Skip to content

Advertisement

  • Research
  • Open Access

Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean

Journal of Inequalities and Applications20162016:176

https://doi.org/10.1186/s13660-016-1113-1

  • Received: 28 March 2016
  • Accepted: 22 June 2016
  • Published:

Abstract

In the article, we prove that the function \(r\mapsto \mathcal{E}(r)/S_{9/2-p, p}(1, r')\) is strictly increasing on \((0, 1)\) for \(p\leq7/4\) and strictly decreasing on \((0, 1)\) for \(p\in [2, 9/4]\), where \(r'=\sqrt{1-r^{2}}\), \(\mathcal{E}(r)=\int_{0}^{\pi/2}\sqrt{1-r^{2}\sin^{2}(t)}\,dt\) is the complete elliptic integral of the second kind, and \(S_{p, q}(a, b)=[q(a^{p}-b^{p})/(p(a^{q}-b^{q}))]^{1/(p-q)}\) is the Stolarsky mean of a and b. As applications, we present several new bounds for \(\mathcal{E}(r)\), the Toader mean \({T}(a,b)=(2/\pi)\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}t+b^{2}\sin^{2}t}\,dt\), and the Toader-Qi mean \(\operatorname{TQ}(a,b)=(2/\pi)\int_{0}^{\pi/2}a^{\cos^{2}\theta}b^{\sin^{2}\theta }d\theta\).

Keywords

  • complete elliptic integral
  • Stolarsky mean
  • Toader mean
  • Toader-Qi mean

MSC

  • 33E05
  • 26D15
  • 26E60

1 Introduction

For \(r\in(0,1)\), \(p,q\in\mathbb{R}\), and \(a, b>0\), the Stolarsky mean \(S_{p, q}(a,b)\) [1] and the complete elliptic integral \(\mathcal{E}(r)\) [2] of the second kind are defined by
$$\begin{aligned}& S_{p, q}(a,b)= \textstyle\begin{cases} [\frac{q (a^{p}-b^{p} )}{p (a^{q}-b^{q} )} ]^{1/(p-q)},& a\neq b, p\neq q, pq\neq0, \\ [\frac{a^{p}-b^{p}}{p(\log a-\log b)} ]^{1/p},& a\neq b, p\neq 0, q=0, \\ [\frac{a^{q}-b^{q}}{q(\log a-\log b)} ]^{1/q},& a\neq b, p=0, q\neq0, \\ \exp (\frac{a^{p}\log a-b^{p}\log b}{a^{p}-b^{p}}-\frac{1}{p} ),& a\neq b, p=q\neq0, \\ \sqrt{ab},& p=q=0, \\ a,& a=b, \end{cases}\displaystyle \end{aligned}$$
(1.1)
$$\begin{aligned}& \mathcal{E}(r)= \int_{0}^{\pi/2}\sqrt{1-r^{2} \sin^{2}(t)}\,dt, \end{aligned}$$
(1.2)
respectively.
It is well known that the Stolarsky mean \(S_{p, q}(a,b)\) is continuous on the domain \(\{(p, q; a, b)|p,q\in\mathbb{R}; a, b>0\}\), symmetric with respect to its parameters p and q or variables a and b, and strictly increasing with respect its parameter p or q and variable a or b. Many classical bivariate means are special cases of the Stolarsky mean \(S_{p, q}(a,b)\), for example,
  • \(S_{1, 0}(a,b)=(a-b)/(\log a-\log b)=L(a,b)\) is the logarithmic mean;

  • \(S_{1, 1}(a,b)=({1}/{e}) ({a^{a}}/{b^{b}} )^{1/(a-b)}=I(a,b)\) is the identic mean;

  • \(S_{2, 1}(a,b)=(a+b)/2=A(a,b)\) is the arithmetic mean;

  • \(S_{3/2, 1/2}(a,b)=(a+\sqrt{ab}+b)/3=\operatorname{He}(a,b)\) is the Heronian mean;

  • \(S_{2p, p}(a,b)= [A (a^{p}, b^{p} ) ]^{1/p}=A_{p}(a,b)\) is the p-order arithmetic mean;

  • \(S_{3p/2, p/2}(a,b)= [\operatorname{He} (a^{p}, b^{p} ) ]^{1/p}=\operatorname{He}_{p}(a, b)\) is the p-order Heronian mean;

  • \(S_{p, 0}(a,b)= [L (a^{p}, b^{p} ) ]^{1/p}=L_{p}(a,b)\) is the p-order logarithmic mean;

  • \(S_{p, p}(a,b)= [I (a^{p}, b^{p} ) ]^{1/p}=I_{p}(a,b)\) is the p-order identric mean.

