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

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Springer Basel
Copyright © 2015 by Springer Basel
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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  • Matrix Young inequalities for the Hilbert–Schmidt norm
    Hirzallah, O; Kittaneh, F

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