Journal of Inequalities and Applications

Impact Factor 0.791

Journal of Inequalities and Applications Cover Image
Research
Open Access

Monotonicity rule for the quotient of two functions and its application

Journal of Inequalities and Applications20172017:106

https://doi.org/10.1186/s13660-017-1383-2

©  The Author(s) 2017

Received: 17 March 2017

Accepted: 26 April 2017

Published: 9 May 2017

Abstract

In the article, we provide a monotonicity rule for the function \([P(x)+A(x)]/[P(x)+B(x)]\), where \(P(x)\) is a positive differentiable and decreasing function defined on \((-R, R)\) (\(R>0\)), and \(A(x)=\sum ^{\infty}_{n=n_{0}}a_{n}x^{n}\) and \(B(x)=\sum^{\infty }_{n=n_{0}}b_{n}x^{n}\) are two real power series converging on \((-R, R)\) such that the sequence \(\{a_{n}/b_{n}\}_{n=n_{0}}^{\infty}\) is increasing (decreasing) with \(a_{n_{0}}/b_{n_{0}}\geq(\leq)\ 1\) and \(b_{n}>0\) for all \(n\geq n_{0}\). As applications, we present new bounds for the complete elliptic integral \(\mathcal{E}(r)=\int_{0}^{\pi /2}\sqrt{1-r^{2}\sin^{2}t}\,dt\) (\(0< r<1\)) of the second kind.

Keywords

monotonicity rule complete elliptic integral absolute error relative error

MSC

26A48 33E05

1 Introduction

The most commonly used monotonicity rule in elementary calculus is that f is increasing (decreasing) on \([a, b]\) if \(f: [a, b]\rightarrow \mathbb{R}\) is continuous on \([a, b]\) and has a positive (negative) derivative on \((a, b)\), and it can be proved easily by the Lagrange mean value theorem. The functions whose monotonicity we prove in this way are usually polynomials, rational functions, or other elementary functions. But we often find that the derivative of a quotient of two functions is quite messy and the process is tedious. Therefore, the improvements, generalizations and refinements of the method for proving monotonicity of quotients have attracted the attention of many researchers.

In 1955, Biernacki and Krzyż [1] (see also [2], Lemma 2.1, [3]) found an important criterion for the monotonicity of the quotient of power series as follows.

Theorem 1.1

[1]

Let \(A(t)=\sum_{k=0}^{\infty }a_{k}t^{k}\) and \(B(t)=\sum_{k=0}^{\infty}b_{k}t^{k}\) be two real power series converging on \((-r,r)\) (\(r>0\)) with \(b_{k}>0\) for all k. If the non-constant sequence \(\{a_{k}/b_{k}\}\) is increasing (decreasing) for all k, then the function \(t\mapsto A(t)/B(t)\) is strictly increasing (decreasing) on \((0,r)\).

In [4], Cheeger et al. presented the monotonicity rule for the quotient of two functions.

Theorem 1.2

[4]

If f and g are positive integrable functions on \(\mathbb{R}\) such that \(f/g\) is decreasing, then the function \(x\mapsto\int_{0}^{x}f(t)\,dt/\int_{0}^{x}g(t)\, dt\) is decreasing.

Unaware of Theorem 1.2, Anderson et al. [5], Lemma 2.2 (see also [6], Theorem 1.25) established l’Hôpital’s monotone rule that can be applied to a wide class of quotients of functions.

Theorem 1.3

[5]

Let \(-\infty< a< b<\infty\), \(f,g:[a,b]\rightarrow{\mathbb{R}}\) be continuous on \([a,b]\) and differentiable on \((a,b)\), and \(g'(x)\neq0\) on \((a,b)\). If \(f^{\prime}(x)/g^{\prime}(x)\) is increasing (decreasing) on \((a,b)\), then so are the functions
$$ \frac{f(x)-f(a)}{g(x)-g(a)}, \qquad\frac{f(x)-f(b)}{g(x)-g(b)}. $$
If \(f^{\prime}(x)/g^{\prime}(x)\) is strictly monotone, then the monotonicity in the conclusion is also strict.

Pinelis [7] provided the following monotonicity theorem.

Theorem 1.4

[7]

Let f and g be differentiable and \(g^{\prime}\) never vanish on an open interval \((a, b)\subset\mathbb{R}\). Then the following statements are true:
  1. (1)

    If \(gg^{\prime}>0\) on \((a,b)\), \(\lim\sup_{x\downarrow a}g^{2}(x)(f(x)/g(x))^{\prime}/|g^{\prime}(x)|\geq0\) and \(f^{\prime }/g^{\prime}\) is increasing on \((a,b)\), then \((f/g)^{\prime}>0\) on \((a,b)\).

     
  2. (2)

    If \(gg^{\prime}>0\) on \((a,b)\), \(\lim\inf_{x\downarrow a}g^{2}(x)(f(x)/g(x))^{\prime}/|g^{\prime}(x)|\leq0\) and \(f^{\prime }/g^{\prime}\) is decreasing on \((a,b)\), then \((f/g)^{\prime}<0\) on \((a,b)\).

     
  3. (3)

    If \(gg^{\prime}<0\) on \((a,b)\), \(\lim\inf_{x\uparrow b}g^{2}(x)(f(x)/g(x))^{\prime}/|g^{\prime}(x)|\leq0\) and \(f^{\prime }/g^{\prime}\) is increasing on \((a,b)\), then \((f/g)^{\prime}<0\) on \((a,b)\).

     
  4. (4)

    If \(gg^{\prime}<0\) on \((a,b)\), \(\lim\sup_{x\uparrow b}g^{2}(x)(f(x)/g(x))^{\prime}/|g^{\prime}(x)|\geq0\) and \(f^{\prime }/g^{\prime}\) is increasing on \((a,b)\), then \((f/g)^{\prime}>0\) on \((a,b)\).

     

Recently, Yang et al. [8], Theorem 1.2, established a more general monotonicity rule for the ratio of two power series.

Theorem 1.5

[8]

Let \(A(t)=\sum_{k=0}^{\infty}a_{k}t^{k}\) and \(B(t)=\sum_{k=0}^{\infty}b_{k}t^{k}\) be two real power series converging on \((-r,r) \) and \(b_{k}>0\) for all k, and \(H_{A, B}=A^{\prime}B/B^{\prime}-A\). Suppose that for certain \(m\in\mathbb{N}\), the non-constant sequence \(\{a_{k}/b_{k}\}\) is increasing (decreasing) for \(0\leq k\leq m\) and decreasing (increasing) for \(k\geq m\). Then the function \(A/B\) is strictly increasing (decreasing) on \((0,r) \) if and only if \(H_{A,B}(r^{-}) \geq(\leq)\ 0\). Moreover, if \(H_{A,B}(r^{-})<(>) 0\), then there exists \(t_{0}\in(0,r) \) such that the function \(A/B\) is strictly increasing (decreasing) on \((0,t_{0}) \) and strictly decreasing (increasing) on \((t_{0},r)\).

