Improvements of Young inequality using the Kantorovich constant

Document Type : Research Paper


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


‎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 $R_k=1-r_k$‎.


Main Subjects

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