New Refinements and Improvements of Jordan’s Inequality

Abstract

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.

New Refinements and Improvements of Some Trigonometric Inequalities Based on Padé Approximant

“…Using the Padé approximant and equation ( 11), we obtain a better approximation of tangent function. Here, we need to pay attention to the value of c j in formula (10). Let c 2j � 0, c 2j− 1 � T(j), and T(j) is given in ( 12); we can obtain the Padé approximant of tan(x).…”

“…Trigonometric inequalities have attracted many researchers. Many improvements of Jordan' inequality [1][2][3][4][5][6][7][8][9][10][11], Kober's inequality [12][13][14][15][16], and Becker-Stark's inequality [4,17,18] have been obtained.…”

Jordan-Type Inequalities and Stratification

“…Since then, many authors have worked on extensions and improvements of Jordan's inequality [1][2][3][4][5][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] . In [8], F. Qi, D.-W. Niu and B.-N. Guo did the elaborate research, summarizing previously discovered improvements and applications of Jordan's inequality, along with related problems.…”