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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