On some Chebyshev type inequalities for thecomplex integral

Abstract

Assume thatfandgare continuous onγ,γ⊂Cis a piecewisesmooth path parametrized byz(t), t∈[a, b]fromz(a) =utoz(b) =wwithw6=u, and thecomplex Chebyshev functionalis defined byDγ(f, g) :=1w−u∫γf(z)g(z)dz−1w−u∫γf(z)dz1w−u∫γg(z)dz.In this paper we establish some bounds for the magnitude of the functionalDγ(f, g)under Lipschitzian assumptions for the functionsfandg,and pro-vide a complex version for the well known Chebyshev inequality.