Fejér-Type Inequalities for Some Classes of Differentiable Functions

Abstract

We let υ be a convex function on an interval [ι1,ι2]⊂R. If ζ∈C([ι1,ι2]), ζ≥0 and ζ is symmetric with respect to ι1+ι22, then υ12∑j=12ιj∫ι1ι2ζ(s)ds≤∫ι1ι2υ(s)ζ(s)ds≤12∑j=12υ(ιj)∫ι1ι2ζ(s)ds. The above estimates were obtained by Fejér in 1906 as a generalization of the Hermite–Hadamard inequality (the above inequality with ζ≡1). This work is focused on the study of right-side Fejér-type inequalities in one- and two-dimensional cases for new classes of differentiable functions υ. In the one-dimensional case, the obtained results hold without any symmetry condition imposed on the weight function ζ. In the two-dimensional case, the right side of Fejer’s inequality is extended to the class of subharmonic functions υ on a disk.