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In this paper, motivated by perturbation theory of operators, we present someupper bounds for $|||f(A)Xg(B)+ X|||$ in terms of $|||\,|AXB|+|X|\,|||$ and$|||f(A)Xg(B)- X|||$ in terms of $|||\,|AX|+|XB|\,|||$, where $A, B$ are$G_{1}$ operators, $|||\cdot|||$ is a unitarily invariant norm and $f, g$ arecertain analytic functions. Further, we find some new upper bounds for the theSchatten $2$-norm of $f(A)X\pm Xg(B)$. Several special cases are discussed aswell.

Unitarily invariant norm inequalities for elementary operators involving $G_{1}$ operators