Improvements of Young inequality using the Kantorovich constant
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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 $ kgeqslant 1: $ $K(h^{frac{1}{2^n}},2)^{r_n} asharp_{nu}b leqslant anabla_{nu} b - sum_{k=0}^{n-1}r_{k}left((a sharp_{frac{m_k}{2^k}} b )^{frac{1}{2}}- (a sharp_{frac{m_k+1}{2^k}}b )^{frac{1}{2}}right)^{2}leqslant K(h^{frac{1}{2^n}},2)^{R_n} asharp_{nu}b,$ where $m_k= [ 2^knu ] $ is the largest integer not greater than $2^knu$, $ r_0=min { nu, 1-nu}, $ $ _{k}=min { 2r_{k-1}, 1-2r_{k-1} } $ and $R_k=1-r_k$.
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Journal title
volume 43 issue 5
pages 1301- 1311
publication date 2017-10-31
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