Refinements, Generalizations, and Applications of Jordan's Inequality and Related Problems

A Note on Jordan, Adamović-Mitrinović, and Cusa Inequalities

“…Some new developments on refinements, generalizations, and applications for the Jordan inequality can be found in [2] and the references therein.…”

“…has attracted the attention of many researchers (see, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14]). The left inequality in (2) was obtained by Mitrinović (see [1, p. 238]), while the right one is due to Huygens (see, e.g., [15]) and it is called Cusa inequality [3,5,6,8,14].…”

“…Several lemmas are given in Section 2. Main results and their proofs are given in Section 3, in which Theorem 7 unifies and generalizes Jordan and Cusa inequalities; Theorem 13 shows that Adamović-Mitrinović and Cusa inequalities (2) can be interpolated by 1 (cos , ) for suitable ; Theorem 18 gives a hyperbolic version of Theorem 7. In Section 4, some new Shafer-Fink type inequalities for inverse sine function and several inequalities for bivariate means are presented, and a simpler but more accurate estimate for sine integral is provided.…”

Refinements of Jordan's inequality

“…Jordan's inequality (1.1) has been refined, generalized, and applied by many mathematicians. It was first extended to the following: 2) and then, it was further refined to inequality (1.3),…”

“…For detailed information, please refer to the expository and survey articles [2] and related references therein.…”