L is a holomorphic line bundle on a compact complex manifold X. The Kodaira dimension of L is defined as the maximal dimension of the image of the map associated to the powers $ mL(m \in N)$. I want to prove the asymptotic estimate

$$ h^0 (X,mL) \leq O(m^{k(L)})$$

I heard that it is an easy consequence of the Schwarz lemma. Maybe it is a similar argument used by Siegel to prove the theorem "the transcendental degree of the meromorphic function field of a compact complex manifold is not bigger than the dimension of the manifold". But I'm afraid to deal with meromorphic mappings and singularities in the image.So I cannot complete the argument myself.

Can somebody tell me how the argument goes?