Open Access

Monotonicity, convexity, and inequalities for the generalized elliptic integrals

Journal of Inequalities and Applications20172017:278

https://doi.org/10.1186/s13660-017-1556-z

Received: 14 August 2017

Accepted: 2 November 2017

Published: 9 November 2017

Abstract

We provide the monotonicity and convexity properties and sharp bounds for the generalized elliptic integrals \(\mathscr{K}_{a}(r)\) and \(\mathscr {E}_{a}(r)\) depending on a parameter \(a\in(0,1)\), which contains an earlier result in the particular case \(a=1/2\).

Keywords

generalized elliptic integrals of the first and second kindsGaussian hypergeometric functionmonotonicityconvexityinequality

MSC

33C0533E0526D20

1 Introduction

For real numbers a, b, and c with \(c\neq0,-1,-2,\ldots \) , the Gaussian hypergeometric function is defined by
$$\begin{aligned} F(a,b;c;x)= {}_{2}F_{1}(a,b;c;x)=\sum _{n=0}^{\infty}\frac {(a,n)(b,n)}{(c,n)}\frac{x^{n}}{n!} \end{aligned}$$
(1.1)
for \(x\in(-1,1)\), where \((a,n)\) denotes the shifted factorial function \((a,n)\equiv a(a+1)\cdots(a+n-1)\), \(n=1,2,\ldots \) , and \((a,0)=1\) for \(a\neq0\). It is well known that the function \(F(a, b; c; x)\) has many important applications in geometric function theory, theory of mean values, and several other contexts, and many classes of elementary functions and special functions in mathematical physics are particular or limiting cases of this function [110].
In what follows, we suppose \(r\in(0,1)\), \(a\in(0,1)\), and \(r'=\sqrt {1-r^{2}}\). The generalized elliptic integrals of the first and second kinds are defined as
$$\begin{aligned} &\mathscr{K}_{a}(r)=\frac{\pi}{2}F \bigl(a,1-a;1;r^{2}\bigr),\qquad\mathscr {K}'_{a}(r)= \mathscr{K}_{a}\bigl(r'\bigr), \end{aligned}$$
(1.2)
$$\begin{aligned} &\mathscr{E}_{a}(r)=\frac{\pi}{2}F\bigl(a-1,1-a;1;r^{2} \bigr),\qquad\mathscr {E}'_{a}(r)=\mathscr{E}_{a} \bigl(r'\bigr). \end{aligned}$$
(1.3)
In the particular case \(a=1/2\), the generalized elliptic integrals \(\mathscr{K}_{a}(r)\) and \(\mathscr{E}_{a}(r)\) reduce to the complete elliptic integrals \(\mathscr{K}(r)\) and \(\mathscr{E}(r)\), respectively. Recently, the Gaussian hypergeometric function and generalized elliptic integrals have been the subject of intensive research [2, 3, 5, 8, 1130].
Anderson, Qiu, and Vamanamurthy [31] considered the monotonicity and convexity of the function
$$\begin{aligned} f(r)=\frac{\mathscr{E}(r)-r^{\prime 2}\mathscr{K}(r)}{r^{2}}\cdot\frac {r^{\prime 2}}{\mathscr{E'}(r)-r^{2}\mathscr{K'}(r)}. \end{aligned}$$
One of the main results of [31] is the following theorem.

Theorem 1.1

The function \(f(r)\) is increasing and convex from \((0,1)\) onto \((\pi /4,4/\pi)\). In particular,
$$\begin{aligned} \frac{\pi}{4}< f(r)< \frac{\pi}{4}+ \biggl(\frac{4}{\pi}- \frac{\pi }{4} \biggr)r \end{aligned}$$
(1.4)
for \(r\in(0,1)\). Both inequalities given in (1.4) are sharp as \(r\rightarrow0\), whereas the second inequality is also sharp as \(r\rightarrow1\).
Alzer and Richards [32] studied the corresponding properties of the additive counterpart
$$\begin{aligned} \Delta(r)&=\frac{\mathscr{E}(r)-r^{\prime 2}\mathscr{K}(r)}{r^{2}}-\frac {\mathscr{E'}(r)-r^{2}\mathscr{K'}(r)}{r^{\prime 2}} \end{aligned}$$
and obtained the following theorem.

