Numerical inequalities for quasi-projective surfaces

Let $V$ be a smooth quasi-projective complex surface with compactification $(X,D)$ and set ${\bar{P}}_1(V):=h^0(X,K_X+D)$, $\bar{q}(V):=h^0(X,{\mathrm{\Omega }}_X^1(\log D))$. We prove that ${\bar{P}}_1(V)\ge \bar{q}(V)-1$ if $V$ has maximal Albanese dimension and ${\bar{P}}_1(V)\ge \frac{1}{6}(\bar{q}(V)-5)$ otherwise. Both bounds are sharp.