Schur complements of sectorial matrices: proof corrections and a sharp mean inequality

This technical note examines proof steps in ‘The Schur complement inequalities involving sectorial matrices’ by Marziyeh Irandegani, Mojtaba Bakherad and Monire Hajmohamadi (Linear and Multilinear Algebra, 2026; DOI: 10.1080/03081087.2026.2751934). An exact positive-definite 2×2 example disproves the interchange of a Schur complement with an integral used in the proofs of Theorems 2.2, 2.6 and 2.9. A separate sectorial example disproves an intermediate norm comparison in the proof of Proposition 3.1. These examples do not disprove the stated main inequalities. A self-contained corrected argument establishes the sharp factor sec²(α) for the matrix-mean inequality and the normalized operator-monotone functional inequality, improving the factors sec⁴(α) stated in the reviewed article. Sharpness is established for every fixed positive block size. A unitarily invariant norm consequence has factor sec³(α), improving sec⁵(α); optimality of the norm factor is not claimed. The final conclusion of the reviewed Proposition 3.1 is explicitly left unresolved. No exhaustive literature-priority claim is made. Files: the technical note in PDF and a ZIP package containing the PDF, editable LaTeX source, README and a Python script using exact rational arithmetic to verify the displayed examples. The reviewed publisher PDF is not redistributed. AI assistance and author responsibility: OpenAI ChatGPT was used for English-language improvement, computational checks and auxiliary mathematical analysis, including assistance with derivations and counterexample verification. The author takes full responsibility for the construction and validity of the counterexamples, the statements of the lemmas and theorems, all proofs, and the final content of this note.