A refinement of Grüss inequality for the complex integral

Abstract

Abstract Assume that f and g are continuous on γ, γ ⊂ 𝔺 is a piecewise smooth path parametrized by z (t), t ∈ [a, b] from z (a) = u to z (b) = w with w ≠ u and the complex Čebyšev functional is defined by 𝒟 γ ( f , g ) : = 1 w - u ∫ γ f ( z ) g ( z ) d z - 1 w - u ∫ γ f ( z ) d z 1 w - u ∫ γ g ( z ) d z . {{\cal D}_\gamma}\left({f,g} \right): = {1 \over {w - u}}\int_\gamma {f\left(z \right)} g\left(z \right)dz - {1 \over {w - u}}\int_\gamma {f\left(z \right)} dz{1 \over {w - u}}\int_\gamma {g\left(z \right)} dz. In this paper we establish some Grüss type inequalities for 𝒟 (f, g) under some complex boundedness conditions for the functions f and g.