Published online by Cambridge University Press: 20 November 2018
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 $\kappa (X)\ge 0$, then ${{K}_{X}}L\,\ge \,2q(X)\,-\,4$, where $q\left( X \right)$ 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 ${{h}^{0}}(L)\,\ge \,2$ , or (3) the case in which $\kappa (X)=2$, $X$ is minimal, ${{h}^{0}}(L)\,=\,1$ , and $L$ satisfies some conditions.