Abstract
In this paper, we obtain a new inequality between the inverse hyperbolic tangent and inverse sine functions, which is a conjecture of Chen-Males̆ević [(Chen in Rev Real Acad Cienc. Exactas Fis. Nat. Ser. A-Mat 114:105, 2020) conjecture 2.1]spsCM.
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1 Introduction
In 2010, Masjed-Jamei [1] studied the relation of inverse tangent function \(\arctan x\) and inverse hyperbolic sine function \(\sinh ^{-1}(x)\) and proved an inequality as follows.
The study related to (1.1) attracted much attention in last decade. At first, Zhu and Males̆ević [3] proved that (1.1) holds for any \(x\in (-\infty , +\infty )\). They also obtained some refinements of (1.1).
Proposition 1.1
[3, Theorem 1.3] For any \(x\in (-\infty ,+\infty )\), we have
Define
By using flexible analysis tools, Zhu and Males̆ević [4] extended (1.2) and (1.3) to general form as follows.
Proposition 1.2
[4, Theorem 1.1] For any \(x\in (-\infty ,+\infty )\), we have
Proposition 1.3
[5, Theorem 2.1] The double inequality
holds for any \(x\in (0,+\infty )\) with best constants 0 and 1/45.
Please see [6, 7] for more generalizations.
Motivated by (1.1)-(1.6), Zhu and Males̆ević [3] also studied the relation of inverse hyperbolic tangent function \(\tanh ^{-1}(x)\) and inverse sine function \(\arcsin x\) as follows.
Proposition 1.4
[3, Theorem 1.4] The inequality
holds for any \(x\in (0,1)\) with the the best power number 2.
Proposition 1.5
[3, Theorem 1.6] The inequality
holds for any \(x\in (0,1)\).
Moreover, by investigating the power series of the following function
L. Zhu [2] obtained the following interesting double inequality of Masjed-Jamei type.
Proposition 1.6
[2, Theorem 1] The double inequality
holds for any \(x\in (0,1)\) with best constants \(-1\) and \(-\frac{1}{45}\).
We gave a now proof of (1.9) in [8] and provided a refinement in [9].
Proposition 1.7
[9, Theorem 1] The double inequality
holds for any \(x\in (0,1)\) with best constants \(-\frac{44}{45}\) and \(-\frac{22}{945}\).
The goal of this paper is to prove a new lower bound of \(\left[ \tanh ^{-1}(x)\right] ^2\), which is a conjecture of Chen-Males̆ević [5, Conjecture 2.1].
Theorem 1.8
If \(x\in (0,1)\), then
Remark 1.9
2 Proof of theorem 1.8
Lemma 2.1
Let
then f(x) is strictly increasing on (0, 1).
Proof
Step 1: Let \(t=\tanh ^{-1}(x)\in (0,+\infty )\), then \(x=\tanh (t)\). Define
In order to prove that f(x) is strictly increasing on (0, 1), we only need to prove F(t) is strictly increasing on \((0,+\infty )\).
Step 2: By direct computation, we have
Then
In order to prove F(t) is strictly increasing on \((0,+\infty )\), it is suffice to prove \(F^{\prime }(t)>0\) for \(t\in (0,+\infty )\), which is equivalent to \(\varphi (t)>0\) for \(t\in (0,+\infty )\).
Step 3: Denote
then
Obviously, \(\varphi _2(t)>0\) on \((0,+\infty )\). And for any \(t\in (0,+\infty )\),
since for any \(n\geqslant 1\), we have
Step 4: Define
and
then \(\varphi (t)>0\) on \((0,+\infty )\) is equivalent to \(\psi (t)>0\) on \((0,+\infty )\). Since
it is suffice to prove that \(\psi ^{\prime }(t)>0\) on \((0,+\infty )\).
Step 5: From
and
we get
and
Then
where
and
Step 6: Let
where
It is easy to check that
and
for any \(n\geqslant 6\). Therefore, \(\theta (t)>0\) for any \(t\in (0,+\infty )\), which implies
The proof of Lemma 2.1 is completed. \(\square \)
Proof of theorem 1.8
By Lemma 2.1 and
we get \(f(x)>0\) for any \(x\in (0,1)\), which implies
The proof is completed. \(\square \)
References
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Chen, X.-D., Nie, L., Huang, W.K.: New inequalities between the inverse hyperbolic tangent and the analogue for corresponding functions. J. Inequal. Appl. (2020). https://doi.org/10.1186/s13660-020-02396-8
Wang, F.: A new proof of a double inequaltiy of Masjed-Jamei type. AIMS Math. 9(4), 8768–8775 (2024). https://doi.org/10.3934/math.2024425
Wang, F.: A refinement of a double inequaltiy of Masjed-Jamei type. Submitted
Zhu, L.: New Masjed Jamei type inequalities for inverse trigonometric and inverse hyperbolic functions. Mathematics 10, 2972 (2022). https://doi.org/10.3390/math10162972
Funding
The author was supported by the Foundation of Hubei Provincial Department of Eduction (No. Q20233003),the Scientific Research Fund of Hubei Provincial Department of Eduction (No. B2022207) and Hubei University of Education, Bigdata Modeling and Intelligent Computing Research Institute.
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Wang, F., Xiao, HY. A proof of Chen-Males̆ević’s conjecture. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 137 (2024). https://doi.org/10.1007/s13398-024-01637-5
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DOI: https://doi.org/10.1007/s13398-024-01637-5