The complete elliptic integral \(\mathcal{E}(r)\) of the second kind can be expressed as
$$ \mathcal{E}(r)=\frac{\pi}{2}F \biggl(-\frac{1}{2}, \frac{1}{2};1;r^{2} \biggr) =\frac{\pi}{2}\sum _{n=0}^{\infty}\frac{ (-{1}/{2} )_{n} ({1}/{2} )_{n}}{(n!)^{2}}r^{2n}, $$
where
$$ F(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}} \frac{x^{n}}{n!}\quad (-1< x< 1) $$
is the Gaussian hypergeometric function, \((a)_{n}=\Gamma(a+n)/\Gamma(a)\) and \(\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\,dt\) (\(x>0\)) is the gamma function. It is well known that \(\mathcal{E}(r)\) is strictly decreasing on \((0, 1)\) and satisfies
$$ \mathcal{E}\bigl(0^{+}\bigr)=\frac{\pi}{2},\qquad \mathcal{E} \bigl(1^{-}\bigr)=1. $$

Recently, the bounds for the complete elliptic integral \(\mathcal{E}(r)\) of the second kind have attracted the interest of many researchers. In particular, many remarkable inequalities for \(\mathcal{E}(r)\) can be found in the literature [311].

Let \(r\in(0, 1)\) and \(r^{\prime}=\sqrt{1-r^{2}}\). Then making use of (1.1) and (1.2) together with the power series formulas we have
$$\begin{aligned}& \frac{2}{\pi}\mathcal{E}(r)-S_{p, q} \bigl(1, r^{\prime} \bigr) \\& \quad = -\frac{p+q-9/2}{96}r^{4}-\frac{p+q-9/2}{128}r^{6} \\& \qquad {} +\frac{8(p+q) (2p^{2}+2q^{2}-5p-5q-550 )+19\text{,}845}{45\times 2^{14}}r^{8}+o \bigl(r^{8} \bigr). \end{aligned}$$
(1.3)
Let \(p+q=9/2\), then (1.3) becomes
$$ \frac{2}{\pi}\mathcal{E}(r)-S_{9/2-p, p} \bigl(1, r^{\prime} \bigr)=\frac{(4p-7)(4p-11)}{5\times 2^{14}}r^{8}+o \bigl(r^{8} \bigr). $$
(1.4)
Motivated by equation (1.4), we discuss the monotonicity of the function \(r\mapsto \mathcal{E}(r)/S_{9/2-p, p}(1, r^{\prime})\) for certain \(p\in\mathbb{R}\), and present several new bounds for the complete elliptic integral of the second kind \(\mathcal{E}(r)\) and the Toader mean
$${T}(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}t+b^{2}\sin^{2}t}\,dt. $$

2 Lemmas

In order to prove our main results we need several lemmas, which we present in this section. Out of the next eight lemmas five were taken from other papers and the last three are due to the authors.

Lemma 2.1

(See [12], Theorem 2)

Let \(r\in(0, 1)\) and \(r^{\prime}=\sqrt{1-r^{2}}\). Then the function
$$ F(r)=\frac{1-{2\mathcal{E}(r)}/{\pi}}{1-S_{5/2, 2}(1, r^{\prime})} $$
is strictly increasing from \((0, 1)\) onto \((1, 25(\pi-2)/(9\pi))\).

Lemma 2.2

(See [13], Theorem 3.1)

Let \(r\in(0, 1)\) and \(r^{\prime}=\sqrt{1-r^{2}}\). Then the function
$$ G(r)=\frac{1-{2\mathcal{E}(r)}/{\pi}}{1-S_{11/4, 7/4}(1, r^{\prime})} $$
is strictly decreasing from \((0, 1)\) onto \((11(\pi-2)/(4\pi), 1)\).

Lemma 2.3

(See [14], equation (3.14), [15], Corollary 1.1)

Let \(c>0\) and \(a, b>0\) with \(a\neq b\). Then the function \(p\mapsto S_{2c-p, p}(a,b)\) is strictly increasing on \((-\infty, c]\) and strictly decreasing on \([c, \infty)\).

Lemma 2.4

(See [15], Corollary 1.2)

Let \(c>0\), \(p\in(0, 2c)\), \(a, b>0\) with \(a\neq b\) and \(\theta(p, c)\) be defined by
$$ \theta(p, c)=\lim_{r\rightarrow0^{+}}S_{2c-p, p}(1, r)= \textstyle\begin{cases} (\frac{2c-p}{p} )^{1/(2p-2c)},& p\neq c, \\ e^{-1/c},& p=c. \end{cases} $$
(2.1)
Then the function \(p\mapsto S_{2c-p, p}(a,b)/\theta(p, c)\) is strictly decreasing on \((0, c]\) and strictly increasing on \([c, 2c)\).

Lemma 2.5

(See [16], Theorem 5)

Let \(c>0\) and \(0< x< y< z\). Then the function \(p\mapsto S_{2c-p, p}(x,y)/S_{2c-p, p}(x,z)\) is strictly decreasing on \((-\infty, c]\) and strictly increasing on \([c, \infty)\).

