Another Version of Young Inequality Applying the Supplemental Young Inequality

Year 2025, Volume: 13 Issue: 2, 134 - 140, 31.10.2025

Leila Nasiri , Mahmood Shakoori

Abstract

In this note, we give the further reverses of the Young type inequalities for non-negative real scalars,
using the supplemental Young inequality
\begin{align*}
a^{\nu}b^{1-\nu} \geqslant \nu a + (1-\nu) b,
\end{align*}
where $a, b \geqslant 0$ and $ \nu \notin [0,1]$.
Making use of them, some matrix inequalities for Hilbert-Schmidt norm and trace norm are deduced.

Keywords

‎Trace norm , Hilbert-Schmidt norm , ‎Reverse Young inequalities

References

• [1] T. Ando, Matrix Young inequality, Oper Theory A dv Appl., 75(1995), 33-38.
• [2] M. Bakherad and M. S. Moslehian, Reverses and variations of Heinz inequality, Linear Multilinear Algebra., 63(2015), n. 10, 1972-1980.
• [3] R. Bhatia, Matrix Analysis, Springer-Verlag, New-York, 1997.
• [4] R. Bhatia, R. Parthasarathy, Positive definite functions and operators inequalities, Bull. London Math. Soc., 32(2000), 214-228.
• [5] R. Bhatia, F. Kittaneh, Notes on matrix arithmetic-geometric mean inequalities, Linear Multilinear Algebra Appl., 308( 2000), 203-211.
• [6] S. Furuichi, et.al., Generalized reverse Young and Heinz inequalities, Bull. Malaysian Math. Sci. Soc., 42(2019), 267-284.
• [7] X. Hu, Young type inequalities for matrices, Journal of East China Normal University, 4(2012), 12-17.
• [8] O. Hirzallah and F. Kittaneh , Matrix Young inequalities for the Hilbert-Schmidt norm, Linear Algebra Appl, 308 (2000), 77-84.
• [9] X. Hu and J. Xue, A note on reveres of Young type inequalities, J. Inequal. Appl., (2015), n. 98. 1-6.
• [10] C. He, L. Zou and S. Qaisar, On improved arithmetic-geometric mean and Heinz inequalities for matrices, J. Math. Ineq., 6(2012), n. 3, 453-459.
• [11] F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl., 361(2010), 262-269.
• [12] F. Kittaneh, On some operator inequalities, Linear Algebra Appl, 208/209(1994), 19-28.
• [13] H. Kosaki, Arithmetic-geometric mean and related inequality for operators, J. Funct. Anal, 156(1998), 429-451.
• [14] L. Nasiri, M. Shakoori and W. Liao, A note on the Young type inequalities, Int. J. Nonline. Anal. Appl., 8 (2017), n. 1, 261-267.
• [15] L. Nasiri and W. Liao, The new reverses of Young type inequalities for numbers, matrices and operators, Oper. Matrices, 12(2018), n. 4, 1063-1071.
• [16] M. Sababheh and D. Choi, A complete refinement of Youngs inequality, J. Math. Anal. Appl., 440(2016), 379-393.
• [17] M. Tominago, Spechts ratio in the Young inequality, Sci. Math. Japon., 5 (2001), 525-530.
• [18] J. L. Wu and J. G. Zhao, Operator inequalities and reverse inequalities related to the Kittaneh and Manasrah inequalities, Linear Multilinear Algebra, 62(2014), n. 7, 884-894.
• [19] H. Zuo, G. Shi and M. Fujii, Refined Young type inequality with Kantorovich constant, J. Math. Inequal, 5, (2011), n. 4, 551-556.
• [20] X. Zhan, Inequalities for unitarily invariant norms, SIAM J. Matrix Anal. Appl., 3(1998), 466-470.