<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we prove that if <italic>a</italic>, <italic>b</italic> > 0 and 0 ≤ α ≤ 1, then for <italic>m</italic> = 1, 2, 3, . . . ,<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0012_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd><mrow><msubsup><mrow><mi>r</mi></mrow><mn>0</mn><mi>m</mi></msubsup><msup><mrow><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>a</mi></mrow><mrow><mfrac><mi>m</mi><mn>2</mn></mfrac></mrow></msup><mo>-</mo><msup><mrow><mi>b</mi></mrow><mrow><mfrac><mi>m</mi><mn>2</mn></mfrac></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mrow><mo>≤</mo><msubsup><mrow><mi>r</mi></mrow><mn>0</mn><mi>m</mi></msubsup><mrow><mo>(</mo><mrow><mfrac><mrow><msup><mrow><mi>b</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfrac><mo>-</mo><mrow><mo>(</mo><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mi>b</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mfrac><mi>m</mi><mn>2</mn></mfrac></mrow></msup></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mo>≤</mo><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>α</mi><mi>a</mi><mo>+</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>α</mi></mrow><mo>)</mo></mrow><mi>b</mi></mrow><mo>)</mo></mrow></mrow></mrow><mi>m</mi></msup><mo>-</mo><msup><mrow><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>a</mi></mrow><mi>α</mi></msup><msup><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mi>m</mi></msup><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math><tex-math>\matrix{ {r_0^m{{\left( {{a^{{m \over 2}}} - {b^{{m \over 2}}}} \right)}^2}} &amp; { \le r_0^m\left( {{{{b^{m + 1}} - {a^{m + 1}}} \over {b - a}} - \left( {m + 1} \right){{\left( {ab} \right)}^{{m \over 2}}}} \right)} \cr {} &amp; { \le {{\left( {\alpha a + \left( {1 - \alpha } \right)b} \right)}^m} - {{\left( {{a^\alpha }{b^{1 - \alpha }}} \right)}^m},} \cr }</tex-math></alternatives></disp-formula>where <italic>r</italic>0 = min{α, 1 – α }. This is a considerable new generalization of two refinements of the Young inequality due to Kittaneh and Manasrah, and Hirzallah and Kittaneh, which correspond to the cases <italic>m</italic> = 1 and <italic>m</italic> = 2, respectively. As applications we give some refined Young type inequalities for generalized euclidean operator radius and the numerical radius of some well-know <italic>f</italic> -connection of operators and refined some Young type inequalities for the traces, determinants, and norms of positive definite matrices.</abstract>

In this paper, we prove that if a, b > 0 and 0 ≤ α ≤ 1, then for m = 1, 2, 3, . . . ,r0m(am2-bm2)2≤r0m(bm+1-am+1b-a-(m+1)(ab)m2)≤(αa+(1-α)b)m-(aαb1-α)m,\matrix{   {r_0^m{{\left( {{a^{{m \over 2}}} - {b^{{m \over 2}}}} \right)}^2}} & { \le r_0^m\left( {{{{b^{m + 1}} - {a^{m + 1}}} \over {b - a}} - \left( {m + 1} \right){{\left( {ab} \right)}^{{m \over 2}}}} \right)}  \cr    {} & { \le {{\left( {\alpha a + \left( {1 - \alpha } \right)b} \right)}^m} - {{\left( {{a^\alpha }{b^{1 - \alpha }}} \right)}^m},}  \cr  }where r0 = min{α, 1 – α }. This is a considerable new generalization of two refinements of the Young inequality due to Kittaneh and Manasrah, and Hirzallah and Kittaneh, which correspond to the cases m = 1 and m = 2, respectively. As applications we give some refined Young type inequalities for generalized euclidean operator radius and the numerical radius of some well-know f -connection of operators and refined some Young type inequalities for the traces, determinants, and norms of positive definite matrices.

A new generalization of two refined Young inequalities and applications

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.

{"article-title":"A new generalization of two refined Young inequalities and applications"}

In this paper, we prove that if a, b > 0 and 0 ≤ α ≤ 1, then for m = 1, 2, 3, . . ....

A new generalization of two refined Young inequalities and applications

Published Online: Oct 02, 2020

Page range: 155 - 167

Received: Jun 01, 2020

Accepted: Jul 10, 2020

DOI: https://doi.org/10.2478/mjpaa-2020-0012

Keywords
Young inequality, Positive semidefinite matrix, Euclidean operator radius, -connection

© 2020 M. A. Ighachane et al., published by Sciendo

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

A new generalization of two refined Young inequalities and applications

Published Online: Oct 02, 2020

Page range: 155 - 167

Received: Jun 01, 2020

Accepted: Jul 10, 2020

DOI: https://doi.org/10.2478/mjpaa-2020-0012

KeywordsYoung inequality, Positive semidefinite matrix, Euclidean operator radius, -connection

© 2020 M. A. Ighachane et al., published by Sciendo

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

Keywords
Young inequality, Positive semidefinite matrix, Euclidean operator radius, -connection