发表机构
Institute of Science, Technology, Engineering and Mathematics (STEM)（科学与技术工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Laplace核正性分析，独立证明了Guo-Zhang-Qi关于digamma幂族对数完全单调性的猜想，并刻画了最优参数阈值。
AI 中文摘要
我们研究由digamma函数构造的幂族的对数完全单调性。其负对数导数的显式Laplace表示将问题归结为连续核的正性。我们在一个显式参数范围内证明了一个严格的核下界，并获得了所有交错对数导数的均匀下界。特别地，这给出了Guo、Zhang和Qi猜想在整个正半轴上性质的独立证明。我们还通过变分最小值刻画了精确的可容许参数阈值，并表明显式的充分下界并非最优。该论证结合了一个辅助digamma核的单调性与正则化对数卷积。附录指出了早期证明中的一个错误辅助不等式，而当前论证独立地确立了其所述结论。
英文摘要
We study logarithmic complete monotonicity of a power family built from the digamma function. An explicit Laplace representation of its negative logarithmic derivative reduces the question to positivity of a continuous kernel. We prove a strict kernel bound on an explicit parameter range and obtain uniform lower bounds for all alternating logarithmic derivatives. In particular, this gives an independent proof of the property conjectured by Guo, Zhang, and Qi on the entire positive half-line. We also characterize the exact admissible parameter threshold by a variational minimum and show that the explicit sufficient bound is not optimal. The argument combines monotonicity of an auxiliary digamma kernel with a regularized logarithmic convolution. An appendix identifies a false auxiliary inequality in an earlier proof, while the present argument independently establishes its stated conclusion.
