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In this paper we prove a converse to the Jensen-Steffensen inequality and two inequalities complementary to the Jensen-Steffensen inequality. We apply so called exp-convex method in order to interpret our results in the form of exponentially convex functions. The outcome is a number of new interesting inequalities as well as some new Cauchy type means.

In this paper we establish some new inequalities for differentiable functions based on concavity and s-convexity. We also prove several Hadamard-type inequalities for products of two convex and s-convex functions.

We present a general closed 4-point quadrature rule based on Euler-type identities. We use this rule to prove a generalization of Hadamard's inequalities for (2r)-convex functions (r ≥ 1).

Several variants of Cebysev's inequality for two monotonic -tuples and also nonnegative -tuples monotonic in the same direction are presented. Immediately after that their refinements of Ostrowski's type are given. Obtained results are used to prove generalizations of discrete Milne's inequality and its converse in which weights satisfy conditions as in the Jensen-Steffensen inequality.

We give several Jensen's type inequalities for (omega1, omega2)-convex functions. Results of Bessenyei and Páles are generalized.

A variant of Jensen–Steffensen's inequality is proved. Necessary and sufficient conditions for the equality in Jensen–Steffensen's inequality are established. Several inequalities involving more than two monotonic functions and generalized quasi-arithmetic means with not only positive weights are proved. It is shown that such generalized quasi-arithmetic means have the same comparability properties as those with positive weights.

We give a survey on the converse Jensen inequality and we show that several recently published inequalities are simple consequences of certain long time known results. We also give a new refinement of the converse Jensen inequality as well as improvements of some related results.

We present improvements of various forms of the Hermite-Hadamard inequality, namely that of Fejèr, Lupaş, Brenner-Alzer and Beesack-Pečarić. It is interesting that these improvements also imply the Hammer-Bullen inequality which deals with a comparison of the left-hand and the right-hand side of the Hermite-Hadamard inequality. These improvements are given in terms of positive linear functionals in the following way. Let E be a nonempty set and L a linear class of functions f:E→ℝ having the properties: (L1) (∀f, g∈L)(∀a, b∈ℝ) af+bg∈L ; (L2) 1∈L (that is if (∀t∈E)f(t)=1 then f∈L) ; (L3) (∀f, g∈L)(min{; ; ; f, g}; ; ; ∈L∧max{; ; ; f, g}; ; ; ∈L) (lattice property). We consider positive normalized linear functionals A:L→ℝ, that is, we assume: (A1) (∀f, g∈L)(∀a, b∈ℝ) A(af+bg)=aA(f)+bA(g) (linearity) ; (A2) (∀f∈L)(f→A(f)) (positivity) ; (A3) A(1)=1. Let L satisfy (L1), (L2) and (L3) on a nonempty set E and let A be a positive normalized linear functional. If f:I→ℝ is a continuous convex function and [a, b]⊆I then for all g∈L such that g(E)⊆[a, b] and f(g)∈L we have A(g)∈[a, b] and f(((pa+qb)/(p+q)))≤A(f(g))≤((pf(a)+qf(b))/(p+q))-A(g)δ_{; ; ; f}; ; ; , where p and q are any nonnegative real numbers such that A(g)=((pa+qb)/(p+q)) and g, δ_{; ; ; f}; ; ; are defined by g=(1/2)1-((|g-((a+b)/2)1|)/(b-a)), δ_{; ; ; f}; ; ; =f(a)+f(b)-2f(((a+b)/2)).

Several inequalities of Jensen’s type for functions convex on the co-ordinates are given. Obtained results generalize the coresponding results of S. S. Dragomir given in [2].

Refinements of generalizedHermite-Hadamard’s inequalities formultivariate g -convex functions are given. Since special instances of the g -convex functions include the r -convex and the logarithmically convex functions, those inequalities also give refinements of the HermiteHadamard’s ineqalities for these families of functions. Mathematics subject classification (2000): 26D15, 26B25.