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It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions f in L-p, then we have the sharp estimateparallel to(I - H)f parallel to(Lp) = 2.It is also shown, via a connection between the operator I - H and Laguerre functions, thatparallel to(1 - alpha)I + Phi(I - H)parallel to(L2 -> L2) = parallel to I - alpha H parallel to(L2 -> L2) = 1 for all a is an element of [ 0, 1].

Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions