Sharp inequality for ${\mathrm{\ell }}_p$ quasi-norm and ${\mathrm{\ell }}_q$-norm with $0<p\le 1$ and $q>1$

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