Abstract
We derive the Levinson type generalization of the Jensen and the converse Jensen inequality for real Stieltjes measure, not necessarily positive. As a consequence, also the Levinson type generalization of the Hermite-Hadamard inequality is obtained. Similarly, we derive the Levinson type generalization of Giaccardi’s inequality. The obtained results are then applied for establishing new mean-value theorems. The results from this paper represent a generalization of several recent results.
Highlights
We will use an analogous technique as in the previous sections to obtain a Levinson type generalization of the Giaccardi inequality for n-tuples p of real numbers which are not necessarily nonnegative
Motivated by the results obtained in previous sections, we define the following linear functionals which, respectively, represent the difference between the right and the left side of inequalities ( . ) and ( . ): J (φ) =
Note that if we set the functions f, g, λ, and μ from our theorems to fulfill the conditions from Jensen’s integral inequality or Jensen-Steffensen’s, or Jensen-Brunk’s, or Jensen-Boas’ inequality, - applying that inequality on the function G which is continuous and convex in both variables - we see that in these cases for all s ∈ [α, c], s ∈ [c, β] inequalities in ( . ) hold, and so from our results we directly get the results from the paper [ ]
Summary
We will use an analogous technique as in the previous sections to obtain a Levinson type generalization of the Giaccardi inequality for n-tuples p of real numbers which are not necessarily nonnegative. ([ ]) Under the conditions from the previous theorem, the following two statements are equivalent: ( ) For every continuous concave function φ : [α, β] → R the reverse inequality in
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