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

Abstract

We study the integral equation $\int_0^m ηρ(η)/(C-η)\,dη= 1$ with $C>m$, where $ρ$ is a $C^1$ probability density on $[0,M]$ vanishing polynomially at $η=M$. Setting $\mathcal{I}^+(m) := \lim_{C \downarrow m}\int_0^m ηρ(η)/(C-η)\,dη$ and $Ω:= \{m \in (0,M) : \mathcal{I}^+(m) > 1\}$, the equation determines $C$ implicitly as a function of $m$ on $Ω$, and our object of study is the dimensionless ratio $β(m) := C(m)/m$. Writing $h(η) := ηρ(η)$, our main theorem establishes openness of $Ω$, $C^1$-smoothness of $β$, a sign formula identifying $β'(m)$ with a positively-weighted integral of $dh/d\lnη$, transfer of monotonicity from $h$ to $β$, and existence of an interior critical point of $β$ when $h$ is unimodal and two technical hypotheses hold. Numerically, $β$ 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.