New sharp Jordan type inequalities and their applications

Abstract

In this paper, we prove that for x ∈ (0, π ∕ 2)

(cos p₀ x)^{1 ∕ p₀} < (sin x ∕ x) < (cos (x ∕ 3 ))³

with the best constants p₀ ≈ 0.34731 and 1 ∕ 3. Moreover, for x ∈ (0, c) ⊆ (0, π ∕ 2) the double inequality

β_p (c)(cos px)^{1 ∕ p} < (sin x ∕ x) < (cos px)^{1 ∕ p}

holds if p ∈ (0, 1 ∕ 3], where β_p (c) = c^-1 (cos pc)^{-1 ∕ p} and 1 are the best possible. Its reverse one holds if p ∈ [1∕ 2, 1). As applications, some precise estimates for sine integral and Catalan constant are given.