Theorem 1.2

The function \(\Delta(r)\) is strictly increasing and strictly convex from \((0,1)\) onto \((\pi/4-1,1-\pi/4)\). Moreover, for all \(r\in(0,1)\), we have
$$\begin{aligned} \frac{\pi}{4}-1+\alpha r< \Delta(r)< \frac{\pi}{4}-1+\beta r \end{aligned}$$
(1.5)
with the best constants \(\alpha=0\) and \(\beta=2-\frac{\pi}{2}\).
It is natural to extend Theorems 1.1 and 1.2 to the generalized elliptic integrals \(\mathscr{K}_{a}(r)\) and \(\mathscr{E}_{a}(r)\). In this paper, we show the monotonicity and convexity of the functions
$$\begin{aligned} f_{a}(r)=\frac{\mathscr{E}_{a}(r)-r^{\prime 2}\mathscr{K}_{a}(r)}{r^{2}}\cdot\frac {r^{\prime 2}}{\mathscr{E'}_{a}(r)-r^{2}\mathscr{K'}_{a}(r)} \end{aligned}$$
(1.6)
and
$$\begin{aligned} g_{a}(r)=\frac{\mathscr{E}_{a}(r)-r^{\prime 2}\mathscr{K}_{a}(r)}{r^{2}}-\frac {\mathscr{E'}_{a}(r)-r^{2}\mathscr{K'}_{a}(r)}{r^{\prime 2}}. \end{aligned}$$
(1.7)
Moreover, we obtain sharp inequalities for them. If \(a=1/2\), then our results return to Theorems 1.1 and 1.2, which are contained in [31] and [32].

2 Preliminaries and lemmas

In this section, we give several formulas and lemmas to establish our main results stated in Section 1. First, let us recall some known results for \(F(a,b;c;x)\).

The following formulas for the hypergeometric function can be found in the literature [3335]:
$$\begin{aligned} F(a,b;a+b+1;x)=(1-x)F(a+1,b+1;a+b+1;x), \end{aligned}$$
(2.1)
the differential formula
$$\begin{aligned} \frac{dF(a,b;c;x)}{dx}=\frac{ab}{c}F(a+1,b+1;c+1;x), \end{aligned}$$
(2.2)
the asymptotic limit
$$\begin{aligned} \lim_{x\rightarrow1^{-}}F(a,b;c;x)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma (c-a)\Gamma(c-b)},\quad c>a+b, \end{aligned}$$
(2.3)
and the contiguous relation
$$\begin{aligned} (\sigma-\rho)F(\alpha,\rho;\sigma+1;z)=\sigma F(\alpha,\rho;\sigma ;z)-\rho F(\alpha,\rho+1;\sigma+1;z), \end{aligned}$$
(2.4)
where \(\Gamma(x)\) is the Euler gamma function.

Lemma 2.1

([2], Lemma 5.2)

Let \(a\in(0,1]\). Then the function \([\mathscr{E}_{a}(r)-r^{\prime 2}\mathscr {K}_{a}(r)]/r^{2}\) is increasing and convex from \((0,1)\) onto \(({\pi a}/{2},[\sin(\pi a)]/[2(1-a)])\).

The following formulas were presented in [2]:
$$\begin{aligned} &\frac{d\mathscr {K}_{a}(r)}{dr}=\frac{2(1-a)(\mathscr {E}_{a}(r)-r^{\prime 2}\mathscr {K}_{a}(r))}{rr^{\prime 2}},\qquad \frac{d\mathscr {E}_{a}(r)}{dr}= \frac{2(1-a)(\mathscr {K}_{a}(r)-{\mathscr {E}_{a}}(r))}{r}, \end{aligned}$$
(2.5)
$$\begin{aligned} &\frac{d(\mathscr {E}_{a}(r)-r^{\prime 2}\mathscr {K}_{a}(r))}{dr}=2ar\mathscr {K}_{a}(r), \qquad\frac{d(\mathscr {K}_{a}(r)-\mathscr {E}_{a}(r))}{dr}= \frac{2(1-a)r \mathscr {E}_{a}(r)}{r^{\prime 2}} . \end{aligned}$$
(2.6)

Lemma 2.2

([2], Lemma 2.3)

Let \(I\subset\mathbb{R}\) be an interval, and let \(f,g:I\rightarrow (0,\infty)\). If both f, g are convex and increasing (decreasing), then the product \(f\cdot g\) is convex.

The following lemma follows from Theorem 1.7 in [1].

Lemma 2.3

For all \(a,b\in(0,\infty)\), the function
$$\begin{aligned} I(x)=(1-x)^{4}F(a,b;a+b;x) \end{aligned}$$
(2.7)
is a strictly decreasing automorphism of \((0,1)\) if and only if \(4ab\leq a+b \).

