An Optimal Double Inequality between Power-Type Heron and Seiffert Means

Abstract

For , the power-type Heron mean and the Seiffert mean of two positive real numbers and are defined by , ; , and , ; , , respectively. In this paper, we find the greatest value and the least value such that the double inequality holds for all with .

1. Introduction

For , the power-type Heron mean and the Seiffert mean of two positive real numbers and are defined by

(1.1)

(1.2)

respectively.

Recently, the means of two variables have been the subject of intensive research [1–15]. In particular, many remarkable inequalities for and can be found in the literature [16–20].

It is well known that is continuous and strictly increasing with respect to for fixed with . Let ,  , ,  , and be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers and with , respectively. Then

(1.3)

For , the power mean of order of two positive numbers and is defined by

(1.4)

The main properties for power mean are given in [21].

In [16], Jia and Cao presented the inequalities

(1.5)

for all with , , and .

Sándor [22] proved that

(1.6)

for all with .

In [19], Seiffert established that

(1.7)

for all with .

The purpose of this paper is to present the optimal upper and lower power-type Heron mean bounds for the Seiffert mean . Our main result is the following Theorem 1.1.

Theorem 1.1.

For all with , one has

(1.8)

and and are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean , respectively.

2. Lemmas

In order to prove our main result, Theorem 1.1, we need two lemmas which we present in this section.

Lemma 2.1.

If and , then

(2.1)

Proof.

For , we clearly see that

(2.2)

Let

(2.3)

Then

(2.4)

and is strictly decreasing in because of for .

Therefore, Lemma 2.1 follows from (2.2)–(2.4) together with the monotonicity of .

Lemma 2.2.

If , , and , then there exists such that for and for .

Proof.

Let ,, , , , , and . Then elaborated computations lead to

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(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.)

(2.22)

(2.23)

From the expression of and Lemma 2.1, we get

(2.24)

From (2.24), we know that is strictly decreasing in . Then (2.22) implies that is strictly decreasing in .

From (2.20) and (2.21) together with the monotonicity of , we clearly see that there exists such that is strictly increasing in and strictly decreasing in .

Inequality (2.17) and (2.18) together with the piecewise monotonicity of imply that there exists such that is strictly increasing in and strictly decreasing in .

The piecewise monotonicity of together with (2.14) and (2.15) leads to the fact that there exists such that is strictly increasing in and strictly decreasing in .

From (2.11) and (2.12) together with the piecewise monotonicity of , we conclude that there exists such that is strictly increasing in and strictly decreasing in .

Equations (2.8) and (2.9) together with the piecewise monotonicity of imply that there exists such that is strictly increasing in and strictly decreasing in .

Therefore, Lemma 2.2 follows from (2.5) and (2.6) together with the piecewise monotonicity of .

3. Proof of Theorem 1.1

Proof of Theorem 1.1.

Without loss of generality, we assume that . We first prove that . Let , then from (1.1) and (1.2) we have

(3.1)

Let

(3.2)

Then simple computations lead to

(3.3)

where . Note that

(3.4)

where

(3.5)

for .

Therefore, follows from (3.1)–(3.5).

Next, we prove that . Let and , then (1.1) and (1.2) lead to

(3.6)

Let

(3.7)

Then simple computations lead to

(3.8)

(3.9)

where . Note that

(3.10)

(3.11)

(3.12)

where

(3.13)

From (3.12) and (3.13) together with Lemma 2.2, we clearly see that there exists such that is strictly increasing in and strictly decreasing in .

Equations (3.9)–(3.11) and the piecewise monotonicity of imply that there exists such that is strictly increasing in and strictly decreasing in . Then from (3.8) we get

(3.14)

for .

Therefore, follows from (3.6) and (3.7) together with (3.14).

At last, we prove that and are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean , respectively.

For any and , from (1.1) and (1.2), one has

(3.15)

(3.16)

where .

Let , making use of Taylor extension, we get

(3.17)

Equations (3.15) and (3.17) together with inequality (3.16) imply that for any , there exist and such that for and for .