Abstract
In this paper, we find the least value r and the greatest value p such that the double inequality holds for all (or ) with , where , and () and are, respectively, the error function, and weighted power mean.
MSC:33B20, 26D15.
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1 Introduction
For , the error function is defined as
It is well known that the error function is odd, strictly increasing on with , strictly concave and strictly log-concave on . For the n th derivation we have the representation
where is a Hermite polynomial.
The error function can be expanded as a power series in the following two ways [1]:
It also can be expressed in terms of incomplete gamma function and a confluent hypergeometric function:
Recently, the error function have been the subject of intensive research. In particular, many remarkable properties and inequalities for the error function can be found in the literature [2–10]. It might be surprising that the error function has applications in heat conduction problems [11, 12].
In [13], Chu proved that the double inequality
holds for all if and only if and .
Mitrinović and Weinacht [14] established that
for all , and proved that the inequality become equality if and only if or .
are the best possible constants such that the double inequality
holds for and all real number (), and the sharp double inequalities
and
hold for all positive real numbers x, y with .
Let , and , , , and () and be, respectively, the weighted arithmetic, geometric, harmonic, and power means of two positive numbers x and y. Then it is well known that the inequalities
hold for all and with .
Very recently, Alzer [17] proved that and are the best possible factors such that the double inequality
holds for all and .
It is natural to ask what are the least value r and the greatest value p such that the double inequality
holds for all (or )? The main purpose of this article is to answer this question.
2 Lemmas
In order to prove our main results we need three lemmas, which we present in this section.
Lemma 2.1 Let and . Then the following statements are true:
-
(1)
if , then for all ;
-
(2)
if , then for all .
Proof Simple computation leads to
for all .
-
(1)
If , then we clearly see that
(2.2)
Therefore, Lemma 2.1(1) follows easily from (2.1) and (2.2).
-
(2)
If , then it is obvious that
(2.3)
Therefore, Lemma 2.1(2) follows from (2.1) and (2.3). □
Lemma 2.2 Let , and . Then the following statements are true:
-
(1)
if , then is strictly concave on ;
-
(2)
if , then is strictly convex on ;
-
(3)
if , then is strictly convex on .
Proof Differentiating leads to
and
where
It follows from (2.6) that
We divide the proof into four cases.
Case 1 . Then from (2.6) and (2.8) together with (2.9) we clearly see that
and is strictly decreasing on .
It follows from the monotonicity of and (2.12) that is strictly increasing on .
The monotonicity of and (2.11) imply that there exists , such that for and for . Therefore, is strictly decreasing on and strictly increasing on .
From the piecewise monotonicity of and (2.10) we clearly see that there exists , such that for and for .
If , then (2.7) leads to , this implies that for . Therefore, (2.5) leads to the conclusion that is strictly concave on .
If , then (2.7) leads to , this implies that for . Therefore, (2.5) leads to the conclusion that is strictly convex on .
Case 2 . Then we clearly see that the function is strictly increasing on with , and
Therefore, Lemma 2.1(1) and (2.13) imply that for . This leads to the conclusion that is strictly decreasing on .
From (2.6) we get
for .
It follows from the monotonicity of and (2.14) that for . Therefore, (2.5) leads to the conclusion that is strictly convex on .
Case 3 . Then we clearly see that the function is strictly decreasing on with , and
It follows from Lemma 2.1(2) and (2.15) that for . This leads to being strictly increasing on .
It follows from (2.6) that
for .
From the monotonicity of and (2.16) we know that for . Therefore, (2.5) leads to the conclusion that is strictly convex on .
Case 4 . Then from (2.6) we clearly see that for . Therefore, is strictly convex on follows easily from (2.5). □
Lemma 2.3 The function is strictly increasing on .
Proof Simple computations lead to
where
We divide the proof into two cases.
Case 1 . Then (2.19) leads to . Therefore, follows from (2.18) and (2.17).
Case 2 . Then from (2.23) we clearly see that . Therefore, follows from (2.17) and (2.18) together with (2.20)-(2.22). □
3 Main results
Theorem 3.1 Let and . Then the double inequality
holds for all if and only if and .
Proof Firstly, we prove that (3.1) holds if and .
If , , then Lemma 2.2(1) leads to
for and .
Let , , and . Then (3.2) leads to the first inequality in (3.1).
Since the function is strictly increasing on R if , it is enough to prove the second inequality in (3.1) is true for .
Let and . Then Lemma 2.2(3) leads to
for and .
Therefore, the second inequality in (3.1) follows from and together with (3.3).
Secondly, we prove that the second inequality in (3.1) implies .
Let . Then the second inequality in (3.1) leads to
It follows from (3.4) that
and
Therefore,
follows from (3.4) and (3.5) together with Lemma 2.3.