The foregoing monotonicity rules have been used very effectively in the study of special functions [9–23], differential geometry [4, 24], probability [25] and approximation theory [26]. The main purpose of the article is to present the monotonicity rule for the function \([P(x)+\sum_{n=n_{0}}^{\infty}a_{n}x^{n}]/[P(x)+\sum _{n=n_{0}}^{\infty}b_{n}x^{n}]\) and to provide new bounds for the complete elliptic integral of the second kind. Some complicated computations are carried out using Mathematica computer algebra system.

2 Monotonicity rule

Theorem 2.1

Let \(P(x)\) be a positive differentiable and decreasing function defined on \((0, r)\) (\(r>0\)), let \(A(x)=\sum _{n=n_{0}}^{\infty}a_{n}x^{n}\) and \(B(x)=\sum_{n=n_{0}}^{\infty}b_{n}x^{n}\) be two real power series converging on \((-r, r)\). If \(a_{n_{0}}/b_{n_{0}}\geq(\leq)\ 1\), \(b_{n}>0\) for all \(n\geq n_{0}\) and the non-constant sequence \(\{ a_{n}/b_{n}\}_{n=n_{0}}^{\infty}\) is increasing (decreasing), then the function \(x\mapsto[P(x)+A(x)]/[P(x)+B(x)]\) is strictly increasing (decreasing) on \((0, r)\).

Proof

Let \(x\in(0, r)\), and
$$ \begin{aligned} &I_{1}=\sum _{n=n_{0}}^{\infty}\bigl[nP(x)-xP^{\prime}(x) \bigr](a_{n}-b_{n})x^{n-1}, \\ &I_{2}=\sum_{n=n_{0}}^{\infty}na_{n}x^{n-1} \sum_{n=n_{0}}^{\infty}b_{n}x^{n} -\sum_{n=n_{0}}^{\infty}a_{n}x^{n} \sum_{n=n_{0}}^{\infty}nb_{n}x^{n-1}. \end{aligned} $$
(2.1)
Then differentiating \([P(x)+A(x)]/[P(x)+B(x)]\) gives
$$\begin{aligned} & \bigl[ P(x)+B(x) \bigr] ^{2} \biggl[ \frac{P(x)+A(x)}{P(x)+B(x)} \biggr] ^{\prime} \\ &\quad= \Biggl( P^{\prime}(x)+\sum_{n=n_{0}}^{\infty}na_{n}x^{n-1} \Biggr) \Biggl( P(x)+\sum_{n=n_{0}}^{\infty}b_{n}x^{n} \Biggr) \\ &\quad\quad{}- \Biggl( P(x)+\sum_{n=n_{0}}^{\infty}a_{n}x^{n} \Biggr) \Biggl( P^{\prime}(x)+\sum_{n=n_{0}}^{\infty}nb_{n}x^{n-1} \Biggr) \\ &\quad=I_{1}+I_{2}. \end{aligned}$$
(2.2)
Note that \(I_{2}\) can be rewritten as
$$\begin{aligned} I_{2}={}&\sum_{j=n_{0}}^{\infty}ja_{j}x^{j-1} \sum_{i=n_{0}}^{\infty}b_{i}x^{i}- \sum_{i=n_{0}}^{\infty}a_{i}x^{i} \sum_{j=n_{0}}^{\infty}jb_{j}x^{j-1} \\ ={}&\sum_{i=n_{0}}^{\infty}\sum _{j=n_{0}}^{\infty}jb_{i}b_{j} \biggl( \frac{a_{j}}{b_{j}}-\frac{a_{i}}{b_{i}} \biggr) x^{i+j-1} \\ ={}&\sum _{j=n_{0}}^{\infty}\sum_{i=n_{0}}^{\infty}ib_{j}b_{i} \biggl( \frac{a_{i}}{b_{i}}-\frac{a_{j}}{b_{j}} \biggr) x^{i+j-1} \\ ={}&\frac{1}{2}\sum_{i=n_{0}}^{\infty}\sum _{j=n_{0}}^{\infty}b_{i}b_{j}(i-j) \biggl( \frac{a_{i}}{b_{i}}-\frac{a_{j}}{b_{j}} \biggr) x^{i+j-1}. \end{aligned}$$
(2.3)
If \(a_{n_{0}}/b_{n_{0}}\geq(\leq)\ 1\), \(b_{n}>0\) for all \(n\geq n_{0}\) and the non-constant sequence \(\{a_{n}/b_{n}\}_{n=n_{0}}^{\infty }\) is increasing (decreasing), then we clearly see that
$$ a_{n}\geq(\leq)\ b_{n} $$
(2.4)
for all \(n\geq n_{0}\) and
$$ \sum_{i=n_{0}}^{\infty}\sum _{j=n_{0}}^{\infty}b_{i}b_{j}(i-j) \biggl( \frac{a_{i}}{b_{i}}-\frac{a_{j}}{b_{j}} \biggr) x^{i+j-1}>(< )\ 0 $$
(2.5)
for all \(x\in(0, r)\).
It follows from \(P(x)\) is a positive differentiable and decreasing function on \((0, r)\) that
$$ nP(x)-xP^{\prime}(x)>0 $$
(2.6)
for all \(x\in(0, r)\).

Therefore, \([(P(x)+A(x))/(P(x)+B(x))]^{\prime}>\) (<) 0 for all \(x\in(0, r)\) follows easily from (2.1)-(2.6), and the proof of Theorem 2.1 is completed. □

