Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means

Abstract

In this paper, we prove three sharp inequalities as follows: $P(a,b)>L_2(a,b)$, $T(a,b)>L_5(a,b)$ and $M(a,b)>L_4(a,b)$ for all $a,b>0$ with $a\ne b$. Here, $L_r(a,b)$, $M(a,b)$, $P(a,b)$ and $T(a,b)$ are the r th generalized logarithmic, Neuman-Sándor, first and second Seiffert means of a and b, respectively.

MSC:26E60.

1 Introduction

The Neuman-Sándor mean $M(a,b)$ [1] and the first and second Seiffert means $P(a,b)$ [2] and $T(a,b)$ [3] of two positive numbers a and b are defined by

(1.1)

(1.2)

and

$$T(a,b)=\{\begin{array}{cc}\frac{a-b}{2arctan(\frac{a-b}{a+b})},\hfill & a\ne b,\hfill \\ a,\hfill & a=b,\hfill \end{array}$$

(1.3)

respectively.

Recently, these means, M, P and T, have been the subject of intensive research. In particular, many remarkable inequalities for M, P and T can be found in the literature [1, 4–8].

The power mean $M_p(a,b)$ of order r of two positive numbers a and b is defined by

$$M_p(a,b)=\{\begin{array}{cc}{(\frac{a^p+b^p}{2})}^{1/p},\hfill & p\ne 0,\hfill \\ \sqrt{ab},\hfill & p=0.\hfill \end{array}$$

The main properties for $M_p(a,b)$ are given in [9]. In particular, the function $p\mapsto M_p(a,b)$ ($a\ne b$) is continuous and strictly increasing on ℝ.

The arithmetic-geometric mean $\mathit{AG}(a,b)$ of two positive numbers a and b is defined as the common limit of sequences $\{a_n\}$ and $\{b_n\}$, which are given by

$$a_0=a,b_0=b,a_{n+1}=\frac{a_n+b_n}{2},b_{n+1}=\sqrt{a_n{b}_n}.$$

Let $H(a,b)=2ab/(a+b)$, $G(a,b)=\sqrt{ab}$, $L(a,b)=(b-a)/(logb-loga)$, $I(a,b)=1/e{(b^b/a^a)}^{1/(b-a)}$, $A(a,b)=(a+b)/2$, and $S(a,b)=\sqrt{(a^2+b^2)/2}$ be the harmonic, geometric, logarithmic, identric, arithmetic and root-square means of two positive numbers a and b with $a\ne b$, respectively. Then it is well known that the inequalities

$$\begin{array}{rcl}H(a,b)& =& M_{-1}(a,b)<G(a,b)=M_0(a,b)<L(a,b)<\mathit{AG}(a,b)<I(a,b)\\ <& A(a,b)=M_1(a,b)<M(a,b)<T(a,b)<S(a,b)=M_2(a,b)\end{array}$$

hold for all $a,b>0$ with $a\ne b$.

For $r>0$ the r th generalized logarithmic mean $L_r(a,b)$ of two positive numbers a and b is defined by

$$L_r(a,b)=L^{1/r}(a^r,b^r)=\{\begin{array}{cc}{[\frac{b^r-a^r}{r(logb-loga)}]}^{1/r},\hfill & a\ne b,\hfill \\ a,\hfill & a=b.\hfill \end{array}$$

(1.4)

It is not difficult to verify that $L_r(a,b)$ is continuous and strictly increasing with respect to $r\in (0,+\mathrm{\infty })$ for fixed $a,b>0$ with $a\ne b$.

In [2, 3] Seiffert proved that the double inequalities

$$L(a,b)<P(a,b)<I(a,b)$$

and

$$A(a,b)<T(a,b)<S(a,b)$$

hold for all $a,b>0$ with $a\ne b$.

The following bounds for the first Seiffert mean $P(a,b)$ in terms of power mean were presented by Jagers in [10]:

$$M_{1/2}<P(a,b)<M_{2/3}(a,b)$$

for all $a,b>0$ with $a\ne b$.

Hästö [11, 12] proved that the function $x\to T(1,x)/M_p(1,x)$ is increasing on $(0,+\mathrm{\infty })$ if $p\le 1$ and found the sharp lower power mean bound for the Seiffert mean $P(a,b)$ as follows:

$$P(a,b)>M_{log2/log\pi }(a,b)$$

for all $a,b>0$ with $a\ne b$.

In [13] the authors presented the following best possible Lehmer mean bounds for the Seiffert means $P(a,b)$ and $T(a,b)$:

$${\overline{L}}_{-1/6}(a,b)<P(a,b)<{\overline{L}}_0(a,b)\text{and}{\overline{L}}_0(a,b)<T(a,b)<{\overline{L}}_{1/3}(a,b)$$

for all $a,b>0$ with $a\ne b$. Here, ${\overline{L}}_p(a,b)=(a^{p+1}+b^{p+1})/(a^p+b^p)$ is the Lehmer mean of a and b.

