Optimal generalized Heronian mean bounds for the logarithmic mean
© Shi et al; licensee Springer. 2012
Received: 22 December 2011
Accepted: 13 March 2012
Published: 13 March 2012
In this article, we establish a double inequality between the generalized Heronian and logarithmic means. The achieved result is inspired by the articles of Lin and Shi et al., and the methods from Janous. The inequalities we obtained improve the existing corresponding results and, in some sense, are optimal.
2010 Mathematics Subject Classification: 26E60.
Recently, the logarithmic mean has been: the subject of intensive research. In particular, many remarkable inequalities for the logarithmic mean can be found in the literature [1–30]. It might be surprising that the logarithmic mean has applications in physics, economics, and even in meteorology [31–33]. In , Kahlig and Matkowski study a variant of Jensen's functional equation involving the logarithmic mean, which appears in a heat conduction problem. A representation of the logarithmic mean as an infinite product and an iterative algorithm for computing the logarithmic mean as the common limit of two sequences of special geometric and arithmetic means are given in . In [34, 35] it is shown that the logarithmic mean can be expressed in terms of Gauss's hypergeometric function 2F1. And, in , Carlson and Gastafson prove that the reciprocal of the logarithmic mean is strictly totally positive, that is, every n × n determinant with elements , where 0 < x1< x2< ... < x n and 0 < y1< y2< ... < y n are positive for all n ≥ 1.
It is well-known that M r (a, b) is continuous and strictly increasing with respect to for fixed a, b > 0 with a ≠ b.
for all a, b > 0 with a ≠ b.
From (1.2) and (1.4) we clearly see that H p ,0(a, b) = M p (a, b) for all and a, b > 0. It easily follows from (1.4) that H p,ω (a, b) is continuous with respect to for fixed a, b > 0 and ω ≥ 0, strictly increasing with respect to for fixed a, b > 0 with a ≠ b and ω ≥ 0, strictly decreasing with respect to ω ≥ 0 for fixed a, b > 0 with a ≠ b and p > 0, and strictly increasing with respect to ω ≥ 0 for fixed a, b > 0 with a ≠ b and p < 0.
for all a, b > 0 with a ≠ b.
The purpose of this article is to find the greatest value p = p(ω) and the least value q = q(ω) such that the double inequality H p,ω (a, b) < L(a, b) < H q,ω (a, b) holds for fixed ω ≥ 0 and all a, b > 0 with a ≠ b.
2 Main result
and and H0,ω(a, b) are the best possible upper and lower generalized Heronian mean bounds of the logarithmic mean L(a, b), respectively.
for t ∈ (1, + ∞).
for t ∈ (1, + ∞).
Therefore, follows from (2.2)-(2.4) and (2.25).
On the other hand, H0,ω(a, b) = M0(a, b) < L(a, b) follows from (1.3).
Next, we prove that H0,ω(a, b) and are the optimal lower and upper generalized Heronian mean bounds of the logarithmic mean L(a, b).
Equations (2.26) and (2.27) imply that for any ω ≥ 0 and there exist sufficiently large X = X(ε, ω) > 1 and sufficiently small δ = δ (ε, ω) > 0; such that H ε,ω (1, x) > L(1, x) for x ∈ (X, +∞) and for x ∈ (0, δ).
Remark 2.1. If we take ω = 0, then inequality (2.1) reduce to inequality (1.3).
Remark 2.2. If we take ω = 4, then the upper bound in inequality (2.1) becomes the upper bound in inequality (1.5).
This study was partly supported by the Natural Science Foundation of China (Grant nos. 11071069, 11171307, 11171105), the Social Science Foundation of China (Grant no. 10BTJ001), the Natural Science Foundation of Hunan Province (Grant no. 09JJ6003), and the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant no. T200924).
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