Lemma 2.6

Let \(r\in(0, 1)\) and \(r^{\prime}=\sqrt{1-r^{2}}\). Then the function
$$ r\mapsto\frac{{2\mathcal{E}(r)}/{\pi}}{S_{11/4, 7/4}(1, r^{\prime})} $$
is strictly increasing from \((0, 1)\) onto \((1, 22/7\pi)\).

Proof

Let \(G(r)\) be defined by Lemma 2.2. Then we clearly see that
$$ \frac{{2\mathcal{E}(r)}/{\pi}}{S_{11/4, 7/4}(1, r^{\prime})}=1+\frac{[1-G(r)][1-S_{11/4, 7/4}(1, r^{\prime})]}{S_{11/4, 7/4}(1, r^{\prime})}, $$
(2.2)
and both the functions \(r\mapsto1-S_{11/4, 7/4}(1, r^{\prime})\) and \(r\mapsto1/S_{11/4, 7/4}(1, r^{\prime})\) are positive and strictly increasing on \((0, 1)\). It follows from (1.1) and (1.2) together with Lemma 2.2 that
$$ \lim_{r\rightarrow0^{+}}\frac{{2\mathcal{E}(r)}/{\pi}}{S_{11/4, 7/4}(1, r^{\prime})}=1,\qquad \lim _{r\rightarrow 1^{-}}\frac{{2\mathcal{E}(r)}/{\pi}}{S_{11/4, 7/4}(1, r^{\prime})}=\frac{22}{7\pi}, $$
(2.3)
and the function \(r\mapsto1-G(r)\) is also positive and strictly increasing on \((0, 1)\).

Therefore, Lemma 2.6 follows from (2.2) and (2.3) together with the monotonicity and positivity of the functions \(r\mapsto1-S_{11/4, 7/4}(1, r^{\prime})\), \(r\mapsto1/S_{11/4, 7/4}(1, r^{\prime})\), and \(r\mapsto1-G(r)\) on \((0, 1)\). □

Lemma 2.7

Let \(r\in(0, 1)\) and \(r^{\prime}=\sqrt{1-r^{2}}\). Then the function
$$ r\mapsto\frac{{2\mathcal{E}(r)}/{\pi}}{S_{5/2, 2}(1, r^{\prime})} $$
is strictly decreasing from \((0, 1)\) onto \((25/8\pi, 1)\).

Proof

Let \(F(r)\) be defined by Lemma 2.1. Then from (1.1) and (1.2) together with Lemma 2.1 we clearly see that
$$\begin{aligned}& \lim_{r\rightarrow0^{+}}\frac{{2\mathcal{E}(r)}/{\pi}}{S_{5/2, 2}(1, r^{\prime})}=1, \qquad \lim _{r\rightarrow 1^{-}}\frac{{2\mathcal{E}(r)}/{\pi}}{S_{5/2, 2}(1, r^{\prime})}=\frac{25}{8\pi}, \end{aligned}$$
(2.4)
$$\begin{aligned}& \frac{{2\mathcal{E}(r)}/{\pi}}{S_{5/2, 2}(1, r^{\prime})}=1-\frac{[F(r)-1][1-S_{5/2, 2}(1, r^{\prime})]}{S_{5/2, 2}(1, r^{\prime})}, \end{aligned}$$
(2.5)
and all the functions \(r\mapsto F(r)-1\), \(r\mapsto1-S_{5/2, 2}(1, r^{\prime})\), and \(r\mapsto1/S_{5/2, 2}(1, r^{\prime})\) are positive and strictly increasing on \((0, 1)\).

Therefore, Lemma 2.7 follows easily from (2.4) and (2.5) together with the monotonicity and positivity of the functions \(r\mapsto F(r)-1\), \(r\mapsto1-S_{5/2, 2}(1, r^{\prime})\), and \(r\mapsto 1/S_{5/2, 2}(1, r^{\prime})\) on \((0, 1)\). □

Lemma 2.8

Let \(c>0\), \(r\in(0, 1)\), and \(r^{\prime}=\sqrt {1-r^{2}}\). Then the function
$$ r\mapsto\frac{S_{2c-p_{0}, p_{0}}(1, r^{\prime})}{S_{2c-p,p}(1, r^{\prime})} $$
is strictly increasing (decreasing) on \((0, 1)\) if \(p< p_{0}\leq c\) (\(p_{0}< p\leq c\)).

