Refinements of Young's integral inequality via fundamental inequalities and mean value theorems for derivatives

“…where C denotes the area showed in Figures 1 to 6 (c) or h(a) < b, h (n+1) (x) is decreasing on [α, β], and n = 2 ; the inequality Proof. This is the outline of the proof of [49,Theorem 3.3].…”

“…Proof. This is the outline of the proof of [49,Theorem 3.5]. Let f (x) and g(x) be nonnegative and convex functions on [µ, ν].…”