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Optimal bounds for Seiffert-like elliptic integral mean by harmonic, geometric, and arithmetic means

Abstract

In this article, we present the optimal bounds for a special elliptic integral mean in terms of the harmonic combinations of harmonic, geometric, and arithmetic means. As consequences, several new bounds for the complete elliptic integral of the second kind are discovered, which are the improvements of many previously known results.

Introduction

For \(r\in (0,1)\), Legendre’s complete elliptic integrals of the first kind \(\mathcal{K}(r)\) and second kind \(\mathcal{E}(r)\) [18] are defined by

$$ \mathcal{K}=\mathcal{K}(r)= \int _{0}^{\pi /2} \bigl(1-r^{2}\sin ^{2} \theta \bigr)^{-1/2}\,d\theta $$

and

$$ \mathcal{E}=\mathcal{E}(r)= \int _{0}^{\pi /2} \bigl(1-r^{2}\sin ^{2} \theta \bigr)^{1/2}\,d\theta , $$

respectively.

It is well known that \(\mathcal{K}(r)\) is strictly increasing from \((0,1)\) onto \((\pi /2,\infty )\) and \(\mathcal{E}(r)\) is strictly decreasing from \((0,1)\) onto \((1,\pi /2)\), they satisfy the derivative formulas

$$\begin{aligned}& \frac{d\mathcal{K}}{dr}=\frac{\mathcal{E}-r^{\prime 2}\mathcal{K}}{rr^{\prime 2}}, \\& \frac{d\mathcal{E}}{dr}=\frac{\mathcal{E}-\mathcal{K}}{r} \end{aligned}$$

and Landen identities

$$ \mathcal{K} \biggl(\frac{2\sqrt{r}}{1+r} \biggr)=(1+r)\mathcal{K}(r), \qquad \mathcal{E} \biggl(\frac{2\sqrt{r}}{1+r} \biggr)= \frac{2\mathcal{E}-r^{\prime 2}\mathcal{K}}{1+r}, $$

where and in what follows we denote \(r'=\sqrt{1-r^{2}}\) for \(r\in (0,1)\).

Let \(a,b>0\) with \(a\neq b\). Then the harmonic mean \(H(a,b)\), geometric mean \(G(a,b)\), arithmetic mean \(A(a,b)\), arithmetic–geometric mean \(AG(a,b)\) [911], and Toader mean \(TD(a,b)\) [1215] are given by

$$\begin{aligned}& H(a,b)=\frac{2ab}{a+b},\qquad G(a,b)=\sqrt{ab},\qquad A(a,b)= \frac{a+b}{2}, \\& AG(a,b)= \frac{\pi }{2\int _{0}^{\pi /2}(a^{2}\cos ^{2}\theta +b^{2}\sin ^{2}\theta )^{-1/2}\,d\theta }= \textstyle\begin{cases} \frac{\pi a}{2\mathcal{K}(\sqrt{1-(b/a)^{2}})}, &a>b, \\ \frac{\pi b}{2\mathcal{K}(\sqrt{1-(a/b)^{2}})},& a< b, \end{cases}\displaystyle \end{aligned}$$
(1.1)

and

$$ TD(a,b)=\frac{2}{\pi } \int _{0}^{\pi /2}\bigl(a^{2}\cos ^{2}\theta +b^{2} \sin ^{2}\theta \bigr)^{1/2}\,d\theta = \textstyle\begin{cases} \frac{2a}{\pi }\mathcal{E}(\sqrt{1-(b/a)^{2}}),& a> b, \\ \frac{2b}{\pi }\mathcal{E}(\sqrt{1-(a/b)^{2}}), &a< b. \end{cases} $$

Recently, the complete elliptic integrals \(\mathcal{K}(r)\) and \(\mathcal{E}(r)\) of the first and second kinds have attracted the attention of many researchers [1622] because they have wide applications in many branches of mathematics including the geometric function theory, differential equations, number theory, and mean value theory. For instance, the perimeter \(\mathcal{L}(a,b)\) of an ellipse with semi-axes a, b and eccentricity \(e=\sqrt{1-b^{2}/a^{2}}\) is given by

$$ \mathcal{L}(a,b)=4 \int _{0}^{\pi /2}\sqrt{a^{2}\cos ^{2}\theta +b^{2} \sin ^{2}\theta }\,d\theta =4a \mathcal{E}(e). $$
(1.2)

Many remarkable inequalities and properties for the complete elliptic integrals \(\mathcal{K}(r)\) and \(\mathcal{E}(r)\) can be found in the literature [2331]. Barnard et al. [32] and Alzer and Qiu [33] proved that \(\lambda =3/2\) and \(\mu =\log 2/\log (\pi /2)\) are the best possible constants such that the double inequality

$$ \frac{\pi }{2} \biggl(\frac{1+r^{\prime \lambda }}{2} \biggr)^{1/\lambda }< \mathcal{E}(r)< \frac{\pi }{2} \biggl(\frac{1+r^{\prime \mu }}{2} \biggr)^{1/\mu } $$
(1.3)

holds for all \(r\in (0,1)\).

