The minimal volume of log surfaces of general type with positive geometric genus

Let $X$ be a smooth projective surface and $Delta$ is a normal crossing curve on $X$ such that $K_X+Delta$ is big. We show that the minimal possible volume of the pair $(X, Delta)$ is $frac{1}{143}$ if its (log) geometric genus is positive. Based on this, we establish a Noether type inequality for stable log surfaces, be they normal or non-normal. In the other direction, we show that, if the volume of $(X, Delta)$ is less than $frac{1}{143}$ then $X$ must be a rational surface and the connected components of $Delta$ are trees of smooth rational curves.