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.
1. Introduction and Main Results
In , the following three proofs for the inequality (1.1) were provided.
Solution by Grinstein
Solution by Marsh
Solution by Konhauser
The substitution transforms the given inequality into , which is a special case of an inequality discussed on [3, pages 105-106] .
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.
As direct consequences of Theorem 1.1, the following inequalities may be derived.
Before proving our theorems, we give several remarks on them.
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.
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.
This paper is a revised version of the preprint .
3. Proofs of Theorems
Now we are in a position to prove our theorems.
Proof of Theorem 1.1.
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 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.
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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