  The canonical degree $C.K_X$ of an integral curve on a smooth projective
surface $X$ is conjecturally bounded from above by an expression of the form
$A(g-1)+B$, where $g$ is the geometric genus of $C$ and $A$, $B$ are constants
depending only on $X$. We prove that this conjecture holds with $A = -1$ under
the assumptions $h^0(X, -K_X) = 0$ and $h^0(X, 2K_X + C) = 0$.
