| Popis: |
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 , 1 2 n ] v \in \left[ {0,{1 \over {{2^n}}}} \right] , we have ( a v b 1 - v ) m + ∑ k = 1 n 2 k - 1 v m ( b m - ( a b 2 k - 1 - 1 ) m 2 k ) 2 ≤ ( v a + ( 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 ∈ [ 2 n - 1 2 n , 1 ] v \in \left[ {{{{2^n} - 1} \over {{2^n}}},1} \right] , we have ( a v b 1 - v ) m + ∑ k = 1 n 2 k - 1 ( 1 - v ) m ( a m - ( b a 2 k - 1 - 1 ) m 2 k ) 2 ≤ ( v a + ( 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 ∈ [ 2 n - 1 2 n , 1 2 ] v \in \left[ {{{{2^n} - 1} \over {{2^n}}},{1 \over 2}} \right] and v ∈ v ∈ [ 1 2 , 2 n + 1 2 n ] 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. |
