关于 Alzer-Berg 问题：一个最优 Bernstein 边界与唯一性猜想

中文摘要 AI 辅助

我们研究了与 Alzer 和 Berg 的完全单调性问题相关的双参数指数族。严格的必要参数界将每一个可能的 Bernstein 函数置于正则化 Laplace 表示的域内。一个定量的正性传递不等式随后给出了一个具有最大可能通用系数的线性充分条件，该系数以精确的单参数临界指数表示。为了描述整个容许区域，我们推导了参数导数的卷积恒等式，并证明了增加任一归一化参数会在非负密度的每个零点处破坏正性。结合均匀尾部估计，这排除了容许参数区间中的间隙，并给出了一个连续、严格单调的最优边界。密度的全局非负性及与零的接触刻画了该边界；其逆决定了第二个参数的完整容许区间。该刻画是隐式的，既不要求接触点的唯一性，也不要求其非退化性。已发表的单参数指数的数值近似与精确结果以及证明中使用的有限有理证书被区分开来。我们还推导了一个变分公式和一个用于验证边界数值包络的严格框架，并提出了一个边界接触猜想，断言在每个内部边界点处唯一性和二次接触。

英文摘要

We study the two-parameter exponential family associated with the complete-monotonicity problem of Alzer and Berg. Strict necessary parameter bounds place every possible Bernstein function in the domain of a regularized Laplace representation. A quantitative positivity-transfer inequality then yields a linear sufficient condition with the largest possible universal coefficient, expressed in terms of the exact one-parameter critical exponent. To describe the whole admissible region, we derive convolution identities for parameter derivatives and prove that increasing either normalized parameter destroys positivity at every zero of a nonnegative density. Combined with uniform tail estimates, this excludes gaps in the admissible parameter intervals and gives a continuous, strictly monotone optimal boundary. Global nonnegativity of the density and contact with zero characterize that boundary; its inverse determines the complete admissible interval for the second parameter. The characterization is implicit and requires neither uniqueness nor nondegeneracy of the contact points. Published numerical approximations of the one-parameter exponent are distinguished from the exact results and from the finite rational certificate used in the proofs. We also derive a variational formula and a rigorous framework for validated numerical enclosure of the boundary, and formulate a boundary-contact conjecture asserting uniqueness and quadratic contact at every interior boundary point.

发表机构

• Institute of Science, Technology, Engineering and Mathematics (STEM)（科学与技术工程学院）

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