In this note, we present monotonicity results of a function involving to the
inverse hyperbolic sine. From these, we derive some inequalities for bounding
the inverse hyperbolic sine.Comment: 3 page

B-N Guo

Bai-Ni Guo

BJ Malešević

F Qi

Feng Qi

GTF de Abreu

J-L Zhao

L Zhu

Qiu-Ming Luo

S-Q Zhang

W Pan

Journal of Inequalities and Applications

English

arXiv

Crossref

Guo et al. Journal of Inequalities and Applications 2013, 2013:536http://www.journalofinequalitiesandapplications.com/content/2013/1/536RESEARCH Open AccessMonotonicity results and inequalities for theinverse hyperbolic sine functionBai-Ni Guo1*, Qiu-Ming Luo2 and Feng Qi1*Correspondence:bai.ni.guo@gmail.com;bai.ni.guo@hotmail.com1School of Mathematics andInformatics, Henan PolytechnicUniversity, Jiaozuo, Henan 028043,ChinaFull list of author information isavailable at the end of the articleAbstractIn the paper, the authors present monotonicity results of a function involving theinverse hyperbolic sine. From these, the authors derive some inequalities forbounding the inverse hyperbolic sine.MSC: Primary 26A48; secondary 26D05; 33B10Keywords: bound; inverse hyperbolic sine; monotonicity; minimumIntroduction andmain resultsIn [, Theorem .], the following inequalities were established: for  ≤ x ≤ r and r > ,the double inequality(a + )xa +√ + x≤ arcsinhx≤ (b + )xb +√ + x()holds true if and only if a≤  andb≥√ + r arcsinh r – rr – arcsinh r . ()The aim of this paper is to elementarily generalize inequality () to monotonicity resultsand to deduce more inequalities.Our results may be stated as the following theorems.Theorem  For θ ∈R, letfθ (x) =θ +√ + xx arcsinhx, x > . (). When θ ≤ , the function fθ (x) is strictly increasing;. When θ > , the function fθ (x) has a unique minimum.As straightforward consequences of Theorem , the following inequalities are inferred.Theorem  Let ≤ x≤ r and r > .©2013 Guo et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.Guo et al. Journal of Inequalities and Applications 2013, 2013:536 Page 2 of 6http://www.journalofinequalitiesandapplications.com/content/2013/1/536. For – < θ ≤ , the double inequality( + θ )xθ +√ + x< arcsinhx≤ (θ +√ + r) arcsinh rrxθ +√ + x()holds true on (, r], where the scalars  + θ and(θ +√ + r) arcsinh rrin () are best possible.. For θ > , the double inequality( – /θ)xθ +√ + x≤ arcsinhx≤ max{ + θ , (θ +√ + r) arcsinh rr} xθ +√ + x()holds true on (, r].RemarksBefore proving our theorems, we give several remarks on them.Remark  Replacing arcsinhx by x in () and () yieldssinhxx <⎧⎨⎩θ+coshx+θ , – < θ ≤ ,θ+coshx(–/θ) , θ > ()andsinhxx >⎧⎨⎩r(θ+coshx)(θ+√+r) arcsinh r, – < θ ≤ ,r(θ+coshx)max{r(+θ ),(θ+√+r) arcsinh r} , θ > ()for x ∈ (, arcsinh r). These can be regarded as Oppenheim-type inequalities for the hyper-bolic sine and cosine functions. For information onOppenheim’s double inequality for thesine and cosine functions, please refer to [], [, Sections . and .] and closely relatedreferences therein.Remark  It is clear that the left-hand side inequality in () recovers the left-hand sideinequality in (), while the right-hand side inequalities in () and () do not include eachother.Remark  Lethx(θ ) = – /θθ +√ + xGuo et al. Journal of Inequalities and Applications 2013, 2013:536 Page 3 of 6http://www.journalofinequalitiesandapplications.com/content/2013/1/536for θ >  and x > . Thenh′x(θ ) =√x +  + θ – θθ(θ +√x + ).Therefore, the function hx(θ ) attains its maximum√√x +  – x{[ √(√ + x – x) + ] – √(√ + x – x)}[ √(√ + x – x) + ][ √(√ + x – x) +√x +  √√x +  – x + ]at the point + √(√ + x – x)√√ + x – x .Combining this with the fact that the functionθ →  + θθ +√ + xis increasing and the functionθ → θ +√ + rθ +√ + xis decreasing, we establish from Theorem  the following best and sharp double inequal-ities:x +√ + x< arcsinhx≤ [( +√ + r)(arcsinh r)/r]x +√ + x()andx √√x +  – x{[ √(√ + x – x) + ] – √(√ + x – x)}[ √(√ + x – x) + ][ √(√ + x – x) +√x +  √√x +  – x + ]< arcsinhx < (√r +  – )(arcsinh r)x(√r +  arcsinh r – r) + (r – arcsinh r)√x + ()for  < x < r.The famous software Mathematica . shows that double inequality () is betterthan ().Remark  By a similar approach to that presented in the next section, we can procuresimilar monotonicity results and inequalities for the inverse hyperbolic cosine and otherinverse trigonometric functions. For more information on this topic, please refer to [,–] and closely related references therein.Remark  We note that Shafer-type inequalities from [] were applied recently in []for obtaining upper and lower bounds on the Gaussian Q-function.Guo et al. Journal of Inequalities and Applications 2013, 2013:536 Page 4 of 6http://www.journalofinequalitiesandapplications.com/content/2013/1/536Proofs of theoremsNow we are in a position to elementarily prove our theorems.Proof of Theorem  Direct diﬀerentiation yieldsf ′θ (x) =x(θ + √x + )[x(θ/√x +  + )θ + /√x + – arcsinhx] x(θ + √x + )hθ (x)andh′θ (x) =x( – θ + θ√x + )√x + (θ√x +  + ) xqx(θ )√x + (θ√x +  + ).The function qx(θ ) has two zerosθ(x) =√ + x –√ + xandθ(x) =√ + x +√ + x .They are strictly increasing and have the bounds – ≤ θ(x) <  and θ(x) ≥  on (,∞).As a result, under the condition θ /∈ (–, ),. when θ ≤ –, the function qx(θ ) and the derivative h′θ (x) are negative, and so thefunction hθ (x) is strictly decreasing on (,∞);. when ≤ θ ≤ , the function qx(θ ) and the derivative h′θ (x) are positive, and so thefunction hθ (x) is strictly increasing on (,∞);. when θ > , the function qx(θ ) and the derivative h′θ (x) have the unique zerox =√θ–θ+θwhich is the unique minimum point of hθ (x).Furthermore, since limx→∞ hθ (x) =∞ for θ ≥  and limx→+ hθ (x) = , it follows that. when θ ≤ –, the function hθ (x) is negative, and so the derivative f ′θ (x) is positive,that is, the function fθ (x) is strictly increasing on (,∞);. when ≤ θ ≤ , the function hθ (x) is positive, and so the derivative f ′θ (x) is alsopositive, accordingly, the function fθ (x) is strictly increasing on (,∞);. when θ > , the function hθ (x) and the derivative f ′θ (x) have a unique zero as asolution of the equationx√x +  – ln(x +√x + )√x +  ln(x +√x + ) – x= θ ,which is the unique minimum point of the function fθ (x) on (,∞).On the other hand, when θ ∈ (–, ), we have[xf ′θ (x)]′ = x[√x +  + (arcsinhx)/x – θ ](x + )/ > Guo et al. Journal of Inequalities and Applications 2013, 2013:536 Page 5 of 6http://www.journalofinequalitiesandapplications.com/content/2013/1/536which means that the function xf ′θ (x) is strictly increasing on (,∞). From the limitlimx→+[xf ′θ (x)] = , it is derived that the function xf ′θ (x) is positive. Hence the functionfθ (x) is strictly increasing on (,∞). The proof of Theorem  is complete. Proof of Theorem  Since limx→+ fθ (x) =  + θ , by Theorem , it is easy to see that  + θ <fθ (x)≤ fθ (r) on (, r] for – < θ ≤ . Inequality () is thus proved.For θ > , the minimum point x ∈ (,∞) satisﬁesarcsinhx =x(θ/√x +  + )θ + /√x + .Therefore, the minimum of the function fθ (x) on (,∞) equals(θ +√ + x) arcsinhxx=x(θ/√x +  + )θ + /√x + ·(θ +√ + x)x=(θ/√x +  + )(θ +√ + x)θ + /√x + =(θ +√ + x)θ√x +  + ≥ ( – θ).From this, it is obtained that( – θ)≤ fθ (x)≤ max{limx→+fθ (x), fθ (r)}for x ∈ (, r], which implies inequality (). The proof of Theorem  is thus completed.Remark  This paper is a slightly revised version of the preprint [].Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAll authors contributed equally to the manuscript and read and approved the ﬁnal manuscript.Author details1School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, Henan 028043, China. 2Department ofMathematics, Chongqing Normal University, Chongqing, 401331, China.AcknowledgementsThe authors appreciate anonymous referees for their valuable comments on and careful corrections to the originalversion 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 underGrant No. KJ120625, and by the Fund of Chongqing Normal University, China under Grant No. 10XLR017 and 2011XLZ07,China.Received: 2 September 2013 Accepted: 24 October 2013 Published: 12 Nov 2013Guo et al. Journal of Inequalities and Applications 2013, 2013:536 Page 6 of 6http://www.journalofinequalitiesandapplications.com/content/2013/1/536References1. Zhu, L: New inequalities of Shafer-Fink type for arc-hyperbolic sine. J. Inequal. Appl. 2008, Article ID 368275 (2008).doi:10.1155/2008/3682752. Qi, F, Luo, Q-M, Guo, B-N: A simple proof of Oppenheim’s double inequality relating to the cosine and sine functions.J. Math. Inequal. 6(4), 645-654 (2012). doi:10.7153/jmi-06-633. Qi, F, Niu, D-W, Guo, B-N: Reﬁnements, generalizations, and applications of Jordan’s inequality and related problems.J. Inequal. Appl. 2009, Article ID 271923 (2009). doi:10.1155/2009/2719234. Guo, B-N, Luo, Q-M, Qi, F: Sharpening and generalizations of Shafer-Fink’s double inequality for the arc sine function.Filomat 27(2), 261-265 (2013). doi:10.2298/FIL1302261G5. Guo, B-N, Qi, F: Sharpening and generalizations of Carlson’s inequality for the arc cosine function. Hacet. J. Math. Stat.39(3), 403-409 (2010)6. Pan, W, Zhu, L: Generalizations of Shafer-Fink-type inequalities for the arc sine function. J. Inequal. Appl. 2009, ArticleID 705317 (2009). doi:10.1155/2009/7053177. Qi, F, Guo, B-N: Sharpening and generalizations of Shafer’s inequality for the arc sine function. Integral TransformsSpec. Funct. 23(2), 129-134 (2012). doi:10.1080/10652469.2011.5645788. Qi, F, Zhang, S-Q, Guo, B-N: Sharpening and generalizations of Shafer’s inequality for the arc tangent function.J. Inequal. Appl. 2009, Article ID 930294 (2009). doi:10.1155/2009/9302949. Zhang, S-Q, Guo, B-N: Monotonicity results and inequalities for the inverse hyperbolic sine. Chin. Q. J. Math. 24(3),384-388 (2009)10. Zhao, J-L, Wei, C-F, Guo, B-N, Qi, F: Sharpening and generalizations of Carlson’s double inequality for the arc cosinefunction. Hacet. J. Math. Stat. 41(2), 201-209 (2012)11. Maleševic´, BJ: An application of λ-method on Shafer-Fink’s inequality. Publ. Elektroteh. Fak. Univ. Beogr., Mat. 8, 90-92(1997)12. de Abreu, GTF: Arbitrarily tight upper and lower bounds on the Gaussian q-function and related functions. In: 2009IEEE International Conference on Communications (ICC 2009), vols. 1-8, pp. 1944-1949. Dresden, Germany, June14-18 (2009). doi:10.1109/ICC.2009.519876213. Qi, F, Guo, B-N: Monotonicity results and bounds for the inverse hyperbolic sine. arXiv:0903.167610.1186/1029-242X-2013-536Cite this article as: Guo et al.:Monotonicity results and inequalities for the inverse hyperbolic sine function. Journalof Inequalities and Applications 2013, 2013:536