Later, Wang and Chu [34] improved the lower bound of (1.3) and proved that the double inequality

$$\begin{aligned}& \frac{\pi [(\alpha +(1-\alpha )r')^{2}+(1-\alpha +\alpha r')^{2} ]^{2}}{(1+r')^{3}}< \mathcal{E}(r) \\& \quad < \frac{\pi }{2} \biggl[ \frac{(\beta +(1-\beta )r')^{2}+(1-\beta +\beta r')^{2}}{2} \biggr]^{1/2} \end{aligned}$$

holds for all \(r\in (0,1)\) with the best possible constants \(\alpha =\frac{4+\sqrt{2}}{8}\) and \(\beta =\frac{1+\sqrt{(4/\pi )-1}}{2}\).

Very recently, Yang et al. [35] found the high accuracy asymptotic bounds for \(\mathcal{E}(r)\) and proved that

$$ \frac{\pi }{2}J\bigl(r'\bigr)- \biggl(\frac{51\pi }{160}-1 \biggr)r^{16}< \mathcal{E}(r)< \frac{\pi }{2}J\bigl(r' \bigr)-\frac{5\pi }{3\times 2^{31}}r^{16} $$

for all \(r\in (0,1)\), where

$$ J(r)=\frac{51r^{2}+20r\sqrt{r}+50r+20\sqrt{r}+51}{16(5r+2\sqrt{r}+5)}. $$

The following Seiffert-like elliptic integral mean

$$\begin{aligned} V(a,b)&= \frac{\pi H(a,b)}{2\mathcal{E} (\frac{ \vert a-b \vert }{a+b} )} = \frac{\pi H(a,b)}{2\mathcal{E} (\sqrt{1-\frac{G^{2}(a,b)}{A^{2}(a,b)}} )} \\ &= \frac{\pi G^{2}(a,b)}{2\int _{0}^{\pi /2}\sqrt{A^{2}(a,b)\cos ^{2}\theta +G^{2} \sin ^{2}\theta }\,d\theta } \end{aligned}$$
(1.4)

was introduced by Witkowski in [36], in which Witkowski investigated the so-called Seiffert-like means

$$ M_{f}(a,b)= \textstyle\begin{cases} \frac{ \vert a-b \vert }{2f(\frac{ \vert a-b \vert }{a+b})}, & a\neq b, \\ a,& a=b, \end{cases} $$

where the function \(f:(0,1)\mapsto \mathbb{R}\) (called Seiffert function) satisfies the double inequality

$$ \frac{x}{1+x}\leq f(x)\leq \frac{x}{1-x}. $$

From (1.3) we clearly see that

$$ r'< \frac{2}{\pi }\mathcal{E}(r)< 1 $$

for \(r\in (0,1)\), which in conjunction with (1.4) gives

$$ H(a,b)< V(a,b)< G(a,b)< A(a,b) $$
(1.5)

for all \(a,b>0\) with \(a\neq b\).

Inspired by (1.5), the main purpose of the article is to find the optimal bounds for \(V(a,b)\) in terms of the harmonic combinations of \(H(a,b)\) and \(G(a,b)\) (or \(H(a,b)\) and \(A(a,b)\)). Our main results are the following Theorems 1.1 and 1.2.

Theorem 1.1

The double inequality

$$ \frac{\alpha _{1}}{H(a,b)}+\frac{1-\alpha _{1}}{G(a,b)}< \frac{1}{V(a,b)}< \frac{\beta _{1}}{H(a,b)}+ \frac{1-\beta _{1}}{G(a,b)} $$
(1.6)

holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha _{1}\leq 1/2\) and \(\beta _{1}\geq 2/\pi =0.6366\ldots \) .

Theorem 1.2

The double inequality

$$ \frac{\alpha _{2}}{H(a,b)}+\frac{1-\alpha _{2}}{A(a,b)}< \frac{1}{V(a,b)}< \frac{\beta _{2}}{H(a,b)}+ \frac{1-\beta _{2}}{A(a,b)} $$
(1.7)

holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha _{2}\leq 2/\pi \) and \(\beta _{2}\geq 3/4\).

To further improve and refine the lower bound in (1.6) and the upper bound in (1.7), we also establish the following Theorems 1.3 and 1.4.

Theorem 1.3

The double inequality

$$\begin{aligned}& \alpha _{3} \biggl[\frac{3}{4H(a,b)}+\frac{1}{4A(a,b)} \biggr]+(1- \alpha _{3}) \biggl[\frac{1}{2H(a,b)}+\frac{1}{2G(a,b)} \biggr]< \frac{1}{V(a,b)} \\& \quad < \beta _{3} \biggl[\frac{3}{4H(a,b)}+\frac{1}{4A(a,b)} \biggr]+(1- \beta _{3}) \biggl[\frac{1}{2H(a,b)}+\frac{1}{2G(a,b)} \biggr] \end{aligned}$$

holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha _{3}\leq 1/4\) and \(\beta _{3}\geq 2(4/\pi -1)=0.5464\ldots \) .

