For a nef and big divisor D on a smooth projective surface S, the inequality $h^{0}$(S;$O_{s}(D)$) ${leq};D^2;+;2$ is well known. For a nef and big canonical divisor KS, there is a better inequality $h^{0}$(S;$O_{s}(K_s)$) ${leq};frac{1}{2}{K_{s}}^{2};+;2$ which is called the Noether inequality. We investigate an inequality $h^{0}$(S;$O_{s}(D)$) ${leq};frac{1}{2}D^{2};+;2$ like Clifford theorem in the case of a curve. We show that this inequality holds except some cases. We show the existence of a counter example for this inequality. We prove also the base-locus freeness of the linear system in the exceptional cases.