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Sharpening and Generalizations of Shafer's Inequality for the Arc Tangent Function
Journal of Inequalities and Applications volume 2009, Article number: 930294 (2009)
Abstract
We sharpen and generalize Shafer's inequality for the arc tangent function. From this, some known results are refined.
1. Introduction and Main Results
In [1], the following elementary problem was posed, showing that for ,
In [2], the following three proofs for the inequality (1.1) were provided.
Solution by Grinstein
Direct computation gives
where
Now is positive for all ; whence is an increasing function.
Since , it follows that for .
Solution by Marsh
It follows from that
The desired result is obtained directly upon integration of the latter inequality with respect to from to for .
Solution by Konhauser
The substitution transforms the given inequality into , which is a special case of an inequality discussed on [3, pages 105106] .
It may be worthwhile to note that the inequality (1.1) is not collected in the authorized monographs [4, 5].
In [4, pages 288289], the following inequalities for the arc tangent function are collected:
where . The inequality (1.5) is better than (1.7).
The aim of this paper is to sharpen and generalize inequalities (1.1) and (1.5).
Our results may be stated as the following theorems.
Theorem 1.1.
For , let
where is a real number.
(1)When , the function is strictly increasing on .
(2)When , the function is strictly decreasing on .
(3)When , the function has a unique minimum on .
As direct consequences of Theorem 1.1, the following inequalities may be derived.
Theorem 1.2.
For ,
For ,
For , the inequality (1.9) is reversed.
Moreover, the constants and in inequalities (1.9) and (1.10) are the best possible.
2. Remarks
Before proving our theorems, we give several remarks on them.
Remark 2.1.
The substitution may transform inequalities in (1.9) and (1.10) into some trigonometric inequalities.
Remark 2.2.
The inequality (1.1) is the special case of the lefthand side inequality in (1.9).
Remark 2.3.
The inequality (1.5) is the special case of the reversed version of the left handside inequality in (1.9).
Remark 2.4.
Let
for and . Direct computation gives
Hence,
(1)when the derivative is negative for ;
(2)when the derivative has a unique zero which is the unique maximum point of for .
Accordingly,
(1)when the function attains its maximum
(2)when the unique zero of equals
and the function attains its maximum
for .
In a word, the sharp lower bounds of (1.10) are
for .
Similarly, the sharp upper bound of (1.10) is
Remark 2.5.
Similar to the deduction of inequalities (2.6) and (2.7), the sharp versions of (1.9) and its reversion are
Remark 2.6.
It is easy to verify that the righthand side inequalities in (2.9) and (2.10) are included in the inequality (2.8).
By the famous software Mathematica, it is revealed that the inequality (2.7) contains (2.6) and the lefthand side inequality in (2.9), and that the inequality (2.7) and the lefthand side inequality in (2.10) are not included in each other.
In conclusion, the following double inequality is the best accurate one:
where denotes.
Remark 2.7.
For possible applications of the double inequality (2.11) in the theory of approximations, the accuracy of bounds in (2.11) for the arc tangent function is described by Figures 1 and 2.
The upper curves in Figures 1 and 2 are, respectively, the graphs of the functions
and the lower curves in Figures 1 and 2 are, respectively, the graphs of the functions
on the interval , where denotes.
These two figures are plotted by the famous software Mathematica 7.0.
Remark 2.8.
The approach below used in the proofs of Theorems 1.1 and 1.2 has been employed in [6–9].
Remark 2.9.
This paper is a revised version of the preprint [10].
3. Proofs of Theorems
Now we are in a position to prove our theorems.
Proof of Theorem 1.1.
Direct calculation gives
Let
then
and the function
has two zeros
Further differentiation yields
This means that the functions and are increasing on . From
it follows that

(1)
when or , the derivative is negative and the function is strictly decreasing on . From
(3.8)
it is deduced that on . Accordingly,
(a)when , the derivative and the function is strictly increasing on ;
(b)when , the derivative is negative and the function is strictly decreasing on ;

(2)
when , the derivative is positive and the function is increasing on . By (3.8), it follows that the function is positive on . Thus, the derivative is positive and the function is strictly increasing on ;

(3)
when , the derivative has a unique zero which is a minimum of on . Hence, by the second limit in (3.8), it may be deduced that
(a)when , the function is negative on , so the derivative is also negative and the function is strictly decreasing on ;
(b)when , the function has a unique zero which is also a unique zero of the derivative , and so the function has a unique minimum of the function on .
On the other hand, the derivative can be rewritten as
and the function
satisfies
When , the derivative is positive and the function is strictly increasing on . Since , the function is positive, and so the derivative is positive, on for . Consequently, when , the function is strictly increasing on . The proof of Theorem 1.1 is complete.
Proof of Theorem 1.2.
Direct calculation yields
By the increasing monotonicity in Theorem 1.1, it follows that for , which can be rewritten as (1.9) for . Similarly, the reversed version of the inequality (1.9) and the righthand side inequality in (1.10) can be procured.
When , the unique minimum point of the function satisfies
and so the minimum of on is
where , as a result, the lefthand side inequality in (1.10) follows. The proof of Theorem 1.2 is complete.
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Acknowledgments
The authors appreciate the anonymous referees for their valuable comments that improve this manuscript. The first author was partially supported by the China Scholarship Council.
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Qi, F., Zhang, S. & Guo, B. Sharpening and Generalizations of Shafer's Inequality for the Arc Tangent Function. J Inequal Appl 2009, 930294 (2009). https://doi.org/10.1155/2009/930294
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Keywords
 Direct Consequence
 Lower Bound
 Direct Calculation
 Minimum Point
 Lower Curve