Abstract
In this paper, we give the explicit formulas for the Neuman means , , , and , and present the best possible upper and lower bounds for these means in terms of the combinations of harmonic mean H, arithmetic mean A, and contraharmonic mean C.
MSC:26E60.
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1 Introduction
Let with . Then the symmetric integral [1] of the first kind is defined as
The degenerate case of , denoted by , plays an important role in the theory of special functions [1, 2], which is given by
For with , the Schwab-Borchardt mean [3–5] of a and b is given by
where and are the inverse cosine and inverse hyperbolic cosine functions, respectively.
Carlson [6] (see also [[7], (3.21)]) proved that
Recently, the Schwab-Borchardt mean has been the subject of intensive research. In particular, many remarkable inequalities for the Schwab-Borchardt mean and its generated means can be found in the literature [3–5, 8–11].
Let , , , , , and be the parameters such that , , , , , , and be, respectively, the harmonic, geometric, arithmetic, quadratic, and contraharmonic means of a and b, , , , . Then Neuman [10] gave the explicit formulas
Very recently, Neuman [12] found a new mean derived from the Schwab-Borchardt mean as follows:
Let , , , , , , , and be the Neuman means. Then Neuman [12] proved that
for all with , and the double inequalities
hold for all with if and only if , , , , , , , and .
Zhang et al. [13] presented the best possible parameters and such that the double inequalities
hold for all with .
In [14], the authors found the greatest values , , , , , , , , and the least values , , , , , , , such that the double inequalities
hold for all with .
The main purpose of this paper is to give the explicit formulas for the Neuman means , , , and , and to present the best possible upper and lower bounds for these means in terms of the combinations of harmonic, arithmetic, and contraharmonic means. Our main results are Theorems 1.1-1.3.
Theorem 1.1 Let , , , , , and be the parameters such that , . Then we have
and
Theorem 1.2 The double inequalities
hold for all with if and only if , , , , , , , and .
Theorem 1.3 The double inequalities
hold for all with if and only if , , , , , , , and .
2 Lemmas
In order to prove our main results we need several lemmas, which we present in this section.
Lemma 2.1 (See [[15], Theorem 1.25])
For , let be continuous on , and be differentiable on , let on . If is increasing (decreasing) on , then so are
If is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2.2 (See [[16], Lemma 1.1])
Suppose that the power series and have the radius of convergence and for all . If the sequence is (strictly) increasing (decreasing) for all , then the function is also (strictly) increasing (decreasing) on .
Lemma 2.3 (See [[12], Theorem 4.1])
If , then
Lemma 2.4 The function
is strictly increasing from onto .
Proof Making use of power series expansion we get
Let
Then
and is strictly increasing for all .
Note that
Therefore, Lemma 2.4 follows easily from Lemma 2.2 and (2.1)-(2.4) together with the monotonicity of the sequence . □
Lemma 2.5 The function
is strictly increasing from onto .
Proof Let and . Then simple computations lead to
and is strictly increasing on .
Note that
Therefore, Lemma 2.5 follows from Lemma 2.1, (2.5), (2.6), and the monotonicity of . □
Lemma 2.6 The function
is strictly decreasing from onto .
Proof Simple computations lead to
Let
Then
and
for all .
Note that
Therefore, Lemma 2.6 follows easily from (2.7)-(2.11) and Lemma 2.2. □
Lemma 2.7 The function
is strictly decreasing on the interval .
Proof Let and . Then simple computations lead to
and
for .
Therefore, Lemma 2.7 follows easily from (2.12) and (2.13) together with Lemma 2.1. □
Lemma 2.8 The function
is strictly decreasing from onto .
Proof Let and . Then simple computations lead to
and
Note that
Therefore, Lemma 2.8 follows from Lemma 2.1 and Lemma 2.7 together with (2.14)-(2.17). □
3 Proofs of Theorems 1.1-1.3
Proof of Theorem 1.1 It follows from (1.1)-(1.3) as we clearly see that
Inequalities (1.8) follow easily from and Lemma 2.3 together with the fact that is a mean of and for . □
Proof of Theorem 1.2 Without loss of generality, we assume that . Let , , , , and be the parameters such that , . Then from (1.4)-(1.7) we have
where the functions and are defined as in Lemmas 2.4 and 2.5, respectively.
Note that
and
Therefore, inequality (1.9) holds for all with if and only if and follows from (3.1) and Lemma 2.4, inequality (1.10) holds for all with if and only if and follows from (3.2) and Lemma 2.5, inequality (1.11) holds for all with if and only if and follows from (3.3) and (3.5) together with Lemma 2.4, and inequality (1.12) holds for all with if and only if and follows from (3.4) and (3.6) together with Lemma 2.5. □
Proof of Theorem 1.3 Without loss of generality, we assume that . Let , , , , and be the parameters such that , . Then from (1.4)-(1.7) we have
and
where the functions and are defined as in Lemmas 2.6 and 2.8, respectively.
Note that
and
Therefore, Theorem 1.3 follows easily from (3.7)-(3.12) together with Lemmas 2.6 and 2.8. □
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Acknowledgements
This research was supported by the Natural Science Foundation of China under Grants 61374086, 11371125, and 11401192, the Natural Science Foundation of Hunan Province under Grant 12C0577, and the Research Foundation of Education Bureau of Hunan Province under Grant 14A026.
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Chen, SB., He, ZY., Chu, YM. et al. Note on certain inequalities for Neuman means. J Inequal Appl 2014, 370 (2014). https://doi.org/10.1186/1029-242X-2014-370
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DOI: https://doi.org/10.1186/1029-242X-2014-370