LAUSR

In this work, Sherman-Steffensen type inequalities for convex functions with not necessarily non-negative coefficients are established by using Steffensen’s conditions. The Brunk, Bellman and Olkin type inequalities are derived as special cases of the Sherman-Steffensen inequality. The superadditivity of the Jensen-Steffensen functional is investigated via Steffensen’s condition for the sequence of the total sums of all entries of the involved vectors of coefficients. Some results of Barić et al. [2] and of Krnić et al. [11] on the monotonicity of the functional are recovered. Finally, a Sherman-Steffensen type inequality is shown for a row graded matrix. 1. Preliminaries and motivation The celebrated weighted Jensen’s inequality [5, 14, 18] claims that if f is a real convex function defined on an interval I ⊂ R, and real coefficients p1, p2, . . . , pn are such that pi ≥ 0, i = 1, 2, . . . ,n, and Pn = p1 + p2 + . . . + pn > 0 (1) then, for any x1, x2, . . . , xn ∈ I, f  1 Pn n ∑ i=1 pixi  ≤ 1 Pn n ∑ i=1 pi f (xi). (2) Theorem A. (Steffensen [21].) Assume that f is a real convex function defined on an interval I ⊂ R. Let w1,w2, . . . ,wn ∈ R. Denote Wi = w1 + w2 + . . . + wi for i = 1, 2, . . . ,n. If Wn ≥Wi ≥ 0, i = 1, 2, . . . ,n, and Wn > 0 (Steffensen’s condition), (3) then, for any monotonic n-tuple x = (x1, x2, . . . , xn) ∈ In, f  1 Wn n ∑ i=1 wixi  ≤ 1 Wn n ∑