Abstract
In this paper, we obtain a general result on the natural approximation of the function \(\left( \arctan x\right) ^{2}-\left( x\mathop {\mathrm{ arcsinh}}\nolimits x\right) /\sqrt{1+x^{2}}\), and prove a conjecture raised by Zhu and Malešević (J Inequal Appl 2019:93, 2019).
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1 Introduction
Masjed-Jamei [1] obtained the following inequality
where \(\ln (x+\sqrt{1+x^{2}})\) is the inverse hyperbolic sinefunction \(\mathop {\mathrm{ arcsinh}}\nolimits x=\sinh ^{-1}x\). In [1] the author conjectured that the above inequality is established in a larger interval \((-\infty ,\infty )\). Recently, the authors of this paper [2] first affirmed Masjed-Jamei’s conjecture, obtained some natural generalizations of this inequality, and pose a conjecture about a natural approach of Masjed-Jamei’s inequality inspired by [3,4,5,6,7,8,9].
Proposition 1.1
([2], Theorem 1.1]) The inequality
holds for all \(x\in (-\infty ,\infty )\), and the power number 2 is the best in (1.2).
Proposition 1.2
([2], Theorem 1.3]) Let \(-\infty<\)\(x<\infty \). Then we have
Conjecture 1.1
([2], Conjecture 8.1]) Let \(x\in {\mathbf {R}},\)\(m\ge 1,\) and \(v_{n}\) be defined by
Then the double inequality
holds.
Using flexible analysis tools this paper obtains a more general conclusion on the natural approximation of the function \(\left( \arctan x\right) ^{2}-\left( x\mathop {\mathrm{arcsinh}}\nolimits x\right) /\sqrt{1+x^{2}}\), and proves the above conjecture.
Theorem 1.1
Let \(k\ge 3\) and \(v_{n}\) be defined by (1.5). Then the function
is decreasing and negative on \((0,+\infty )\) when k is an even number, and is increasing and positive on \((0,+\infty )\) when k is an odd number. In particular, for \(m\ge 1\),
- (i)
the inequality
$$\begin{aligned} \left( \arctan x\right) ^{2}-\frac{x\ln \left( x+\sqrt{1+x^{2}}\right) }{\sqrt{1+x^{2}}} \le \sum _{n=3}^{2m+2}\left( -1\right) ^{n}v_{n}x^{2n} \end{aligned}$$(1.8)holds for all \(x\in [0,+\infty )\), the constant \(v_{2m+2}\) is best possible in (1.8);
- (ii)
the inequality
$$\begin{aligned} \sum _{n=3}^{2m+1}\left( -1\right) ^{n}v_{n}x^{2n}\le \left( \arctan x\right) ^{2}-\frac{x\ln \left( x+\sqrt{1+x^{2}}\right) }{\sqrt{1+x^{2}}} \end{aligned}$$(1.9)holds for all \(x\in [0,+\infty )\), the constant \(-v_{2m+1}\) is best possible in (1.9).
Obviously, the Conjecture 1.1 is from Theorem 1.1 immediately.
2 Lemmas
Lemma 2.1
Let \(\left| x\right| <1\). Then
Lemma 2.2
Let \(\left| x\right| <1,\) and \(v_{n}\) be defined by (1.5). Then
and \(v_{n}>0\) for \(n\ge 3\).
Lemma 2.3
Let \(v_{n}\) be defined by (1.5), and
Then \(\gamma _{k}=0\) for \(k\ge 3\).
Lemma 2.4
Let \(v_{n}\) be defined by (1.5). For \(k\ge 3\),
3 Proof of Lemma 2.1
Let
Then \(p(0)=0,\) and
Differentiating (3.2) gives
Since p(x) is an odd function, we can express it as a power series
Differentiation of (3.4) yields
Substituting the series of \(p^{\prime }(x)\) and p(x) into (3.3) yields
Equating coefficients of \(x^{2n+2},\) we can obtain
Then
and
The last equation is true due to
The proof of Lemma 2.1 is complete.
