A new majorization between functions, polynomials, and operator inequalities |
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Authors: | Mitsuru Uchiyama |
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Institution: | Department of Mathematics, Fukuoka University of Education, Munakata, Fukuoka 811-4192, Japan |
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Abstract: | Let P+ be the set of all non-negative operator monotone functions defined on 0,∞), and put . Then and . For a function and a strictly increasing function h we write if is operator monotone. If and and if and , then . We will apply this result to polynomials and operator inequalities. Let and be non-increasing sequences, and put for t≧a1 and for t≧b1. Then v+?u+ if m≦n and : in particular, for a sequence of orthonormal polynomials, (pn-1)+?(pn)+. Suppose 0<r,p and s=0 or 1≦s≦1+p/r. Then 0≦A≦B implies for 0<α≦r/(p+r). |
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Keywords: | Matrix order Lö wner-Heinz inequality Operator monotone function Pick function Orthogonal polynomial Majorization |
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