Monotonicity results and inequalities for the inverse hyperbolic sine function

Springer - Publisher Connector

Guo et al. Journal of Inequalities and Applications 2013, 2013:536
http://www.journalofinequalitiesandapplications.com/content/2013/1/536
RESEARCH Open Access
Monotonicity results and inequalities for the
inverse hyperbolic sine function
Bai-Ni Guo1*, Qiu-Ming Luo2 and Feng Qi1
*Correspondence:
bai.ni.guo@gmail.com;
bai.ni.guo@hotmail.com
1School of Mathematics and
Informatics, Henan Polytechnic
University, Jiaozuo, Henan 028043,
China
Full list of author information is
available at the end of the article
Abstract
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: Primary 26A48; secondary 26D05; 33B10
Keywords: bound; inverse hyperbolic sine; monotonicity; minimum
Introduction andmain results
In [, Theorem .], the following inequalities were established: for  ≤ x ≤ r and r > ,
the double inequality
(a + )x
a +
√
 + x
≤ arcsinhx≤ (b + )x
b +
√
 + x
()
holds true if and only if a≤  and
b≥
√
 + r arcsinh r – r
r – arcsinh r . ()
The aim of this paper is to elementarily generalize inequality () to monotonicity results
and to deduce more inequalities.
Our results may be stated as the following theorems.
Theorem  For θ ∈R, let
fθ (x) =
θ +
√
 + x
x arcsinhx, x > . ()
. When θ ≤ , the function fθ (x) is strictly increasing;
. When θ > , the function fθ (x) has a unique minimum.
As straightforward consequences of Theorem , the following inequalities are inferred.
Theorem  Let ≤ x≤ r and r > .
©2013 Guo et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-
tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Guo et al. Journal of Inequalities and Applications 2013, 2013:536 Page 2 of 6
http://www.journalofinequalitiesandapplications.com/content/2013/1/536
. For – < θ ≤ , the double inequality
( + θ )x
θ +
√
 + x
< arcsinhx≤ (θ +
√
 + r) arcsinh r
r
x
θ +
√
 + x
()
holds true on (, r], where the scalars  + θ and
(θ +
√
 + r) arcsinh r
r
in () are best possible.
. For θ > , the double inequality
( – /θ)x
θ +
√
 + x
≤ arcsinhx
≤ max
{
 + θ , (θ +
√
 + r) arcsinh r
r
} x
θ +
√
 + x
()
holds true on (, r].
Remarks
Before proving our theorems, we give several remarks on them.
Remark  Replacing arcsinhx by x in () and () yields
sinhx
x <
⎧⎨
⎩
θ+coshx
+θ , – < θ ≤ ,
θ+coshx
(–/θ) , θ > 
()
and
sinhx
x >
⎧⎨
⎩
r(θ+coshx)
(θ+
√
+r) arcsinh r
, – < θ ≤ ,
r(θ+coshx)
max{r(+θ ),(θ+√+r) arcsinh r} , θ > 
()
for x ∈ (, arcsinh r). These can be regarded as Oppenheim-type inequalities for the hyper-
bolic sine and cosine functions. For information onOppenheim’s double inequality for the
sine and cosine functions, please refer to [], [, Sections . and .] and closely related
references therein.
Remark  It is clear that the left-hand side inequality in () recovers the left-hand side
inequality in (), while the right-hand side inequalities in () and () do not include each
other.
Remark  Let
hx(θ ) =
 – /θ
θ +
√
 + x
Guo et al. Journal of Inequalities and Applications 2013, 2013:536 Page 3 of 6
http://www.journalofinequalitiesandapplications.com/content/2013/1/536
for θ >  and x > . Then
h′x(θ ) =

√
x +  + θ – θ
θ(θ +
√
x + )
.
Therefore, the function hx(θ ) attains its maximum
√√x +  – x{[ 
√
(
√
 + x – x) + ] – 
√
(
√
 + x – x)}
[ 
√
(
√
 + x – x) + ][ 
√
(
√
 + x – x) +
√
x +  
√√
x +  – x + ]
at the point
 + 
√
(
√
 + x – x)
√√ + x – x .
Combining this with the fact that the function
θ →  + θ
θ +
√
 + x
is increasing and the function
θ → θ +
√
 + r
θ +
√
 + x
is decreasing, we establish from Theorem  the following best and sharp double inequal-
ities:
x
 +
√
 + x
< arcsinhx≤ [( +
√
 + r)(arcsinh r)/r]x
 +
√
 + x
()
and
x 
√√
x +  – x{[ 
√
(
√
 + x – x) + ] – 
√
(
√
 + x – x)}
[ 
√
(
√
 + x – x) + ][ 
√
(
√
 + x – x) +
√
x +  
√√
x +  – x + ]
< arcsinhx < (
√
r +  – )(arcsinh r)x
(
√
r +  arcsinh r – r) + (r – arcsinh r)
√
x + 
()
for  < x < r.
The famous software Mathematica . shows that double inequality () is better
than ().
Remark  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 [,
–] and closely related references therein.
Remark  We note that Shafer-type inequalities from [] were applied recently in []
for obtaining upper and lower bounds on the Gaussian Q-function.
Guo et al. Journal of Inequalities and Applications 2013, 2013:536 Page 4 of 6
http://www.journalofinequalitiesandapplications.com/content/2013/1/536
Proofs of theorems
Now we are in a position to elementarily prove our theorems.
Proof of Theorem  Direct diﬀerentiation yields
f ′θ (x) =

