# A generalization of two refined Young inequalities

A generalization of two refined Young inequalities We prove that if $$a,b>0$$ a , b > 0 and $$0\le \nu \le 1$$ 0 ≤ ν ≤ 1 , then for $$m=1,2,3,\ldots$$ m = 1 , 2 , 3 , … , we have \begin{aligned} \left( a^{\nu }b^{1-\nu }\right) ^{m}+r_{0}^{m}\left( a^{\frac{m}{2}}-b^{\frac{m}{2}}\right) ^{2}\le \Big ( \nu a+(1-\nu )b\Big ) ^{m}, \end{aligned} a ν b 1 - ν m + r 0 m a m 2 - b m 2 2 ≤ ( ν a + ( 1 - ν ) b ) m , where $$r_{0}=\min \left\{ \nu ,1-\nu \right\}$$ r 0 = min ν , 1 - ν . This is a considerable generalization of two refinements of the classical Young inequality due to Kittaneh and Manasrah, and Hirzallah and Kittaneh, which correspond to the cases $$m=1$$ m = 1 and $$m=2$$ m = 2 , respectively. As applications of this inequality, we give refined Young-type inequalities for the traces, determinants, and norms of positive definite matrices. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# A generalization of two refined Young inequalities

, Volume 19 (4) – Feb 4, 2015
12 pages

/lp/springer_journal/a-generalization-of-two-refined-young-inequalities-9rIM0IZogn
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-015-0326-8
Publisher site
See Article on Publisher Site

### Abstract

We prove that if $$a,b>0$$ a , b > 0 and $$0\le \nu \le 1$$ 0 ≤ ν ≤ 1 , then for $$m=1,2,3,\ldots$$ m = 1 , 2 , 3 , … , we have \begin{aligned} \left( a^{\nu }b^{1-\nu }\right) ^{m}+r_{0}^{m}\left( a^{\frac{m}{2}}-b^{\frac{m}{2}}\right) ^{2}\le \Big ( \nu a+(1-\nu )b\Big ) ^{m}, \end{aligned} a ν b 1 - ν m + r 0 m a m 2 - b m 2 2 ≤ ( ν a + ( 1 - ν ) b ) m , where $$r_{0}=\min \left\{ \nu ,1-\nu \right\}$$ r 0 = min ν , 1 - ν . This is a considerable generalization of two refinements of the classical Young inequality due to Kittaneh and Manasrah, and Hirzallah and Kittaneh, which correspond to the cases $$m=1$$ m = 1 and $$m=2$$ m = 2 , respectively. As applications of this inequality, we give refined Young-type inequalities for the traces, determinants, and norms of positive definite matrices.

### Journal

PositivitySpringer Journals

Published: Feb 4, 2015

### References

• Matrix Young inequalities for the Hilbert–Schmidt norm
Hirzallah, O; Kittaneh, F

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