On a convolution inequality of Saitoh

Let F-1, F-2,..., F-j,... be in the class L(loc)(R(+)) of locally integrable functions on R(+) = (0, infinity). Define the convolution product Pi(j=1)(m)*F-j inductively by [Pi(j=1)(2)*F-j](x) = (F-1 * F-2)(x) = integral(0)(x) F-1(y)F-2(x - y) dy and Pi(j=1)(m)*F-j = [Pi(j=1)(m-1)*F-j]*F-m for m > 2. The inequality [GRAPHICS] is obtained for each p, 1 < p < infinity. Further, the constant [(m-1)l](1-p) is shown to be the best possible, and the nonzero extremal functions are determined.