x
(
θ + √
x + 
)[x(θ/√x +  + )
θ + /
√
x + 
– arcsinhx
]
 x
(
θ + √
x + 
)
hθ (x)
and
h′θ (x) =
x( – θ + θ
√
x + )√
x + (θ
√
x +  + )
 x
qx(θ )√
x + (θ
√
x +  + )
.
The function qx(θ ) has two zeros
θ(x) =
√
 + x –
√
 + x

and
θ(x) =
√
 + x +
√
 + x
 .
They are strictly increasing and have the bounds – ≤ θ(x) <  and θ(x) ≥  on (,∞).
As a result, under the condition θ /∈ (–, ),
. when θ ≤ –, the function qx(θ ) and the derivative h′θ (x) are negative, and so the
function hθ (x) is strictly decreasing on (,∞);
. when ≤ θ ≤ , the function qx(θ ) and the derivative h′θ (x) are positive, and so the
function hθ (x) is strictly increasing on (,∞);
. when θ > , the function qx(θ ) and the derivative h′θ (x) have the unique zero
x =
√
θ–θ+
θ
which is the unique minimum point of hθ (x).
Furthermore, since limx→∞ hθ (x) =∞ for θ ≥  and limx→+ hθ (x) = , it follows that
. when θ ≤ –, the function hθ (x) is negative, and so the derivative f ′θ (x) is positive,
that is, the function fθ (x) is strictly increasing on (,∞);
. when ≤ θ ≤ , the function hθ (x) is positive, and so the derivative f ′θ (x) is also
positive, accordingly, the function fθ (x) is strictly increasing on (,∞);
. when θ > , the function hθ (x) and the derivative f ′θ (x) have a unique zero as a
solution of the equation
x
√
x +  – ln(x +
√
x + )√
x +  ln(x +
√
x + ) – x
= θ ,
which is the unique minimum point of the function fθ (x) on (,∞).
On the other hand, when θ ∈ (–, ), we have
[
xf ′θ (x)
]′ = x[
√
x +  + (arcsinhx)/x – θ ]
(x + )/ > 
Guo et al. Journal of Inequalities and Applications 2013, 2013:536 Page 5 of 6
http://www.journalofinequalitiesandapplications.com/content/2013/1/536
which means that the function xf ′θ (x) is strictly increasing on (,∞). From the limit
limx→+[xf ′θ (x)] = , it is derived that the function xf ′θ (x) is positive. Hence the function
fθ (x) is strictly increasing on (,∞). The proof of Theorem  is complete. 
Proof of Theorem  Since limx→+ fθ (x) =  + θ , by Theorem , it is easy to see that  + θ <
fθ (x)≤ fθ (r) on (, r] for – < θ ≤ . Inequality () is thus proved.
For θ > , the minimum point x ∈ (,∞) satisﬁes
arcsinhx =
x(θ/
√
x +  + )
θ + /
√
x + 
.
Therefore, the minimum of the function fθ (x) on (,∞) equals
(θ +
√
 + x) arcsinhx
x
=
x(θ/
√
x +  + )
θ + /
√
x + 
·
(θ +
√
 + x)
x
=
(θ/
√
x +  + )(θ +
√
 + x)
θ + /
√
x + 
=
(θ +
√
 + x)
θ
√
x +  + 
≥ 
(
 – 
θ
)
.
From this, it is obtained that

(
 – 
θ
)
≤ fθ (x)≤ max
{
lim
x→+
fθ (x), fθ (r)
}
for x ∈ (, r], which implies inequality (). The proof of Theorem  is thus completed.

Remark  This paper is a slightly revised version of the preprint [].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the ﬁnal manuscript.
Author details
1School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, Henan 028043, China. 2Department of
Mathematics, Chongqing Normal University, Chongqing, 401331, China.
Acknowledgements
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.
Received: 2 September 2013 Accepted: 24 October 2013 Published: 12 Nov 2013
Guo et al. Journal of Inequalities and Applications 2013, 2013:536 Page 6 of 6
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Cite this article as: Guo et al.:Monotonicity results and inequalities for the inverse hyperbolic sine function. Journal
of Inequalities and Applications 2013, 2013:536


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