Refinements of Jordan's inequality
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A method of sharpening Jordan's inequality proposed by Li-Li would be improved. Increasing lower bounds and decreasing upper bounds for strengthened Jordan's inequality can be constructed and the errors of lower-upper bounds for strengthened Jordan's inequality can be estimated.
(2010) Mathematics Subject Classification: 26D05; 26D15.
KeywordsJordan's inequality lower bound upper bound
For detailed information, please refer to the expository and survey articles  and related references therein.
In [3, Theorem 2.1] or [2, (2.26)], a new method of sharpening Jordan's inequality was established by Li-Li which shows that one can obtain strengthened Jordan's inequalities from old ones. This result may be stated as follows.
The right-hand side inequality in (1.3) is slightly stronger than the right-hand side inequality in (1.7) at but less than it at x = 0. These two lower bounds for sin x/x can not be included each other. In fact, this obstacle can be improved and we can obtain better results (see (2.9)). Moreover, we can estimate the errors of lower bounds and upper bounds for strengthened Jordan's inequality (see (2.12)).
Corollary 2.2 leads us to know that the lower bounds and upper bounds for sin x/x are increasing and decreasing with respect to positive integers m, respectively, and we obtain the double inequality (2.9). Corollary 2.4, equality (2.12), leads us to know that the errors of lower-upper bounds for strengthened Jordan's inequality can be estimated and we give two examples such as (2.14) and (2.16) to estimate errors.
2 Lower-upper bounds for Jordan's inequality
With the help of Theorem 1.1, we find the important relations among lower-upper bounds for strengthened Jordan's inequality.
(2.6) is nonnegative and increasing so that we obtain (2.2) and (2.4) and we complete the proof.
If we set g2(x) = sin x/x in Theorem 2.1, then we would obtain important results in the following corollary.
Proof. It follows from Theorem 2.1, if replacing g1(x) and g2(x) by g(x) and L g (x), respectively, then we have . Repeating (2.1) in this inequality, we get that is increasing with respect to positive integers m. A similar way, repeating (2.2), we have that is decreasing with respect to positive integers m. With the help of Theorem 1.1, we have L g (x) ≤ sin x/x ≤ U g (x). Replacing g(x) by L g (x), we have . Repeating this method, we obtain (2.9) and we complete our proof.
The following lemma will be used in the next corollary.
L g (0) = 1 and U g (0) = 1.
By approximation, we set L g (0) = 1 and U g (0) = 1 and we get our desired results.
The errors of lower-upper bounds for strengthened Jordan's inequality can be estimated in the following corollary.
It is clear that the error in (2.16) is much smaller than that in (2.14), and it seems to us that and are convergent to sin x/x uniformly. However, we have not proved our conjecture yet.
The author expresses his sincere thanks to the referees for careful reading of the article and several helpful suggestions.
- 2.Qi F, Niu D-W, Guo B-N: Refinements, generalizations, and applications of Jordan's inequality and related problems. J Inequal Appl 2009., 52: 2009(Article ID 271923)Google Scholar
- 3.Li J-L, Li Y-L: On the strengthened Jordan's inequality. J Inequal Appl 2007., 8: 2007(Article ID 74328)Google Scholar
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