\(^\S \)Department of Mathematics, Huzhou Teachers College, Xueshi Str. no. 1, 313000, Huzhou, China, e-mail: wmk000@126.com, qiuyefang861013@126.com,
chuyuming2005@yahoo.com.cn.

\(^\ast \)The work of the last author has been supported by the Natural Science Foundation of China (Grant No. 11071069), the Natural Science Foundation of Zhejiang Province (Grant No. Y7080106) and the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant No. T200924)

1 Introduction

For \(p\in \mathbb {R}\), the one-parameter mean \(J_{p}(a,b)\), arithmetic mean \(A(a,b)\), and harmonic mean \(H(a,b)\) of two positive real numbers \(a\) and \(b\) are defined by

\(A(a,b)=\tfrac {a+b}{2}\), and \(H(a,b)=\tfrac {2ab}{a+b}\), respectively.

Recently, the one-parameter mean \(J_{p}(a,b)\) has been the subject of intensive research. In particular, many remarkable inequalities and properties for the one-parameter mean \(J_{p}\) can be found in the literature [1–7].

It is well-known that the one-parameter mean \(J_{p}(a,b)\) is continuous and strictly increasing with respect to \(p\in \mathbb {R}\) for fixed \(a,b{\gt}0\) with \(a\neq b\) [5]. Many mean values are special case of the one-parameter mean, for example

and

For \(r\in \mathbb {R}\), the power mean \(M_{r}(a,b)\) of order \(r\) of two positive numbers \(a\) and \(b\) is defined by

The main properties of the power mean are given in [8]. In particular, \(M_{r}(a,b)\) is continuous and strictly increasing with respect to \(r\in \mathbb {R}\) for fixed \(a,b{\gt}0\) with \(a\neq b\).

In [9], Alzer and Janous established the following sharp double inequality (see also [9, p. 350])

for all \(a,b{\gt}0\) with \(a\neq b\).

In [10], Mao proved

for all \(a,b{\gt}0\), and \(M_{\tfrac {1}{3}}(a,b)\) is the best possible lower power mean bound for the sum \(\tfrac {1}{3}J_{1}(a,b)+\tfrac {2}{3}J_{-\tfrac {1}{2}}(a,b)\).

The purpose of this paper is to answer the question: For \(\alpha \in (0,1)\), what are the greatest value \(r_{1}\) and the least value \(r_{2}\) such that the double inequality

holds for all \(a,b{\gt}0\) with \(a\neq b\)?

2 Lemmas

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

Then simple computations lead to

for \(t{\gt}1\).

Therefore, Lemma 1 follows from (4)–(6).

Then simple computations lead to

for \(t{\gt}1\).

Therefore, Lemma 2 follows from (8)–(10).

3 Main Result

for \(\alpha \in (0,1)\) and all \(a,b{\gt}0\) with \(a\neq b\).

Without loss of generality, we assume that \(a{\gt}b\) and take \(t={\tfrac {a}{b}}{\gt}1\). We divide the proof into three cases.

Case 1. If \(\alpha =\tfrac {1}{3}\), then from (1) we have

Therefore, inequality (11) follows from (12) and Lemma 1.

Case 2. If \(\alpha =\tfrac {2}{3}\), then (1) leads to

Therefore, inequality (11) follows from (13) and Lemma 2.

Case 3. If \(\alpha \in (0,\tfrac {1}{3})\cup (\tfrac {1}{3},\tfrac {2}{3})\cup (\tfrac {2}{3},1)\), then (1) implies that

Let

then \(f(t)\) can be rewritten as

where

Note that

and

We divide the proof into two subcases.

Subcase 1. If \(\alpha \in (0,\tfrac {1}{3})\cup (\tfrac {2}{3},1)\), then from (23) we clearly see that \({f_{1}}'''(t){\gt}0\) for \(t\in (1,\infty )\). Then (18) and (20) together with (22) imply that \(f_{1}(t){\gt}0\) for \(t\in (1,\infty )\). Note that

for \(t{\gt}1\).

Therefore, inequality (11) follows from (14)–(16) and (24) together with the fact that \(f_{1}(t){\gt}0\) for \(t\in (1,\infty )\).

Subcase 2. If \(\alpha \in (\tfrac {1}{3},\tfrac {2}{3})\), then from (23) we clearly see that \({f_{1}}'''(t){\lt}0\) for \(t\in (1,\infty )\). Then (18) and (20) together with (22) imply that \(f_{1}(t){\lt}0\) for \(t\in (1,\infty )\). Note that

for \(t{\gt}1\).

Therefore, inequality (11) follows from (14)–(16) and (25) together with the fact that \(f_{1}(t){\lt} 0\) for \(t\in (1,\infty )\).

Next, we prove that

for \(\alpha \in (0,1)\) and all \(a,b{\gt}0\) with \(a\neq b\).

Without loss of generality, we assume that \(a{\gt}b\). Let \(t={\tfrac {a}{b}}{\gt}1\), then from (1) we have

Let

then \(F(t)\) can be rewritten as

where

Note that

and

for \(t\in (1,\infty )\) and \(\alpha \in (0,1)\).

Therefore, inequality (26) follows from (27)–(29), (31), (33) and (34).

At last, we prove that \(J_{3\alpha -2}(a,b)\) and \(J_{\tfrac {\alpha }{2-\alpha }}(a,b)\) are the best possible lower and upper one-parameter mean bounds for the sum \(\alpha A(a,b)+(1-\alpha )H(a,b)\), respectively.

For any \(0{\lt}\varepsilon {\lt}\tfrac {\alpha }{2-\alpha }\) and \(x{\gt}0\), from (1) one has

For any \(\varepsilon {\gt}0\), \(\varepsilon \neq 1-3\alpha \), \(\varepsilon \neq 2-3\alpha \) and \(x{\gt}0\), let \(x\rightarrow 0\), making use of (1) and the Taylor expansion one has

Inequality (35) implies that for any \(0{\lt}\varepsilon {\lt}\tfrac {\alpha }{2-\alpha }\) there exists \(X=X(\alpha ,\varepsilon ){\gt}1\) such that \({\alpha A(x,1)+(1-\alpha )H(x,1)}{\gt}J_{\tfrac {\alpha }{2-\alpha }-\varepsilon }(x,1)\) for \(x\in (X,\infty )\), and inequality (36) implies that for any \(\varepsilon {\gt}0\), \(\varepsilon \neq 1-3\alpha \) and \(\varepsilon \neq 2-3\alpha \) there exists \(\delta =\delta (\alpha ,\varepsilon ){\gt}0\) such that \(J_{3\alpha -2+\varepsilon }(1+x,1){\gt}\alpha A(1+x,1)+(1-\alpha )H(1+x,1)\) for \(x\in (0,\delta )\).