Theorem 1.4

The double inequality

$$\begin{aligned}& \biggl[\frac{3}{4H(a,b)}+\frac{1}{4A(a,b)} \biggr]^{\alpha _{4}} \biggl[ \frac{1}{2H(a,b)}+\frac{1}{2G(a,b)} \biggr]^{1-\alpha _{4}} \\& \quad < \frac{1}{V(a,b)} \\& \quad < \biggl[\frac{3}{4H(a,b)}+\frac{1}{4A(a,b)} \biggr]^{\beta _{4}} \biggl[ \frac{1}{2H(a,b)}+\frac{1}{2G(a,b)} \biggr]^{1-\beta _{4}} \end{aligned}$$

holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha _{4}\leq 1/4\) and \(\beta _{4}\geq [\log (4/\pi )]/\log (3/2)=0.5957\ldots \) .

Lemmas

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

Lemma 2.1

(See [1, Theorem 1.25])

Let \(-\infty < a< b<\infty \), and \(f,g:[a,b]\rightarrow \mathbb{R}\) be continuous and differentiable on \((a,b)\) such that \(f(a)=g(a)=0\) or \(f(b)=g(b)=0\). Assume that \(g'(x)\neq 0\) for each \(x\in (a,b)\). If \(f'/g'\) is (strictly) increasing (decreasing) on \((a,b)\), then so is \(f/g\).

Lemma 2.2

The functions

  1. (i)

    \(r\mapsto (\mathcal{E}-r^{\prime 2}\mathcal{K})/r^{2}\) is strictly increasing from \((0,1)\) onto \((\pi /4,1)\);

  2. (ii)

    \(r\mapsto (\mathcal{K}-\mathcal{E})/r^{2}\) is strictly increasing from \((0,1)\) onto \((\pi /4,\infty )\);

  3. (iii)

    \(r\mapsto (\mathcal{E}^{2}-r^{\prime 2}\mathcal{K}^{2})/r^{4}\) is strictly increasing from \((0,1)\) onto \((\pi ^{2}/32,1)\);

  4. (iv)

    \(r\mapsto [(1+r^{\prime 2})\mathcal{K}-2\mathcal{E})/r^{4}\) is strictly increasing from \((0,1)\) onto \((\pi /16,\infty )\);

  5. (v)

    \(r\mapsto \varrho (r)=[(1-r')(3+r')]/r^{2}\) is strictly increasing from \((0,1)\) onto \((2,3)\);

  6. (vi)

    \(r\mapsto \rho (r)=(1-r')\mathcal{E}/r^{2}\) is strictly increasing from \((0,1)\) onto \((\pi /4,1)\).

Proof

Parts (i)–(iv) can be found in [1, Theorem 3.21 (1) and Exercise 3.43 (11), (16), (29)].

For part (v), \(\varrho (r)\) can be rewritten as

$$ \varrho (r)=\frac{(1-r')(3+r')}{r^{2}}=1+\frac{2}{1+r'}, $$

which gives the monotonicity of \(\varrho (r)\). Note that \(\varrho (0^{+})=2\) and \(\varrho (1^{-})=3\).

For part (vi), differentiating \(\rho (r)\) and making use of part (iii), we get

$$ \rho '(r)=\frac{r(1-r')}{r'(\mathcal{E}+r'\mathcal{K})}\cdot \frac{\mathcal{E}^{2}-r^{\prime 2}\mathcal{K}^{2}}{r^{4}}>0. $$

This in conjunction with \(\rho (0^{+})=\pi /4\) and \(\rho (1^{-})=1\) gives the desired result. □

Lemma 2.3

The function

$$ \varphi (r)=\frac{r'[(1+r^{\prime 2})\mathcal{E}-2r^{\prime 2}\mathcal{K}]}{r^{4}} $$

is strictly decreasing from \((0,1)\) onto \((0,3\pi /16)\).

Proof

Differentiating \(\varphi (r)\) yields

$$ \varphi '(r)= \frac{(3r^{4}-11r^{2}+8)\mathcal{K}+(7r^{2}-8)\mathcal{E}}{r'r^{5}}= \frac{\varphi _{1}(r)}{r'r^{5}}, $$
(2.1)

where

$$ \varphi _{1}(r)=\bigl(3r^{4}-11r^{2}+8\bigr) \mathcal{K}+\bigl(7r^{2}-8\bigr)\mathcal{E}. $$

Simple computations lead to

$$\begin{aligned}& \varphi _{1}(0)=0, \end{aligned}$$
(2.2)
$$\begin{aligned}& \varphi _{1}^{\prime }(r)=-9r^{5} \biggl[ \frac{(1+r^{\prime 2})\mathcal{K}-2\mathcal{E}}{r^{4}} \biggr]. \end{aligned}$$
(2.3)

Therefore, Lemma 2.3 follows easily from (2.1)–(2.3) and Lemma 2.2(iv) together with \(\varphi (0^{+})=3\pi /16\) and \(\varphi (1^{-})=0\). □

Lemma 2.4

The function

$$ \phi (r)=\frac{r'[(3+r^{\prime 2})\mathcal{K}-(4+r^{2})\mathcal{E}]}{r^{4}} $$

is strictly decreasing from \((0,1)\) onto \((0,3\pi /8)\).

