  Let $X$ be a smooth projective surface over the complex number field and let
$L$ be a nef-big divisor on $X$. Here we consider the following conjecture; If
the Kodaira dimension $\kappa(X)\geq 0$, then $K_{X}L\geq 2q(X)-4$, where
$q(X)$ is the irregularity of $X$. In this paper, we prove that this conjecture
is true if (1) the case in which $\kappa(X)=0$ or 1, (2) the case in which
$\kappa(X)=2$ and $h^{0}(L)\geq 2$, or (3) the case in which $\kappa(X)=2$, $X$
is minimal, $h^{0}(L)=1$, and $L$ satisfies some conditions.
