The Young inequality and the ${\mit \Delta }_2$-condition
Volume 94 / 2002
Colloquium Mathematicum 94 (2002), 221-223
MSC: Primary 26D07.
DOI: 10.4064/cm94-2-4
Abstract
If $\varphi : [0,\infty )\to [0,\infty )$ is a convex function with $\varphi (0)=0$ and conjugate function $\varphi ^*$, the inequality $xy \le \varepsilon \varphi (x) + C_\varepsilon \ \varphi ^*(y)$ is shown to hold true for every $\varepsilon \in (0,\infty )$ if and only if $\varphi ^*$ satisfies the ${\mit \Delta }_2$-condition.