Proof

Let

$$\begin{aligned}& \phi _{1}(r)=\bigl(3r^{4}-16r^{2}+16\bigr) \mathcal{K}-8\bigl(2-r^{2}\bigr)\mathcal{E}, \\& \phi _{2}(r)=\bigl(8-7r^{2}\bigr)\mathcal{E}- \bigl(1-r^{2}\bigr) \bigl(8-3r^{2}\bigr)\mathcal{K}. \end{aligned}$$

Then simple computations lead to

$$\begin{aligned}& \phi _{1}(0)=\phi _{2}(0)=0, \end{aligned}$$
(2.4)
$$\begin{aligned}& \phi '(r)=-\frac{\phi _{1}(r)}{r'r^{5}}, \end{aligned}$$
(2.5)
$$\begin{aligned}& \phi '_{1}(r)=\frac{3r}{r^{\prime 2}}\phi _{2}(r), \end{aligned}$$
(2.6)
$$\begin{aligned}& \phi '_{2}(r)=9r^{5} \biggl[ \frac{(1+r^{\prime 2})\mathcal{K}-2\mathcal{E}}{r^{4}} \biggr]. \end{aligned}$$
(2.7)

Therefore, Lemma 2.4 follows easily from (2.4)–(2.7) and Lemma 2.2(iv) together with \(\phi (0^{+})=3\pi /8\) and \(\phi (1^{-})=0\). □

Proofs of Theorems 1.11.4

In this section, we assume that \(a>b>0\) because all the bivariate means \(H(a,b)\), \(G(a,b)\), \(A(a,b)\), and \(V(a,b)\) are symmetric and homogeneous of degree one.

Proof of Theorem 1.1

Let \(r=(a-b)/(a+b)\in (0,1)\). Then from (1.1) and (1.4) we obtain

$$ \begin{aligned}&H(a,b)=A(a,b) \bigl(1-r^{2}\bigr),\quad G(a,b)=A(a,b)\sqrt{1-r^{2}}, \\ &V(a,b)=A(a,b) \frac{\pi (1-r^{2})}{2\mathcal{E}}. \end{aligned}$$
(3.1)

From (3.1), inequality (1.6) can be rewritten as

$$ \frac{\frac{1}{V(a,b)}-\frac{1}{G(a,b)}}{\frac{1}{H(a,b)}-\frac{1}{G(a,b)}} = \frac{\frac{2\mathcal{E}}{\pi r^{\prime 2}}-\frac{1}{r'}}{\frac{1}{r^{\prime 2}}-\frac{1}{r'}}=1-f(r), $$
(3.2)

where

$$ f(r)=\frac{1-2\mathcal{E}/\pi }{1-r'}. $$

Let \(f_{1}(r)=1-2\mathcal{E}/\pi \) and \(f_{2}(r)=1-r'\). Then we clearly see that \(f(r)=f_{1}(r)/f_{2}(r)\) and \(f_{1}(0)=f_{2}(0)=0\), and simple computations lead to

$$\begin{aligned}& \frac{f'_{1}(r)}{f'_{2}(r)}=\frac{2}{\pi } \frac{r'(\mathcal{K}-\mathcal{E})}{r^{2}}, \end{aligned}$$
(3.3)
$$\begin{aligned}& \biggl[\frac{r'(\mathcal{K}-\mathcal{E})}{r^{2}} \biggr]'=- \frac{2[(1+r^{\prime 2})\mathcal{K}-2\mathcal{E}]}{\pi r'r^{3}}. \end{aligned}$$
(3.4)

Lemma 2.1 and Lemma 2.2(iv) together with (3.3) and (3.4) lead to the conclusion that \(f(r)\) is strictly decreasing on \((0,1)\). Note that

$$ f\bigl(0^{+}\bigr)=\frac{1}{2},\qquad f\bigl(1^{-} \bigr)=1-\frac{2}{\pi }. $$
(3.5)

Therefore, Theorem 1.1 follows from (3.2) and (3.5) together with the monotonicity of \(f(r)\). □

Proof of Theorem 1.2

Let \(r=(a-b)/(a+b)\in (0,1)\). Then it follows from (3.1) that

$$ \frac{\frac{1}{V(a,b)}-\frac{1}{A(a,b)}}{\frac{1}{H(a,b)}-\frac{1}{A(a,b)}} =\frac{\frac{2\mathcal{E}}{\pi r^{\prime 2}}-1}{\frac{1}{r^{\prime 2}}-1}=1-g(r), $$
(3.6)

where

$$ g(r)=\frac{1-2\mathcal{E}/\pi }{r^{2}}. $$

Let \(g_{1}(r)=1-2\mathcal{E}/\pi \) and \(g_{2}(r)=r^{2}\). Then elementary computations lead to

$$\begin{aligned}& g(r)=\frac{g_{1}(r)}{g_{2}(r)}, \qquad g_{1}(0)=g_{2}(0)=0, \end{aligned}$$
(3.7)
$$\begin{aligned}& \frac{g'_{1}(r)}{g'_{2}(r)}=\frac{1}{\pi } \frac{\mathcal{K}-\mathcal{E}}{r^{2}}. \end{aligned}$$
(3.8)

Lemma 2.1 and Lemma 2.2(ii) together with (3.7) and (3.8) lead to the conclusion that \(g(r)\) is strictly increasing on \((0,1)\). Note that

$$ g\bigl(0^{+}\bigr)=\frac{1}{4}, \qquad g\bigl(1^{-} \bigr)=1-\frac{2}{\pi }. $$
(3.9)

