engleski

Extensions of the Hermite-Hadamard inequality with applications

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. <theorem/>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)).

Hermite-Hadamard inequality; Fejér inequality; Hammer-Bullen inequality; linear functionals

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