Abstract
A two-point Padé approximant method is presented for refining some remarkable trigonometric inequalities including the Jordan inequality, Kober inequality, Becker–Stark inequality, and Wu–Srivastava inequality. Simple proofs are provided. It shows to achieve better approximation results than those of prevailing methods.
Highlights
We present a two-point Padé-approximant-based method [1] for refining the rational bounds of several trigonometric inequalities, and provide a method for proving the refined bounds
It can be verified that ∀x ∈ [0, π/2], c5(x) x6(161x2 – 495)(x2 – 33) 4725x2 + 210x4 – x6)(x2 + 20)(3x4
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Summary
Trigonometric inequalities have caused interest of a lot of researchers, they analyzed the Wilker inequality [6,7,8,9,10,11, 14, 16,17,18,19], Jordan inequality [3, 5, 15, 20, 21], Shafer–Fink inequality [12], Becker–Stark inequalities [13], and so on. Bercu provided a Padé-approximant-based method and obtained the following inequalities [2]. We present a two-point Padé-approximant-based method [1] for refining the rational bounds of several trigonometric inequalities, and provide a method for proving the refined bounds.
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