Therefore, Theorem 1.2 follows easily from (3.6) and (3.9) together with the monotonicity of \(g(r)\). □

Proof of Theorem 1.3

Let \(r=(a-b)/(a+b)\in (0,1)\). Then from (3.1) we get

$$\begin{aligned}& \frac{\frac{1}{V(a,b)}- [\frac{1}{2H(a,b)}+\frac{1}{2G(a,b)} ]}{ [\frac{3}{4H(a,b)}+\frac{1}{4A(a,b)} ] - [\frac{1}{2H(a,b)}+\frac{1}{2G(a,b)} ]} \\& \quad = \frac{\frac{2\mathcal{E}}{\pi r^{\prime 2}}-(\frac{1}{2r^{\prime 2}}+\frac{1}{2r'})}{(\frac{3}{4r^{\prime 2}}+\frac{1}{4})-(\frac{1}{2r^{\prime 2}}+\frac{1}{2r'})}=1-h(r), \end{aligned}$$
(3.10)

where

$$ h(r)=\frac{3+r^{\prime 2}-8\mathcal{E}/\pi }{(1-r')^{2}}. $$

Let \(h_{1}(r)=3+r^{\prime 2}-8\mathcal{E}/\pi \), \(h_{2}(r)=(1-r')^{2}\), \(h_{3}(r)=4(\mathcal{K}-\mathcal{E})/(\pi r^{2})-1\), and \(h_{4}(r)=1/r'-1\). Then we clearly see that \(h_{1}(0)=h_{2}(0)=h_{3}(0)=h_{4}(0)=0\). Simple computations lead to

$$\begin{aligned}& h(r)=\frac{h_{1}(r)}{h_{2}(r)}, \qquad \frac{h'_{1}(r)}{h'_{2}(r)}= \frac{h_{3}(r)}{h_{4}(r)}, \end{aligned}$$
(3.11)
$$\begin{aligned}& \frac{h'_{3}(r)}{h'_{4}(r)}= \frac{4r'[(1+r^{\prime 2})\mathcal{E}-2r^{\prime 2}\mathcal{K}]}{\pi r^{4}}= \frac{4}{\pi }\varphi (r), \end{aligned}$$
(3.12)

where \(\varphi (r)\) is defined in Lemma 2.3.

Lemmas 2.1 and 2.3 together with (3.11) and (3.12) lead to the conclusion that \(h(r)\) is strictly decreasing on \((0,1)\). Moreover, by Taylor’s formula, one has

$$ h\bigl(0^{+}\bigr)=\lim_{r\to 0^{+}} \frac{(1+r')^{2}[3r^{4}/16+o(r^{4})]}{r^{4}}= \frac{3}{4},\qquad h\bigl(1^{-}\bigr)=3- \frac{8}{\pi }. $$
(3.13)

Therefore, Theorem 1.3 follows easily from (3.10) and (3.13) together with the monotonicity of \(h(r)\). □

Proof of Theorem 1.4

Let \(r=(a-b)/(a+b)\in (0,1)\). Then it follows from (3.1) that

$$\begin{aligned}& \frac{\log [\frac{1}{V(a,b)} ]-\log [\frac{1}{2H(a,b)}+\frac{1}{2G(a,b)} ]}{\log [\frac{3}{4H(a,b)}+\frac{1}{4A(a,b)} ]-\log [\frac{1}{2H(a,b)}+\frac{1}{2G(a,b)} ]} \\& \quad = \frac{\log [(2\mathcal{E})/(\pi r^{\prime 2}) ]-\log [(1+r')/(2r^{\prime 2}) ]}{\log [(3+r^{\prime 2})/(4r^{\prime 2}) ]-\log [(1+r')/(2r^{\prime 2}) ]}:=1-j(r), \end{aligned}$$
(3.14)

where

$$ j(r)= \frac{\log [(3+r^{\prime 2})/4 ]-\log [(2\mathcal{E})/\pi ]}{\log [(3+r^{\prime 2})/4 ]-\log [(1+r')/2 ]}. $$

Let \(j_{1}(r)=\log [(3+r^{\prime 2})/4 ]-\log [(2\mathcal{E})/ \pi ]\) and \(j_{2}(r)=\log [(3+r^{\prime 2})/4 ]-\log [(1+r')/2 ]\). Then elaborated computations lead to

$$\begin{aligned}& j(r)=\frac{j_{1}(r)}{j_{2}(r)}= \frac{j_{1}(r)-j_{1}(0)}{j_{2}(r)-j_{2}(0)}, \end{aligned}$$
(3.15)
$$\begin{aligned}& \frac{j'_{1}(r)}{j'_{2}(r)}= \frac{r'[(4-r^{2})\mathcal{K}-(4+r^{2})\mathcal{E}]}{(1-r')^{2}(r'+3)\mathcal{E}}= \frac{\phi (r)}{\varrho (r)\rho (r)}, \end{aligned}$$
(3.16)

where \(\varrho (r)\), \(\rho (r)\), and \(\phi (r)\) are defined as in Lemma 2.2(v), (vi) and Lemma 2.4, respectively.