In [14, 15] the authors proved that the inequalities

and

$${\alpha }_{3}T(a,b)+(1-{\alpha }_3)G(a,b)<A(a,b)<{\beta }_{3}T(a,b)+(1-{\beta }_3)G(a,b)$$

hold for all $a,b>0$ with $a\ne b$ if and only if ${\alpha }_1\le (4-\pi )/[(\sqrt{2}-1)\pi ]$, ${\beta }_1\ge 2/3$, ${\alpha }_2\le 2/3$, ${\beta }_2\ge 4-2log\pi /log2$, ${\alpha }_3\le 3/5$ and ${\beta }_3\ge \pi /4$.

For all $a,b>0$ with $a\ne b$, the following inequalities can be found in [16, 17]:

$$L(a,b)=L_1(a,b)<\mathit{AG}(a,b)<L_{3/2}(a,b)<M_{1/2}(a,b).$$

Neuman and Sándor [1] established that

$$P(a,b)<A(a,b)<M(a,b)<T(a,b)$$

and

$$\frac{\pi }{2}P(a,b)>A(a,b)>arcsinh(1)M(a,b)>\frac{\pi }{4}T(a,b)$$

for all $a,b>0$ with $a\ne b$. In particular, the Ky Fan inequalities

$$\frac{G(a,b)}{G(a^{\prime },b^{\prime })}<\frac{L(a,b)}{L(a^{\prime },b^{\prime })}<\frac{P(a,b)}{P(a^{\prime },b^{\prime })}<\frac{A(a,b)}{A(a^{\prime },b^{\prime })}<\frac{M(a,b)}{M(a^{\prime },b^{\prime })}<\frac{T(a,b)}{T(a^{\prime },b^{\prime })}$$

hold for all $0<a,b\le 1/2$ with $a\ne b$, $a^{\prime }=1-a$ and $b^{\prime }=1-b$.

It is the aim of this paper to find the best possible generalized logarithmic mean bounds for the Neuman-Sándor and Seiffert means.

2 Lemmas

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

Lemma 2.1 The inequality

$${\left(\frac{x^4-1}{4logx}\right)}^{1/2}<\frac{x^2-1}{4arctanx-\pi }$$

(2.1)

holds for all $x>1$.

Proof Let

$$f(x)=\frac{x^4-1}{4logx}-{\left(\frac{x^2-1}{4arctanx-\pi }\right)}^2.$$

(2.2)

Then $f(x)$ can be rewritten as

$$f(x)=\frac{(x^4-1)f_1(x)}{4{(4arctanx-\pi )}^{2}logx},$$

(2.3)

where

$$f_1(x)={(4arctanx-\pi )}^2-\frac{4(x^2-1)logx}{x^2+1}.$$

Simple computations lead to

(2.4)

(2.5)

where

(2.6)

(2.7)

where

(2.8)

(2.9)

(2.10)

and

$$f_3^{‴}(x)=-\frac{8}{x^3}(x+1)(x-1)(3x^4+2x^2+3)<0$$

(2.11)

for $x>1$.

Inequality (2.11) implies that $f_3^{″}(x)$ is strictly decreasing in $[1,+\mathrm{\infty })$, then equation (2.10) leads to the conclusion that $f_3^{\prime }(x)$ is strictly decreasing in $[1,+\mathrm{\infty })$.

From equations (2.4)-(2.9) and the monotonicity of $f_3^{\prime }(x)$, we clearly see that

$$f_1(x)<0$$

(2.12)

for $x>1$.

Therefore, inequality (2.1) follows from equations (2.2) and (2.3) together with inequality (2.12). □

Lemma 2.2 The inequality

$${\left(\frac{x^5-1}{5logx}\right)}^{1/5}<\frac{x-1}{2arctan\frac{x-1}{x+1}}$$

(2.13)

holds for all $x>1$.

Proof Let

$$g(x)=\frac{1}{5}log\left(\frac{x^5-1}{5logx}\right)-log\left(\frac{x-1}{2arctan\frac{x-1}{x+1}}\right).$$

(2.14)

Then simple computations lead to

(2.15)

(2.16)

where

(2.17)

(2.18)

where

(2.19)

(2.20)

(2.21)

(2.22)

(2.23)

where

(2.24)

(2.25)

Let

$$\begin{array}{rcl}{g}_4(x)& =& 10(-3,780x^4-3,360x^3+1,170x^2+2,900x+950-120x^{-1}-6x^{-2}\\ -20x^{-3}-30x^{-4}+120x^{-5}-30x^{-6}-360x^{-7}-630x^{-8})logx\\ +2,055x^4-29,720x^3-1,845x^2+18,060x+8,510-2,500x^{-1}\\ -1,040x^{-2}+190x^{-4}-2,260x^{-5}+445x^{-6}+5,220x^{-7}+1,845x^{-8}.\end{array}$$

Then

(2.26)

(2.27)

where

(2.28)

for $x>1$.

