Extensions of interpolation between the arithmetic-geometric mean inequality for matrices

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 $n\times n$ matrices, then \begin{align*} \|AXB^*\|^2\leq\|f_1(A^*A)Xg_1(B^*B)\|\,\|f_2(A^*A)Xg_2(B^*B)\|, \end{align*} where $f_1,f_2,g_1,g_2$ are non-negative continues functions such that $f_1(t)f_2(t)=t$ and $g_1(t)g_2(t)=t\,\,(t\geq0)$. We also obtain the inequality \begin{align*} \left|\left|\left|AB^*\right|\right|\right|^2\nonumber&\leq \left|\left|\left|p(A^*A)^{\frac{m}{p}}+ (1-p)(B^*B)^{\frac{s}{1-p}}\right|\right|\right|\,\left|\left|\left|(1-p)(A^*A)^{\frac{n}{1-p}}+ p(B^*B)^{\frac{t}{p}}\right|\right|\right|, \end{align*} in which $m,n,s,t$ are real numbers such that $m+n=s+t=1$, $|||\cdot|||$ is an arbitrary unitarily invariant norm and $p\in[0,1]$.