Reverses of the Jensen inequality in terms of first derivative and applications
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REVERSES OF THE JENSEN INEQUALITY IN TERMS OF FIRST DERIVATIVE AND APPLICATIONS S.S. Dragomir
Received: 26 April 2012 / Revised: 28 May 2012 / Accepted: 6 June 2012 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract Two new reverses of the celebrated Jensen integral inequality for convex functions with applications for means, the Hölder inequality and f -divergence measures in information theory are given. Keywords Jensen’s inequality · Measurable functions · Lebesgue integral · Divergence measures · f -Divergence Mathematics Subject Classification (2000) Primary 26D15 · 26D20 · Secondary 94A05 1 Introduction Let (Ω, A, μ) be a measurable space consisting of a set Ω, a σ -algebra A of parts of Ω and a countably additive and positive measure μ on A with values in R ∪ {∞}. For a μmeasurable function w: Ω → R, with w(x) ≥ 0 for μ-a.e. (almost every) x ∈ Ω, consider the Lebesgue space w(x)f (x) dμ(x) < ∞ . Lw (Ω, μ) := f : Ω → R, f is μ-measurable and Ω
For simplicity of notation we write everywhere in the sequel Ω w dμ instead of Ω w(x) dμ(x). In order to provide a reverse of the celebrated Jensen’s integral inequality for convex functions, Dragomir [13] obtained the following result:
B
S.S. Dragomir ( ) Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia e-mail: [email protected] url: http://rgmia.org/dragomir S.S. Dragomir School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa
S.S. DRAGOMIR
Theorem 1 Let Φ : [m, M] ⊂ R → R be a differentiable convex function on (m, M) and f : Ω → [m, M] so that Φ ◦ f, f , Φ ◦ f , (Φ ◦ f )f ∈ Lw (Ω, μ), where w ≥ 0 μ-a.e. (almost everywhere) on Ω with Ω w dμ = 1. Then we have the inequality
w(Φ ◦ f ) dμ − Φ
0≤
wf dμ
Ω
w Φ ◦ f f dμ −
≤ Ω
1 ≤ Φ (M) − Φ (m) 2
Ω
w Φ ◦ f dμ
Ω
Ω
wf dμ Ω
w f − wf dμ dμ.
(1.1)
Ω
For a generalization of the first inequality in (1.1) without the differentiability assumption and the derivative Φ replaced with a selection ϕ from the subdifferential ∂Φ, see the paper [26] by C.P. Niculescu. Remark 1 If μ(Ω) < ∞ and Φ ◦ f , f , Φ ◦ f , (Φ ◦ f )f ∈ L(Ω, μ), then we have the inequality: 1 f dμ μ(Ω) Ω Ω
1 1 1 ≤ f dμ Φ ◦ f f dμ − Φ ◦ f dμ · μ(Ω) Ω μ(Ω) Ω μ(Ω) Ω 1 1 f − 1 dμ. ≤ Φ (M) − Φ (m) f dμ 2 μ(Ω) Ω μ(Ω) Ω
0≤
1 μ(Ω)
(Φ ◦ f ) dμ − Φ
(1.2)
Corollary 1 Let Φ : [m, M] → R be a differentiable convex function on (m, M). If xi ∈
[m, M] and wi ≥ 0 (i = 1, . . . , n) with Wn := ni=1 wi = 1, then one has the counterpart of Jensen’s weighted discrete inequality: 0≤
n
wi Φ(xi ) − Φ
n
i=1
≤
n
wi xi
i=1
wi Φ (xi )xi −
i=1
n
wi Φ (xi )
i=1
n
wi xi
i=1
n n 1 ≤ Φ (M) − Φ (m) wi xi − wj xj . 2 i=1 j =1
(1.3)
Remark 2 We notice that the inequality between the first and the se
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