Proof

We clearly see that it suffices to prove that the function
$$ r^{\prime}\mapsto\frac{S_{2c-p_{0}, p_{0}}(1, r^{\prime})}{S_{2c-p,p}(1, r^{\prime})} $$
is strictly decreasing (increasing) on \((0, 1)\) if \(p< p_{0}\leq c\) (\(p_{0}< p\leq c\)).
Let \(r_{1}^{\prime}, r_{2}^{\prime}\in(0, 1)\) with \(r_{1}^{\prime}< r_{2}^{\prime}\). Then \(1<1/r_{2}^{\prime}<1/r_{1}^{\prime}\) and Lemma 2.5 leads to
$$ \frac{S_{2c-p_{0}, p_{0}}(1, 1/r_{2}^{\prime})}{S_{2c-p_{0},p_{0}}(1, 1/r_{1}^{\prime})}< (>)\, \frac{S_{2c-p, p}(1, 1/r_{2}^{\prime})}{S_{2c-p,p}(1, 1/r_{1}^{\prime})} $$
or
$$ \frac{S_{2c-p_{0}, p_{0}}(1, 1/r_{2}^{\prime})}{S_{2c-p,p}(1, 1/r_{2}^{\prime})}< (>)\, \frac{S_{2c-p_{0}, p_{0}}(1, 1/r_{1}^{\prime})}{S_{2c-p,p}(1, 1/r_{1}^{\prime})} $$
(2.6)
if \(p< p_{0}\leq c\) (\(p_{0}< p\leq c\)). The homogeneity of degree 1 for the Stolarsky mean and (2.6) give the desired result. □

3 Main results

Theorem 3.1

Let \(p\in(-\infty, 9/4]\), \(r\in(0, 1)\), \(r^{\prime}=\sqrt{1-r^{2}}\), and
$$ R_{p}(r)=\frac{{2\mathcal{E}(r)}/{\pi}}{S_{9/2-p, p}(1, r^{\prime})}. $$
(3.1)
Then we have
  1. (1)

    the function \(r\mapsto R_{p}(r)\) is strictly increasing on \((0, 1)\) if and only if \(p\in(-\infty, 7/4]\);

     
  2. (2)

    the function \(r\mapsto R_{p}(r)\) is strictly decreasing on \((0, 1)\) if \(p\in[2, 9/4]\).

     

Proof

(1) If the function \(r\mapsto R_{p}(r)\) is strictly increasing on \((0, 1)\), then we clearly see that
$$ \lim_{r\rightarrow0^{+}} \frac{{d(\log R_{p}(r))}/{dr}}{8r^{7}}\geq0. $$
(3.2)
From (1.1), (1.2), (1.4), and (3.1) one has
$$\begin{aligned}& \lim_{r\rightarrow0^{+}}S_{9/2-p, p}\bigl(1, r^{\prime}\bigr)=\lim_{r\rightarrow 0^{+}}\frac{2}{\pi} \mathcal{E}(r)=\lim_{r\rightarrow 0^{+}}R_{p}(r)=1, \end{aligned}$$
(3.3)
$$\begin{aligned}& \frac{2}{\pi}\mathcal{E}(r)-S_{9/2-p, p}\bigl(1, r^{\prime}\bigr)= \frac{ (p-{7}/{4} ) (p-{11}/{4} )}{5\times 2^{10}}r^{8}+o\bigl(r^{8}\bigr). \end{aligned}$$
(3.4)
It follows from (3.1), (3.3), (3.4), and L’Hôspital’s rule that
$$\begin{aligned}& \lim_{r\rightarrow0^{+}}\frac{{d(\log R_{p}(r))}/{dr}}{{d ({2\mathcal{E}(r)}/{\pi}-S_{9/2-p, p}(1, r^{\prime}) )}/{dr}} \\& \quad =\lim_{r\rightarrow0^{+}}\frac{\log R_{p}(r)}{{2\mathcal{E}(r)}/{\pi}-S_{9/2-p, p}(1, r^{\prime})} \\& \quad =\lim_{r\rightarrow0^{+}}\frac{\log R_{p}(r)}{(R_{p}(r)-1)S_{9/2-p, p}(1, r^{\prime})}=1, \end{aligned}$$
(3.5)
$$\begin{aligned}& \lim_{r\rightarrow 0^{+}}\frac{{d ({2\mathcal{E}(r)}/{\pi}-S_{9/2-p, p}(1, r^{\prime}) )}/{dr}}{8r^{7}} \\& \quad =\lim_{r\rightarrow 0^{+}}\frac{{2\mathcal{E}(r)}/{\pi}-S_{9/2-p,p}(1,r')}{r^{8}} \\& \quad =\frac{ (p-{7}/{4} ) (p-{11}/{4} )}{5\times 2^{10}}. \end{aligned}$$
(3.6)
Note that
$$\begin{aligned} \frac{{d(\log R_{p}(r))}/{dr}}{8r^{7}} =&\frac{{d(\log R_{p}(r))}/{dr}}{{d ({2\mathcal{E}(r)}/{\pi}-S_{9/2-p, p}(1, r^{\prime}) )}/{dr}} \\ &{}\times\frac{{d ({2\mathcal{E}(r)}/{\pi}-S_{9/2-p, p}(1, r^{\prime}) )}/{dr}}{8r^{7}}. \end{aligned}$$
(3.7)
It follows from (3.2), (3.5)-(3.7) that
$$ \biggl(p-\frac{7}{4} \biggr) \biggl(p-\frac{11}{4} \biggr)\geq0. $$
(3.8)

Therefore, \(p\in(-\infty, 7/4]\) follows from \(p\in(-\infty, 9/4]\) and (3.8).