Lemma 2.1, Lemma 2.2(v), (vi), and Lemma 2.4 together with (3.15) and (3.16) lead to the conclusion that \(j(r)\) is strictly decreasing on \((0,1)\). Moreover, by L’Hôpital’s rule we get

$$ j\bigl(0^{+}\bigr)=\lim_{r\to 0^{+}}\frac{j'_{1}(r)}{j'_{2}(r)}= \frac{3}{4},\qquad j\bigl(1^{-}\bigr)=\frac{\log (3\pi )-3\log 2}{\log 3-\log 2}. $$
(3.17)

Therefore, Theorem 1.4 follows easily from (3.14) and (3.17) together with the monotonicity of \(j(r)\). □

As a consequence of Theorems 1.11.4, we can derive the following Corollary 3.1 immediately.

Corollary 3.1

Let \(l(r)=(1+r)/2\) and \(u(r)=(3+r^{2})/4\). Then the double inequalities

$$\begin{aligned}& \frac{\pi }{2}l\bigl(r'\bigr)< \mathcal{E}(r)< 1+ \biggl( \frac{\pi }{2}-1 \biggr)r', \\& 1+ \biggl(\frac{\pi }{2}-1 \biggr)r^{\prime 2}< \mathcal{E}(r)< \frac{\pi }{2}u\bigl(r'\bigr), \\& \frac{\pi }{2} \biggl[\frac{u(r')}{4}+\frac{3l(r')}{4} \biggr]< \mathcal{E}(r)< \frac{\pi }{2} \bigl[\sigma u\bigl(r'\bigr)+(1- \sigma )l\bigl(r'\bigr) \bigr], \\& \frac{\pi }{2}u\bigl(r'\bigr)^{1/4}l \bigl(r'\bigr)^{3/4}< \mathcal{E}(r)< \frac{\pi }{2}u \bigl(r'\bigr)^{ \tau }l\bigl(r' \bigr)^{1-\tau } \end{aligned}$$

hold for all \(r\in (0,1)\), where \(\sigma =2(4/\pi -1)\) and \(\tau = [\log (4/\pi ) ]/\log (3/2)\) are given in Theorems 1.3and 1.4, respectively.

In order to compare the lower and upper bounds in Corollary 3.1, we provide Theorem 3.2 as follows.

Theorem 3.2

The double inequality

$$ \max_{r\in ({0,1} )} \biggl\{ {1+ \biggl({ \frac{\pi }{2}-1} \biggr){{r'}^{2}}, \frac{\pi }{{32}}{{ \bigl({3+r'} \bigr)}^{2}}} \biggr\} < \mathcal{E} (r )< \frac{\pi }{4}r' \bigl({3- r'} \bigr)+{ \bigl({1- r'} \bigr)^{2}} $$

holds for all \(r\in ({0,1} )\).

Proof

We clearly see that the function

$$ r\mapsto \frac{u(r)}{l(r)}=\frac{1}{2} \biggl(r+1+\frac{4}{r+1}-2 \biggr) $$

is strictly decreasing on \((0,1)\). Therefore, \(u(r)/l(r)\in (1,3/2)\) and

$$ \frac{u(r')}{4}+\frac{3l(r')}{4}>l\bigl(r'\bigr),\qquad \sigma u \bigl(r'\bigr)+(1-\sigma )l\bigl(r'\bigr)< u \bigl(r'\bigr). $$
(3.18)

It is well known that

$$ \frac{u(r')}{4}+\frac{3l(r')}{4}>u\bigl(r' \bigr)^{1/4}l\bigl(r'\bigr)^{3/4}. $$
(3.19)

It is not difficult to verify that the functions \(1+ (\frac{\pi }{2}-1 )r^{\prime 2}\) and \(\frac{\pi }{2} [\frac{u(r')}{4}+\frac{3l(r')}{4} ]\) are not comparable on \((0,1)\) due to

$$ \frac{1+ (\frac{\pi }{2}-1 )r^{\prime 2}-\frac{\pi }{2} [\frac{u(r')}{4}+\frac{3l(r')}{4} ]}{r^{2}} \rightarrow \textstyle\begin{cases} 1-\frac{3\pi }{8}< 0,& r\rightarrow 0^{+}, \\ 1-\frac{9\pi }{32}>0,& r\rightarrow 1^{-}. \end{cases} $$

This in conjunction with (3.18) and (3.19) implies that

$$\begin{aligned}& \max_{r\in (0,1)} \biggl\{ \frac{\pi }{2}l\bigl(r' \bigr), 1+ \biggl(\frac{\pi }{2}-1 \biggr)r^{\prime 2}, \frac{\pi }{2} \biggl[\frac{u(r')}{4}+\frac{3l(r')}{4} \biggr], \frac{\pi }{2}u \bigl(r'\bigr)^{1/4}l\bigl(r' \bigr)^{3/4} \biggr\} \\& \quad =\max_{r\in ({0,1} )} \biggl\{ {1+ \biggl({\frac{\pi }{2}- 1} \biggr){{r'}^{2}}, \frac{\pi }{{32}} {{ \bigl({3+r'} \bigr)}^{2}}} \biggr\} . \end{aligned}$$

We now claim that

$$ s(x)=\sigma x^{1-\tau }+(1-\sigma )x^{-\tau }< 1 $$
(3.20)

for \(x\in (1,3/2)\). Indeed, differentiating \(s(x)\) yields

$$ s'(x)=\sigma (1-\tau )x^{-1-\tau } \biggl[x- \frac{\tau (1-\sigma )}{\sigma (1-\tau )} \biggr], $$

which together with \(\tau (1-\sigma )/[\sigma (1-\tau )]=1.223\ldots \) enables us to know that \(s(x)\) is convex on \((1,3/2)\). Therefore, inequality (3.20) follows from \(s(1)=s(3/2)=1\).

