Young's Inequality

Young's Inequality provides a way to relate the product of two numbers to their individual properties. It states that for any positive numbers \(a\) and \(b\), and for certain constants \(p\) and \(q\) satisfying \(1/p + 1/q = 1\), the product \(ab\) can be bounded by a weighted sum of powers of \(a\) and \(b\). Specifically, \(ab \leq \frac{a^p}{p} + \frac{b^q}{q}\). This inequality is useful in analysis for estimating the size of products in terms of sums, especially in areas like calculus and functional analysis.