@article { author = {Khosravi, M. and Sheikh Hosseini, A.}, title = {Improvements of Young inequality using the Kantorovich constant}, journal = {Bulletin of the Iranian Mathematical Society}, volume = {43}, number = {5}, pages = {1301-1311}, year = {2017}, publisher = {Iranian Mathematical Society (IMS)}, issn = {1017-060X}, eissn = {1735-8515}, doi = {}, 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 $ 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 $R_k=1-r_k$‎.}, keywords = {Heinz mean‎,‎Hilbert-Schmidt norm‎,‎Kantorovich constant‎,‎Young inequality‎}, url = {http://bims.iranjournals.ir/article_1025.html}, eprint = {http://bims.iranjournals.ir/article_1025_461e366823aee2c9e7465c996b81443d.pdf} }