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.