On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities

Abstract

We determine the best positive constants p and q such that (sinh x/x) p < x/ sin x < (sinh x/x) q. Some applications for Wilker's type inequalities are given. Mathematics subject classification (2010): 26D05, 26D07, 26D99. Keywords and phrases: inequalities; trigonometric functions; hyperbolic functions; means and their inequalities. R E F E R E N C E S [1] R. KLÉNKL´KLÉN, M. VISURI AND M. VUORINEN, On Jordan type inequalities for hyperbolic functions, J.

Wilker and Huygens-Type Inequalities Involving Gudermannian and the Inverse Gudermannian Functions

“…B. Wilker [23] and C. Huygens [5], respectively. Several proofs of these results can be found in mathematical literature (see, e.g., [4,8,9,17,20,22,24,25,31,32,33,30,34,35,36] and the references therein).…”

“…They read as follows 2 < sinh x x 2 + tanh x x (3) and 3 < 2 sinh x x + tanh x x (4) (x = 0). For the proofs of these results the interested reader is referred to [35] and [17], respectively. Generalizations of inequalities (1) - (4) are known.…”

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

“…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]. In [16], the following open problem was proposed: for each > 0, there are greatest value = ( ) and least value = ( ) such that the double inequality sin 1 + cos < < sin 1 + cos (3) holds for all ∈ (0, /2).…”

“…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].…”

Sharpness of Wilker and Huygens type inequalities

“…Neuman and Sándor [16] have pointed out that (3) implies (1). In [17], Zhu established some new inequalities of the Huygens type for trigonometric and hyperbolic functions.…”