Young's Inequality: Let \(f\) be a function defined and differentiable on the open interval \(I\), and let \(a \in I\). Then, for any \(b \in I\), the follo...
Young's Inequality:
Let (f) be a function defined and differentiable on the open interval (I), and let (a \in I). Then, for any (b \in I), the following inequality holds:
Holder's Inequality:
Let (f) and (g) be functions defined and differentiable on the open interval (I). Then, for any (a \in I), the following inequality holds:
Examples:
Young's Inequality:
Consider the function (f(x) = x^2). Then, for any (a \in I), we have:
Therefore, Young's inequality implies that (f(a) - f(b) \le 2|ab| ) for all (a, b \in I).
Holder's Inequality:
Consider the functions (f(x) = x) and (g(x) = x^2). Then, for any (a \in I), we have:
Therefore, Holder's inequality implies that (|a - a^2| \le |a| |a - 1| ) for all (a \in I)