The Guo-Zhang-Qi conjecture and positive Laplace kernels for a digamma power family

Abstract

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.