A lower bound for KXL of quasi-polarized surfaces (X,L) with non-negative Kodaira dimension

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 k(X) greater than or equal to 0, then KXL greater tha n or equal to 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 k(X) = 0 or 1, (2) the case in which k(X) = 2 and h(0)(L) greater than or equal to 2, or (3) the case in which k(X) = 2, X is minimal, h(0)(L) = 1, and L satisfi es some conditions.