3 Bounds for the complete elliptic integral of the second kind

For \(r\in(0,1)\), Legendre’s complete elliptic integral [27] of the second kind is given by
$$ \mathcal{E}(r)= \int_{0}^{\pi/2}\sqrt{1-r^{2}\sin ^{2}(t)}\,dt. $$
It is well known that \(\mathcal{E}(0^{+})=\pi/2\), \(\mathcal {E}(1^{-})=1\), and \(\mathcal{E}(r)\) is the particular case of the Gaussian hypergeometric function
$$ F(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}} \frac{x^{n}}{n!}\quad(-1< x< 1), $$
where \((a)_{n}=\Gamma(a+n)/\Gamma(a)\) and \(\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\,dt\) (\(x>0\)) is the gamma function. Indeed, we have
$$ \mathcal{E}(r)=\frac{\pi}{2}F \biggl( -\frac{1}{2}, \frac{1}{2};1;r^{2} \biggr) =\frac{\pi}{2}\sum _{n=0}^{\infty}\frac{ ( -\frac{1}{2} ) _{n} ( \frac{1}{2} ) _{n}}{(n!)^{2}}r^{2n}. $$
(3.1)
Recently, the bounds for the complete elliptic integral \(\mathcal {E}(r)\) of the second kind have been the subject of intensive research. In particular, many remarkable inequalities for \(\mathcal{E}(r)\) can be found in the literature [28–41]. Vuorinen [42] conjectured that the inequality
$$ \mathcal{E}(r)\geq\frac{\pi}{2} \biggl( \frac {1+{r^{\prime}}^{3/2}}{2} \biggr) ^{2/3} $$
(3.2)
holds for all \(r\in(0, 1)\), where, and in what follows, \(r^{\prime }=(1-r^{2})^{1/2}\). Inequality (3.2) was proved by Barnard et al. in [43].
Very recently, the accurate bounds for \(\mathcal{E}(r)\) in terms of the Stolarsky mean \(S_{p, q}(1, r^{\prime})\) were given in [44, 45]:
$$\begin{aligned}& \frac{\pi}{2}S_{11/4, 7/4}\bigl(1, r^{\prime}\bigr)< \mathcal{E}(r)< \frac{11}{7}S_{11/4, 7/4}\bigl(1, r^{\prime}\bigr), \end{aligned}$$
(3.3)
$$\begin{aligned}& \frac{25}{16}S_{5/2, 2}\bigl(1, r^{\prime}\bigr)< \mathcal{E}(r)< \frac{\pi}{2}S_{5/2, 2}\bigl(1, r^{\prime}\bigr), \end{aligned}$$
(3.4)
where \(S_{p, q}(a,b)=[q(a^{p}-b^{p})/(p(a^{q}-b^{q}))]^{1/(p-q)}\).

In this section, we shall use Theorem 2.1 to present new bounds for the complete elliptic integral \(\mathcal{E}(r)\) of the second kind. In order to prove our main result, we need three lemmas, which we present in this section.

Lemma 3.1

see [46], Lemma 7

Let \(n\in\mathbb{N}\) and \(m\in\mathbb{N}\cup\{0\}\) with \(n>m\), \(a_{i}\geq0\) for all \(0\leq i\leq n\), \(a_{n}a_{m}>0\) and
$$ P_{n}(t)=-\sum_{i=0}^{m}a_{i}t^{i}+ \sum_{i=m+1}^{n}a_{i}t^{i}. $$
Then there exists \(t_{0}\in(0, \infty)\) such that \(P_{n}(t_{0})=0\), \(P_{n}(t)<0\) for \(t\in(0, t_{0})\) and \(P_{n}(t)>0\) for \(t\in(t_{0}, \infty)\).

Lemma 3.2

see [47, 48]

The double inequality
$$ \frac{1}{(x+a)^{1-a}}< \frac{\Gamma(x+a)}{\Gamma(x+1)}< \frac{1}{x^{1-a}} $$
holds for all \(x>0\) and \(a\in(0, 1)\).

Lemma 3.3

Let \(n\in\mathbb{N}\), \(p_{1}(n)\), \(p_{2}(n)\), \(p_{3}(n)\) and \(w_{n}\) be defined by
$$\begin{aligned}& p_{1}(n)=125n^{4}+1\text{,}482n^{3}+1\text{,}601n^{2}-7\text{,}532n+1\text{,}944, \end{aligned}$$
(3.5)
$$\begin{aligned}& p_{2}(n)=4\text{,}375n^{6}+213\text{,}247n^{5}+1\text{,}696\text{,}697n^{4}-4\text{,}433\text{,}167n^{3} \\& \qquad \quad {}-1\text{,}571\text{,}832n^{2}+4\text{,}662\text{,}360n-1\text{,}555\text{,}200, \end{aligned}$$
(3.6)
$$\begin{aligned}& p_{3}(n)=625n^{6}+22\text{,}583n^{5}-304\text{,}413n^{4}+234\text{,}181n^{3} \\& \qquad \quad {}+1\text{,}906\text{,}328n^{2}-2\text{,}918\text{,}064n+675\text{,}360, \end{aligned}$$
(3.7)
$$\begin{aligned}& w_{n}=-648np_{1}(n) \biggl( \frac{1}{4} \biggr) _{n-2}+p_{2}(n) \biggl( \frac{1}{2} \biggr) _{n-3}+\frac{5np_{3}(n)}{2n-5} \biggl( \frac{3}{4} \biggr) _{n-3}, \end{aligned}$$
(3.8)
respectively. Then \(w_{n}\geq0\) for all \(n\geq3\).

