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

Abstract

We improve the Jordan, Adamović-Mitrinović, and Cusa inequalities. As applications, several new Shafer-Fink type inequalities for inverse sine function and bivariate means inequalities are established, and a new estimate for sine integral is given.

Sharp Inequalities for Trigonometric Functions

“…where the equality is due to the fact that 1 is the unique root of (42). Therefore, we get the right inequality in (41) and the first inequality in (44). We clearly see that 1 is the best possible constant.…”

“…where the exponents 3/2, 1/ ln 2 and coefficients 1, 2 √ 2/ in (43) are the best possible constants and so is 1 ≈ 0.6505536 in (44).…”

“…which is also a new mean, and it satisfies the double inequality < < . There are many inequalities involving means , , , , and ; we quote [15,20,25,27,[31][32][33][34][35][36][37][38][39][40][41][42][43][44]. Inequalities for Sándor's mean can be found in [20, pages 86-93] and [19,Section 6].…”

Sharpened versions of Mitrinović-Adamović, Lazarević and Wilkers inequalities for trigonometric and hyperbolic functions

“…A hyperbolic analogue of inequality (1.1) was presented by Lazarević [5], which is stated as follows: 2) where x = 0, and the exponent 3 is the least possible. A number of generalizations, improvements and applications relating to Mitrinović-Adamović's inequality (1.1) and Lazarević's inequality (1.2) can be found in the literature [4,7,9,12,15,19,20,21,22,26]. Among these investigations, we remark here that Wu and Baricz [15] dealt with the generalizations of inequalities (1.1) and (1.2) and obtained two excellent results, as follows: …”

Several improvements of Mitrinovic-Adamovic and Lazarevics inequalities with applications to the sharpening of Wilker-type inequalities

“…Some generalizations, improvements and variants of the Mitrinović-Adamović and Lazarević's inequalities can be found in the literature [3,6,7,8,9,13,17,18,19,20,21,23].…”