Further refinements of the Heinz inequality

The celebrated Heinz inequality asserts that 2|||A1/2XB1/2|||⩽|||AνXB1-ν+A1-νXBν|||⩽|||AX+XB|||2|||A1/2XB1/2|||⩽|||AνXB1-ν+A1-νXBν|||⩽|||AX+XB||| for X∈B(H)X∈B(H), A,B∈B(H)+A,B∈B(H)+, every unitarily invariant norm |||·||||||·||| and ν∈[0,1]ν∈[0,1]. In this paper, we present several improvement of the Heinz inequality by using the convexity of the function F(ν)=|||AνXB1-ν+A1-νXBν|||F(ν)=|||AνXB1-ν+A1-νXBν|||, some integration techniques and various refinements of the Hermite–Hadamard inequality. In the setting of matrices we prove thatAα+β2XB1-α+β2+A1-α+β2XBα+β2⩽1|β-α|∫αβAνXB1-ν+A1-νXBνdν⩽12AαXB1-α+A1-αXBα+AβXB1-β+A1-βXBβ,for real numbers α, β.