1. Introduction

For , the generalized logarithmic mean and power mean with parameter of two positive numbers and are defined by

(1.1)

and

(1.2)

respectively. It is well known that both means are continuous and increasing with respect to for fixed and . Recently, both means have been the subject of intensive research. In particular, many remarkable inequalities involving and can be found in the literature [1–9]. Let

(1.3)

, and be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers and , respectively. Then

(1.4)

for all .

In [10], Carlson proved that

(1.5)

for all with .

The following inequality is due to Sándor [11, 12]:

(1.6)

In [13], Lin established the following results: (1) implies that for all with ; (2) implies that for all with ; (3) implies that there exist such that ; (4) implies that there exist such that . Hence the question was answered: what are the least value and the greatest value such that the inequality holds for all with .

Pittenger [14] established that

(1.7)

for all , where

(1.8)

Here, and are sharp and inequality (1.7) becomes equality if and only if or or . The case reduces to Lin's results [13]. Other generalizations of Lin's results were given by Imoru [15].

Recently, some monotonicity results of the ratio between generalized logarithmic means were established in [16–18].

The aim of this paper is to prove the following Theorem 1.1.

Theorem 1.1.

Let and with , then

(1) for ;

(2) for , and for , moreover, in each case, the bound for the sum is optimal.

2. Proof of Theorem 1.1

In order to prove our Theorem 1.1 we need a lemma, which we present in this section.

Lemma 2.1.

For and one has

(1)If , then for ;

(2)If , then for , for , and for .

Proof.

  1. (1)

    If , then we clearly see that

    (2.1)

for .

If , then

(2.2)

for .

Therefore, Lemma 2.1() follows from (2.1) and (2.2).

  1. (2)

    If , then

    (2.3)

Therefore, Lemma 2.1(2) follows from (2.3).

Proof of Theorem 1.1.

Proof.

() If , then (1.1) leads to

(2.4)

() We divide the proof into two cases.

Case 1.

or . From inequalities (1.5) and (1.6) we clearly see that

(2.5)

for , and

(2.6)

for .

Case 2.

. Without loss of generality, we assume that . Let , then (1.1) leads to

(2.7)

Let , then simple computations yield

(2.8)
(2.9)

where

(2.10)

Note that

(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)

where is defined as in Lemma 2.1.

We divide the proof into five subcases.

Subcase 2 A.

. From (2.18) and Lemma 2.1() we clearly see that for and for , then we know that is strictly decreasing in and strictly increasing in . Now from the monotonicity of and (2.17) together with the fact that we clearly see that for , then from (2.7)–(2.15) and for we get for .

Subcase 2 B.

. Then (2.18) and Lemma 2.1(1) lead to

(2.19)

for .

From (2.7)–(2.17) and (2.19) together with the fact that for we know that for .

Subcase 2 C.

. Then (2.18) and Lemma 2.1(1) imply that

(2.20)

for .

From (2.7)–(2.17), (2.20) and for we know that for .

Subcase 2 D.

. Then (2.19) again yields, and for follows from (2.7)–(2.17) and (2.19) together with .

Subcase 2 E.

. Then (2.20) is also true, and for follows from (2.7)–(2.17), (2.20) and the fact that .

Next, we prove that the bound for the sum is optimal in each case. The proof is divided into six cases.

Case 1.

. For any and , then (1.1) leads to

(2.21)

where

Let making use of Taylor expansion, one has

(2.22)

Equations (2.21) and (2.22) imply that for any , there exists , such that for any and .

Case 2.

. For any and , from (1.1) we have

(2.23)

where

Let making use of Taylor expansion, one has

(2.24)

Equations (2.23) and (2.24) imply that for any , there exists , such that for and .

Case 3.

. For and , we get

(2.25)

where

Let making use of Taylor expansion, one has

(2.26)

Equations (2.25) and (2.26) imply that for any and any , there exists , such that for .

Case 4.

. For any and , we get

(2.27)

where

Let using Taylor expansion we have

(2.28)

Equations (2.27) and (2.28) show that for any and any , there exists , such that for .

Case 5.

. For any and , we have

(2.29)

where

Let making use of Taylor expansion we get

(2.30)

Equations (2.29) and (2.30) imply that for any and any , there exists , such that for .

Case 6.

. For any and , we get

(2.31)

where

Let , using Taylor expansion we have

(2.32)

From (2.31) and (2.32) we know that for any and any , there exists , such that for .

At last, we propose two open problems as follows.

Open Problem 1

What is the least value such that the inequality

(2.33)

holds for and all with ?

Open Problem 2

What is the greatest value such that the inequality

(2.34)

holds for and all with ?