1. Introduction

Lazarević [1] (or see Mitrinović [2]) gives us the following result.

Theorem 1.1.

Let . Then

(1.1)

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

(1.2)

holds if and only if .

Moreover, the inequality (1.1) can be extended as follows.

Theorem 1.3.

Let or , and . Then

(1.3)

holds if and only if .

2. Three Lemmas

Lemma 2.1 (see [5–8]).

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 .

Lemma 2.2 (see [9–11]).

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

(3.1)

where , and .

We compute

(3.2)

where

(3.3)

and .

We obtain results in two cases.

  1. (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.

  2. (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

(3.4)

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])

(4.1)

Ostle and Terwilliger [17] (or see Leach and Sholander [18], Zhu [16]) gave bounds for in terms of and as follows:

(4.2)

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

(4.3)

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:

(4.4)