Minimax theorems for lower semicontinuous quadratically minorized functions

We prove minimax theorem for lower semicontinuous quadratically minorized functions in Hilbert spaces. To this aim we use the framework of $\Phi$-convexity theory and sufficient and necessary conditions for the minimax equality for $\Phi$-convex functions. The conditions we propse are expressed in therms of convex hull of proximal subdifferentials. As a corollary we obtain minimax theorem for prox-bouneded, paraconvex, weakly convex and semiconvex functions.