Jordan’s Inequality: Refinements, Generalizations, Applications and Related Problems

This is an expository article. Some developments on refinements, generalizations, applications of Jordan’s inequality and related problems, including some estimates for three classes of complete elliptic integrals and several proofs of Wilker’s inequality, are summarized. 1. Refinements of Jordan’s inequality 1.1. Jordan’s inequality. The well-known Jordan’s inequality (see [2, 9], [5, p. 143], [23, p. 269] and [27, p. 33]) reads that 2 π ≤ sinx x < 1 (1.1) for 0 < |x| ≤ π2 . The equality in (1.1) is valid if and only if x = π 2 . Note that the origin of Jordan’s inequality is not found in the references listed in this paper. So, it is unknown that why inequality (1.1) is due to Jordan and to which Jordan. 1.2. Kober’s inequality. In [23, pp. 274–275], an inequality due to Kober [20, p. 22] was given: 1− 2 π x ≤ cosx ≤ 1− x 2 π , x ∈ [ 0, π 2 ] . (1.2) In [21] and [22, p. 313], it was given that for x ∈ [0, π], cosx ≤ 1− 2 π2 x. (1.3) The left hand side inequalities in (1.1) and (1.2) are equivalent, since they can be deduced from each other via the transformation x→ π2 − x. 1.3. Redheffer’s inequality. In [44, 45], it was proposed that sinx x ≥ π 2 − x π2 + x2 , x 6= 0. (1.4) In [49], inequality (1.4) was proved as follows. For x ≥ 1, 1− x 1 + x2 − sin(πx) πx = 1− x 1 + x2 + sin[π(x− 1)] π(x− 1) · x− 1 x ≤ 1− x 2 1 + x2 + x− 1 x = − (1− x) 2