Abstract
We give the best lower bound for the weighted Jensen's discrete inequality with ordered variables applied to a convex function , in the case when the lower bound depends on , weights, and two given variables. Furthermore, under the same conditions, we give some sharp lower bounds for the weighted AM-GM inequality and AM-HM inequality.
Highlights
Let x {x1, x2, . . . , xn} be a sequence of real numbers belonging to an interval I, and let p {p1, p2, . . . , pn} be a sequence of given positive weights associated to x and satisfying p1 p2 · · · pn 1
We need to prove that pkf xk pk 1f xk 1 · · · pnf xn − f p1x1 p2x2 · · · pnxn 3.11
Xk pk xk pk 1xk 1 Rk pnxn, Yk p1x1 · · · pk−1xk−1 Rkxk, The inequality 3.14 follows from Lemma 2.2, since xk, X ∈ Yk, Xk and RkXk Yk Rkxk X
Summary
Let x {x1, x2, . . . , xn} be a sequence of real numbers belonging to an interval I, and let p {p1, p2, . . . , pn} be a sequence of given positive weights associated to x and satisfying p1 p2 · · · pn 1. Xn} be a sequence of real numbers belonging to an interval I, and let p {p1, p2, . Pn} be a sequence of given positive weights associated to x and satisfying p1 p2 · · · pn 1. If f is a convex function on I, the well-known discrete Jensen’s inequality 1 states that. The refinement of Jensen’s inequality was proven in 2 , as a consequence of its Theorem 2.1, part ii. We will establish that the best lower bound Lp,f xi, xk of Jensen’s difference Δ f, p, x for x1 ≤ · · · ≤ xi ≤ · · · ≤ xk ≤ · · · ≤ xn.
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