On some Gruss' type inequalities for the complex integral

Assume that f and g are continuous on.,.. C 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 C. ebysev functional is defined by D. (f, g) := 1 w - u . f (z) g (z) dz - 1 w - u . f (z) dz 1 w - u . g (z) dz. In this paper we establish some bounds for the magnitude of the functional D. ( f, g) under various assumptions for the functions f and g and provide a complex version for the well known Gruss inequality.