From Wikipedia, I have learned that the Fenchel conjugate of $f(x)=\frac{|x|^p}{p}$, where $0<p<\infty$, is
\begin{equation} f^\ast(x^\ast)=\frac{1}{q}|x^\ast|^{q}, 1<q<\infty, \text{where}\; \frac{1}{p}+\frac{1}{q}=1. \end{equation}
From the definition of the Fenchel conjugate, I know
\begin{equation} f^\ast(x^\ast)=\underset{x \in X}{\sup} \{\langle x^\ast, x \rangle - f(x)\}= \underset{x \in X}{\sup} \left\{\langle x^\ast, x \rangle - \frac{|x|^p}{p} \right\} \end{equation}
After that I am stuck and couldn't figure out how $f^\ast(x^\ast) = \frac{1}{q}|x^\ast|^{q}$.
