Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. (Q1412591)

The author proves certain inequalities of Hadamard type for convex mappings. We quote the following result: Let \(| f'|\) be convex on \([a,b]\). Then \[ \Biggl| M_f(a, b)- f\Biggl({a+b\over 2}\Biggr)\Biggr|\leq {b-a\over 8} (| f'(a)|+| f'(b)|) \] (\(M_f(a,b)\) denotes the integral means of \(f\) on \([a,b]\)). Some applications for special means of two arguments are also pointed out. However, these results are not compared to the existing relations in the theory of means.