Structural properties of the implicit function defined by an integral self-consistency equation

Abstract

We study the integral equation ${\int }_0^m\hat{I}\cdot \ddot{I}(\hat{I}\cdot )/(C-\hat{I}\cdot )d\hat{I}\cdot =1$ with $C>m$, where $\ddot{I}$ is a $C^1$ probability density on $[0,M]$ vanishing polynomially at $\hat{I}\cdot =M$. Setting ${\mathcal{I}}^{+}(m):={\lim }_{C\downarrow m}{\int }_0^m\hat{I}\cdot \ddot{I}(\hat{I}\cdot )/(C-\hat{I}\cdot )d\hat{I}\cdot$ and $\hat{I}\text{c◯}:=\{m\in (0,M):{\mathcal{I}}^{+}(m)>1\}$, the equation determines $C$ implicitly as a function of $m$ on $\hat{I}\text{c◯}$, and our object of study is the dimensionless ratio ${\hat{I}}^2(m):=C(m)/m$. Writing $h(\hat{I}\cdot ):=\hat{I}\cdot \ddot{I}(\hat{I}\cdot )$, our main theorem establishes openness of $\hat{I}\text{c◯}$, $C^1$-smoothness of ${\hat{I}}^2$, a sign formula identifying $β'(m)$ with a positively-weighted integral of $dh/d\ln \hat{I}\cdot$, transfer of monotonicity from $h$ to ${\hat{I}}^2$, and existence of an interior critical point of ${\hat{I}}^2$ when $h$ is unimodal and two technical hypotheses hold. Numerically, ${\hat{I}}^2$ has a single critical point in seven log-concave test densities (mostly Beta-type), in support of a separate uniqueness conjecture. A bimodal density that violates both unimodality and log-concavity exhibits three critical points; this shows that dropping the two hypotheses jointly admits multiple critical points, but does not separate their roles.