The concept of superquadratic functions in several variables, as a generalization of the same concept in one variable is introduced. Analogous results to results obtained for convex functions in one and several variables are presented. These include refinements of Jensen's inequality and its counterpart, and of SlaterPečarić's inequality.

Let (μ,) be a probability measure space. We consider the so-called ' Jensen gap' J(ϕ, μ, f) = ϕ(f (s)) dμ(s)-ϕ f (s) dμ(s) for some classes of functions ϕ. Several new estimates and equalities are derived and compared with other results of this type. Especially the case when ϕ has a Taylor expansion is treated and the corresponding discrete results are pointed out.

Abstract. For n ∈ Z + we consider the difference

Abstract. We prove the inequality l2lv\l ~ *-\lv\\ -^llYdl ~ ^il*ol f°r tne difference of the first two eigenvalues of one-dimensional Schrödinger operators The gap between consecutive eigenvalues of Schrödinger operators has been the object of considerable attention recently (see [1][2][3][4][5] and many others).In this note we use the same method established in [1]. We consider the two Schrödinger operators H0 = -4-z + V0(x), and Hx = -j-y + Vx(x), both acting on L2(0, n) with Dirichlet boundary conditions and with both V0 and Vx symmetric with respect to x = n/2 and in l'(0, n). Let (kx,ux) and (k2, u2) be the first two eigenvalues together with their associated eigenfunctions of ZZj, and let (px,vx) and (p2, v2) be the corresponding quantities for H0. We will use the following lemma, which is part of Proposition 1 in [1].Lemma. Let H0 and Hx be as described above; thenwhere u = (vx/v2)u2.Proof. See the proof of inequality (7) in Proposition 1 [1].Definition. A potential F is a double-well potential on the closed interval I if there are cx < c2 < c3 € I such that V is nonincreasing for x < cx and c2 < x < c3 and is nondecreasing otherwise.

Abstract. In this paper it is shown that the inequality known in the literature as Alzer's inequality (1993), has already been known since 1975. and is due to Jan van de Lune. A review of different methods in proving Van de Lune -Alzer's inequality and generalizations in a several directions, is given. It is shown how some results and proofs can be corrected, refined and extended. New results, inspired by the generalization of Van de Lune -Alzer's inequality for increasing convex sequences presented by N. Elezović and J. Pečarić, are obtained. (2000): 26A51, 26D15, 26D20. Mathematics subject classification