Refinements of Jordan's inequality

Analytic Number Theory, Approximation Theory, and Special Functions

2014

The study of quasiconformal maps led the first two authors in their joint work with M. K. Vamanamurthy to formulate open problems or questions involving special functions [14,16]. During the past two decades, many authors have contributed to the solution of these problems. However, most of the problems posed in [14] are still open.The present paper is the third in a series of surveys by the first two authors, the previous papers [19,21] being written jointly with the late M. K. Vamanamurthy. The aim of this series of surveys is to review the results motivated by the problems in [14,16] and related developments during the past two decades. In the first of these we studied classical special functions, and in the next we focused on special functions occurring in the distortion theory of quasiconformal maps. Regretfully, Vamanamurthy passed away in 2009, and the remaining authors dedicate the present work as a tribute to his memory. For an update to the bibliographies of [19] and [21] the reader is referred to [23].In 1993 the following monotone rule was derived [17, Lemma 2.2]. Though simple to state and easy to prove by means of the Cauchy Mean Value Theorem, this l'Hôpital Monotone Rule (LMR) has had wide application to special functions by many authors. Vamanamurthy was especially skillful in the application of this rule. We here quote the rule as it was restated in [20, Theorem 2].

Section: Generalizations Of Jordan's Inequalitymentioning

confidence: 99%

“…They also found a method for obtaining new bounds from old for (sin x)/x, but M.-K. Kuo [117] gave an example to show that the new bounds are not necessarily stronger. In 2003 C.-P. Chen et al [77], using mathematical induction and infinite products, found analogs of the Redheffer inequality for cos x:…”

Section: Redheffermentioning

confidence: 99%

Topics in Special Functions III

Anderson

Вуоринен

Analytic Number Theory, Approximation Theory, and Special Functions

2014

“…has been studied in a large number of literature works, and many refinements have been presented [1][2][3][4][5][6]. Zhang et al [7] gave the polynomial bounds of degree one:…”

Section: Introductionmentioning

confidence: 99%

New Polynomial Bounds for Jordan’s and Kober’s Inequalities Based on the Interpolation and Approximation Method

2019

Mathematics

“…The Jordan's inequality plays an important role in many areas of pure and applied mathematics. Many improvements and refinements of Jordan's inequality were presented in the recent period [2][3][4][5][6][7]. There are some sharp lower and upper bounds for the sinc(x) function by using polynomial degrees from 1 to 4.…”

Section: Introductionmentioning

confidence: 99%

New Refinements and Improvements of Jordan’s Inequality

2018

Mathematics

The polynomial bounds of Jordan’s inequality, especially the cubic and quartic polynomial bounds, have been studied and improved in a lot of the literature; however, the linear and quadratic polynomial bounds can not be improved very much. In this paper, new refinements and improvements of Jordan’s inequality are given. We present new lower bounds and upper bounds for strengthened Jordan’s inequality using polynomials of degrees 1 and 2. Our bounds are tighter than the previous results of polynomials of degrees 1 and 2. More importantly, we give new improvements of Jordan’s inequality using polynomials of degree 5, which can achieve much tighter bounds than those previous methods.