Equation (2.27) and inequality (2.28) lead to the conclusion that $g_4(x)$ is strictly decreasing in $[1,+\mathrm{\infty })$. Then equation (2.26) implies that

$$g_4(x)<0$$

(2.29)

for $x>1$.

Inequalities (2.25) and (2.29) imply that $g_3^{\prime }(x)<0$. Then equation (2.24) shows that

$$g_3(x)<0$$

(2.30)

for $x>1$.

From equations (2.17)-(2.23) and inequality (2.30), we clearly see that

$$g_1(x)<0$$

(2.31)

for $x>1$.

Therefore, inequality (2.13) follows easily from equations (2.14)-(2.16) and inequality (2.31). □

Lemma 2.3 The inequality

$${arcsinh}^4\left(\frac{x-1}{x+1}\right)-\frac{{(x-1)}^{4}logx}{4(x^4-1)}<0$$

(2.32)

holds for all $x>1$.

Proof Let

$$h(x)=log\left[{arcsinh}^4\left(\frac{x-1}{x+1}\right)\right]-log\left[\frac{{(x-1)}^{4}logx}{4(x^4-1)}\right].$$

(2.33)

Then simple computations lead to

(2.34)

(2.35)

where

(2.36)

(2.37)

where

(2.38)

(2.39)

where

(2.40)

(2.41)

(2.42)

(2.43)

where

(2.44)

(2.45)

(2.46)

where

$$\begin{array}{rcl}{h}_6(x)& =& -8(105x^{10}+21x^9-45x^8+20x^7+3x^6-36x^5+3x^4+20x^3\\ -45x^2+21x+105)logx-(x^2-1)(463x^8+346x^7-14x^6\\ +298x^5+6x^4+298x^3-14x^2+346x+463)\\ <& 0\end{array}$$

(2.47)

for all $x>1$.

Equations (2.45) and (2.46) together with inequality (2.47) imply that $h_5(x)$ is strictly decreasing in $[1,+\mathrm{\infty })$. Then equation (2.44) leads to

$$h_5(x)<0$$

(2.48)

for all $x>1$.

From equations (2.36)-(2.43) and inequality (2.48), we clearly see that

$$h_1(x)<0$$

(2.49)

for all $x>1$.

Therefore, inequality (2.32) follows from equations (2.33)-(2.35) and inequality (2.49). □

3 Main results

Theorem 3.1 The inequality

$$P(a,b)>L_2(a,b)$$

holds for all $a,b>0$ with $a\ne b$, and $L_2(a,b)$ is the best possible lower generalized logarithmic mean bound for the first Seiffert mean $P(a,b)$.

Proof From (1.2) and (1.4), we clearly see that both $P(a,b)$ and $L_r(a,b)$ are symmetric and homogenous of degree one. Without loss of generality, we assume that $b=1$ and $a=x^2>1$. Then (1.2) and (1.4) lead to

$$L_2(x^2,1)-P(x^2,1)={\left(\frac{x^4-1}{4logx}\right)}^{1/2}-\frac{x^2-1}{4arctanx-\pi }.$$

(3.1)

Therefore, $P(x^2,1)>L_2(x^2,1)$ follows from Lemma 2.1 and equation (3.1).

Next, we prove that $L_2(a,b)$ is the best possible lower generalized logarithmic mean bound for the first Seiffert mean $P(a,b)$.

For any $ϵ>0$ and $x>0$, from (1.2) and (1.4), one has

$$L_{2+ϵ}(1+x,1)-P(1+x,1)={\left[\frac{{(1+x)}^{2+ϵ}-1}{(2+ϵ)log(1+x)}\right]}^{1/(2+ϵ)}-\frac{x}{4arctan\sqrt{1+x}-\pi }.$$

(3.2)

Letting $x\to 0$ and making use of Taylor expansion, we get

(3.3)

Equations (3.2) and (3.3) imply that for any $ϵ>0$, there exists ${\delta }_1={\delta }_1(ϵ)>0$ such that $L_{2+ϵ}(1+x,1)>P(1+x,1)$ for $x\in (0,{\delta }_1)$. □

Remark 3.1 It follows from (1.2) and (1.4) that

$$\underset{x\to +\mathrm{\infty }}{lim}\frac{P(x,1)}{L_{\lambda }(x,1)}=\underset{x\to +\mathrm{\infty }}{lim}\frac{{\lambda }^{1/\lambda }(1-1/x){log}^{1/\lambda }x}{\pi {(1-x^{-\lambda })}^{1/\lambda }}=+\mathrm{\infty }$$

(3.4)

for all $\lambda >0$.

