- Research Article
- Open Access
Generalized Lazarevic's Inequality and Its Applications—Part II
- Ling Zhu1Email author
https://doi.org/10.1155/2009/379142
© Ling Zhu. 2009
- Received: 21 July 2009
- Accepted: 30 November 2009
- Published: 24 December 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.
Keywords
- Lower Bound
- Concise Proof
1. Introduction
Lazarević [1] (or see Mitrinović [2]) gives us the following result.
Theorem 1.1.
. 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.
, and
. Then
holds if and only if
.
Moreover, the inequality (1.1) can be extended as follows.
Theorem 1.3.
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
.
- (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.
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:
Authors’ Affiliations
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