A remark on the canonical degree of curves on smooth projective surface

The canonical degree $C\mathrm{.}{K}_X$ 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 $h^0(X,2K_X+C)=0$.