Lemma 2.4

The function
$$\begin{aligned} J(r)=\frac{(1-r^{2}) (\mathscr{E}_{a}(r)-((4a-1)r^{2}+1)\mathscr {K}_{a}(r) )}{4a(1-a)r^{3}} \end{aligned}$$
(2.8)
is increasing from \((0,1)\) onto \((-\infty,0)\).

Proof

Let
$$\begin{aligned} f_{1}(x)=(1-x) \bigl(F(a-1,1-a;1;x)-\bigl((4a-1)x+1\bigr)F(a,1-a;1;x) \bigr). \end{aligned}$$
By the series expansion for \(F(a,b;c;x)\) we have
$$\begin{aligned} f_{1}(x) &=(1-x) \Biggl(\sum_{n=0}^{\infty} \frac{(a-1,n)(1-a,n)}{n!}\cdot\frac {x^{n}}{n!}- \bigl((4a-1)x+1 \bigr)\sum _{n=0}^{\infty}\frac {(a,n)(1-a,n)}{n!}\cdot\frac{x^{n}}{n!} \Biggr) \\ &=(1-x) \Biggl(\sum_{n=0}^{\infty} \biggl( \frac{(a-1,n)(1-a,n)}{n!}-\frac {(a,n)(1-a,n)}{n!} \biggr)\cdot\frac{x^{n}}{n!} \\ &\quad {} -\sum _{n=0}^{\infty}\frac{(4a-1)(a,n)(1-a,n)}{n!}\cdot \frac {x^{n+1}}{n!} \Biggr) \\ &=(1-x)\sum_{n=1}^{\infty}\frac {(a,n-1)(1-a,n-1)}{n!n!} \bigl(-4an^{2}+an\bigr)x^{n} \\ &=\sum_{n=1}^{\infty}\frac {(a,n-1)(1-a,n-1)}{n!n!} \bigl(-4an^{2}+an\bigr)x^{n} \\ &\quad {}-\sum _{n=1}^{\infty}\frac {(a,n-1)(1-a,n-1)}{n!n!}\bigl(-4an^{2}+an \bigr)x^{n+1} \\ &=-3ax \\ &\quad {}+\sum_{n=2}^{\infty}\frac{(a,n-2)(1-a,n-2)}{n!n!} an \bigl(\bigl(4n+4a^{2}-4a-6\bigr)n+a+2-a^{2} \bigr)x^{n}. \end{aligned}$$
(2.9)
By the definition of the generalized elliptic integrals of the first and second kinds (1.2) we have
$$\begin{aligned} J(r)&=\frac{\frac{\pi}{2}f_{1}(r^{2})}{4a(1-a)r^{3}} \\ &=\frac{\pi}{8a(1-a)} \Biggl(-\frac{3a}{r}\\ &\quad {}+\sum _{n=2}^{\infty}\frac {(a,n-2)(1-a,n-2)}{n!n!}an \bigl( \bigl(4n+4a^{2}-4a-6\bigr)n+a+2-a^{2} \bigr)r^{2n-3} \Biggr). \end{aligned}$$
Since \(0< a<1\), \(n\geq2\), we have \((4n+4a^{2}-4a-6)n+a+2-a^{2}>0\), and hence \(J(r)\) is an increasing function on \((0,1)\). From this formula it is easy to see that \(\lim_{r\rightarrow0^{+}}J(r)=-\infty\). By Lemma 2.3 we have that \(\lim_{r\rightarrow1^{-1}}J(r)=0\). □

Lemma 2.5

([6], Lemma 2.1)

For \(-\infty< a< b<\infty\), let \(f,g: [a,b]\rightarrow R\) be continuous on \([a, b]\) and differentiable on \((a, b)\). Let \(g'(x) \neq0\) on \((a, b)\). If \(f'(x)/g'(x)\) is increasing (decreasing) on \((a, b)\), then so are
$$\begin{aligned} \frac{f(x)-f(a)}{g(x)-g(a)} \quad\textit{and}\quad\frac{f(x)-f(b)}{g(x)-g(b)}. \end{aligned}$$
If \(f'(x)/g'(x)\) is strictly monotone, then the monotonicity in the conclusion is also strict.

3 Main results and proofs

In this section, we present and prove two main theorems.

