\(^\ast \) School of Teacher Education, Huzhou Teachers College, Huzhou 313000, Zhejiang, China, e-mail: xwf212@hutc.zj.cn.

\(^\S \)Department of Mathematics, Huzhou Teachers College, Huzhou 313000, Zhejiang, China, e-mail: chuyuming@hutc.zj.cn.

\(^\bullet \) The work of the first author has been supported by the Natural Science Foundation of Zhejiang Province (Grant no. Y7080185) and the work of the second author has been supported by the Natural Science Foundation (Grant no. 11071069) and the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant no. T200924).

1 Introduction

The logarithmic mean \(L(a,b)\) and identric mean \(I(a,b)\) of two positive real values \(a\) and \(b\) are defined by

and

respectively. In the recent past, both mean values have been the subject of intensive research. In particular, many remarkable inequalities for logarithmic mean or identric mean can be found in the literature [1–25]. It might be surprising that the logarithmic mean has applications in physics, economics, and even in meteorology [26–28]. In [26] the authors study a variant of Jensen’s functional equation involving \(L(a,b)\), which appears in a heat conduction problem.

The power mean \(M_{p}(a,b)\) of order \(p\) is defined by

If we denote by \(A(a,b)=\tfrac {1}{2}(a+b)\), \(G(a,b)=\sqrt{ab}\) and \(H(a,b)=\tfrac {2ab}{a+b}\) the arithmetic mean, geometric mean and harmonic mean of two positive numbers \(a\) and \(b\), respectively, then it is well-known that

and all inequalities are strict for \(a\neq b\).

In [9, 12, 30] the authors present bounds for \(L(a,b)\) in terms of \(G(a,b)\) and \(A(a,b)\).

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

The following companion of (1.3) provides inequalities for the geometric and arithmetic means of \(L(a,b)\) and \(I(a,b)\). A proof can be found in [7].

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

The following bounds for \(L(a,b)\), \(I(a,b)\), \((L(a,b)I(a,b))^{\tfrac {1}{2}}\), and \(\tfrac {L(a,b)+I(a,b)}{2}\) in terms of power means are proved in [4, 6–8, 10, 24, 30].

and

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

The main purpose of this paper is to answer the questions: What are the best possible parameters \(\alpha _{1},\alpha _{2},\beta _{1}\) and \(\beta _{2}\), such that \(\alpha _{1}A(a,b)+(1-\alpha _{1})H(a,b)\leq L(a,b)\leq \beta _{1}A(a,b)+(1-\beta _{1})H(a,b)\) and \(\alpha _{2}A(a,b)+(1-\alpha _{2})H(a,b)\leq I(a,b)\leq \beta _2A(a,b)+(1-\beta _{2})H(a,b)\) hold for all \(a,b{\gt}0\)?

2 Lemmas

By simple computation we have

and

Equation (2.4) leads to \(g'''(t){\gt}0\) for \(t\in (1,+\infty )\), then \(g''(t)\) is strictly increasing in \((1,+\infty )\). Hence \(g(t){\gt}0 \) for \(t\in (1,+\infty )\) follows from the monotonicity of \(g''(t)\) and (2.1)–(2.3).

By elementary computation we get

and

Equation (2.9) leads to \(g^{(4)}(t){\gt}0\) for \(t\in (1,+\infty )\), then \(g'''(t)\) is strictly increasing in \((1,+\infty )\). Hence \(g(t){\gt}0\) for \(t\in (1,+\infty )\) follows from the monotonicity of \(g'''(t)\) and (2.5)–(2.8).

Elementary computation yields

Equation (2.14) implies that \(g^{(4)}(t){\lt}0\) for \(t\in (1,+\infty )\), then \(g'''(t)\) is strictly decreasing in \((1,+\infty )\).

From (2.13) and the monotonicity of \(g'''(t)\) we clearly see that there exists \(t_{1}\in (1,+\infty )\), such that \(g'''(t){\gt}0\) for \(t\in (1,t_{1})\) and \(g'''(t){\lt}0\) for \(t\in (t_{1},+\infty )\). Hence we know that \(g''(t)\) is strictly increasing in \([1,t_{1})\) and strictly decreasing in \([t_{1},+\infty )\).

