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

Abstract

<jats:title>Abstract</jats:title>
<jats:p>In this paper, we prove that if <jats:italic>a</jats:italic>, <jats:italic>b</jats:italic> &gt; 0 and 0 ≤ <jats:italic>v</jats:italic> ≤ 1. Then for all positive integer <jats:italic>m</jats:italic>
</jats:p>
<jats:p>(1) - For <jats:italic>v</jats:italic> ∈ <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_001.png"/>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline">
<m:mrow>
<m:mi>v</m:mi>
<m:mo>∈</m:mo>
<m:mrow>
<m:mo>[</m:mo>
<m:mrow>
<m:mn>0</m:mn>
<m:mo>,</m:mo>
<m:mfrac>
<m:mn>1</m:mn>
<m:mrow>
<m:msup>
<m:mrow>
<m:mn>2</m:mn>
</m:mrow>
<m:mi>n</m:mi>
</m:msup>
</m:mrow>
</m:mfrac>
</m:mrow>
<m:mo>]</m:mo>
</m:mrow>
</m:mrow>
</m:math>
<jats:tex-math>v \in \left[ {0,{1 \over {{2^n}}}} \right]</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, we have
<jats:disp-formula>
<jats:alternatives>
<jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_002.png"/>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block">
<m:mrow>
<m:msup>
<m:mrow>
<m:mrow>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:msup>
<m:mrow>
<m:mi>a</m:mi>
</m:mrow>
<m:mi>v</m:mi>
</m:msup>
<m:msup>
<m:mrow>
<m:mi>b</m:mi>
</m:mrow>
<m:mrow>
<m:mn>1</m:mn>
<m:mo>-</m:mo>
<m:mi>v</m:mi>
</m:mrow>
</m:msup>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
<m:mi>m</m:mi>
</m:msup>
<m:mo>+</m:mo>
<m:munderover>
<m:mo>∑</m:mo>
<m:mrow>
<m:mi>k</m:mi>
<m:mo>=</m:mo>
<m:mn>1</m:mn>
</m:mrow>
<m:mi>n</m:mi>
</m:munderover>
<m:mrow>
<m:msup>
<m:mrow>
<m:mn>2</m:mn>
</m:mrow>
<m:mrow>
<m:mi>k</m:mi>
<m:mo>-</m:mo>
<m:mn>1</m:mn>
</m:mrow>
</m:msup>
<m:msup>
<m:mrow>
<m:mi>v</m:mi>
</m:mrow>
<m:mi>m</m:mi>
</m:msup>
<m:msup>
<m:mrow>
<m:mrow>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:msqrt>
<m:mrow>
<m:msup>
<m:mrow>
<m:mi>b</m:mi>
</m:mrow>
<m:mi>m</m:mi>
</m:msup>
</m:mrow>
</m:msqrt>
<m:mo>-</m:mo>
<m:mroot>
<m:mrow>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mi>a</m:mi>
<m:msup>
<m:mrow>
<m:mi>b</m:mi>
</m:mrow>
<m:mrow>
<m:mn>2</m:mn>
<m:mi>k</m:mi>
<m:mo>-</m:mo>
<m:mn>1</m:mn>
</m:mrow>
</m:msup>
<m:mo>-</m:mo>
<m:mn>1</m:mn>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
<m:mi>m</m:mi>
</m:mrow>
<m:mrow>
<m:msup>
<m:mrow>
<m:mn>2</m:mn>
</m:mrow>
<m:mi>k</m:mi>
</m:msup>
</m:mrow>
</m:mroot>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
<m:mn>2</m:mn>
</m:msup>
<m:mo>≤</m:mo>
<m:msup>
<m:mrow>
<m:mrow>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mi>v</m:mi>
<m:mi>a</m:mi>
<m:mo>+</m:mo>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mn>1</m:mn>
<m:mo>-</m:mo>
<m:mi>v</m:mi>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
<m:mi>b</m:mi>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
<m:mi>m</m:mi>
</m:msup>
<m:mo>.</m:mo>
</m:mrow>
</m:mrow>
</m:math>
<jats:tex-math>{\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}.}</jats:tex-math>
</jats:alternatives>
</jats:disp-formula>
</jats:p>
<jats:p>(2) - For <jats:italic>v</jats:italic> ∈ <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_003.png"/>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline">
<m:mrow>
<m:mi>v</m:mi>
<m:mo>∈</m:mo>
<m:mrow>
<m:mo>[</m:mo>
<m:mrow>
<m:mfrac>
<m:mrow>
<m:msup>
<m:mrow>
<m:mn>2</m:mn>
</m:mrow>
<m:mi>n</m:mi>
</m:msup>
<m:mo>-</m:mo>
<m:mn>1</m:mn>
</m:mrow>
<m:mrow>
<m:msup>
<m:mrow>
<m:mn>2</m:mn>
</m:mrow>
<m:mi>n</m:mi>
</m:msup>
</m:mrow>
</m:mfrac>
<m:mo>,</m:mo>
<m:mn>1</m:mn>
</m:mrow>
<m:mo>]</m:mo>
</m:mrow>
</m:mrow>
</m:math>
<jats:tex-math>v \in \left[ {{{{2^n} - 1} \over {{2^n}}},1} \right]</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, we have
<jats:disp-formula>
<jats:alternatives>
<jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_004.png"/>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block">
<m:mrow>
<m:msup>
<m:mrow>
<m:mrow>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:msup>
<m:mrow>
<m:mi>a</m:mi>
</m:mrow>
<m:mi>v</m:mi>
</m:msup>
<m:msup>
<m:mrow>
<m:mi>b</m:mi>
</m:mrow>
<m:mrow>
<m:mn>1</m:mn>
<m:mo>-</m:mo>
<m:mi>v</m:mi>
</m:mrow>
</m:msup>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
<m:mi>m</m:mi>
</m:msup>
<m:mo>+</m:mo>
<m:munderover>
<m:mo>∑</m:mo>
<m:mrow>
<m:mi>k</m:mi>
<m:mo>=</m:mo>
<m:mn>1</m:mn>
</m:mrow>
<m:mi>n</m:mi>
</m:munderover>
<m:mrow>
<m:msup>
<m:mrow>
<m:mn>2</m:mn>
</m:mrow>
<m:mrow>
<m:mi>k</m:mi>
<m:mo>-</m:mo>
<m:mn>1</m:mn>
</m:mrow>
</m:msup>
<m:msup>
<m:mrow>
<m:mrow>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mn>1</m:mn>
<m:mo>-</m:mo>
<m:mi>v</m:mi>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
<m:mi>m</m:mi>
</m:msup>
<m:msup>
<m:mrow>
<m:mrow>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:msqrt>
<m:mrow>
<m:msup>
<m:mrow>
<m:mi>a</m:mi>
</m:mrow>
<m:mi>m</m:mi>
</m:msup>
</m:mrow>
</m:msqrt>
<m:mo>-</m:mo>
<m:mroot>
<m:mrow>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mi>b</m:mi>
<m:msup>
<m:mrow>
<m:mi>a</m:mi>
</m:mrow>
<m:mrow>
<m:mn>2</m:mn>
<m:mi>k</m:mi>
<m:mo>-</m:mo>
<m:mn>1</m:mn>
</m:mrow>
</m:msup>
<m:mo>-</m:mo>
<m:mn>1</m:mn>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
<m:mi>m</m:mi>
</m:mrow>
<m:mrow>
<m:msup>
<m:mrow>
<m:mn>2</m:mn>
</m:mrow>
<m:mi>k</m:mi>
</m:msup>
</m:mrow>
</m:mroot>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
<m:mn>2</m:mn>
</m:msup>
<m:mo>≤</m:mo>
<m:msup>
<m:mrow>
<m:mrow>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mi>v</m:mi>
<m:mi>a</m:mi>
<m:mo>+</m:mo>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mn>1</m:mn>
<m:mo>-</m:mo>
<m:mi>v</m:mi>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
<m:mi>b</m:mi>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
<m:mi>m</m:mi>
</m:msup>
<m:mo>,</m:mo>
</m:mrow>
</m:mrow>
</m:math>
<jats:tex-math>{\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},}</jats:tex-math>
</jats:alternatives>
</jats:disp-formula>
we also prove two similar inequalities for the cases <jats:italic>v</jats:italic> ∈ <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_005.png"/>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline">
<m:mrow>
<m:mi>v</m:mi>
<m:mo>∈</m:mo>
<m:mrow>
<m:mo>[</m:mo>
<m:mrow>
<m:mfrac>
<m:mrow>
<m:msup>
<m:mrow>
<m:mn>2</m:mn>
</m:mrow>
<m:mi>n</m:mi>
</m:msup>
<m:mo>-</m:mo>
<m:mn>1</m:mn>
</m:mrow>
<m:mrow>
<m:msup>
<m:mrow>
<m:mn>2</m:mn>
</m:mrow>
<m:mi>n</m:mi>
</m:msup>
</m:mrow>
</m:mfrac>
<m:mo>,</m:mo>
<m:mfrac>
<m:mn>1</m:mn>
<m:mn>2</m:mn>
</m:mfrac>
</m:mrow>
<m:mo>]</m:mo>
</m:mrow>
</m:mrow>
</m:math>
<jats:tex-math>v \in \left[ {{{{2^n} - 1} \over {{2^n}}},{1 \over 2}} \right]</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> and <jats:italic>v</jats:italic> ∈ <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_006.png"/>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline">
<m:mrow>
<m:mi>v</m:mi>
<m:mo>∈</m:mo>
<m:mrow>
<m:mo>[</m:mo>
<m:mrow>
<m:mfrac>
<m:mn>1</m:mn>
<m:mn>2</m:mn>
</m:mfrac>
<m:mo>,</m:mo>
<m:mfrac>
<m:mrow>
<m:msup>
<m:mrow>
<m:mn>2</m:mn>
</m:mrow>
<m:mi>n</m:mi>
</m:msup>
<m:mo>+</m:mo>
<m:mn>1</m:mn>
</m:mrow>
<m:mrow>
<m:msup>
<m:mrow>
<m:mn>2</m:mn>
</m:mrow>
<m:mi>n</m:mi>
</m:msup>
</m:mrow>
</m:mfrac>
</m:mrow>
<m:mo>]</m:mo>
</m:mrow>
</m:mrow>
</m:math>
<jats:tex-math>v \in \left[ {{1 \over 2},{{{2^n} + 1} \over {{2^n}}}} \right]</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. 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 <jats:italic>p</jats:italic>-norms of positive τ-measurable operators.</jats:p>