Theorem 3.1

The function \(f_{a}(r)\) in (1.6) is increasing and convex from \((0,1)\) onto \((\frac{ \pi a(1-a)}{\sin(\pi a)}, [4]\frac{\sin(\pi a)}{\pi a(1-a)} )\). In particular,
$$\begin{aligned} \frac{\pi a(1-a)}{\sin(\pi a)}+\alpha r< f_{a}(r)< \frac{\pi a(1-a)}{\sin (\pi a)}+ \biggl( \frac{\sin(\pi a)}{\pi a(1-a)}-\frac{\pi a(1-a)}{\sin (\pi a)} \biggr)r \end{aligned}$$
(3.1)
for \(r\in(0,1)\) with the best constant \(\alpha=0\), \(\beta=\frac{\sin(\pi a)}{\pi a(1-a)}-\frac{\pi a(1-a)}{\sin(\pi a)}\). These two inequalities are sharp as \(r\rightarrow0\), whereas the second inequality is sharp as \(r\rightarrow1\).

Proof

Let
$$\begin{aligned} f^{1}_{a}(r)=\frac{\mathscr{E}_{a}(r)-r^{\prime 2}\mathscr{K}_{a}(r)}{r^{2}}. \end{aligned}$$
Then
$$\begin{aligned} f_{a}(r)=\frac{\mathscr{E}_{a}(r)-r^{\prime 2}\mathscr{K}_{a}(r)}{r^{2}}\cdot\frac {r^{\prime 2}}{\mathscr{E'}_{a}(r)-r^{2}\mathscr{K'}_{a}(r)}=f^{1}_{a}(r) \cdot\frac {1}{f^{1}_{a}(r')}. \end{aligned}$$
By Lemma 2.1, \(f_{a}^{1}(r)\), \(1/f^{1}_{a}(r')\) are positive increasing functions on \((0,1)\), and hence \(f_{a}(r)\) is also an increasing function on \((0,1)\). Since \(f_{a}^{1}(r)\) is a convex function by Lemma 2.1, the desired convexity of \(f_{a}(r)\) will follow from Lemma 2.2 if we prove that \(1/f^{1}_{a}(r')\) is a convex function on \((0,1)\).
According to (2.6), we have
$$\begin{aligned} \biggl(\frac{1}{f_{a}^{1}(r)} \biggr)'= \biggl(\frac{r^{2}}{\mathscr {E}_{a}(r)-r^{\prime 2}\mathscr{K}_{a}(r)} \biggr)'=\frac{g_{1}(r)}{g_{2}(r)}, \end{aligned}$$
where
$$\begin{aligned} g_{1}(r)=2 \bigl(\mathscr{E}_{a}(r)-\mathscr{K}_{a}(r)+(1-a)r^{2} \mathscr {K}_{a}(r) \bigr),\qquad g_{2}(r)=\frac{ (\mathscr{E}_{a}(r)-r^{\prime 2}\mathscr{K}_{a}(r) )^{2}}{r}. \end{aligned}$$
Obviously, \(g_{1}(0^{+})=0\). By Lemma 2.1 we get \(g_{2}(0^{+})=0\). Moreover,
$$\begin{aligned} \frac{g'_{1}(r)}{g'_{2}(r)}=\frac{4a(1-a)r^{3}}{(1-r^{2}) (\mathscr {E}_{a}(r)-((4a-1)r^{2}+1)\mathscr{K}_{a}(r) )}=\frac{1}{J(r)}, \end{aligned}$$
where \(J(r)\) is defined by (2.8). Hence, by Lemma 2.4 and Lemma 2.5, \(({1}/{f_{a}^{1}(r)} )'\) is decreasing, so that \(({1}/{f_{a}^{1}(r')} )'\) is increasing, and \({1}/{f_{a}^{1}(r')}\) is convex on \((0,1)\). □

Theorem 3.2

The function \(g_{a}(r)\) in (1.7) is strictly increasing and strictly convex from \((0,1)\) onto \((\frac{\pi a}{2}-\frac{\sin(\pi a)}{2(1-a)},\frac{\sin(\pi a)}{2(1-a)}-\frac{\pi a}{2} )\). Moreover, for all \(r\in(0,1)\), we have
$$\begin{aligned} \frac{\pi a}{2}-\frac{\sin(\pi a)}{2(1-a)}+\alpha r< g_{a}(r)< \frac{\pi a}{2}-\frac{\sin(\pi a)}{2(1-a)}+\beta r \end{aligned}$$
(3.2)
with the best constants \(\alpha=0\) and \(\beta=\frac{\sin(\pi a)}{1-a}-\pi a\). These two inequalities are sharp as \(r\rightarrow0\), whereas the second inequality is sharp as \(r\rightarrow1\).

