  In this work, several sharp bounds for the \v{C}eby\v{s}ev functional
involving various type of functions are proved. In particular, for the
\v{C}eby\v{s}ev functional of two absolutely continuous functions whose first
derivatives are both convex, convex and belong to $L_p$-spaces, convex and
bounded variation, convex and Lipschitz mappings new sharp bounds are
presented. Other related results regarding two convex and concave functions are
given.
