ABSTRACT

In the present paper, we answer the question: for fixed, what are the greatest value and the least value such that the double inequality holds for all with? where for, the one-parameter mean, arithmetic mean and geometric mean of two positive real numbers and are defined by and, respectively.

1. Introduction

For, the one-parameter mean, arithmetic mean and geometric mean of two positive real numbers and are defined by

(1)

and, respectively.

It is well-known that the one-parameter mean is continuous and strictly increasing with respect to for fixed with. Many means are special cases of the one-parameter mean, for example:

is the arithmetic mean, is the Heronian mean, is the geometric mean, and is the harmonic mean.

The one-parameter mean and its inequalities have been studied intensively, see [1-6].

The purpose of this paper is to answer the question: for, what are the greatest value and the least value such that the double inequality

holds for all with?

2. Main Result

The main result of this paper is the following theorem.

Theorem 2.1. Let. Then for any with, we have 1)

for2)

for3)

for.

The numbers and in 2) and 3) are optimal.

In order to prove Theorem 2.1, we need a preliminary lemma.

Lemma 2.1. For, one has

(2)

Proof. Simple calculations lead to

(3)

(4)

(5)

(6)

(2) follows from (3)-(6).

Proof of Theorem 2.1. Without loss of generality we assume and take We first consider the case. 1) follows from

From now on we assume Let then (1) leads to

(7)

where

Simple calculations lead to

(8)

where

(9)

(10)

(11)

where

(12)

(13)

where

(14)

(15)

(16)

We shall distinguish between two cases.

Case 1.. The left-hand side inequality of 2)

for follows from Lemma 2.1 because in this case

for all. In the sequel we assume.

We clearly see from (16) that

Thus is strictly decreasing for and strictly increasing for. (2.14) yields then for and for. The same reasoning applies to and as well, and noticing (13) and (12), one has

This result together with (11) implies

Thus is strictly decreasing for

and strictly increasing for The same reasoning applies to

and as well, and applying (8)-(10), we derive

Since for and for, then we know from (7) that

This implies the left-hand side of 2) and the right-hand side of 3).

Case 2.. From (14) we know that

From (13) we know that for and for. This implies is strictly decreasing for and strictly increasing for. From (12) we know

Therefore

(11) implies has the same property asthus is strictly decreasing for and strictly increasing for. The same reasoning applies to, and as well, and noticing (9) and (8), one has

which together with (7) implies

This implies the right-hand side of 2) and the left-hand side of 3).

We are now in the position to prove the constants

and are optimal.

For any (positive or negative, with sufficiently small) we consider the case. (12)

implies

By the continuity of, there exists such that

By (11), as the same property as. The same reasoning applies to, , and as well, and noticing (10)-(8), we know has the same property as. By (7) one has

This proves the optimality for.

To prove the optimality for in the right-hand side of 2) and the left-hand side of 3), we notice from

that there exists such that

for and and

for This ends the proof of Theorem 2.1.

3. Acknowledgements

This paper is supported by NSF of Hebei Province (A2011201011).

REFERENCES