# Improvements of Young inequality using the Kantorovich constant

Document Type: Research Paper

Authors

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

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 $R_k=1-r_k$‎.

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### History

• Receive Date: 25 December 2015
• Revise Date: 26 May 2016
• Accept Date: 27 May 2016