It follows from (3.20) and \(1< u(r)/l(r)<3/2\) that

$$\begin{aligned}& \sigma u(r)+(1-\sigma )l(r)-u(r)^{\tau }l(r)^{1-\tau } \\& \quad =u(r)^{\tau }l(r)^{1-\tau } \biggl[\sigma \biggl(\frac{u(r)}{l(r)} \biggr)^{1- \tau } +(1-\sigma ) \biggl(\frac{l(r)}{u(r)} \biggr)^{\tau }-1 \biggr] \\& \quad =u(r)^{\tau }l(r)^{1-\tau } \bigl[s \bigl(u(r)/l(r) \bigr)-1 \bigr]< 0. \end{aligned}$$
(3.21)

Moreover, it is not difficult to verify that

$$ \frac{\pi }{2} \bigl[\sigma u\bigl(r'\bigr)+(1-\sigma )l \bigl(r'\bigr) \bigr]- \biggl[1+ \biggl(\frac{\pi }{2}-1 \biggr)r' \biggr] =- \biggl(1-\frac{\pi }{4} \biggr)r'\bigl(1-r'\bigr)< 0. $$

This in conjunction with (3.18) and (3.21) implies that

$$\begin{aligned}& \min_{r\in (0,1)} \biggl\{ 1+ \biggl(\frac{\pi }{2}-1 \biggr)r', \frac{\pi }{2}u\bigl(r'\bigr), \frac{\pi }{2} \bigl[\sigma u\bigl(r'\bigr)+(1-\sigma )l \bigl(r'\bigr) \bigr], \frac{\pi }{2}u\bigl(r' \bigr)^{\tau }l\bigl(r'\bigr)^{1-\tau } \biggr\} \\& \quad =\frac{\pi }{4}r' \bigl({3-r'} \bigr)+{ \bigl({1- r'} \bigr)^{2}}. \end{aligned}$$

 □

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References

  1. Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)

    MATH  Google Scholar 

  2. Zhao, T.-H., Wang, M.-K., Zhang, W., Chu, Y.-M.: Quadratic transformation inequalities for Gaussian hypergeometric function. J. Inequal. Appl. 2018, Article ID 251 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  3. Chu, Y.-M., Zhao, T.-H.: Convexity and concavity of the complete elliptic integrals with respect to Lehmer mean. J. Inequal. Appl. 2015, Article ID 396 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  4. Zhao, T.-H., Wang, M.-K., Chu, Y.-M.: On the bounds of the perimeter of an ellipse. Acta Math. Sci. 42B(2), 491–501 (2022)

    Article  Google Scholar 

  5. Zhao, T.-H., Wang, M.-K., Chu, Y.-M.: A sharp double inequality involving generalized complete elliptic integral of the first kind. AIMS Math. 5(5), 4512–4528 (2020)

    MathSciNet  Article  Google Scholar 

  6. Zhao, T.-H., Shi, L., Chu, Y.-M.: Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114(2), Article ID 96 (2020)

    MATH  Article  Google Scholar 

  7. Alzer, H., Richards, K.: Inequalities for the ratio of complete elliptic integrals. Proc. Am. Math. Soc. 145(4), 1661–1670 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  8. Qian, W.-M., Wang, M.-K., Xu, H.-Z., Chu, Y.-M.: Approximations for the complete elliptic integral of the second kind. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 115(2), Article ID 88 (2021)

    MathSciNet  MATH  Article  Google Scholar 

  9. Wang, M.-K., Hong, M.-Y., Shen, Z.-H., Chu, Y.-M.: Inequalities for generalized trigonometric and hyperbolic functions with one parameter. J. Math. Inequal. 14(1), 1–21 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  10. Borwein, J.M., Borwein, P.B.: Pi and the AGM. Wiley, New York (1987)

    MATH  Google Scholar 

  11. Zhao, T.-H., Zhou, B.-C., Wang, M.-K., Chu, Y.-M.: On approximating the quasi-arithmetic mean. J. Inequal. Appl. 2019, Article ID 42 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  12. Toader, G.: Some mean values related to the arithmetic-geometric mean. J. Math. Anal. Appl. 218(2), 358–368 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  13. Zhao, T.-H., Chu, Y.-M., Zhang, W.: Optimal inequalities for bounding Toader mean by arithmetic and quadratic means. J. Inequal. Appl. 2017, Article ID 26 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  14. Song, Y.-Q., Zhao, T.-H., Chu, Y.-M., Zhang, X.-H.: Optimal evaluation of a Toader-type mean by power mean. J. Inequal. Appl. 2015, Article ID 408 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  15. Chu, H.-H., Zhao, T.-H., Chu, Y.-M.: Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means. Math. Slovaca 70(5), 1097–1112 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  16. Zhao, T.-H., He, Z.-Y., Chu, Y.-M.: Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals. Comput. Methods Funct. Theory 21, 413–426 (2021)

