Refinements of the Lower Bounds of the Jensen Functional
نویسندگان
چکیده
and Applied Analysis 3 where Pj j ∑ i 1 pi, j 1, . . . , n. 1.7 Lemma 1.6. Let f be a convex function on I, p a positive n-tuple such that Pn ∑n i 1 pi 1 and x1, x2, . . . , xn ∈ I, n ≥ 3 such that x1 ≤ x2 ≤ · · · ≤ xn. For fixed x1, x2, . . . , xk, where k 2, 3, . . . , n− 1, the Jensen functional J x,p, f defined in 1.2 is minimal when xk xk 1 · · · xn−1 xn, that is, J ( x,p, f ) ≥ k−1 ∑ i 1 pif xi Qkf xk − f ( k−1 ∑ i 1 pixi Qkxk ) , 1.8
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