Further refinements of the Heinz inequality

The celebrated Heinz inequality asserts that $ 2|||A^{1/2}XB^{1/2}|||\leq |||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}|||\leq |||AX+XB|||$for$X \in $\mathbb{B}$(\mathscr{H})$,$A,B\in +$,everyunitarilyinvariantnorm$|||\cdot|||$and$\nu \in [0,1]$. In this paper, we present several improvement of the Heinz inequality by using the convexity of the function $F(\nu )=\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }{A}^{\nu }X{B}^{1-\nu }+A^{1-\nu }X{B}^{\nu }\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }$, some integration techniques and various refinements of the Hermite–Hadamard inequality. In the setting of matrices we prove that \begin{eqnarray*} &&\hspace{-0.5cm}\left|\left|\left|A^{\frac{\alpha+\beta}{2}}XB^{1-\frac{\alpha+\beta}{2}}+A^{1-\frac{\alpha+\beta}{2}}XB^{\frac{\alpha+\beta}{2}}\right|\right|\right|\leq\frac{1}{|\beta-\alpha|} \left|\left|\left|\int_{\alpha}^{\beta}\left(A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\right)d\nu\right|\right|\right|\nonumber\ &&\qquad\qquad\leq \frac{1}{2}\left|\left|\left|A^{\alpha}XB^{1-\alpha}+A^{1-\alpha}XB^{\alpha}+A^{\beta}XB^{1-\beta}+A^{1-\beta}XB^{\beta}\right|\right|\right|,, \end{eqnarray*} for real numbers $\alpha ,\beta$.