Finally, we prove that the first inequality in (3.1) implies .
Let . Then the first inequality in (3.1) leads to
for .
It follows from (3.6) that
where
If , then from (3.10) we know that there exists a small , such that for . Therefore, is strictly decreasing on .
The monotonicity of together with (3.8) and (3.9) imply that there exists , such that is strictly increasing on
It follows from the monotonicity of and (3.7) that there exists , such that for , this contradicts with (3.6). □
Theorem 3.2 Let and . Then the double inequality
holds for all if and only if and .
Proof Firstly, we prove that inequality (3.11) holds if and . Since the first inequality in (3.11) is true if , thus we only need to prove the second inequality in (3.11).
It follows from the monotonicity of the function with respect to t that we only need to prove the second inequality in (3.11) holds for .
Let and . Then Lemma 2.2(2) leads to
for and .
Therefore, the second inequality in (3.11) follows from and together with (3.12).
Secondly, we prove that the second inequality in (3.11) implies .
Let and . Then the second inequality in (3.11) leads to
It follows from (3.13) that
and
Therefore,
follows from (3.13) and (3.14) together with Lemma 2.3.
Finally, we prove that the first inequality in (3.11) implies . We divide the proof into two cases.
Case 1 . Then for any fixed one has
and
which contradicts with the first inequality in (3.11).
Case 2 . Let , , and . Then the first inequality in (3.11) leads to
It follows from (3.15) that
and
Note that
It follows from (3.17) and (3.18) that there exists a large enough , such that for , hence is strictly increasing on .
From the monotonicity of and (3.16) we conclude that there exists a large enough , such that for , this contradicts with (3.15). □
References
Oldham K, Myland J, Spanier J: An Atlas of Functions. Springer, New York; 2009.
Alzer H, Baricz Á: Functional inequalities for the incomplete gamma function. J. Math. Anal. Appl. 2012,385(1):167-178. 10.1016/j.jmaa.2011.06.032
Baricz Á: Mills’ ratio: monotonicity patterns and functional inequalities. J. Math. Anal. Appl. 2008,340(2):1362-1370. 10.1016/j.jmaa.2007.09.063
Dominici D: Some properties of the inverse error function. Contemp. Math. 2008, 457: 191-203.
Gawronski W, Müller J, Reinhard M: Reduced cancellation in the evaluation of entire functions and applications to the error function. SIAM J. Numer. Anal. 2007,45(6):2564-2576. 10.1137/060669589
Morosawa S:The parameter space of error functions of the form . In Complex Analysis and Potential Theory. World Scientific, Hackensack; 2007:174-177.
Fisher B, Al-Sirehy F, Telci M: Convolutions involving the error function. Int. J. Appl. Math. 2003,13(4):317-326.
Alzer H: On some inequalities for the incomplete gamma function. Math. Comput. 1997,66(218):771-778. 10.1090/S0025-5718-97-00814-4
Chapman SJ: On the non-universality of the error function in the smoothing of Stokes discontinuities. Proc. R. Soc. Lond. Ser. A 1996,452(1953):2225-2230. 10.1098/rspa.1996.0118
Weideman JAC: Computation of the complex error function. SIAM J. Numer. Anal. 1994,31(5):1497-1518. 10.1137/0731077
Kharin SN: A generalization of the error function and its application in heat conduction problems. 176. In Differential Equations and Their Applications. ‘Nauka’ Kazakh. SSR, Alma-Ata; 1981: 51-59. (in Russian)
Chaudhry MA, Qadir A, Zubair SM: Generalized error functions with applications to probability and heat conduction. Int. J. Appl. Math. 2002,9(3):259-278.
Chu JT: On bounds for the normal integral. Biometrika 1955, 42: 263-265. 10.2307/2333443
Mitrinović DS, Weinacht RJ: Problems and solutions: solutions of advanced problems: 5555. Am. Math. Mon. 1968,75(10):1129-1130. 10.2307/2315774
Alzer H: Functional inequalities for the error function. Aequ. Math. 2003,66(1-2):119-127. 10.1007/s00010-003-2683-9
Alzer H: Functional inequalities for the error function II. Aequ. Math. 2009,78(1-2):113-121. 10.1007/s00010-009-2963-0
Alzer H: Error function inequalities. Adv. Comput. Math. 2010,33(3):349-379. 10.1007/s10444-009-9139-2
Acknowledgements
This research was supported by the Natural Science Foundation of China under Grants 61174076, 61374086, 11371125, and 11401191, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004. The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions.
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Chu, YM., Li, YM., Xia, WF. et al. Best possible inequalities for the harmonic mean of error function. J Inequal Appl 2014, 525 (2014). https://doi.org/10.1186/1029-242X-2014-525
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DOI: https://doi.org/10.1186/1029-242X-2014-525