Two weighted-mean proofs for conjectures of Jian Liu (2022)

Two complementary research notes prove Conjectures 3.3 and 3.4 in Jian Liu's 2022 article A beautiful inequality chain for a triangle. The notes explicitly use the established Theorem 1.1 of the same source. Reciprocal exradii and normalized powers of paired altitudes, internal angle bisectors, medians or symmedians reduce the statements to two-point weighted means. Concavity, convexity and monotonicity of scalar powers establish every stated parameter range and determine equality exactly when b = c. The first note also gives an exact counterexample to the negative-exponent region of the printed Conjecture 3.1 and identifies the correct direction of that chain. The proofs are analytic and are presented in full in the two PDFs. This is a preprint and has not undergone independent peer review. The two PDFs provide the complete research notes. Use of AI tools. ChatGPT was used solely for language editing. All mathematical ideas, arguments, calculations, implementations and code are the authors' own work. The authors are responsible for the content.