The monotonicity of \(g''(t)\) and (2.12) imply that there exists \(t_{2}\in (1,+\infty )\), such that \(g''(t){\gt}0\) for \(t\in (1,t_{2})\) and \(g''(t){\lt}0\) for \(t\in (t_{2},+\infty )\). Hence we know that \(g'(t)\) is strictly increasing in \([1,t_{2})\) and strictly decreasing in \([t_{2},+\infty )\).

From (2.11) and the monotonicity of \(g'(t)\) we clearly see that there exists \(t_{3}\in (1,+\infty )\), such that \(g'(t){\gt}0\) for \(t\in (1,t_{3})\) and \(g'(t){\lt}0\) for \(t\in (t_{3},+\infty )\). Hence we conclude that \(g(t)\) is strictly increasing in \([1,t_{3})\) and strictly decreasing in \([t_{3},+\infty )\).

The monotonicity of \(g(t)\) and (2.10) imply that there exists \(\lambda \in (1,+\infty )\), such that \(g(t){\gt}0\) for \(t\in (1,\lambda )\) and \(g(t){\lt}0\) for \(t\in (\lambda ,+\infty )\).

3 Main Results

If \(a=b\), then \(\alpha _{1}A(a,b)+(1-\alpha _{1})H(a,b)= L(a,b)= \beta _{1}A(a,b)+(1-\beta _{1})H(a,b)=a\) for all \(\alpha _{1},\beta _{1}\in \mathbb {R}\). Next, we assume that \(a\neq b\).

Firstly, we prove that \(H(a,b){\lt}L(a,b){\lt}\tfrac {2}{3}A(a,b)+\tfrac {1}{3}H(a,b)\). From (1.3) we know that \(H(a,b){\lt}L(a,b)\) is true, so we only need to prove that \(L(a,b){\lt}\tfrac {2}{3}A(a,b)+\tfrac {1}{3}H(a,b)\).

Without loss of generality, we assume that \(a{\gt}b\). Let \(t=\tfrac {a}{b}{\gt}1\), then simple computation leads to

Therefore, \(L(a,b){\lt}\tfrac {2}{3}A(a,b)+\tfrac {1}{3}H(a,b)\) follows from Lemma 2.1 and (3.1).

Secondly, we prove that the parameters \(\alpha _{1}\leq 0\) and \(\beta _{1}\geq \tfrac {2}{3}\) cannot be improved.

For any \(0{\lt}\varepsilon {\lt}1\) and \(0{\lt}x{\lt}1\), from (1.1) we have

Equation (3.2) implies that for any \(0{\lt}\varepsilon {\lt}1\), there exists \(0{\lt}\delta =\delta (\varepsilon ){\lt}1\), such that \(\varepsilon A(1,x)+(1-\varepsilon )H(1,x){\gt}L(1,x)\) for \(x\in (0,\delta )\). Hence the parameter \(\alpha _{1}\leq 0\) cannot be improved.

Next, for any \(0{\lt}\varepsilon {\lt}1\) and \(0{\lt}x{\lt}1\), from (1.1) we get

where \(h(x)=2x(x+2)-[(\tfrac {2}{3}-\varepsilon )x^{2}+4x+4]\log (1+x)\).

Let \(x\to 0\) and using Taylor expansion we obtain

Equations (3.3) and (3.4) imply that for any \(0{\lt}\varepsilon {\lt}1\), there exists \(0{\lt}\delta =\delta (\varepsilon ){\lt}1\), such that \(L(1+x,1){\gt}(\tfrac {2}{3}-\varepsilon )A(1+x,1)+(\tfrac {1}{3}+\varepsilon )H(1+x,1) \) for \(x\in (0,\delta )\). Hence the parameter \(\beta _{1}\geq \tfrac {2}{3}\) cannot be improved.

If \(a=b\), then \(\alpha _{2}A(a,b)+(1-\alpha _{2})H(a,b)=I(a,b)=\beta _{2}A(a,b)+(1-\beta _{2})H(a,b)=a\) for all \(\alpha _{2},\beta _{2}\in \mathbb {R}\). Next, we assume that \(a\neq b\).

