Research Article
BibTex RIS Cite
Year 2019, Volume: 48 Issue: 4, 951 - 958, 08.08.2019

Abstract

References

  • [1] G. Aghamollaei and A. Sheikh Hosseini, Some numerical radius inequalities with positive definite functions, Bull. Iranian Math. Soc. 41 (4), 889-900, 2015.
  • [2] T. Ando and K. Okubo, Induced norms of the Schur multiplication operator, Linear Algebra Appl. 147, 181-199, 1991.
  • [3] K.M.R. Audenaert, A characterization of anti-Lowner function, Proc. Amer. Math. Soc. 139 (12), 4217-4223, 2011.
  • [4] M. Bakherad and F. Kittaneh, Numerical Radius Inequalities Involving Commutators of G1 Operators, Complex Anal. Oper. Theory 13 (4), 1557-1567, 2019.
  • [5] M. Bakherad and M.S. Moslehian, Reverses and variations of Heinz inequality, Linear Multilinear Algebra 63 (10), 1972-1980, 2015.
  • [6] R. Bhatia and Ch. Davis, More matrix forms of the arithmetic-geometric mean inequality, SIAM J. Matrix Anal. Appl. 14 (1), 132-136, 1993.
  • [7] M. Erfanian Omidvar, M.S. Moslehian and A. Niknam, Some numerical radius inequalities for Hilbert space operators, Involve 2 (4), 469-476, 2009.
  • [8] J. Fujii, M. Fujii, Y. Seo and H. Zuo, Recent developments of matrix versions of the arithmetic-geometric mean inequality. Ann. Funct. Anal. 7 (1), 102-117, 2016.
  • [9] M. Fujii, J. Mićić Hot, J. Pečarić and Y. Seo, Recent developments of Mond-Pečarić method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. II., Monographs in Inequalities 4. Zagreb: Element, 2012.
  • [10] M. Fujii, Y. Seo and H. Zuo, Zhan’s inequality on A-G mean inequalities. Linear Algebra Appl. 470, 241-251, 2015.
  • [11] K.E. Gustafson and D.K.M. Rao, Numerical Range, The Field of Values of Linear Operators and Matrices, Springer, New York, 1997.
  • [12] O. Hirzallah, F. Kittaneh and Kh. Shebrawi, Numerical radius inequalities for certain 2 × 2 operator matrices, Integral Equations Operator Theory 71 (1), 129-147, 2011.
  • [13] M.K. Kwong, Some results on matrix monotone functions, Linear Algebra Appl. 118, 129-153, 1989.
  • [14] H. Najafi, Some results on Kwong functions and related inequalities, Linear Algebra Appl. 439 (9), 2634-2641, 2013.
  • [15] G. Ramesh, On the numerical radius of a quaternionic normal operator, 2 (1), 78-86, 2017.
  • [16] T. Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Studia Math. 178, 83-89, 2007.
  • [17] X. Zhan, Inequalities for unitarily invariant norms, SIAM J. Matrix Anal. Appl. 20 (2), 466-470, 1999.
  • [18] F. Zhang, Matrix Theory, Second edition, Springer, New York, 2011.

Some generalized numerical radius inequalities involving Kwong functions

Year 2019, Volume: 48 Issue: 4, 951 - 958, 08.08.2019

Abstract

We prove several numerical radius inequalities involving positive semidefinite matrices via the Hadamard product and Kwong functions. Among other inequalities, it is shown that if   $X$ is an arbitrary $n\times n$ matrix and $A,B$ are positive semidefinite, then
\[ \omega(H_{f,g}(A))\leq k\, \omega(AX+XA), \]
 which is equivalent to
\[\omega\big(H_{f,g}(A,B)\pm H_{f,g}(B,A)\big)\leq k'\,\left\{\omega((A+B)X+X(A+B))+\omega((A-B)X-X(A-B))\right\},\]
 where  $f$ and $g$ are two continuous functions on $(0,\infty)$ such that $h(t)={f(t)\over g(t)}$ is Kwong, $k=\max\left\{{f(\lambda)g(\lambda)\over \lambda}: {\lambda\in\sigma(A)}\right\}$ and $k'=\max\left\{{f(\lambda)g(\lambda)\over \lambda}: {\lambda\in\sigma(A)\cup\sigma(B)}\right\}$.