Proof

Let \(p_{0}(n)\), \(p_{4}(n)\), \(p_{5}(n)\), \(\alpha_{n}\) and \(\beta_{n}\) be defined by
$$\begin{aligned}& p_{0}(n)=625n^{4}-2\text{,}018n^{3}+3\text{,}985n^{2}-4\text{,}032n+1\text{,}080, \end{aligned}$$
(3.9)
$$\begin{aligned}& p_{4}(n)=875n^{7}-81\text{,}128n^{6}-2\text{,}341\text{,}894n^{5}-10\text{,}120\text{,}928n^{4}+19\text{,}839\text{,}719n^{3} \\& \qquad \quad {}-22\text{,}615\text{,}904n^{2}+73\text{,}667\text{,}340n-71\text{,}971\text{,}200, \end{aligned}$$
(3.10)
$$\begin{aligned}& p_{5}(n)=250n^{6}-1\text{,}834n^{5}+391\text{,}275n^{4}+975\text{,}000n^{3}-7\text{,}770\text{,}415n^{2} \\& \quad \qquad {}+7\text{,}980\text{,}844n+298\text{,}140, \end{aligned}$$
(3.11)
$$\begin{aligned}& \alpha_{n}=\frac{p_{4}(n)}{n(n+1)} \biggl( \frac{1}{2} \biggr) _{n-3}, \end{aligned}$$
(3.12)
$$\begin{aligned}& \beta_{n}=\frac{10(n-1)p_{5}(n)}{(2n-3)(2n-5)} \biggl( \frac{3}{4} \biggr)_{n-3}, \end{aligned}$$
(3.13)
respectively.
Then from (3.5)-(3.13) and elaborated computations we get
$$\begin{aligned}& w_{3}=w_{4}=w_{5}=w_{6}=0, \end{aligned}$$
(3.14)
$$\begin{aligned}& w_{7}=\frac{2\text{,}848\text{,}355}{16}>0, \end{aligned}$$
(3.15)
$$\begin{aligned}& \frac{w_{n+1}}{(n+1)p_{1}(n+1)}-\frac{(n-7/4)w_{n}}{np_{1}(n)} \\& \quad = \frac{p_{2}(n+1)}{(n+1)p_{1}(n+1)} \biggl( \frac{1}{2} \biggr) _{n-2}- \frac{(n-7/4)p_{2}(n)}{np_{1}(n)} \biggl( \frac{1}{2} \biggr) _{n-3} \\& \quad \quad {}+\frac{5p_{3}(n+1)}{(2n-3)p_{1}(n+1)} \biggl( \frac{3}{4} \biggr) _{n-2}- \frac{5(n-7/4)p_{3}(n)}{(2n-5)p_{1}(n)} \biggl( \frac{3}{4} \biggr) _{n-3} \\& \quad = \biggl[ \frac{(n-5/2)p_{2}(n+1)}{(n+1)p_{1}(n+1)}-\frac {(n-7/4)p_{2}(n)}{np_{1}(n)} \biggr] \biggl( \frac{1}{2} \biggr) _{n-3} \\& \quad \quad {}+ \biggl[ \frac{5(n-9/4)p_{3}(n+1)}{(2n-3)p_{1}(n+1)}-\frac {5(n-7/4)p_{3}(n)}{(2n-5)p_{1}(n)} \biggr] \biggl( \frac{3}{4} \biggr) _{n-3} \\& \quad = \frac{p_{0}(n+1)p_{4}(n)}{4n(n+1)p_{1}(n)p_{1}(n+1)} \biggl( \frac {1}{2} \biggr) _{n-3} \\& \quad \quad {}+\frac{5(n-1)p_{0}(n+1)p_{5}(n)}{2(2n-3)(2n-5)p_{1}(n)p_{1}(n+1)} \biggl( \frac{3}{4} \biggr) _{n-3} \\& \quad = \frac{p_{0}(n+1)}{4p_{1}(n)p_{1}(n+1)} ( \alpha_{n}+\beta_{n} ) , \end{aligned}$$
(3.16)
$$\begin{aligned}& \frac{\alpha_{n}}{\beta_{n}}=\frac {(2n-3)(2n-5)p_{4}(n)}{10n(n-1)(n+1)p_{5}(n)} \frac{ ( \frac{1}{2} ) _{n-3}}{ ( \frac{3}{4} ) _{n-3}}, \end{aligned}$$
(3.17)
$$\begin{aligned}& \frac{\alpha_{7}}{\beta_{7}}=-\frac{7\text{,}313\text{,}056}{7\text{,}313\text{,}875}, \end{aligned}$$
(3.18)
$$\begin{aligned}& \frac{10 ( \frac{3}{4} ) _{n-3}}{ ( \frac{1}{2} ) _{n-1}} \biggl( \frac{\alpha_{n+1}}{\beta_{n+1}}-\frac{\alpha_{n}}{\beta_{n}} \biggr) \\& \quad = \frac{(2n-1)(2n-3)(n-5/2)p_{4}(n+1)}{n(n+1)(n+2)(n-9/4)p_{5}(n+1)}-\frac{(2n-3)(2n-5)p_{4}(n)}{n(n-1)(n+1)p_{5}(n)} \\& \quad = -\frac{(2n-3)(2n-5)(n-3)(n-4) (125n^{4}+1\text{,}982n^{3}+6\text{,}797n^{2}+616n-2\text{,}380 )}{n(4n-9)(n-1)(n+1)(n+2)p_{5}(n)p_{5}(n+1)} \\& \quad \quad {} \times\bigl(1\text{,}750n^{8}-802\text{,}264n^{7}-54\text{,}353\text{,}513n^{6}+811\text{,}227\text{,}431n^{5}-4\text{,}748\text{,}597\text{,}075n^{4} \\& \quad \quad {}+21\text{,}568\text{,}863\text{,}989n^{3}-80\text{,}185\text{,}046\text{,}202n^{2} \\& \quad \quad {}+144\text{,}028\text{,}552\text{,}644n-85\text{,}225\text{,}499\text{,}160 \bigr) \\& \quad = -\frac{(2n-3)(2n-5)(n-3)(n-4) ( 125n^{4}+1\text{,}982n^{3}+6\text{,}797n^{2}+616n-2\text{,}380 ) }{n(4n-9)(n-1)(n+1)(n+2)p_{5}(n)p_{5}(n+1)} \\& \quad \quad {} \times\bigl[-420\text{,}331\text{,}641\text{,}120-714\text{,}515\text{,}222\text{,}844(n-7)-475\text{,}749\text{,}995\text{,}856(n-7)^{2} \\& \quad \quad {}-152\text{,}526\text{,}681\text{,}341(n-7)^{3}-25\text{,}642\text{,}525\text{,}865(n-7)^{4} \\& \quad \quad {}-2\text{,}263\text{,}535\text{,}771(n-7)^{5} \\& \quad \quad {}-91\text{,}263\text{,}449(n-7)^{6}-704\text{,}264(n-7)^{7}+1\text{,}750(n-7)^{8} \bigr]. \end{aligned}$$
(3.19)
From Lemma 3.1 and (3.19) together with the facts that \(p_{5}(n)>0\) and \(p_{5}(n+1)>0\) for \(n\geq7\), we clearly see that there exists \(n_{0}>7\) such that the sequence \(\{\alpha_{n}/\beta _{n}\}_{n=7}^{\infty}\) is increasing for \(7\leq n\leq n_{0}\) and decreasing for \(n\geq n_{0}\), which implies that
$$ \frac{\alpha_{n}}{\beta_{n}}\geq \min \biggl\{ \frac{\alpha _{7}}{\beta_{7}}, \lim _{n\rightarrow\infty}\frac{\alpha_{n}}{\beta_{n}} \biggr\} . $$
(3.20)
It follows from Lemma 3.2 that
$$ \frac{\Gamma ( \frac{3}{4} ) ( n-\frac{5}{2} ) ^{-1/4}}{\Gamma ( \frac {1}{2} ) } < \frac{ ( \frac{1}{2} ) _{n-3}}{ ( \frac{3}{4} ) _{n-3}}=\frac{\Gamma ( \frac{3}{4} ) }{\Gamma ( \frac{1}{2} ) }\frac {\Gamma ( n-\frac{13}{4}+\frac{3}{4} ) }{\Gamma ( n-\frac{13}{4}+1 ) } < \frac{\Gamma ( \frac{3}{4} ) ( n-\frac{13}{4} ) ^{-1/4}}{\Gamma ( \frac{1}{2} ) } $$
(3.21)
for all \(n\geq7\).
From (3.10), (3.11), (3.17), (3.18), (3.20) and (3.21) we get
$$ \frac{\alpha_{n}}{\beta_{n}}\geq\frac{\alpha_{7}}{\beta_{7}}>-1 $$
(3.22)
for all \(n\geq7\).

