<abstract xmlns="http://www.w3.org/1999/xhtml">

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

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.

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

Department of Mathematics, Faculy of Sciences-Semlalia

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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}}}}...

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

Publié en ligne: 28 déc. 2020

Pages: 214 - 226

Reçu: 16 sept. 2020

Accepté: 02 déc. 2020

DOI: https://doi.org/10.2478/mjpaa-2021-0015

Mots clés
AM-GM inequality, Young’s inequality, Determinants, Trace, -norms, τ-mesurable operators

© 2020 Mohamed Amine Ighachane et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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

Publié en ligne: 28 déc. 2020

Pages: 214 - 226

Reçu: 16 sept. 2020

Accepté: 02 déc. 2020

DOI: https://doi.org/10.2478/mjpaa-2021-0015

Mots clésAM-GM inequality, Young’s inequality, Determinants, Trace, -norms, τ-mesurable operators

© 2020 Mohamed Amine Ighachane et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Mots clés
AM-GM inequality, Young’s inequality, Determinants, Trace, -norms, τ-mesurable operators