= ,,,

Therefore, by using starltlartl arguments [see Cacoullos arid Papatharm siou (1989)], the quadratic OX" 26X (where O = o%[(hl(Y))" - Var [h(X)] 2 0, 6 = a2J3 [hl(l')] -- Cov[X, h (X)] ) is normegativr for all X , and thus 6 = 0. Hence, takiiig into account (5.2.2) we coriclude that for any bomided fimctiori I I ;

Finally, by using similar argurrierits ancl the results of Theorern 5.2.2, a. converse of Theorem 5.3.3 and Lemma 5.2.l(iii) car1 be easily estahlishrd:

(ii) P'or e~ierg a c . junction g v i i l l ~ tderivntiw g', Var[g(Xj] 1 v ~ ~ [ x ] I E [ ( ~ ' ( Y ) ) ~ ] .