Special cases of the Jacobian conjecture

The famous Jacobian conjecture asks if a morphism $f:K[x,y]\to K[x,y]$ having an invertible Jacobian is invertible ($K$ is a characteristic zero field). We show that if one of the following three equivalent conditions is satisfied, then $f$ is invertible: $K[f(x),f(y)][x+y]$ is normal; $K[x,y]$ is flat over $K[f(x),f(y)][x+y]$; $K[f(x),f(y)][x+y]$ is separable over $K[f(x),f(y)]$.