The following extremal problem is solved: Let $O$ be an operator whose lowest-mass coupling is to the two particles $A$ and $B$ and is given by a form factor $F$ which is analytic in a complex plane cut along a section $L$ of the real line. Given values of $F$, at points not on $L$, and the partial-wave amplitudes for $\mathrm{AB}$ scattering in the channels with the same quantum numbers as $O$, over some portion of $L$, what is the optimal lower bound for a given positively weighted integral over $L$ of the spectral function of $O$, consistent with the elastic and inelastic unitarity relations on $L$? The solution involves a system of two inhomogeneous singular integral equations of the Muskhelishvili type, which can be reduced to a singular integral equation of the Fredholm type. The results are applied to establish an upper bound of 0.25 for the nucleon renormalization constant, using the strong-coupling constant and $\ensuremath{\pi}N$ scattering data in the ${P}_{11}$ and ${S}_{11}$ channels up to a c.m. energy of 1.7 GeV. The bound indicates that the nucleon is at least 75% composite.