Therefore, Lemma 3.3 follows easily from (3.14)-(3.16) and (3.22) together with the facts that \(p_{1}(n)>0\), \(p_{0}(n+1)>0\) and \(p_{1}(n+1)>0\) for \(n\geq7\). □

Theorem 3.4

The double inequality
$$ \frac{40(\pi-2)}{29}J\bigl(r^{\prime}\bigr)-\frac{51\pi-160}{58}< \mathcal{E}(r)< \frac{\pi}{2}J\bigl(r^{\prime}\bigr) $$
(3.23)
holds for all \(r\in(0, 1)\), where
$$ J\bigl(r^{\prime}\bigr)=\frac{51{r^{\prime}}^{2}+20r^{\prime}\sqrt {r^{\prime}}+50r^{\prime}+20\sqrt{r^{\prime}}+51}{16(5r^{\prime }+2\sqrt{r^{\prime}}+5)}. $$
(3.24)

Proof

Let \(r\in(0, 1)\), \(x=r^{2}\), \(P(x)\), \(f_{1}(r)\), \(f_{2}(r)\) and \(F(r)\) be defined by
$$\begin{aligned}& P(x)=1\text{,}536-1\text{,}248x, \end{aligned}$$
(3.25)
$$\begin{aligned}& f_{1}(r)= 2\text{,}592\bigl(2-r^{2}\bigr) \bigl(1-r^{2} \bigr)^{3/4}+15\bigl(425r^{4}-624r^{2}+192\bigr) \bigl(1-r^{2}\bigr)^{1/2} \\& \quad \quad {}-50\bigl(r^{4}-96r^{2}+96\bigr) \bigl(1-r^{2} \bigr)^{1/4}-105r^{4}+3\text{,}600r^{2}+2\text{,}880, \end{aligned}$$
(3.26)
$$\begin{aligned}& f_{2}(r)=625r^{4}-384r^{2}+384, \end{aligned}$$
(3.27)
$$\begin{aligned}& F(r)=\frac{1-J(r^{\prime})}{1-2\mathcal{E}(r)/\pi}, \end{aligned}$$
(3.28)
respectively.
Then from (3.1) and (3.24)-(3.28) we have
$$\begin{aligned}& f_{1}(r)= 2\text{,}880+3\text{,}600r^{2}-105r^{4}+2\text{,}592 \Biggl( 2 \sum_{n=0}^{\infty}\frac{ ( -\frac{3}{4} ) _{n}}{n!}r^{2n}- \sum_{n=1}^{\infty}\frac{ ( -\frac{3}{4} ) _{n-1}}{(n-1)!}r^{2n} \Biggr) \\& \hphantom{f_{1}(r)=}+15 \Biggl( 425\sum_{n=2}^{\infty} \frac{ ( -\frac{1}{2} ) _{n-2}}{(n-2)!}r^{2n}- 624\sum_{n=1}^{\infty} \frac{ ( -\frac{1}{2} ) _{n-1}}{(n-1)!}r^{2n} +192\sum_{n=0}^{\infty} \frac{ ( -\frac{1}{2} ) _{n}}{(n!}r^{2n} \Biggr) \\& \hphantom{f_{1}(r)=}-50 \Biggl( \sum_{n=2}^{\infty} \frac{ ( -\frac{1}{4} ) _{n-2}}{(n-2)!}r^{2n}- 96\sum_{n=1}^{\infty} \frac{ ( -\frac{1}{4} ) _{n-1}}{(n-1)!}r^{2n} +96\sum_{n=0}^{\infty} \frac{ ( -\frac{1}{4} ) _{n}}{n!}r^{2n} \Biggr) \\& \hphantom{f_{1}(r)} = 6\text{,}144-7\text{,}680r^{2}-105r^{4}+2\text{,}592\sum _{n=2}^{\infty}\frac{ ( n-\frac{7}{2} ) ( -\frac{3}{4} ) _{n-1}}{n!}r^{2n} \\& \hphantom{f_{1}(r)=}-15\sum_{n=2}^{\infty}\frac{ ( 7n^{2}-367n-720 ) ( -\frac{1}{2} ) _{n-2}}{n!}r^{2n} \\& \hphantom{f_{1}(r)=}-50\sum_{n=2}^{\infty}\frac{ ( n^{2}-121n+270 ) ( -\frac{1}{4} ) _{n-2}}{n!}r^{2n} \\& \hphantom{f_{1}(r)} = 6\text{,}144-7\text{,}680r^{2}+11\text{,}248r^{4}-1\text{,}944\sum _{n=3}^{\infty}\frac{ ( n-\frac{7}{2} ) ( \frac{1}{4} ) _{n-2}}{n!}r^{2n} \\& \hphantom{f_{1}(r)=}+\frac{15}{2}\sum_{n=3}^{\infty} \frac{ ( 7n^{2}-367n-720 ) ( \frac{1}{2} ) _{n-3}}{n!}r^{2n} \\& \hphantom{f_{1}(r)=}+\frac{25}{2}\sum_{n=3}^{\infty} \frac{ ( n^{2}-121n+270 ) ( \frac{3}{4} ) _{n-3}}{n!}r^{2n}, \\& 16f_{2}(r)-f_{1}(r) \\& \quad =1\text{,}536r^{2}-1\text{,}248r^{4}- \frac{25}{2}\sum_{n=3}^{\infty} \frac{ ( n^{2}-121n+270 ) ( \frac{3}{4} ) _{n-3}}{n!}r^{2n} \\& \quad \quad {}-\frac{15}{2}\sum_{n=3}^{\infty} \frac{ ( 7n^{2}-367n-720 ) ( \frac{1}{2} ) _{n-3}}{n!}r^{2n} +1\text{,}944\sum _{n=3}^{\infty} \frac{ ( n-\frac{7}{2} ) ( \frac{1}{4} ) _{n-2}}{n!}r^{2n} \\& \quad = 1\text{,}536r^{2}-1\text{,}248r^{4}+\sum_{n=3}^{\infty}u_{n}r^{2n}, \end{aligned}$$
where
$$\begin{aligned}& u_{n}=\frac{972(2n-7)}{n!} \biggl( \frac{1}{4} \biggr) _{n-2} \\& \quad \quad {}-\frac{15(7n^{2}-367n-720)}{2n!} \biggl( \frac{1}{2} \biggr) _{n-3} -\frac{25(n^{2}-121n+270)}{2n!} \biggl( \frac{3}{4} \biggr) _{n-3}, \\& 1-\frac{2}{\pi}\mathcal{E}(r) \\& \quad =-\sum_{n=1}^{\infty} \frac{ ( -\frac{1}{2} ) _{n} ( \frac{1}{2} ) _{n}}{(n!)^{2}}r^{2n} =\sum _{n=1}^{\infty} \frac{ ( \frac{1}{2} ) _{n-1} ( \frac{1}{2} ) _{n}}{2(n!)^{2}}r^{2n}, \\& 16f_{2}(r) \biggl( 1-\frac{2}{\pi}\mathcal{E}(r) \biggr) \\& \quad = 8 \bigl( 625r^{4}-384r^{2}+384 \bigr) \sum _{n=1}^{\infty}\frac{ ( \frac{1}{2} ) _{n-1} ( \frac{1}{2} ) _{n}}{(n!)^{2}}r^{2n} \\& \quad = 1\text{,}536r^{2}-1\text{,}248r^{4}+\sum_{n=3}^{\infty}v_{n}r^{2n}, \end{aligned}$$
(3.29)
where
$$ v_{n}=\frac{8p_{0}(n)}{(n!)^{2}} \biggl( \frac{1}{2} \biggr) _{n-3} \biggl( \frac{1}{2} \biggr) _{n-2} $$
(3.30)
and \(p_{0}(n)\) is defined by (3.9).
$$\begin{aligned}& J\bigl(r^{\prime}\bigr)={} \frac{(51{r^{\prime}}^{2}+20r^{\prime}\sqrt {r^{\prime}}+50r^{\prime}+20\sqrt{r^{\prime}}+51) (5r^{\prime}-2\sqrt{r^{\prime}}+5)(25{r^{\prime}}^{2}-46r^{\prime }+25)}{16(5r^{\prime}+2\sqrt{r^{\prime}}+5)(5r^{\prime}-2\sqrt {r^{\prime}}+5)(25{r^{\prime}}^{2}-46r^{\prime}+25)} \\& \hphantom{J\bigl(r^{\prime}\bigr) }= \frac{f_{1}(r)}{16f_{2}(r)}, \\& F(r)=\frac{16f_{2}(r)-f_{1}(r)}{16f_{2}(r) ( 1-\frac{2}{\pi}\mathcal {E}(r) ) } =\frac{P(x)+\sum_{n=2}^{\infty}u_{n+1}x^{n}}{P(x)+\sum _{n=2}^{\infty}v_{n+1}x^{n}}. \end{aligned}$$
(3.31)
It follows from (3.9), (3.29), (3.30) and elaborated computations that
$$\begin{aligned}& \frac{u_{3}}{v_{3}}=1, \end{aligned}$$
(3.32)
$$\begin{aligned}& u_{n+1}-\frac{v_{n+1}}{v_{n}}u_{n}=-\frac{15(2n-5)}{8(n+1)p_{0}(n)(n+1)!}w_{n}, \end{aligned}$$
(3.33)
where \(w_{n}\) is defined by (3.8).
It is not difficult to verify that
$$ p_{0}(n)=625n^{4}-2\text{,}018n^{3}+3\text{,}985n^{2}-4\text{,}032n+1\text{,}080>0 $$
(3.34)
for all \(n\geq3\).
From Lemma 3.3, (3.30), (3.33) and (3.34) we know that
$$ v_{n}>0 $$
(3.35)
for all \(n\geq3\), and the sequence \(\{u_{n}/v_{n}\}_{n=3}^{\infty}\) is decreasing.
Equation (3.25) implies that
$$ P(x)>0 $$
(3.36)
for \(x\in(0, 1)\), and \(P(x)\) is decreasing on \((0, 1)\).
It follows from Theorem 2.1, (3.31) and (3.32) together with the monotonicity of the sequence \(\{u_{n}/v_{n}\} _{n=3}^{\infty}\) and the function \(P(x)\) on \((0, 1)\) that the function \(F(r)\) is strictly decreasing on \((0, 1)\) and
$$ \lim_{r\rightarrow1^{-}}F(r)< F(r)< \lim_{r\rightarrow0^{+}}F(r) $$
(3.37)
for all \(r\in(0, 1)\).
Note that (3.24), (3.28) and (3.31) lead to the conclusion that
$$ \lim_{r\rightarrow1^{-}}F(r)=\frac{1-J(0^{+})}{1-2\mathcal {E}(1^{-})/\pi}=\frac{1-51/80}{1-2/\pi}= \frac{29\pi}{80(\pi-2)}, \qquad\lim_{r\rightarrow0^{+}}F(r)=1. $$
(3.38)

