截断 Stieltjes 积分方程的唯一峰值

We study maxima of curves defined by normalized, truncated Stieltjes integrals. For a positive profile h, strict decrease of its elasticity xh'(x)/h(x) on its declining side gives a unique nondegenerate maximum at every level for which the curve exists. This proves a uniqueness conjecture of Viakhirev and makes its additional existence hypotheses unnecessary. We treat every positive power of the kernel, on a finite interval or the positive half-line: all positive levels occur for powers at least one, whereas a uniquely maximized boundary integral gives the exact level threshold for smaller powers. We determine the movement and limiting positions of the peak. For log-normal densities, the normalized polynomial cutoffs proposed by Viakhirev have maximizing locations and heights converging at rate O(R^-1) to limits independent of the cutoff exponent. In contrast, smooth strictly unimodal profiles can produce arbitrarily many critical points.