References

  • [1] G. Aghamollaei and A. Sheikh Hosseini, Some numerical radius inequalities with positive definite functions, Bull. Iranian Math. Soc. 41 (4), 889-900, 2015.
  • [2] T. Ando and K. Okubo, Induced norms of the Schur multiplication operator, Linear Algebra Appl. 147, 181-199, 1991.
  • [3] K.M.R. Audenaert, A characterization of anti-Lowner function, Proc. Amer. Math. Soc. 139 (12), 4217-4223, 2011.
  • [4] M. Bakherad and F. Kittaneh, Numerical Radius Inequalities Involving Commutators of G1 Operators, Complex Anal. Oper. Theory 13 (4), 1557-1567, 2019.
  • [5] M. Bakherad and M.S. Moslehian, Reverses and variations of Heinz inequality, Linear Multilinear Algebra 63 (10), 1972-1980, 2015.
  • [6] R. Bhatia and Ch. Davis, More matrix forms of the arithmetic-geometric mean inequality, SIAM J. Matrix Anal. Appl. 14 (1), 132-136, 1993.
  • [7] M. Erfanian Omidvar, M.S. Moslehian and A. Niknam, Some numerical radius inequalities for Hilbert space operators, Involve 2 (4), 469-476, 2009.
  • [8] J. Fujii, M. Fujii, Y. Seo and H. Zuo, Recent developments of matrix versions of the arithmetic-geometric mean inequality. Ann. Funct. Anal. 7 (1), 102-117, 2016.
  • [9] M. Fujii, J. Mićić Hot, J. Pečarić and Y. Seo, Recent developments of Mond-Pečarić method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. II., Monographs in Inequalities 4. Zagreb: Element, 2012.
  • [10] M. Fujii, Y. Seo and H. Zuo, Zhan’s inequality on A-G mean inequalities. Linear Algebra Appl. 470, 241-251, 2015.
  • [11] K.E. Gustafson and D.K.M. Rao, Numerical Range, The Field of Values of Linear Operators and Matrices, Springer, New York, 1997.
  • [12] O. Hirzallah, F. Kittaneh and Kh. Shebrawi, Numerical radius inequalities for certain 2 × 2 operator matrices, Integral Equations Operator Theory 71 (1), 129-147, 2011.
  • [13] M.K. Kwong, Some results on matrix monotone functions, Linear Algebra Appl. 118, 129-153, 1989.
  • [14] H. Najafi, Some results on Kwong functions and related inequalities, Linear Algebra Appl. 439 (9), 2634-2641, 2013.
  • [15] G. Ramesh, On the numerical radius of a quaternionic normal operator, 2 (1), 78-86, 2017.
  • [16] T. Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Studia Math. 178, 83-89, 2007.
  • [17] X. Zhan, Inequalities for unitarily invariant norms, SIAM J. Matrix Anal. Appl. 20 (2), 466-470, 1999.
  • [18] F. Zhang, Matrix Theory, Second edition, Springer, New York, 2011.
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Mojtaba Bakherad 0000-0003-0323-6310

Publication Date August 8, 2019
Published in Issue Year 2019 Volume: 48 Issue: 4

Cite

APA Bakherad, M. (2019). Some generalized numerical radius inequalities involving Kwong functions. Hacettepe Journal of Mathematics and Statistics, 48(4), 951-958.
AMA Bakherad M. Some generalized numerical radius inequalities involving Kwong functions. Hacettepe Journal of Mathematics and Statistics. August 2019;48(4):951-958.
Chicago Bakherad, Mojtaba. “Some Generalized Numerical Radius Inequalities Involving Kwong Functions”. Hacettepe Journal of Mathematics and Statistics 48, no. 4 (August 2019): 951-58.
EndNote Bakherad M (August 1, 2019) Some generalized numerical radius inequalities involving Kwong functions. Hacettepe Journal of Mathematics and Statistics 48 4 951–958.
IEEE M. Bakherad, “Some generalized numerical radius inequalities involving Kwong functions”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 4, pp. 951–958, 2019.
ISNAD Bakherad, Mojtaba. “Some Generalized Numerical Radius Inequalities Involving Kwong Functions”. Hacettepe Journal of Mathematics and Statistics 48/4 (August 2019), 951-958.
JAMA Bakherad M. Some generalized numerical radius inequalities involving Kwong functions. Hacettepe Journal of Mathematics and Statistics. 2019;48:951–958.
MLA Bakherad, Mojtaba. “Some Generalized Numerical Radius Inequalities Involving Kwong Functions”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 4, 2019, pp. 951-8.
Vancouver Bakherad M. Some generalized numerical radius inequalities involving Kwong functions. Hacettepe Journal of Mathematics and Statistics. 2019;48(4):951-8.