Improvements of Young inequality using the Kantorovich constant

Document Type : Research Paper

Authors

Department of‎ ‎Pure Mathematics‎, ‎Faculty of Mathematics and Computer‎, ‎Shahid Bahonar University of Kerman‎, ‎Kerman‎, ‎Iran.

Abstract

‎Some improvements of Young inequality and its reverse for positive‎ ‎numbers with Kantorovich constant $K(t‎, ‎2)=\frac{(1+t)^2}{4t}$‎ ‎are given‎. ‎Using these inequalities some operator inequalities and‎ ‎Hilbert-Schmidt norm versions for matrices are proved‎. ‎In‎ ‎particular‎, ‎it is shown that if $a‎, ‎b$ are positive numbers and‎ ‎$0 \leqslant \nu \leqslant 1,$ then for all integers $ k\geqslant‎ ‎1‎: ‎$‎
‎$K(h^{\frac{1}{2^n}},2)^{r_n} a\sharp_{\nu}b \leqslant a\nabla_{\nu} b‎ - ‎\sum_{k=0}^{n-1}r_{k}\left((a \sharp_{\frac{m_k}{2^k}} b‎ ‎)^{\frac{1}{2}}‎- ‎(a \sharp_{\frac{m_k+1}{2^k}}b‎ ‎)^{\frac{1}{2}}\right)^{2}\leqslant K(h^{\frac{1}{2^n}},2)^{R_n} a\sharp_{\nu}b,$
‎where $m_k= [ 2^k\nu ] $ is the largest integer not greater than‎ ‎$2^k\nu$‎, ‎$ r_0=\min \{ \nu‎, ‎1-\nu\}‎, ‎$ $  _{k}=\min \{ 2r_{k-1}‎, ‎1-2r_{k-1} \} $ and $R_k=1-r_k$‎.

Keywords

Main Subjects

References

M. Bakherad, M. Krnic and M. S. Moslehian, Reverse Young-type inequalities for matrices and operators, Rocky Mountain J. Math. 46 (2016), no. 4, 1089--1105.
C. Conde, Young type inequalities for positive operators, Ann. Funct. Anal. 4 (2013), no. 2, 144--152.
T. Furuta, Invitation to Linear Operators: Form Matrix to Bounded Linear Operators on a Hilbert Space, Taylor and Francis, 2002.
T. Furuta and M. Yanagide, Generalized means and convexity of inversion for positive operators, Amer. Math. Monthly 105 (1998) 258--259.
C.J. He and L.M. Zou, Some inequalities involving unitarily invariant norms, Math. Inequal. Appl. 12 (2012), no. 4, 767--776.
O. Hirzallah and F. Kittaneh, Matrix Young inequalities for the Hilbert-Schmidt norm, Linear Algebra Appl. 308 (2000), no. 1-3, 77--84.
F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl. 361 (2010), no. 1, 262--269.
F. Kittaneh and Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Linear Multilinear Algebra 59 (2011), no. 9, 1031--1037.
W. Liao and J. Wu, Improved Young and Heinz inequalities with the Kantorovich constant, J. Math. Inequal. 10 (2016), no. 2, 559--570.
W. Liao, J. Wu and J. Zhao, New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math. 19 (2015), no. 2, 467--479.
A. Salemi and A. Sheikh Hosseini, On reversing of the modified Young inequality , Ann. Funct. Anal. 5 (2014), no. 1, 70--76.
J.Wu and J. Zhao, Operator inequalities and reverse inequalities related to the Kittaneh-Manasrah inequalities, Linear Multilinear Algebra 62 (2014), no. 7, 884--894.
H.L. Zuo, G.H. Shi and M. Fujii, Refined Young inequality with Kantorovich constant, J. Math. Inequal. 5 (2011), no. 4, 551--556.
Bulletin of the Iranian Mathematical Society
Volume 43, Issue 5
October 2017
Pages 1301-1311

Files

History

  • Receive Date: 25 December 2015
  • Revise Date: 26 May 2016
  • Accept Date: 27 May 2016

Share

How to cite

Statistics