The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean

Abstract

We find the greatest value and least value such that the double inequality holds for all with . Here , , and denote the arithmetic, harmonic, and Seiffert's means of two positive numbers and , respectively.

1. Introduction

For with the Seiffert's mean was introduced by Seiffert [1] as follows:

(1.1)

Recently, the inequalities for means have been the subject of intensive research [2–11]. In particular, many remarkable inequalities for the Seiffert's mean can be found in the literature [12–17]. The Seiffert's mean can be rewritten as (see [14, ()])

(1.2)

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

(1.3)

In [1], Seiffert proved that

(1.4)

for all with .

Later, Seiffert [18] established that

(1.5)

for all with .

In [19], Sándor proved that

(1.6)

for all with .

The following bounds for the Seiffert's mean in terms of the power mean were presented by Jagers in [17]:

(1.7)

for all with .

Hästö [13] found the sharp lower power bound for the Seiffert's mean as follows:

(1.8)

for all with .

The purpose of this paper is to find the greatest value and the least value such that the double inequality holds for all with .

2. Main Result

Theorem 2.1.

The double inequality holds for all with if and only if and .

Proof.

Firstly, we prove that

(2.1)

(2.2)

for all with .

Without loss of generality, we assume . Let and . Then (1.1) leads to

(2.3)

Let

(2.4)

Then simple computations lead to

(2.5)

(2.6)

(2.7)

where

(2.8)

We divide the proof into two cases.

Case 1.

If , then it follows from (2.8) that

(2.9)

for .

Therefore, inequality (2.1) follows from (2.3)–(2.5) and (2.7) together with (2.9).

Case 2.

If , then from (2.8) we have

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

(2.19)

(2.20)

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

From (2.24) and (2.25) we clearly see that for , hence is strictly decreasing in . It follows from (2.22) and (2.23) together with the monotonicity of that there exists such that for and for , hence is strictly increasing in and strictly decreasing in .

From (2.19) and (2.20) together with the monotonicity of we know that there exists such that for ) and for , hence, is strictly increasing in and strictly decreasing in .

From (2.16) and (2.17) together with the monotonicity of we clearly see that there exists such that is strictly increasing in and strictly decreasing in . It follows from (2.13) and (2.14) together with the monotonicity of that there exists such that is strictly increasing in and strictly decreasing in . Then (2.7), (2.10) and (2.11) imply that there exists such that is strictly increasing in and strictly decreasing in .

Note that (2.6) becomes

(2.26)

for .

It follows from (2.5) and (2.26) together with the monotonicity of that

(2.27)

for .

Therefore, inequality (2.2) follows from (2.3) and (2.4) together with (2.27).

Secondly, we prove that is the best possible upper convex combination bound of arithmetic and harmonic means for the Seiffert's mean .

For any and , from (1.1) we have

(2.28)

where

(2.29)

It follows from (2.29) that

(2.30)

(2.31)

If , then (2.31) leads to

(2.32)

From (2.32) and the continuity of we clearly see that there exists such that

(2.33)

for . Then (2.30) and (2.33) imply that

(2.34)

for .

Therefore, for follows from (2.28) and (2.34).

Finally, we prove that is the best possible lower convex combination bound of arithmetic and harmonic means for the Seiffert's mean .

For , then from (1.1) one has

(2.35)

Inequality (2.35) implies that for any there exists such that for .