LAUSR

In this paper, we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities, it is shown that if A, B, X are n×n$n\times n$ matrices, then ∥AXB∗∥2≤∥f1(A∗A)Xg1(B∗B)∥∥f2(A∗A)Xg2(B∗B)∥, $$\begin{aligned} \bigl\Vert AXB^{*} \bigr\Vert ^{2}\leq \bigl\Vert f_{1} \bigl(A^{*}A\bigr)Xg_{1}\bigl(B^{*}B\bigr) \bigr\Vert \bigl\Vert f_{2}\bigl(A^{*}A\bigr)Xg_{2}\bigl(B^{*}B\bigr) \bigr\Vert , \end{aligned}$$ where f1$f_{1}$, f2$f_{2}$, g1$g_{1}$, g2$g_{2}$ are non-negative continuous functions such that f1(t)f2(t)=t$f_{1}(t)f_{2}(t)=t$ and g1(t)g2(t)=t$g_{1}(t)g_{2}(t)=t$ (t≥0$t\geq0$). We also obtain the inequality |||AB∗|||2≤|||p(A∗A)mp+(1−p)(B∗B)s1−p||||||(1−p)(A∗A)n1−p+p(B∗B)tp|||, $$\begin{aligned} \bigl\vert \!\bigl\vert \!\bigl\vert AB^{*} \bigr\vert \!\bigr\vert \!\bigr\vert ^{2} &\leq \bigl\vert \!\bigl\vert \!\bigl\vert p \bigl(A^{*}A\bigr)^{\frac{m}{p}}+ (1-p) \bigl(B^{*}B\bigr)^{\frac {s}{1-p}} \bigr\vert \!\bigr\vert \!\bigr\vert \bigl\vert \!\bigl\vert \!\bigl\vert (1-p) \bigl(A^{*}A\bigr)^{\frac{n}{1-p}}+ p\bigl(B^{*}B \bigr)^{\frac{t}{p}} \bigr\vert \!\bigr\vert \!\bigr\vert , \end{aligned}$$ in which m, n, s, t are real numbers such that m+n=s+t=1$m+n=s+t=1$, |||⋅|||$\vert \!\vert \!\vert \cdot \vert \!\vert \!\vert $ is an arbitrary unitarily invariant norm and p∈[0,1]$p\in[0,1]$.