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Some new estimates for the di¤erence between the integral mean of a function and its mean over a subinterval are established and new applications for special means and probability density functions are also given.
Some nontrivial increasing functions will be interpolated in the Hadamard's inequality for r-convex function in Carnot groups. The methods are more natural and allow us to extend the condition Γ 2 (G) ∩ C w H,r (G) to the condition C(G) ∩ C w H,r (G). A continuous analogue and extension of Rado's formulae for convex and concave functions, Bull.