Let D be an integrally closed domain with quotient field K, * be a star operation on D, X, Y be indeterminates over D,

and

. Let b be the b-operation on R, and let

be the star operation on D defined by

. Finally, let Kr(R, b) (resp., Kr(D,

)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X, Y). In this paper, we show that Kr(R, b)

Kr(D,

) and Kr(R, b) is a kfr with respect to K(Y) and X in the notion of [2]. We also prove that Kr(R, b) = Kr(D,

) if and only if D is a

. As a corollary, we have that if D is not a

, then Kr(R, b) is an example of a kfr with respect to K(Y) and X but not a Kronecker function ring with respect to K(Y) and X.