Bounds for the Unweighted Jensen’s Gap of Absolutely Continuous Functions

Silvestru Sever Dragomir ()
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Silvestru Sever Dragomir: Victoria University

A chapter in Geometry and Non-Convex Optimization, 2025, pp 51-69 from Springer

Abstract: Abstract Let Φ : I → ℂ $$\Phi :I\rightarrow \mathbb {C}$$ be an absolutely continuous function on m , M ⊂ I ̊ $$\left [ m,M\right ] \subset \mathring {I}$$ , the interior of I and h : a , b → m , M $$h:\left [ a,b\right ] \rightarrow \left [ m,M\right ] $$ absolutely continuous on a , b $$\left [ a,b\right ] $$ and such that The image displays a mathematical formula involving the essential supremum. The formula is: \\n\\n\[\\n\| h' \|_{[a,b], \infty} := \text{ess sup}_{t \in [a,b]} |h'(t)| γ $$\Gamma >\gamma $$ such that condition γ ≤ Φ ′ y ≤ Γ for a.e. y ∈ m , M $$\displaystyle \gamma \leq \Phi ^{\prime }\left ( y\right ) \leq \Gamma \text{ for a.e. }y\in \left [ m,M\right ] $$ holds, then 1 b − a ∫ a b Φ ∘ h t dt − Φ 1 b − a ∫ a b h t dt $$\displaystyle \left \vert \frac {1}{b-a}\int _{a}^{b}\left ( \Phi \circ h\right ) \left ( t\right ) dt-\Phi \left ( \frac {1}{b-a}\int _{a}^{b}h\left ( t\right ) dt\right ) \right \vert $$ ≤ 1 8 b − a Γ − γ h ′ a , b , ∞ . $$\displaystyle \leq \frac {1}{8}\left ( b-a\right ) \left ( \Gamma -\gamma \right ) \left \Vert h^{\prime }\right \Vert { }_{\left [ a,b\right ] ,\infty }. $$ Moreover, if Φ ′ $$\Phi ^{\prime }$$ is absolutely continuous on m , M $$\left [ m,M\right ] $$ with Φ ′′ m , M , ∞ : = esssup x ∈ a , b Φ ′′ x

Keywords: Jensen’s inequality; Measurable functions; Lebesgue integral; Discrete inequalities (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-031-87057-6_3

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DOI: 10.1007/978-3-031-87057-6_3

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