Next, we prove that the function \(r\mapsto R_{p}(r)\) is strictly increasing on \((0, 1)\) if \(p\in(-\infty, 7/4]\). We divide the proof into two cases.

Case 1: \(p=7/4\). Then the desired result follows directly from Lemma 2.6.

Case 2: \(p<7/4\). Then \(R_{p}(r)\) can be rewritten as
$$\begin{aligned} R_{p}(r)&=\frac{{2\mathcal{E}(r)}/{\pi}}{S_{9/2-7/4, 7/4}(1, r^{\prime})}\times\frac{S_{9/2-7/4, 7/4}(1, r^{\prime})}{S_{9/2-p, p}(1, r^{\prime})} \\ &=\frac{{2\mathcal{E}(r)}/{\pi}}{S_{11/4, 7/4}(1, r^{\prime})}\times\frac{S_{9/2-7/4, 7/4}(1, r^{\prime})}{S_{9/2-p, p}(1, r^{\prime})}. \end{aligned}$$
(3.9)

Therefore, the function \(r\mapsto R_{p}(r)\) is strictly increasing on \((0, 1)\) follows from Lemmas 2.6 and 2.8 together with (3.9).

(2) If \(p\in[2, 9/4]\), then we divide the proof into two cases.

Case 1: \(p=2\). Then the desired result follows directly from Lemma 2.7.

Case 2: \(p\in(2, 9/4]\). Then \(R_{p}(r)\) can be expressed as
$$\begin{aligned} R_{p}(r)&=\frac{{2\mathcal{E}(r)}/{\pi}}{S_{9/2-2, 2}(1, r^{\prime})}\times\frac{S_{9/2-2, 2}(1, r^{\prime})}{S_{9/2-p, p}(1, r^{\prime})} \\ &=\frac{{2\mathcal{E}(r)}/{\pi}}{S_{5/2, 2}(1, r^{\prime})}\times\frac{S_{9/2-2, 2}(1, r^{\prime})}{S_{9/2-p, p}(1, r^{\prime})}. \end{aligned}$$
(3.10)

Therefore, the function \(r\mapsto R_{p}(r)\) is strictly decreasing on \((0, 1)\) follows from Lemmas 2.7 and 2.8 together with (3.10). □

From (1.1), (1.2), and Theorem 3.1 we get Corollary 3.2 immediately.

Corollary 3.2

Let \(r\in(0, 1)\), \(r^{\prime}=\sqrt{1-r^{2}}\), and \(\theta(p, c)\) be defined by (2.1). Then the double inequality
$$ S_{9/2-p, p}\bigl(1, r^{\prime}\bigr)< \frac{2}{\pi}\mathcal{E}(r)< \frac{2}{\pi \theta(p, 9/4)}S_{9/2-p, p}\bigl(1, r^{\prime}\bigr) $$
holds for all \(r\in(0, 1)\) and \(p\in(0, 7/4]\), and the double inequality
$$ \frac{2}{\pi\theta(p, 9/4)}S_{9/2-p, p}\bigl(1, r^{\prime}\bigr)< \frac{2}{\pi}\mathcal{E}(r)< S_{9/2-p, p}\bigl(1, r^{\prime}\bigr) $$
holds for all \(r\in(0, 1)\) and \(p\in[2, 9/4]\).

Letting \(p=9/8, 3/2, 7/4; 2, 9/4\) and making use of (2.1), Lemmas 2.3 and 2.4, then Corollary 3.2 leads to Corollary 3.3.

Corollary 3.3

Let \(r\in(0, 1)\) and \(r^{\prime}=\sqrt{1-r^{2}}\). Then the inequalities
$$\begin{aligned} \operatorname{He}_{9/4}\bigl(1, r^{\prime}\bigr)&< A_{3/2} \bigl(1, r^{\prime}\bigr)< S_{11/4, 7/4}\bigl(1, r^{\prime}\bigr)< \frac{2}{\pi}\mathcal{E}(r) \\ &< \frac{22}{7\pi}S_{11/4, 7/4}\bigl(1, r^{\prime}\bigr)< \frac{2^{5/3}}{\pi}A_{3/2}\bigl(1, r^{\prime}\bigr)< \frac{2\times 3^{4/9}}{\pi}\operatorname{He}_{9/4}\bigl(1, r^{\prime}\bigr) \end{aligned}$$
and
$$ \frac{2e^{4/9}}{\pi}I_{9/4}\bigl(1, r^{\prime}\bigr)< \frac{25}{8\pi}S_{5/2, 2}\bigl(1, r^{\prime}\bigr)< \frac{2}{\pi}\mathcal{E}(r)< S_{5/2, 2}\bigl(1, r^{\prime} \bigr)< I_{9/4}\bigl(1, r^{\prime}\bigr) $$
hold for all \(r\in(0, 1)\).

