The paper extends homogeneous functional calculus on vector lattices. It is shown that the function of elements of a relatively uniformly complete vector lattice can naturally be defined if the positively homogeneous function is defined on some conic set and is continuous on some subcone. An interplay between Minkowski duality and homogeneous functional calculus is considered. The quasilinearization method for proving convexity inequalities in vector lattices is develop and general forms of some classical and new inequalities are proved. This machinery is applied to compute and estimate homogeneous functions of linear and bilinear regular operators on vector lattices. Homogeneous functions on vector lattices of continuous or measurable sections of bundles of Banach lattices is also considered.