Norm Inequalities via Convex and Log-Convex Functions

CONDE, CRISTIAN; MINCULETE, NICUSOR; MORADI, HAMID REZA; SABABHEH, MOHAMMAD

MEDITERRANEAN JOURNAL OF MATHEMATICS

BIRKHAUSER VERLAG AG

Lugar: BASEL; Año: 2023 vol. 20

1660-5446

In this paper, we study the norm and skew angular distancesin a normed space X , where convex functions are used to obtain refinements and reverses of some outstanding results in the literature. Forexample, in this regard, we show that if a, b ∈ X are non-zero and ifp, q > 0 are such that 1p + 1q = 1, then2λ prar + qrbr2 −pa + qb2r≤ pr−1ar + qr−1br − a + br≤ 2μ prar + qrbr2 −pa + qb2r,where r ≥ 1, λ = min {1/p, 1/q} and μ = max {1/p, 1/q}. Then weexplain how this result extends some known results in the literature.Many other related results will be also shown. Then, with the theme ofconvexity, we employ a log-convex approach on certain matrix functionsto obtain improvements and new sights of some matrix inequalities, including possible bounds of AtXB1−t, where A, B are positive definitematrices, X is an arbitrary matrix, · is a unitarily invariant norm and0 ≤ t ≤ 1. Many other results involving matrix and scalar log-convexfunctions will be presented too.