- Research Article
- Open Access
Sharpening and Generalizations of Shafer's Inequality for the Arc Tangent Function
© Feng Qi et al. 2009
- Received: 18 March 2009
- Accepted: 7 July 2009
- Published: 4 August 2009
We sharpen and generalize Shafer's inequality for the arc tangent function. From this, some known results are refined.
- Direct Consequence
- Lower Bound
- Direct Calculation
- Minimum Point
- Lower Curve
In , the following three proofs for the inequality (1.1) were provided.
Solution by Grinstein
Now is positive for all ; whence is an increasing function.
Since , it follows that for .
Solution by Marsh
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 105-106] .
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.
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.
For , the inequality (1.9) is reversed.
Moreover, the constants and in inequalities (1.9) and (1.10) are the best possible.
Before proving our theorems, we give several remarks on them.
The substitution may transform inequalities in (1.9) and (1.10) into some trigonometric inequalities.
The inequality (1.1) is the special case of the left-hand side inequality in (1.9).
The inequality (1.5) is the special case of the reversed version of the left hand-side inequality in (1.9).
(1)when the derivative is negative for ;
(2)when the derivative has a unique zero which is the unique maximum point of for .
It is easy to verify that the right-hand 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 left-hand side inequality in (2.9), and that the inequality (2.7) and the left-hand side inequality in (2.10) are not included in each other.
where denotes .
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.
on the interval , where denotes .
This paper is a revised version of the preprint .
Now we are in a position to prove our theorems.
Proof of Theorem 1.1.
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 ;
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 ;
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 .
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.
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 right-hand side inequality in (1.10) can be procured.
where , as a result, the left-hand side inequality in (1.10) follows. The proof of Theorem 1.2 is complete.
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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