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# Further generalized refinement of Young’s inequalities for τ -mesurable operators

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In this paper, we prove that if a, b > 0 and 0 ≤ v ≤ 1. Then for all positive integer m

(1) - For v $v∈[ 0,12n ]$ v \in \left[ {0,{1 \over {{2^n}}}} \right] , we have $(avb1-v)m+∑k=1n2k-1vm(bm-(ab2k-1-1)m2k)2≤(va+(1-v)b)m.$ {\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{v^m}{{\left( {\sqrt {{b^m}} - \root {{2^k}} \of {\left( {a{b^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m}.}

(2) - For v $v∈[ 2n-12n,1 ]$ v \in \left[ {{{{2^n} - 1} \over {{2^n}}},1} \right] , we have $(avb1-v)m+∑k=1n2k-1(1-v)m(am-(ba2k-1-1)m2k)2≤(va+(1-v)b)m,$ {\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{{\left( {1 - v} \right)}^m}{{\left( {\sqrt {{a^m}} - \root {{2^k}} \of {\left( {b{a^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m},} we also prove two similar inequalities for the cases v $v∈[ 2n-12n,12 ]$ v \in \left[ {{{{2^n} - 1} \over {{2^n}}},{1 \over 2}} \right] and v $v∈[ 12,2n+12n ]$ v \in \left[ {{1 \over 2},{{{2^n} + 1} \over {{2^n}}}} \right] . These inequalities provides a generalization of an important refinements of the Young inequality obtained in 2017 by S. Furuichi. As applications we shall give some refined Young type inequalities for the traces, determinants, and p-norms of positive τ-measurable operators.

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