4 Proof of Lemma 2.2
First, by (2.1) we have
Second, we have
Integrating two sides of (4.2) on [0, x] we can obtain
where \(v_{n}\) is defined by (1.5). The proof of \(v_{n}>0\)\( \left( n\ge 3\right) \) can be found in \(\left[ 2\right] \).
5 Proof of Lemma 2.3
The fact \(v_{1}=v_{2}=0\) and \(v_{3}=1/45\) is directly derived from (1.5). Then \(\gamma _{3}=45v_{3}-1=0.\)
Below we assume \(k\ge 4.\) For convenience, we can order
Substituting the expression of \(v_{k}\) defined by (1.5) to the left-hand side of the formula (1.6), we can easy verify that
6 Proof of Lemma 2.4
We first verify the first inequality, which is equivalent to
that is,
We use mathematical induction to prove (6.1). Obviously, the formula (6.1) holds for \(k=3\). Assuming that (6.1) holds for \(k=m\), we have
Next, we prove that (6.1) is valid for \(k=m+1\). By (6.2) we have
In order to complete the proof of (6.1) it suffices to show that
which is
Using mathematical induction again to prove (6.3). First, we can see that (6.3) holds for \(m=3\). Second, assuming that (6.3) holds for \(m=k\), we have
Via the inequality above, we have
In order to prove that (6.3) is true for \(m=k+1\) it suffices to show that
that is,
The last inequality holds due to
Then we turn to the proof of second inequality. The desired one is equivalent to that
that is,
or
By (6.1) we have
In order to complete the proof of (6.4) it suffices to show that
which is true due to
7 Proof of Theorem 1.1
Let \(\arctan x=t,x\in (-\infty ,\infty )\). Then \(x=\tan t,t\in (-\pi /2,\pi /2)\). Because the functions involved in this section are all even functions, we only assume \(x>0\), which is corresponding to \(t\in (0,\pi /2)\). Since
we examine the properties of the function \(F_{k}(t)\)\((k\ge 3)\) as follows.
Computing to give
where
Then
where
Computing to get
by Lemm 2.3, where
By Lemmas 2.2 and 2.4 respectively, we have \(\lambda >0\), and \(\mu ,\zeta >0\). So we have
which leads to that \(f_{k}^{\prime }(t)<0\) for all \(t\in (0,\pi /2)\) when k is an even number and \(f_{k}^{\prime }(t)>0\) for all \(t\in (0,\pi /2)\) when k is an odd number. Noticing these facts \(F_{k}(0^{+})=g_{k}(0)=f_{k}(0)=0,\) by differential method we get the conclusion that \(F_{k}(t)\) is decreasing and negative on \((0,\pi /2)\) when k is an even number, and is increasing and positive on \((0,\pi /2)\) when k is an odd number. Because the transformation \(x=\tan t\) increases monotonically on the interval \((0,\pi /2) \), the function \(G_{k}(x)\) involved in Theorem 1.1 has the same properties as \(F_{k}(t)\), that is to say, \(G_{k}(x)\) is decreasing and negative on \((0,+\infty )\) when k is an even number and is increasing and positive on \((0,+\infty )\) when k is an odd number.
In view of
by Lemma 2.2, the proof of Theorem 1.1 is complete.
Remark 7.1
We know that the inequality (1.1) can be traced back to the generalization of the famous Cauchy-Schwarz inequality, which can be found in [10] and the references cited therein.
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Acknowledgements
The authors are grateful to anonymous referees for their careful corrections to and valuable comments on the original version of this paper. The first author was supported by the National Natural Science Foundation of China (No. 61772025). The second author was supported in part by the Serbian Ministry of Education, Science and Technological Development, under projects ON 174032 and III 44006.
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Zhu, L., Malešević, B. Natural approximation of Masjed-Jamei’s inequality. RACSAM 114, 25 (2020). https://doi.org/10.1007/s13398-019-00735-z
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DOI: https://doi.org/10.1007/s13398-019-00735-z