Therefore, Theorem 3.4 follows from (3.28), (3.37) and (3.38). □

Remark 3.5

Let
$$\begin{aligned}& \lambda_{1}(r)=\frac{\pi}{2}S_{11/4,7/4}\bigl(1, r^{\prime}\bigr), \qquad\mu_{1}(r)=\frac{11}{7}S_{11/4,7/4} \bigl(1, r^{\prime}\bigr), \end{aligned}$$
(3.39)
$$\begin{aligned}& \lambda_{2}(r)=\frac{25}{16}S_{5/2,2}\bigl(1, r^{\prime}\bigr), \qquad\mu_{2}(r)=\frac{\pi}{2}S_{5/2,2} \bigl(1, r^{\prime}\bigr), \end{aligned}$$
(3.40)
$$\begin{aligned}& \lambda(r)=\frac{40(\pi-2)}{29}J\bigl(r^{\prime}\bigr)-\frac{51\pi-160}{58}, \qquad\mu(r)=\frac{\pi}{2}J\bigl(r^{\prime}\bigr), \end{aligned}$$
(3.41)
where \(J(r^{\prime})\) is defined by (3.24). Then simple computations lead to
$$\begin{aligned}& \lambda_{1}\bigl(1^{-}\bigr)=\frac{7\pi}{22}=0.999597 \ldots, \qquad\mu_{1}\bigl(0^{+}\bigr)= \frac{11}{7}=1.571428\ldots, \end{aligned}$$
(3.42)
$$\begin{aligned}& \lambda_{2}\bigl(0^{+}\bigr)=\frac{25}{16}=1.5625, \qquad\mu_{2}\bigl(1^{-}\bigr)=\frac{2\pi}{5}=1.256637 \ldots, \end{aligned}$$
(3.43)
$$\begin{aligned}& \lambda\bigl(0^{+}\bigr)=\frac{\pi}{2}=\mu\bigl(0^{+} \bigr)=1.5707963\ldots, \end{aligned}$$
(3.44)
$$\begin{aligned}& \lambda\bigl(1^{-}\bigr)=1, \qquad\mu\bigl(1^{-}\bigr)= \frac{51\pi}{160}=1.00138\ldots. \end{aligned}$$
(3.45)

From (3.3), (3.4), (3.23) and (3.39)-(3.45) we clearly see that there exists small enough \(\delta\in(0, 1)\) such that the lower bound given in (3.23) for \(\mathcal{E}(r)\) is better than the lower bound given in (3.3) for \(r\in(\delta, 1-\delta)\), the lower bound given in (3.23) for \(\mathcal{E}(r)\) is better than the lower bound given in (3.4) for \(r\in(0, \delta)\), the upper bound given in (3.23) for \(\mathcal{E}(r)\) is better than the upper bound given in (3.3) for \(r\in(0, \delta)\), and the upper bound given in (3.23) for \(\mathcal{E}(r)\) is better than the upper bound given in (3.4) for \(r\in(\delta, 1-\delta)\).

Corollary 3.6

Let \(J(r^{\prime})\) be defined by (3.24). Then the double inequality
$$ \frac{\pi}{2}J\bigl(r^{\prime}\bigr)- \biggl( \frac{51\pi}{160}-1 \biggr) < \mathcal{E}(r)< \frac{\pi}{2}J\bigl(r^{\prime}\bigr) $$
(3.46)
holds for all \(r\in(0, 1)\).

