In this paper the authors extend Petrović's inequality for coordinated $m$-convex functions in the plane and also find Lagrange type and Cauchy type mean value theorems for Petrović's inequality for $m$-convex functions and coordinated $m$-convex functions. The authors consider functional due to Petrović's inequality in plane and discuss its properties for certain class of coordinated log-$m$-convex functions.