Corollary 3.4

Let \(p\in(-\infty, 9/4]\), \(r\in(0, 1)\) and \(r^{\prime}=\sqrt{1-r^{2}}\). Then the inequality
$$ \frac{2}{\pi}\mathcal{E}(r)>S_{9/2-p, p}\bigl(1, r^{\prime}\bigr) $$
(3.11)
holds for all \(r\in(0, 1)\) if and only if \(p\in(-\infty, 7/4]\), and inequality (3.11) is reversed if \(p\in[2, 9/4]\).

Proof

From Lemma 2.3 and Corollary 3.2 we clearly see that inequality (3.11) holds for all \(r\in(0, 1)\) if \(p\in(-\infty, 7/4]\), and inequality (3.11) is reversed if \(p\in[2, 9/4]\).

If inequality (3.11) holds for all \(r\in(0, 1)\), then (3.4) leads to the conclusion that \(p\in(-\infty, 7/4]\). □

Let \(p=-9/2, -9/4, 0, 9/8, 3/2, 7/4; 2, 9/4\). Then Lemma 2.3 and Corollary 3.4 lead to Corollary 3.5.

Corollary 3.5

Let \(r\in(0, 1)\) and \(r^{\prime}=\sqrt{1-r^{2}}\). Then the inequalities
$$\begin{aligned} A_{9/2}^{1/3}\bigl(1, r^{\prime}\bigr)G^{2/3} \bigl(1, r^{\prime}\bigr)&< G^{1/2}\bigl(1, r^{\prime}\bigr) \operatorname{He}_{9/2}^{1/2}\bigl(1, r^{\prime} \bigr)< L_{9/2}\bigl(1, r^{\prime}\bigr)< \operatorname{He}_{9/4} \bigl(1, r^{\prime}\bigr) \\ & < A_{3/2}\bigl(1,r^{\prime}\bigr)< S_{11/4, 7/4}\bigl(1, r^{\prime}\bigr)< \frac{2}{\pi}\mathcal{E}(r)< S_{5/2, 2}\bigl(1, r^{\prime}\bigr)< I_{9/4}\bigl(1, r^{\prime}\bigr) \end{aligned}$$
hold for all \(r\in(0, 1)\), where \(G(a,b)=\sqrt{ab}\) is the geometric mean of a and b.
The Toader mean \({T}(a,b)\) [17] of two positive real numbers a and b is defined by
$$ {T}(a,b)=\frac{2}{\pi} \int_{0}^{\pi/2}\sqrt{a^{2} \cos^{2}t+b^{2}\sin^{2}t}\,dt = \textstyle\begin{cases} \frac{2}{\pi}\mathcal{E} (\sqrt{1- (\frac{b}{a} )^{2}} ), &a>b, \\ \frac{2}{\pi}\mathcal{E} (\sqrt{1- (\frac{a}{b} )^{2}} ), &a< b, \\ a, &a=b. \end{cases} $$
(3.12)

From (3.12) we clearly see that all the results given in Corollaries 3.2-3.5 can be restated by the Toader mean \({T}(a, b)\).

Remark 3.6

Let \(\theta(p, c)\) be defined by (2.1). Then the double inequality
$$ S_{9/2-p, p}(a,b)< {T}(a,b)< \frac{2}{\pi\theta(p, 9/4)}S_{9/2-p, p}(a,b) $$
holds for all \(a, b>0\) with \(a\neq b\) and \(p\in(0, 7/4]\), and the double inequality
$$ \frac{2}{\pi\theta(p, 9/4)}S_{9/2-p, p}(a,b)< {T}(a,b)< S_{9/2-p, p}(a,b) $$
holds for all \(a, b>0\) with \(a\neq b\) and \(p\in[2, 9/4]\).

Remark 3.7

The inequalities
$$\begin{aligned}& \operatorname{He}_{9/4}(a,b)< A_{3/2}(a,b)< S_{11/4, 7/4}(a,b)< {T}(a,b) \\& \hphantom{\operatorname{He}_{9/4}(a,b)}< \frac{22}{7\pi}S_{11/4, 7/4}(a,b)< \frac{2^{5/3}}{\pi}A_{3/2}(a,b)< \frac{2\times 3^{4/9}}{\pi}\operatorname{He}_{9/4}(a,b), \\& \frac{2e^{4/9}}{\pi}I_{9/4}(a,b)< \frac{25}{8\pi}S_{5/2, 2}(a,b)< {T}(a,b)< S_{5/2, 2}(a,b)< I_{9/4}(a,b), \\& A_{9/2}^{1/3}(a, b)G^{2/3}(a, b)< G^{1/2}(a, b)\operatorname{He}_{9/2}^{1/2}(a, b)< L_{9/2}(a, b)< \operatorname{He}_{9/4}(a, b) \\& \hphantom{A_{9/2}^{1/3}(a, b)G^{2/3}(a, b)}< A_{3/2}(a, b)< S_{11/4, 7/4}(a, b)< {T}(a, b)< S_{5/2, 2}(a, b)< I_{9/4}(a, b), \\& L_{3/2}(a, b)< \operatorname{He}_{3/4}(a, b)< A_{1/2}(a, b)< S_{11/12, 7/12}(a,b) \\& \hphantom{L_{3/2}(a, b)}< {T}_{1/3}(a, b)< S_{5/6, 2/3}(a, b)< I_{3/4}(a, b) \end{aligned}$$
(3.13)
hold for all \(a, b>0\) with \(a\neq b\), where \(T_{1/3}(a,b)=T^{3}(a^{1/3}, b^{1/3})\).

Remark 3.8

Let \(p\in(-\infty, 9/4]\). Then the inequality
$$ {T}(a,b)>S_{9/2-p, p}(a,b) $$
(3.14)
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(p\in(-\infty, 7/4]\), and inequality (3.14) is reversed if \(p\in[2, 9/4]\).
The Toader-Qi mean \(\operatorname{TQ}(a,b)\) [17, 18] and Gauss arithmetic-geometric mean \(\operatorname{AGM}(a,b)\) [19] of two positive real numbers a and b are, respectively, given by
$$ \operatorname{TQ}(a,b)=\frac{2}{\pi} \int_{0}^{\pi/2}a^{\cos^{2}\theta}b^{\sin^{2}\theta }\, d \theta $$
and
$$ \operatorname{AGM}(a,b)=\lim_{n\rightarrow\infty}a_{n}=\lim _{n\rightarrow \infty}b_{n}, $$
where \(a_{n}\) and \(b_{n}\) (\(n\geq1\)) are defined by
$$ a_{1}=a, \qquad b_{1}=b, \qquad a_{n+1}= \frac{a_{n}+b_{n}}{2},\qquad b_{n+1}=\sqrt{a_{n}b_{n}}. $$
For all \(a, b>0\) with \(a\neq b\), Yang et al. [20] proved that
$$\begin{aligned} \operatorname{AGM}(a,b)&< L^{3/4}(a,b)A^{1/4}(a,b)< L_{3/2}(a,b)< \sqrt {L(a,b)I(a,b)} \\ & < \operatorname{He}_{3/4}(a,b)< A_{1/2}(a,b)< I_{3/4}(a,b), \end{aligned}$$
(3.15)
and the following inequalities can be found in Remark 3.6 of [21]:
$$\begin{aligned} L(a,b)&< \operatorname{AGM}(a,b)< L^{3/4}(a,b)A^{1/4}(a,b)< \operatorname{TQ}(a,b) \\ & < A_{1/2}(a,b)< {T}_{1/3}(a,b)< I_{3/4}(a,b). \end{aligned}$$
(3.16)
In [22, 23], the authors proved that the double inequality
$$ L_{3/2}(a,b)< \operatorname{TQ}(a,b)< \sqrt{L(a,b)I(a,b)} $$
(3.17)
holds for all \(a, b>0\) with \(a\neq b\).

Remark 3.9

It follows from (3.13) and (3.15)-(3.17) that
$$\begin{aligned} L(a,b)&< \operatorname{AGM}(a,b)< L^{3/4}(a,b)A^{1/4}(a,b)< L_{3/2}(a,b) \\ &< \operatorname{TQ}(a,b)< \sqrt{L(a,b)I(a,b)}< \operatorname{He}_{3/4}(a,b)< A_{1/2}(a,b) \\ &< S_{11/12,7/12}(a,b)< {T}_{1/3}(a,b)< S_{5/6, 2/3}(a,b)< I_{3/4}(a,b) \end{aligned}$$
for all \(a, b>0\) with \(a\neq b\).

Remark 3.10

Unfortunately, in the article we cannot present the monotonicity conclusion of the function \(r\rightarrow R_{p}(r)\) given by (3.1) on the interval \((0, \infty)\) for \(p\in(7/4, 2)\cup(9/4, \infty)\), we leave it as an open problem to the reader.

