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.

1 Introduction

For $$r\in (0,1)$$, Legendreâ€™s complete elliptic integrals of the first kind $$\mathcal{K}(r)$$ and second kind $$\mathcal{E}(r)$$ [1â€“8] 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 $$A(a,b)$$, arithmeticâ€“geometric mean $$AG(a,b)$$ [9â€“11], and Toader mean $$TD(a,b)$$ [12â€“15] 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 [16â€“22] 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 [23â€“31]. 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 $$\beta _{1}\geq 2/\pi =0.6366\ldots$$â€‰.

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 $$\beta _{3}\geq 2(4/\pi -1)=0.5464\ldots$$â€‰.

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 $$\beta _{4}\geq [\log (4/\pi )]/\log (3/2)=0.5957\ldots$$â€‰.

2 Lemmas

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

Lemma 2.1

(See [1, TheoremÂ 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)â€“(iv) can be found in [1, TheoremÂ 3.21 (1) and ExerciseÂ 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.â€ƒâ–¡

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

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

\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, LemmaÂ 2.3 follows easily from (2.1)â€“(2.3) and LemmaÂ 2.2(iv) together with $$\varphi (0^{+})=3\pi /16$$ and $$\varphi (1^{-})=0$$.â€ƒâ–¡

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}

\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, LemmaÂ 2.4 follows easily from (2.4)â€“(2.7) and LemmaÂ 2.2(iv) together with $$\phi (0^{+})=3\pi /8$$ and $$\phi (1^{-})=0$$.â€ƒâ–¡

3 Proofs of Theorems 1.1â€“1.4

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 TheoremÂ 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)

LemmaÂ 2.1 and LemmaÂ 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, TheoremÂ 1.1 follows from (3.2) and (3.5) together with the monotonicity of $$f(r)$$.â€ƒâ–¡

Proof of TheoremÂ 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)

LemmaÂ 2.1 and LemmaÂ 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, TheoremÂ 1.2 follows easily from (3.6) and (3.9) together with the monotonicity of $$g(r)$$.â€ƒâ–¡

Proof of TheoremÂ 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 LemmaÂ 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 Taylorâ€™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, TheoremÂ 1.3 follows easily from (3.10) and (3.13) together with the monotonicity of $$h(r)$$.â€ƒâ–¡

Proof of TheoremÂ 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 LemmaÂ 2.2(v), (vi) and LemmaÂ 2.4, respectively.

LemmaÂ 2.1, LemmaÂ 2.2(v), (vi), and LemmaÂ 2.4 together with (3.15) and (3.16) lead to the conclusion that $$j(r)$$ is strictly decreasing on $$(0,1)$$. Moreover, by Lâ€™HÃ´pitalâ€™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, TheoremÂ 1.4 follows easily from (3.14) and (3.17) together with the monotonicity of $$j(r)$$.â€ƒâ–¡

As a consequence of Theorems 1.1â€“1.4, we can derive the following CorollaryÂ 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 CorollaryÂ 3.1, we provide TheoremÂ 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}

â€ƒâ–¡

Not applicable.

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

Corresponding author

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