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Monotonicity results and inequalities for the inverse hyperbolic sine function
Journal of Inequalities and Applications volume 2013, Article number: 536 (2013)
In the paper, the authors present monotonicity results of a function involving the inverse hyperbolic sine. From these, the authors derive some inequalities for bounding the inverse hyperbolic sine.
MSC:26A48, 26D05, 33B10.
Introduction and main results
In [, Theorem 1.9], the following inequalities were established: for and , the double inequality
holds true if and only if and
The aim of this paper is to elementarily generalize inequality (1) to monotonicity results and to deduce more inequalities.
Our results may be stated as the following theorems.
Theorem 1 For , let
When , the function is strictly increasing;
When , the function has a unique minimum.
As straightforward consequences of Theorem 1, the following inequalities are inferred.
Theorem 2 Let and .
For , the double inequality(4)
holds true on , where the scalars and
in (4) are best possible.
For , the double inequality(5)
holds true on .
Before proving our theorems, we give several remarks on them.
Remark 1 Replacing arcsinhx by x in (4) and (5) yields
for . These can be regarded as Oppenheim-type inequalities for the hyperbolic sine and cosine functions. For information on Oppenheim’s double inequality for the sine and cosine functions, please refer to , [, Sections 1.7 and 7.6] and closely related references therein.
Remark 2 It is clear that the left-hand side inequality in (4) recovers the left-hand side inequality in (1), while the right-hand side inequalities in (1) and (4) do not include each other.
Remark 3 Let
for and . Then
Therefore, the function attains its maximum
at the point
Combining this with the fact that the function
is increasing and the function
is decreasing, we establish from Theorem 2 the following best and sharp double inequalities:
The famous software Mathematica 7.0 shows that double inequality (9) is better than (8).
Remark 4 By a similar approach to that presented in the next section, we can procure similar monotonicity results and inequalities for the inverse hyperbolic cosine and other inverse trigonometric functions. For more information on this topic, please refer to [2, 4–10] and closely related references therein.
Proofs of theorems
Now we are in a position to elementarily prove our theorems.
Proof of Theorem 1 Direct differentiation yields
The function has two zeros
They are strictly increasing and have the bounds and on . As a result, under the condition ,
when , the function and the derivative are negative, and so the function is strictly decreasing on ;
when , the function and the derivative are positive, and so the function is strictly increasing on ;
when , the function and the derivative have the unique zero which is the unique minimum point of .
Furthermore, since for and , it follows that
when , the function is negative, and so the derivative is positive, that is, the function is strictly increasing on ;
when , the function is positive, and so the derivative is also positive, accordingly, the function is strictly increasing on ;
when , the function and the derivative have a unique zero as a solution of the equation
which is the unique minimum point of the function on .
On the other hand, when , we have
which means that the function is strictly increasing on . From the limit , it is derived that the function is positive. Hence the function is strictly increasing on . The proof of Theorem 1 is complete. □
Proof of Theorem 2 Since , by Theorem 1, it is easy to see that on for . Inequality (4) is thus proved.
For , the minimum point satisfies
Therefore, the minimum of the function on equals
From this, it is obtained that
for , which implies inequality (5). The proof of Theorem 2 is thus completed. □
Remark 6 This paper is a slightly revised version of the preprint .
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The authors appreciate anonymous referees for their valuable comments on and careful corrections to the original version of this manuscript. This work was partially supported by the NSF Project of Chongqing City under Grant No. CSTC2011JJA00024, by the Research Project of Science and Technology of Chongqing Education Commission under Grant No. KJ120625, and by the Fund of Chongqing Normal University, China under Grant No. 10XLR017 and 2011XLZ07, China.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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Guo, BN., Luo, QM. & Qi, F. Monotonicity results and inequalities for the inverse hyperbolic sine function. J Inequal Appl 2013, 536 (2013). https://doi.org/10.1186/1029-242X-2013-536
- inverse hyperbolic sine