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

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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 matrices, then align* AXB*²≤f₁(A*A)Xg₁(B*B)f₂(A*A)Xg₂(B*B), align* where f₁,f₂,g₁,g₂ are non-negative continues functions such that f₁(t)f₂(t)=t and g₁(t)g₂(t)=t(t≥0). We also obtain the inequality align* |||AB*|||²&≤ |||p(A*A)ᵐ/ᵖ+ (1-p)(B*B)ˢ/¹⁻ᵖ||||||(1-p)(A*A)ⁿ/¹⁻ᵖ+ p(B*B)ᵗ/ᵖ|||, align* in which m,n,s,t are real numbers such that m+n=s+t=1, |||·||| is an arbitrary unitarily invariant norm and p∈[0,1].