    MathSciNet  MATH  Article  Google Scholar 

  17. Xu, H.-Z., Qian, W.-M., Chu, Y.-M.: Sharp bounds for the lemniscatic mean by the one-parameter geometric and quadratic means. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 116(1), Article ID 21 (2022)

    MathSciNet  MATH  Article  Google Scholar 

  18. Alzer, H., Richards, K.C.: A concavity property of the complete elliptic integral of the first kind. Integral Transforms Spec. Funct. 31(9), 758–768 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  19. Zhao, T.-H., Bhayo, B.A., Chu, Y.-M.: Inequalities for generalized Grötzsch ring function. Comput. Methods Funct. Theory (2021). https://doi.org/10.1007/s40315-021-00415-3

    Article  Google Scholar 

  20. Yin, L., Qi, F.: Some inequalities for complete elliptic integrals. Appl. Math. E-Notes 14, 193–199 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Zhao, T.-H., Wang, M.-K., Chu, Y.-M.: Concavity and bounds involving generalized elliptic integral of the first kind. J. Math. Inequal. 15(2), 701–724 (2021)

    MathSciNet  MATH  Article  Google Scholar 

  22. Zhao, T.-H., Wang, M.-K., Chu, Y.-M.: Monotonicity and convexity involving generalized elliptic integral of the first kind. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 115(2), Article ID 46 (2021)

    MathSciNet  MATH  Article  Google Scholar 

  23. Chu, Y.-M., Qiu, Y.-F., Wang, M.-K.: Hölder mean inequalities for the complete elliptic integrals. Integral Transforms Spec. Funct. 23(7), 521–527 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  24. Guo, B.-N., Qi, F.: Some bounds for the complete elliptic integrals of the first and second kinds. Math. Inequal. Appl. 14(2), 323–334 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Zhao, T.-H., Qian, W.-M., Chu, Y.-M.: On approximating the arc lemniscate functions. Indian J. Pure Appl. Math. (2021). https://doi.org/10.1007/s13226-021-00016-9

    Article  Google Scholar 

  26. Zhao, T.-H., Wang, M.-K., Hai, G.-J., Chu, Y.-M.: Landen inequalities for Gaussian hypergeometric function. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 116(1), Article ID 53 (2022)

    MathSciNet  MATH  Article  Google Scholar 

  27. Alzer, H.: Sharp inequalities for the complete elliptic integral of the first kind. Math. Proc. Camb. Philos. Soc. 124(2), 309–314 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  28. Zhao, T.-H., Qian, W.-M., Chu, Y.-M.: Sharp power mean bounds for the tangent and hyperbolic sine means. J. Math. Inequal. 15(4), 1459–1472 (2021)

    MathSciNet  MATH  Article  Google Scholar 

  29. Anderson, G.D., Duren, P., Vamanamurthy, M.K.: An inequality for complete elliptic integrals. J. Math. Anal. Appl. 182(1), 257–259 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  30. Zhao, T.-H., Shen, Z.-H., Chu, Y.-M.: Sharp power mean bounds for the lemniscate type means. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 115(4), Article ID 175 (2021)

    MathSciNet  MATH  Article  Google Scholar 

  31. Zhao, T.-H., He, Z.-Y., Chu, Y.-M.: On some refinements for inequalities involving zero-balanced hypergeometric function. AIMS Math. 5(6), 6479–6495 (2020)

    MathSciNet  Article  Google Scholar 

  32. Barnard, R.W., Pearce, K., Richards, K.C.: An inequality involving the generalized hypergeometric function and the arc length of an ellipse. SIAM J. Math. Anal. 31(3), 693–699 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  33. Alzer, H., Qiu, S.-L.: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172(2), 289–312 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  34. 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)

    MathSciNet  MATH  Article  Google Scholar 

  35. Yang, Z.-H., Chu, Y.-M., Zhang, W.: High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl. Math. Comput. 348, 552–564 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  36. Witkowski, A.: On Seiffert-like means. J. Math. Inequal. 9(4), 1071–1092 (2015)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

Funding

The work was supported by the Key Project of the Scientific Research of Zhejiang Open University in 2019 (Grant no. XKT-19Z02) and the Natural Science Foundation of the Department of Education of Zhejiang Province in 2020 (Grant no. Y202043179).

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FZ: conceptualization, computation, writing–original draft, writing–review and editing. WQ: problem statement, conceptualization, methodology, computation, writing–original draft, supervision, and funding acquisition. HZX: computation, writing–review and editing. All authors read and approved the final manuscript.

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Correspondence to Hui Zuo Xu.

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Zhang, F., Qian, W. & Xu, H.Z. Optimal bounds for Seiffert-like elliptic integral mean by harmonic, geometric, and arithmetic means. J Inequal Appl 2022, 33 (2022). https://doi.org/10.1186/s13660-022-02768-2

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  • DOI: https://doi.org/10.1186/s13660-022-02768-2

MSC

  • 26D15
  • 33E05

Keywords

  • Seiffert-like mean
  • Complete elliptic integral
  • Harmonic mean
  • Geometric mean
  • Arithmetic mean