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

<p>In this paper we show that, if that the function <italic>f</italic> : [0, ∞) → 𝔾 is operator monotone in [0, ∞) then there exist <italic>b</italic> ≥ 0 and a positive measure <italic>m</italic> on [0, ∞) such that <disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2022-0008_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>[</mo> <mrow><mi>f</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>-</mo><mi>f</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow> <mo>]</mo></mrow><mrow><mo>(</mo><mrow><mi>B</mi><mo>-</mo><mi>A</mi></mrow><mo>)</mo></mrow><mo>=</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>=</mo><mi>b</mi><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>B</mi><mo>-</mo><mi>A</mi></mrow><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup><mo>+</mo><mrow><msubsup><mo>∫</mo><mn>0</mn><mo>∞</mo></msubsup><mrow><msup><mrow><mi>s</mi></mrow><mn>2</mn></msup><mrow><mo>[</mo> <mrow><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msup><mrow><mrow><mrow><mo>[</mo> <mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>t</mi></mrow><mo>)</mo></mrow><mi>A</mi><mo>+</mo><mi>t</mi><mi>B</mi><mo>+</mo><mi>s</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mrow><mi>B</mi><mo>-</mo><mi>A</mi></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow></mrow><mn>2</mn></msup><mi>d</mi><mi>t</mi></mrow></mrow></mrow> <mo>]</mo></mrow><mi>d</mi><mi>m</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr></mtable></mrow></math>
<tex-math>\matrix{   {\left[ {f\left( B \right) - f\left( A \right)} \right]\left( {B - A} \right) = } \hfill  \cr    { = b{{\left( {B - A} \right)}^2} + \int_0^\infty  {{s^2}\left[ {\int_0^1 {{{\left[ {{{\left( {\left( {1 - t} \right)A + tB + s} \right)}^{ - 1}}\left( {B - A} \right)} \right]}^2}dt} } \right]dm\left( s \right)} } \hfill  \cr  }</tex-math>
</alternatives>
</disp-formula> for all <italic>A, B</italic> > 0. Some necessary and sufficient conditions for the operators <italic>A, B</italic> > 0 such that the inequality <disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2022-0008_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mi>B</mi><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mi>A</mi><mo>≥</mo><mi>f</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mi>B</mi><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mi>A</mi></mrow></math>
<tex-math>f\left( B \right)B + f\left( A \right)A \ge f\left( A \right)B + f\left( B \right)A</tex-math>
</alternatives>
</disp-formula>
holds for any operator monotone function <italic>f</italic> on [0, ∞) are also given.</p>
</abstract>

In this paper we show that, if that the function f : [0, ∞) → 𝔾 is operator monotone in [0, ∞) then there exist b ≥ 0 and a positive measure m on [0, ∞) such that 


[ f(B)-f(A) ](B-A)==b(B-A)2+∫0∞s2[ ∫01[ ((1-t)A+tB+s)-1(B-A) ]2dt ]dm(s)
\matrix{   {\left[ {f\left( B \right) - f\left( A \right)} \right]\left( {B - A} \right) = } \hfill  \cr    { = b{{\left( {B - A} \right)}^2} + \int_0^\infty  {{s^2}\left[ {\int_0^1 {{{\left[ {{{\left( {\left( {1 - t} \right)A + tB + s} \right)}^{ - 1}}\left( {B - A} \right)} \right]}^2}dt} } \right]dm\left( s \right)} } \hfill  \cr  }

 for all A, B > 0. Some necessary and sufficient conditions for the operators A, B > 0 such that the inequality 


f(B)B+f(A)A≥f(A)B+f(B)A
f\left( B \right)B + f\left( A \right)A \ge f\left( A \right)B + f\left( B \right)A


holds for any operator monotone function f on [0, ∞) are also given.

Some inequalities for operator monotone functions

Annals of West University of Timisoara - Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Some inequalities for operator monotone functions"}

In this paper we show that, if that the function f : [0, ∞) → 𝔾 is operator monotone in [0, ∞) then there exist b ≥ 0 and a positive measure m on [0, ∞) such...

Some inequalities for operator monotone functions

Published Online: Jun 20, 2022

Page range: 100 - 110

DOI: https://doi.org/10.2478/awutm-2022-0008

Keywords
Operator monotone functions, Integral inequalities, Operator inequality

© 2022 Silvestru Sever Dragomir, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Some inequalities for operator monotone functions

Published Online: Jun 20, 2022

Page range: 100 - 110

DOI: https://doi.org/10.2478/awutm-2022-0008

KeywordsOperator monotone functions, Integral inequalities, Operator inequality

© 2022 Silvestru Sever Dragomir, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Operator monotone functions, Integral inequalities, Operator inequality