Proof

Let \(F(r)\) be defined by (3.28) and
$$ A(r)=J\bigl(r^{\prime}\bigr)-\frac{2}{\pi}\mathcal{E}(r). $$
(3.47)
Then we clearly see that
$$\begin{aligned}& A\bigl(0^{+}\bigr)=J\bigl(1^{-}\bigr)-\frac{2}{\pi} \mathcal{E}\bigl(0^{+}\bigr)=0, \qquad A\bigl(1^{-}\bigr)=J \bigl(0^{+}\bigr)-\frac{2}{\pi}\mathcal{E}\bigl(1^{-} \bigr)=\frac{51\pi-160}{80\pi}, \end{aligned}$$
(3.48)
$$\begin{aligned}& A(r)= \biggl[ 1-\frac{2}{\pi}\mathcal{E}(r) \biggr] \biggl[ 1- \frac{1-J(r^{\prime})}{1-\frac{2}{\pi}\mathcal{E}(r)} \biggr] = \biggl[ 1-\frac{2}{\pi}\mathcal{E}(r) \biggr] \bigl[1-F(r)\bigr]. \end{aligned}$$
(3.49)

From (3.49) and the proof of Theorem 3.4 we know that \(F(r)\) is strictly decreasing on \((0, 1)\) and \(A(r)\) is strictly increasing on \((0, 1)\). Therefore, inequality (3.46) follows from (3.47) and (3.48) together with the monotonicity of \(A(r)\) on the interval \((0, 1)\). □

Corollary 3.7

Let \(J(r^{\prime})\) be defined by (3.24). Then the double inequality
$$ \frac{80}{51}J\bigl(r^{\prime}\bigr)< \mathcal{E}(r)< \frac{\pi}{2}J \bigl(r^{\prime}\bigr) $$
(3.50)
holds for all \(r\in(0, 1)\).

Proof

Let \(A(r)\) be defined by (3.47) and
$$ B(r)=\frac{J(r^{\prime})}{\frac{2}{\pi}E(r)}. $$
(3.51)
Then we clearly see that
$$\begin{aligned}& B\bigl(0^{+}\bigr)=\frac{J(1^{-})}{\frac{2}{\pi}E(0^{+})}=1, \qquad B\bigl(1^{-} \bigr)=\frac{J(0^{+})}{\frac{2}{\pi}E(1^{-})}=\frac{51\pi}{160}, \end{aligned}$$
(3.52)
$$\begin{aligned}& B(r)=\frac{1}{\frac{2}{\pi}\mathcal{E}(r)} \biggl[ J\bigl(r^{\prime }\bigr)-\frac{2}{\pi} \mathcal{E}(r) \biggr] +1=\frac{A(r)}{\frac{2}{\pi}\mathcal{E}(r)}+1. \end{aligned}$$
(3.53)

From (3.53) and the proof of Corollary 3.6 we know that both \(A(r)\) and \(B(r)\) are strictly increasing on \((0, 1)\). Therefore, inequality (3.50) follows from (3.51) and (3.52) together with the monotonicity of \(B(r)\) on the interval \((0, 1)\). □

Remark 3.8

From Corollaries 3.6 and 3.7 we have
$$ \biggl\vert \mathcal{E}(r)-\frac{\pi}{2}J\bigl(r^{\prime}\bigr) \biggr\vert < \frac{51\pi}{160}-1=0.001382\ldots,\qquad\biggl\vert \frac{\mathcal{E}(r)-\frac{\pi}{2}J(r^{\prime})}{\mathcal{E}(r)} \biggr\vert < \frac{51\pi}{160}-1=0.001382\ldots\,. $$
for all \(r\in(0, 1)\), which implies that both the absolute and relative errors using \(\pi J(r^{\prime})/2\) to approximate \(\mathcal{E}(r)\) are less than 0.14%.

4 Conclusions

In this paper, we find a monotonicity rule for the function \([P(x)+\sum _{n=n_{0}}^{\infty}a_{n}x^{n}]/[P(x)+\sum_{n=n_{0}}^{\infty }b_{n}x^{n}]\). As applications, we present new bounds for the complete elliptic integral \(\mathcal{E}(r)=\int _{0}^{\pi/2}\sqrt{1-r^{2}\sin^{2}t}\,dt\) (\(0< r<1\)) of the second kind, and we show that our bounds are sharper than the previously known bounds for some \(r\in(0, 1)\).

Declarations

Acknowledgements

The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 61374086, 11371125, 11401191), the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11626101) and the Natural Science Foundation of the Department of Education of Zhejiang Province (Grant No. Y201635325).

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)
Department of Mathematics, Huzhou University
(2)
School of Distance Education, Huzhou Broadcast and TV University
(3)
Friedman Brain Institute, Icahn School of Medicine at Mount Sinai

