Two upper bounds for the Erdős--Hooley Delta-function

For integer $n⩾1$ and real $u$, let $\mathrm{\Delta }(n,u):=\mathrm{\mid }\{d:d\mid n,{\mathrm{e}}^u<d⩽{\mathrm{e}}^{u+1}\}\mathrm{\mid }$. The Erdős–Hooley Delta-function is then defined by $\mathrm{\Delta }(n):={\max }_{u\in \mathbb{R}}\mathrm{\Delta }(n,u)\mathrm{.}$ We improve the current upper bounds for the average and normal orders of this arithmetic function.