Abstract
The main purpose of this paper is to present two methods of sharpening Jordan's inequality. The first method shows that one can obtain new strengthened Jordan's inequalities from old ones. The other method shows that one can sharpen Jordan's inequality by choosing proper functions in the monotone form of L'Hopital's rule. Finally, we improve a related inequality proposed early by Redheffer.
Highlights
The well-known Jordan’s inequality states that sin x/x ≥ 2/π (x ∈ (0,π/2]) holds with equality if and only if x = π/2. It plays an important role in many areas of pure and applied mathematics
In a recent paper [10], the first author established an identity which states that the function sin x/x is a power series of (π2 − 4x2) with positive coefficients for all x = 0
Motivated by the previous research on Jordan’s inequality, in this paper, we present two methods of sharpening Jordan’s inequality
Summary
The well-known Jordan’s inequality states that sin x/x ≥ 2/π (x ∈ (0,π/2]) holds with equality if and only if x = π/2 (see [1]). It plays an important role in many areas of pure and applied mathematics. This inequality was first extended to the following: sin x x. It was further extended to the following:. Which holds with equality if and only if x = π/2 (see [2,3,4]). The monotone form of L’Hopital’s rule (see [5, Lemma 5.1]) has been successfully used by Zhu [6, 7] and Wu and Debnath [8, 9] to sharpen
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