Bogomolov–Miyaoka–Yau inequality for symplectic four-manifolds

Let $X$ be a closed, connected, oriented, smooth four-dimensional manifold with $b^+(X)>1$. A symplectic four-manifold is a four-manifold equipped with a symplectic form. Bogomolov–Miyaoka–Yau conjecture for symplectic four-manifolds. If $X$ is symplectic, then

$$c_1(X)^2\leq 3c_2(X).$$

This is presented as a well-known special case implied by the Seiberg–Witten version, using Taubes's result that symplectic four-manifolds with $b^+(X)>1$ have nonzero Seiberg–Witten invariants. Its resolution is not specified in the supplied text.