Firstly, we prove that \(\tfrac {2}{{\rm e}}A(a,b)+(1-\tfrac {2}{{\rm e}})H(a,b){\lt} I(a,b)\) and the parameter \(\alpha _{2}\leq \tfrac {2}{{\rm e}}\) cannot be improved.

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

Let \(f(t)=\log t^{\tfrac {t}{t-1}}-\log \left[(t+1)+({\rm e}-2)\tfrac {2t}{t+1}\right] \), then elementary computation yields

and

where \(g(t)=-[t^{3}+(2{\rm e}-1)t^{2}+(2{\rm e}-1)t+1]\log t+(2{\rm e}-2)t^{3}+(-2{\rm e}+6)t^{2}+(2{\rm e}-6)t-2{\rm e}+2\).

From (3.7) and Lemma 2.3 we know that there exists \(\lambda \in (1,+\infty )\), such that \(f(t)\) is strictly increasing in \((1,\lambda )\) and strictly decreasing in \((\lambda ,+\infty )\). Then (3.6) and the monotonicity of \(f(t)\) imply that \(f(t){\gt}0\) for \(t\in (1,+\infty )\), and from (3.5) we know that \(I(a,b){\gt}\tfrac {2}{{\rm e}}A(a,b)+(1-\tfrac {2}{{\rm e}})H(a,b)\) for \(a,b{\gt}0\) with \(a\neq b\).

Next, we prove that the parameter \(\alpha _{2}\leq \tfrac {2}{{\rm e}}\) cannot be improved.

For any \(0{\lt}\varepsilon {\lt}1\) and \(0{\lt}x{\lt}1\), from (1.2) we have

Equation (3.8) implies that for any \(0{\lt}\varepsilon {\lt}1\), there exists \(0{\lt}\delta =\delta (\varepsilon ){\lt}1\), such that \((\tfrac {2}{{\rm e}}+\varepsilon )A(1,x)+(1-\tfrac {2}{{\rm e}}-\varepsilon )H(1,x){\gt}I(1,x)\) for \(x\in (0,\delta )\). Hence the parameter \(\alpha _{2}\leq \tfrac {2}{{\rm e}}\) cannot be improved.

Secondly, we prove that \(I(a,b){\lt}\tfrac {5}{6}A(a,b)+\tfrac {1}{6}H(a,b)\) and the parameter \(\beta _{2}\geq \tfrac {5}{6}\) cannot be improved.

Let \(t=\tfrac {a}{b}{\gt}1\), then from (1.2) we have

Let \(f(t)=\log \left[ \tfrac {5t^{2}+14t+5}{12(t+1)}\right]-\log \left(\tfrac {1}{{\rm e}}t^{\tfrac {t}{t-1}}\right)\), then

and

where \(g(t)=(5t^{3}+19t^{2}+19t+5)\log t-14t^{3}-6t^{2}+6t+14\).

From Lemma 2.2 and (3.11) together with (3.10) we clearly see that \(f(t){\gt}0\) for \(t\in (1,+\infty )\). Hence from (3.9) we know that \(\tfrac {5}{6}A(a,b)+\tfrac {1}{6}H(a,b){\gt}I(a,b)\) for \(a,b{\gt}0\) with \(a\neq b\).

Next, we prove that the parameter \(\beta _{2}\geq \tfrac {5}{6}\) cannot be improved.

For any \(0{\lt}\varepsilon {\lt}1\) and \(0{\lt}x{\lt}1\), from (1.2) we get

where \(h(x)=\tfrac {2}{{\rm e}}(2+x)(1+x)^{\tfrac {1+x}{x}}-(\tfrac {5}{6}-\varepsilon )x^{2}-4x-4\).

Let \(x\to 0\) and using Taylor expansion we obtain

Equations (3.12) and (3.13) imply that for any \(0{\lt}\varepsilon {\lt}1\), there exists \(0{\lt}\delta =\delta (\varepsilon ){\lt}1\), such that \(I(1+x,1){\gt}(\tfrac {5}{6}-\varepsilon )A(1+x,1)+(\tfrac {1}{6}+\varepsilon )H(1+x,1)\) for \(x\in (0,\delta )\). Hence the parameter \(\beta _{2}\geq \tfrac {5}{6}\) cannot be improved.