Refinements of Young’s Integral Inequality via Fundamental Inequalities and Mean Value Theorems for Derivatives1

In the first part of this paper, we mainly review several refinements of Young’s integral inequality via several mean value theorems, such as Lagrange’s and Taylor’s mean value theorems of Lagrange’s and Cauchy’s type remainders, and 195via several fundamental inequalities, such as Čebyšev’s integral inequality, Hermite-Hadamard’s type integral inequalities, Hölder’s integral inequality, and Jensen’s discrete and integral inequalities, in terms of higher-order derivatives and their norms, and simply survey several applications of several refinements of Young’s integral inequality.