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We establish necessary conditions for the existence of solutions to various systems of partial differential inequalities in the plane. The obtained conditions provide new Hermite-Hadamard-type inequalities for differentiable functions in the plane. In particular, we obtain a refinement of an inequality due to Dragomir [ On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane , Taiwan. J. Math.  5  (2001), no. 4, 775–788] for convex functions on the coordinates.

On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane