LAUSR

Using a refinement of the classical Young inequality, we refine some inequalities of operators including the function ωpωp, where ωpωp is defined for p⩾1p⩾1 and operators T1,…,Tn∈B(H)T1,…,Tn∈B(H) by ωp(T1,…,Tn):=sup‖x‖=1⁡(∑i=1n|〈Tix,x〉|p)1p. Among other things, we show that if T1,…,Tn∈B(H)T1,…,Tn∈B(H) and p≥q≥1p≥q≥1 with 1p+1q=1, then 1n‖∑i=1nTi‖2≤ωp(|T1|,…,|Tn|)ωq(|T1⁎|,…,|Tn⁎|)1n‖∑i=1nTi‖2≤1p‖∑i=1n|Ti|p‖+1q‖∑i=1n|Ti⁎|q‖−inf‖x‖=‖y‖=1⁡δ(x,y), where δ(x,y)=1p(∑i=1n〈|Ti|x,x〉p−∑i=1n〈|Ti⁎|y,y〉q)2.