Sharp inequality for $\ell_p$ quasi-norm and $\ell_q$-norm with $0<p\leq 1$ and $q>1$

A sharp inequality for $\ell_p$ quasi-norm with $0<p\leq 1$ and $\ell_q$-norm with $q>1$ is derived, which shows that the difference between $\|\textbf{\textit{x}}\|_p$ and $\|\textbf{\textit{x}}\|_q$ of an $n$-dimensional signal $\textbf{\textit{x}}$ is upper bounded by the difference between the maximum and minimum absolute value in $\textbf{\textit{x}}$. The inequality could be used to develop new $\ell_p$-minimization algorithms.