Equation (3.4) implies that $\lambda >0$ such that $L_{\lambda }(a,b)>P(a,b)$ for all $a,b>0$ does not exist.

Theorem 3.2 The inequality

$$T(a,b)>L_5(a,b)$$

holds for all $a,b>0$ with $a\ne b$, and $L_5(a,b)$ is the best possible lower generalized logarithmic mean bound for the second Seiffert mean $T(a,b)$.

Proof Without loss of generality, we assume that $b=1$ and $a=x>1$. Then (1.3) and (1.4) lead to

$$L_5(x,1)-T(x,1)={\left(\frac{x^5-1}{5logx}\right)}^{1/5}-\frac{x-1}{2arctan\frac{x-1}{x+1}}.$$

(3.5)

Therefore, $T(x,1)>L_5(x,1)$ follows from Lemma 2.2 and equation (3.5).

Next, we prove that $L_5(a,b)$ is the best possible lower generalized logarithmic mean bound for the second Seiffert mean $T(a,b)$.

For any $ϵ>0$ and $x>0$, from (1.3) and (1.4), one has

$$L_{5+ϵ}(1+x,1)-T(1+x,1)={\left[\frac{{(1+x)}^{5+ϵ}-1}{(5+ϵ)log(1+x)}\right]}^{1/(5+ϵ)}-\frac{x}{2arctan\frac{x}{2+x}}.$$

(3.6)

Letting $x\to 0$ and making use of Taylor expansion, we have

(3.7)

Equations (3.6) and (3.7) imply that for any $ϵ>0$, there exists ${\delta }_2={\delta }_2(ϵ)>0$ such that $L_{5+ϵ}(1+x,1)>T(1+x,1)$ for $x\in (0,{\delta }_2)$. □

Remark 3.2 It follows from (1.3) and (1.4) that

$$\underset{x\to +\mathrm{\infty }}{lim}\frac{T(x,1)}{L_{\mu }(x,1)}=\underset{x\to +\mathrm{\infty }}{lim}\frac{2{\mu }^{1/\mu }(1-1/x){log}^{1/\mu }x}{\pi {(1-x^{-\mu })}^{1/\mu }}=+\mathrm{\infty }$$

(3.8)

for all $\mu >0$.

Equation (3.8) implies that $\mu >0$ such that $L_{\mu }(a,b)>T(a,b)$ for all $a,b>0$ does not exist.

Theorem 3.3 The inequality

$$M(a,b)>L_4(a,b)$$

holds for all $a,b>0$ with $a\ne b$, and $L_4(a,b)$ is the best possible lower generalized logarithmic mean bound for the Neuman-Sándor mean $M(a,b)$.

Proof Without loss of generality, we assume that $b=1$ and $a=x>1$. Then (1.1) and (1.4) lead to

(3.9)

Therefore, $M(x,1)>L_4(x,1)$ follows from Lemma 2.3 and equation (3.9).

Next, we prove that $L_4(a,b)$ is the best possible lower generalized logarithmic mean bound for the Neuman-Sándor mean $M(a,b)$.

For any $ϵ>0$ and $x>0$, from (1.1) and (1.4), one has

$$L_{4+ϵ}(1+x,1)-T(1+x,1)={\left[\frac{{(1+x)}^{4+ϵ}-1}{(4+ϵ)log(1+x)}\right]}^{1/(4+ϵ)}-\frac{x}{2arcsinh(\frac{x}{2+x})}.$$

(3.10)

Letting $x\to 0$ and making use of Taylor expansion, we have

(3.11)

Equations (3.10) and (3.11) imply that for any $ϵ>0$, there exists ${\delta }_3={\delta }_3(ϵ)>0$ such that $L_{4+ϵ}(1+x,1)>M(1+x,1)$ for $x\in (0,{\delta }_3)$. □

Remark 3.3 It follows from (1.1) and (1.4) that

$$\underset{x\to +\mathrm{\infty }}{lim}\frac{M(x,1)}{L_{\nu }(x,1)}=\underset{x\to +\mathrm{\infty }}{lim}\frac{{\nu }^{1/\nu }(1-1/x){log}^{1/\nu }x}{2arcsinh(1){(1-x^{-\nu })}^{1/\nu }}=+\mathrm{\infty }$$

(3.12)

for all $\nu >0$.

Equation (3.12) implies that $\nu >0$ such that $L_{\nu }(a,b)>M(a,b)$ for all $a,b>0$ does not exist.