Using Furuta's inequality, we can get that
if $1
0,A_2\>B>0$ and
$A_1^t\natural _{\frac{1-t}{p-t}}A_2^p\
(B^r{A_2}^pB^r)^{\frac{\a+2r}{p+2r}}.$$
holds for any $0\<\a\<\min \{2p-1,t\}$ and $ r\>0$.
We can also get that if $1\0,A_2\>B>0$ and
$A_1^t\natural _{\frac{2p-t}{p-t}}A_2^p\
(B^r{A_2}^pB^r)^{\frac{\a+2r}{p+2r}}.$$
holds for any $0\<\a\<2p$ and $ r\>0$.