In this note we prove that if A and $B^*$ are subnormal operators and is a bounded linear operator such that AX - XB is a Hilbert-Schmidt operator, then f(A)X - Xf(B) is also a Hilbert-Schmidt operator and $${parallel}f(A)X;-;Xf(B){parallel}_2;leq;L{parallel}AX;-;XB{parallel}_2$$, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and $X;{in};cal{L}(cal{H})$ is such that SX - XT belongs to a norm ideal (J, ${parallel};{cdot};{parallel}_J$) and prove that f(S)X - Xf(T) $in$ J and ${parallel}f(S)X;-;Xf(T){parallel}_J;leq;C{parallel}SX;-;XT{parallel}_J$, for f in a certain class of functions.