On a convolution inequality of Saitoh

Let $F_1,F_2,\dotsc,F_j,\dotsc$ be in the class $L_{{\operatorname{loc}}}(\mathbb{R}_+)$ of locally integrable functions on $\mathbb{R}_+=(0,\infty)$. Define the convolution product $\prod^m_{j=1}*F_j$ inductively by $[\prod^2_{j=1}*F_j](x) =(F_1*F_2)(x)=\int^x_0 F_1(y)F_2(x-y)\,dy$ and $\prod^m_{j=1} *F_j=[\prod^{m-1}_{j=1}*F_j]*F_m$ for $m>2$. The inequality

is obtained for each $p$, $1<p<\infty$. Further, the constant $[(m-1)!]^{1-p}$ is shown to be the best possible, and the nonzero extremal functions are determined.