BOUNDS FOR THE NORMALIZED JENSEN-MERCER FUNCTIONAL

We introduce the normalized Jensen-Mercer functional M-n(f, x, p) = f(a) + f(b) - Sigma(n)(i=1) pif(x(i)) - f (a+b - Sigma(n)(i=1) p(i)x(i)) and establish the inequalities of type MMn(f, x, q) >= M-n(f, x, p) >= mM(n)(f, x, q), where f is a convex function, x =(x(1), ... , x(n)) and m and M are real numbers satisfying certain conditions. We prove them for the case when p and q are nonnegative n-tuples and when p and q satisfy the conditions for the Jensen-Steffensen inequality. We also give their integral versions in both cases.