References

  1. Biernacki, M, Krzyż, J: On the monotonicity of certain functionals in the theory of analytic functions. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 9, 135-147 (1955) MATHGoogle Scholar
  2. Ponnusamy, S, Vuorinen, M: Asymptotic expansions and inequalities for hypergeometric functions. Mathematika 44(2), 278-301 (1997) MathSciNetView ArticleMATHGoogle Scholar
  3. Heikkala, V, Vamanamurthy, MK, Vuorinen, M: Generalized elliptic integrals. Comput. Methods Funct. Theory 9(1), 75-109 (2009) MathSciNetView ArticleMATHGoogle Scholar
  4. Cheeger, J, Gromov, M, Taylor, M: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17(1), 15-53 (1982) MathSciNetView ArticleMATHGoogle Scholar
  5. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Inequalities for quasiconformal mappings in space. Pac. J. Math. 160(1), 1-18 (1993) MathSciNetView ArticleMATHGoogle Scholar
  6. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Conformal Invariants, Inequalities, and Quasiconformal Maps. John Wiley & Sons, New York (1997) MATHGoogle Scholar
  7. Pinelis, I: L’Hospital type rules for oscillation, with applications. JIPAM. J. Inequal. Pure Appl. Math. 2(3), Article ID 33 (2001) MathSciNetMATHGoogle Scholar
  8. Yang, Z-H, Chu, Y-M, Wang, M-K: Monotonicity criterion for the quotient of power series with applications. J. Math. Anal. Appl. 428(1), 587-604 (2015) MathSciNetView ArticleMATHGoogle Scholar
  9. Balasubramanian, R, Ponnusamy, S, Vuorinen, M: Functional inequalities for the quotients of hypergeometric functions. J. Math. Anal. Appl. 218(1), 256-268 (1998) MathSciNetView ArticleMATHGoogle Scholar
  10. Anderson, GD, Qiu, S-L, Vamanamurthy, MK, Vuorinen, M: Generalized elliptic integrals and modular equations. Pac. J. Math. 192(1), 1-37 (2000) MathSciNetView ArticleGoogle Scholar
  11. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Topics in special functions. In: Papers on Analysis. Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 83, pp. 5-26. Univ. Jyväskylä, Jyväskylä (2001) Google Scholar
  12. Alzer, H, Qiu, S-L: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172(2), 289-312 (2004) MathSciNetView ArticleMATHGoogle Scholar
  13. Baricz, Á: Landen-type inequality for Bessel functions. Comput. Methods Funct. Theory 5(2), 373-379 (2005) MathSciNetView ArticleMATHGoogle Scholar
  14. Zhu, L: Sharpening Jordan’s inequality and the Yang Le inequality. Appl. Math. Lett. 19(3), 240-243 (2006) MathSciNetView ArticleMATHGoogle Scholar
  15. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Generalized convexity and inequalities. J. Math. Anal. Appl. 355(2), 1294-1308 (2007) MathSciNetView ArticleMATHGoogle Scholar
  16. Zhu, L: New inequalities for hyperbolic functions and their applications. J. Inequal. Appl. 2012, Article ID 303 (2012) MathSciNetView ArticleMATHGoogle Scholar
  17. Yang, Z-H: New sharp bounds for identric mean in terms of logarithmic mean and arithmetic mean. J. Math. Inequal. 6(4), 533-543 (2012) MathSciNetView ArticleMATHGoogle Scholar
  18. Yang, Z-H, Jiang, Y-L, Song, Y-Q, Chu, Y-M: Sharp inequalities for trigonometric functions. Abstr. Appl. Anal. 2014, Article ID 601839 (2014) MathSciNetGoogle Scholar
  19. Baricz, Á: Landen inequalities for special functions. Proc. Am. Math. Soc. 142(9), 3059-3066 (2014) MathSciNetView ArticleMATHGoogle Scholar
  20. Wang, MK, Chu, Y-M, Song, Y-Q: Asymptotical formulas for Gaussian and generalized hypergeometric functions. Appl. Math. Comput. 276, 44-60 (2016) MathSciNetGoogle Scholar
  21. Yang, Z-H, Chu, Y-M: On approximating the modified Bessel function of the second kind. J. Inequal. Appl. 2017, Article ID 41 (2017) MathSciNetView ArticleMATHGoogle Scholar
  22. Wang, M-K, Chu, Y-M: Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. 37B(3), 607-622 (2017) View ArticleGoogle Scholar
  23. Wang, M-K, Li, Y-M, Chu, Y-M: Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J. doi:https://doi.org/10.1007/s11139-017-9888-3
  24. Chavel, I: Riemannian Geometry - A Modern Introduction. Cambridge University Press, Cambridge (1993) MATHGoogle Scholar
  25. Pinelis, I: L’Hospital type rules for monotonicity: applications to probability inequalities for sums of bounded random variables. J. Inequal. Pure Appl. Math. 3(1), Article ID 7 (2002) MathSciNetMATHGoogle Scholar
  26. Pinelis, I: Monotonicity properties of the relative error of a Padé approximation for Mills ratio. JIPAM. J. Inequal. Pure Appl. Math. 3(2), Article ID 20 (2002) MathSciNetMATHGoogle Scholar
  27. Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U. S. Government Printing Office, Washington (1964) MATHGoogle Scholar
  28. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23(2), 512-524 (1992) MathSciNetView ArticleMATHGoogle Scholar
  29. 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 (2001) MathSciNetMATHGoogle Scholar
  30. Chu, Y-M, Wang, M-K, Qiu, S-L: Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Soc. 122(1), 41-51 (2012) View ArticleMATHGoogle Scholar
  31. Chu, Y-M, Wang, M-K: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3-4), 223-229 (2012) MathSciNetView ArticleMATHGoogle Scholar
  32. 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(7), 1177-1184 (2012) MathSciNetView ArticleMATHGoogle Scholar
  33. 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) MathSciNetView ArticleMATHGoogle Scholar
  34. Wang, M-K, Chu, Y-M: Asymptotical bounds for the complete elliptic integrals of the second kind. J. Math. Anal. Appl. 402(1), 119-126 (2013) MathSciNetView ArticleMATHGoogle Scholar
  35. Song, Y-Q, Jiang, W-D, Chu, Y-M, Yan, D-D: Optimal bounds for Toader mean in terms of arithmetic and contraharmonic means. J. Math. Inequal. 7(4), 751-757 (2013) MathSciNetView ArticleMATHGoogle Scholar
  36. Wang, M-K, Chu, Y-M, Jiang, Y-P, Qiu, S-L: Bounds of the perimeter of an ellipse using arithmetic, geometric and harmonic means. Math. Inequal. Appl. 17(1), 101-111 (2014) MathSciNetMATHGoogle Scholar
  37. Li, Y, Qi, F: Some inequalities for complete elliptic integrals. Appl. Math. E-Notes 14, 193-199 (2014) MathSciNetMATHGoogle Scholar
  38. Hua, Y, Qi, F: The best bounds for Toader mean in terms of the centroidal and arithmetic means. Filomat 28(4), 775-780 (2014) MathSciNetView ArticleMATHGoogle Scholar
  39. Hua, Y, Qi, F: A double inequality for bounding Toader mean by the centroidal mean. Proc. Indian Acad. Sci. Math. Sci. 124(4), 527-531 (2014) MathSciNetView ArticleMATHGoogle Scholar
  40. Jiang, W-D, Qi, F: A double inequality for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean. Publ. Inst. Math. 99(113), 237-242 (2016) MathSciNetView ArticleGoogle Scholar
  41. Yang, Z-H, Chu, Y-M, Zhang, W: Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean. J. Inequal. Appl. 2016, Article ID 176 (2016) MathSciNetView ArticleMATHGoogle Scholar
  42. 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
  43. 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) MathSciNetView ArticleMATHGoogle Scholar
  44. Yang, Z-H, Chu, Y-M, Zhang, W: Accurate approximations for the complete elliptic integral of the second kind. J. Math. Anal. Appl. 438(2), 875-888 (2016) MathSciNetView ArticleMATHGoogle Scholar
  45. Yang, Z-H, Chu, Y-M, Zhang, X-H: Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind. J. Nonlinear Sci. Appl. 10(3), 929-936 (2017) View ArticleGoogle Scholar
  46. Yang, Z-H, Chu, Y-M, Tao, X-J: A double inequality for the trigamma function and its applications. Abstr. Appl. Anal. 2014, Article ID 702718 (2014) MathSciNetGoogle Scholar
  47. Wendel, JG: Note on the gamma function. Am. Math. Mon. 55, 563-564 (1948) MathSciNetView ArticleGoogle Scholar
  48. Qi, F: Bounds for the ratio of two gamma functions. J. Inequal. Appl. 2010, Article ID 493058 (2010) MathSciNetView ArticleMATHGoogle Scholar

Copyright

© The Author(s) 2017