Let $A = K[x_1,\dots, x_n]$ be the polynomial ring over a field $K$ and $P \subset A$ a homogeneous prime ideal with no linear form. Let $\mathrm{reg}(P)$ denote the Castelnuovo–Mumford regularity of $P$ and $e(A/P)$ the multiplicity of $A/P$. It will be shown that if $P$ possesses an initial ideal with no embedded prime ideal, then the regularity of $P$ satisfies the inequality $\mathrm{reg}(P) \leq  e(A/P)-\mathrm{codim}(A/P) + 1$, where $\mathrm{codim}(A/P)$ is the codimension of $A/P$.