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Generalized Lazarevic's Inequality and Its Applications—Part II
Journal of Inequalities and Applications volume 2009, Article number: 379142 (2009)
Abstract
A generalized Lazarevic's inequality is established. The applications of this generalized Lazarevic's inequality give some new lower bounds for logarithmic mean.
1. Introduction
Lazarević [1] (or see Mitrinović [2]) gives us the following result.
Theorem 1.1.
Let . Then
holds if and only if .
Recently, the author of this paper gives a new proof of the inequality (1.1) in [3] and extends the inequality (1.1) to the following result in [4].
Theorem 1.2.
Let , and . Then
holds if and only if .
Moreover, the inequality (1.1) can be extended as follows.
Theorem 1.3.
Let or , and . Then
holds if and only if .
2. Three Lemmas
Let be two continuous functions which are differentiable on . Further, let on . If is increasing (or decreasing) on , then the functions and are also increasing (or decreasing) on .
Let and be real numbers, and let the power series and be convergent for . If for and if is strictly increasing (or decreasing) for then the function is strictly increasing (or decreasing) on .
Lemma 2.3.
Let and . Then the function strictly increases as increases.
3. A Concise Proof of Theorem 1.3
Let , where , and . Then
where , and .
We compute
where
and .
We obtain results in two cases.

(a)
Let , then and . Let for we have that and is decreasing for so is decreasing for and is decreasing on by Lemma 2.2. Hence is decreasing on and is decreasing on by Lemma 2.1. Thus is decreasing on by Lemma 2.1.

(b)
Let , then . Let for we have that and is decreasing for so is decreasing for and is decreasing on by Lemma 2.2. Hence is increasing on and is decreasing on by Lemma 2.1. Thus is decreasing on by Lemma 2.1.
Since
the proof of Theorem 1.3 is complete.
4. Some New Lower Bounds for Logarithmic Mean
Assuming that and are two different positive numbers, let , , and be the arithmetic, geometric, and logarithmic means, respectively. It is well known that (see [2, 12–16])
Ostle and Terwilliger [17] (or see Leach and Sholander [18], Zhu [16]) gave bounds for in terms of and as follows:
Without loss of generality, let and , then . Replacing with in (1.3), we obtain the following new results for three classical means.
Theorem 4.1.
Let or , and and be two positive numbers such that . Then
holds if and only if .
Now letting in inequality (4.3) be , and , respectively, by Theorem 4.1 and Lemma 2.3 we have the following inequalities:
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Zhu, L. Generalized Lazarevic's Inequality and Its Applications—Part II. J Inequal Appl 2009, 379142 (2009). https://doi.org/10.1155/2009/379142
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DOI: https://doi.org/10.1155/2009/379142