Declarations

Acknowledgements

The research was supported by the Natural Science Foundation of China under Grants 61374086, 11371125, and 11401191.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China
(2)
Albert Einstein College of Medicine, Yeshiva University, New York, NY, 10033, United States

References

  1. Stolarsky, KB: Generalizations of the logarithmic mean. Math. Mag. 48, 87-92 (1975) View ArticleMATHMathSciNetGoogle Scholar
  2. Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington (1964) MATHGoogle Scholar
  3. Vuorinen, M: Hypergeometric functions in geometric function theory. In: Special Functions and Differential Equations (Madras, 1997), pp. 119-126. Allied Publ., New Delhi (1998) Google Scholar
  4. Barnard, RW, Pearce, K, Richards, KC: An inequality involving the generalized hypergeometric function and the arc length of an ellipse. SIAM J. Math. Anal. 31(3), 693-699 (2000) View ArticleMATHMathSciNetGoogle Scholar
  5. Barnard, RW, Pearce, K, Richards, KC: A monotonicity property involving \({}_{3}F_{2}\) and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32(2), 403-419 (2000) View ArticleMATHMathSciNetGoogle Scholar
  6. Alzer, H, Qiu, S-L: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172(2), 289-312 (2004) View ArticleMATHMathSciNetGoogle Scholar
  7. Chu, Y-M, Wang, M-K, Qiu, S-L, Jiang, Y-P: Bounds for complete elliptic integrals of the second kind with applications. Comput. Math. Appl. 63(2), 1177-1184 (2012) View ArticleMATHMathSciNetGoogle Scholar
  8. Chu, Y-M, Wang, M-K: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3-4), 223-229 (2012) View ArticleMATHMathSciNetGoogle Scholar
  9. Wang, M-K, Chu, Y-M, Qiu, S-L, Jiang, Y-P: Bounds for the perimeter of an ellipse. J. Approx. Theory 164(7), 928-937 (2012) View ArticleMATHMathSciNetGoogle Scholar
  10. Wang, M-K, Chu, Y-M: Asymptotical bounds for complete elliptic integrals of the second kind. J. Math. Anal. Appl. 402(1), 119-126 (2013) View ArticleMATHMathSciNetGoogle Scholar
  11. Hua, Y, Qi, F: The best bounds for Toader mean in terms of the centroidal and arithmetic means. Filomat 28(4), 775-780 (2014) View ArticleMathSciNetGoogle Scholar
  12. Yang, Z-H: Very accurate approximations for the elliptic integrals of the second kind in terms of Stolarksy means. arXiv:1508.05513 [math.CA]
  13. Yang, Z-H, Chu, Y-M, Zhang, W: Accurate approximation for the complete elliptic integral of the second kind. J. Math. Anal. Appl. 438(2), 875-888 (2016) View ArticleMATHMathSciNetGoogle Scholar
  14. Leach, EB, Sholander, MC: Extended mean values. Am. Math. Mon. 85(2), 84-90 (1978) View ArticleMATHMathSciNetGoogle Scholar
  15. Yang, Z-H: The log-convexity of another class of one-parameter means and its applications. Bull. Korean Math. Soc. 49(1), 33-47 (2012) View ArticleMATHMathSciNetGoogle Scholar
  16. Losonczi, L: Ratio of Stolarsky means: monotonicity and comparison. Publ. Math. (Debr.) 75(1-2), 221-238 (2009) MATHMathSciNetGoogle Scholar
  17. Toader, G: Some mean values related to the arithmetic-geometric mean. J. Math. Anal. Appl. 218(2), 358-368 (1998) View ArticleMATHMathSciNetGoogle Scholar
  18. Qi, F, Shi, X-T, Liu, F-F, Yang, Z-H: A double inequality for an integral mean in terms of the exponential and logarithmic means. ResearchGate (2015). doi:10.13140/RG.2.1.2353.6800 Google Scholar
  19. Borwein, JM, Borwein, PB: Pi and the AGM. Wiley, New York (1987) MATHGoogle Scholar
  20. Yang, Z-H, Song, Y-Q, Chu, Y-M: Sharp bounds for the arithmetic-geometric mean. J. Inequal. Appl. 2014, Article ID 192 (2014) View ArticleMATHMathSciNetGoogle Scholar
  21. Yang, Z-H, Chu, Y-M: On approximating the modified Bessel function of the first kind and Toader-Qi mean. J. Inequal. Appl. 2016, Article ID 40 (2016) View ArticleMATHMathSciNetGoogle Scholar
  22. Yang, Z-H, Chu, Y-M, Song, Y-Q: Sharp bounds for Toader-Qi mean in terms of logarithmic and identric means. Math. Inequal. Appl. 19(2), 721-730 (2016) MATHMathSciNetGoogle Scholar
  23. Yang, Z-H, Chu, Y-M: A sharp lower bounds for Toader-Qi mean with applications. J. Funct. Spaces 2016, Article ID 4165601 (2016) MATHMathSciNetGoogle Scholar

Copyright

© Yang et al. 2016

Advertisement