‎‎Let $C_{\\\\\\\\\\\\\\\\||.\\\\\\\\\\\\\\\\||}$ be an ideal of compact operators with symmetric norm $\\\\\\\\\\\\\\\\||.\\\\\\\\\\\\\\\\||$‎. ‎In this paper‎, ‎we extend the van Hemmen--Ando norm inequality for arbitrary bounded operators as follows‎: ‎If $f$ is an operator monotone function on $[0,\\\\\\\\\\\\\\\\infty)$ and $S$ and $T$ are bounded operators in $\\\\\\\\\\\\\\\\mathbb{B}(\\\\\\\\\\\\\\\\mathscr{H})$ such that ${\\\\\\\\\\\\\\\\rm{sp}}(S),{\\\\\\\\\\\\\\\\rm{sp}}(T) \\\\\\\\\\\\\\\\subseteq \\\\\\\\\\\\\\\\Gamma_a=\\\\\\\\\\\\\\\\{z\\\\\\\\\\\\\\\\in \\\\\\\\\\\\\\\\mathbb{C} \\\\\\\\\\\\\\\\ | \\\\\\\\\\\\\\\\ {\\\\\\\\\\\\\\\\rm{re}}(z)\\\\\\\\\\\\\\\\geq a\\\\\\\\\\\\\\\\}$‎, ‎then‎ ‎$$\\\\\\\\\\\\\\\\||f(S)X-Xf(T)\\\\\\\\\\\\\\\\|| \\\\\\\\\\\\\\\\leq f^{\\\\\\\\\\\\\\\'}(a) \\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\||SX-XT\\\\\\\\\\\\\\\\||,$$‎ ‎for each $X\\\\\\\\\\\\\\\\in C_{\\\\\\\\\\\\\\\\||.\\\\\\\\\\\\\\\\||}$‎. ‎In particular‎, ‎if ${\\\\\\\\\\\\\\\\rm{sp}}(S)‎, ‎{\\\\\\\\\\\\\\\\rm{sp}}(T) \\\\\\\\\\\\\\\\subseteq \\\\\\\\\\\\\\\\Gamma_a$‎, ‎then‎ ‎$$\\\\\\\\\\\\\\\\||S^r X-XT^r\\\\\\\\\\\\\\\\|| \\\\\\\\\\\\\\\\leq r a^{r-1} \\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\||SX-XT\\\\\\\\\\\\\\\\||,$$‎ ‎for each $X\\\\\\\\\\\\\\\\in C_{\\\\\\\\\\\\\\\\||.\\\\\\\\\\\\\\\\||}$ and for each $0\\\\\\\\\\\\\\\\leq r\\\\\\\\\\\\\\\\leq 1$‎.

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