Ta članek temelji na klasičnih pojmih bilinearnih matričnih neenakosti (BMI) in parcialne konveksnosti ter raziskuje parcialno konveksnost za nekomutativne funkcije. Dokazano je, da nekomutativne ...racionalne funkcije, ki so parcialno konveksne, dopuščajo nove realizacije oblike metulja, ki zahtevajo kvadratne korene. Obravnavan je tudi pojem xy-konveksnosti, okrepitev parcialne konveksnosti, ki se pojavlja v povezavi z BMI. Podana je karakterizacija xy-konveksnih polinomov.
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