Fejér-Type Inequalities for Some Classes of Differentiable Functions

We let υ be a convex function on an interval [ ι 1 , ι 2 ] ⊂ R . If ζ ∈ C ( [ ι 1 , ι 2 ] ) , ζ ≥ 0 and ζ is symmetric with respect to ι 1 + ι 2 2 , then υ 1 2 ∑ j = 1 2 ι j ∫ ι 1 ι 2 ζ ( s ) d s ≤ ∫ ι 1 ι 2 υ ( s ) ζ ( s ) d s ≤ 1 2 ∑ j = 1 2 υ ( ι j ) ∫ ι 1 ι 2 ζ ( s ) d s . 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.