Proof

Let
$$\begin{aligned} M_{a}(r)=\frac{\pi}{2r^{2}} \bigl(F\bigl(a-1,1-a;1;r^{2} \bigr)-r^{\prime 2}F\bigl(a,1-a;1;r^{2}\bigr) \bigr). \end{aligned}$$
By the series expansion for \(F(a,b;c;x)\) we obtain
$$\begin{aligned} M_{a}(r)=\frac{a\pi}{2}F\bigl(a,1-a;2;r^{2} \bigr). \end{aligned}$$
(3.3)
Then
$$\begin{aligned} g_{a}(r)=M_{a}(r)-M_{a} \bigl(r'\bigr)=\frac{a\pi}{2} \bigl(F\bigl(a,1-a;2;r^{2} \bigr)-F\bigl(a,1-a;2;1-r^{2}\bigr) \bigr). \end{aligned}$$
(3.4)
Using the differentiation formula (2.2), we have
$$\begin{aligned} g'_{a}(r)&=\frac{a^{2}(1-a)\pi}{2}r \bigl(F \bigl(a+1,2-a;3;r^{2}\bigr)+F\bigl(a+1,2-a;3;1-r^{2}\bigr) \bigr), \end{aligned}$$
(3.5)
$$\begin{aligned} g''_{a}(r)&=\frac{a^{2}(1-a)\pi}{2} \biggl(F \bigl(a+1,2-a;3;r^{2}\bigr)+F\bigl(a+1,2-a;3;1-r^{2}\bigr) \\ &\quad{}+\frac{2(a+1)(2-a)}{3}r^{2} \bigl(F\bigl(a+2,3-a;4;r^{2} \bigr)-F\bigl(a+2,3-a;4;1-r^{2}\bigr) \bigr) \biggr) . \end{aligned}$$
(3.6)
By formula (2.1),we get
$$\begin{aligned} F\bigl(a+2,3-a;4;1-r^{2}\bigr)=\frac{1}{r^{2}}F \bigl(a+1,2-a;4;1-r^{2}\bigr). \end{aligned}$$
(3.7)
Using the contiguous relation (2.4), we take \(\alpha=a+1\), \(\rho=2-a\), \(\sigma=3\), and \(z=1-r^{2}\) and obtain
$$\begin{aligned} &(a+1)F\bigl(a+1,2-a;4;1-r^{2}\bigr)\\ &\quad =3F\bigl(a+1,2-a;3;1-r^{2} \bigr)-(2-a)F\bigl(a+1,3-a;4;1-r^{2}\bigr). \end{aligned}$$
Hence, it follows from (3.6), (3.7), and the last formula that
$$\begin{aligned} &\frac{2}{a^{2}(1-a)\pi }g''_{a}(r) \\ &\quad =F \bigl(a+1,2-a;3;r^{2}\bigr)+F\bigl(a+1,2-a;3;1-r^{2}\bigr) \\ &\qquad{}+\frac{2(a+1)(2-a)}{3}r^{2} \biggl(F\bigl(a+2,3-a;4;r^{2} \bigr)-\frac {F(a+2,3-a;4;1-r^{2})}{r^{2}} \biggr) \\ &\quad =F\bigl(a+1,2-a;3;r^{2}\bigr)+\frac {2(a+1)(2-a)r^{2}}{3}F \bigl(a+2,3-a;4;r^{2}\bigr) \\ &\qquad{}+(2a-3)F\bigl(a+1,2-a;3;1-r^{2}\bigr)+\frac {2(2-a)^{2}}{3}F \bigl(a+1,3-a;4;1-r^{2}\bigr) \\ &\quad >1+(2a-3)F\bigl(a+1,2-a;3;1-r^{2}\bigr)+\frac {2(2-a)^{2}}{3}F \bigl(a+1,3-a;4;1-r^{2}\bigr). \end{aligned}$$
By the series expansion for \(F(a,b;c;x)\) we have
$$\begin{aligned} &(2a-3)F\bigl(a+1,2-a;3;1-r^{2}\bigr)+\frac {2(2-a)^{2}}{3}F \bigl(a+1,3-a;4;1-r^{2}\bigr) \\ &\quad =\sum_{n=0}^{\infty} \biggl(\frac {2(a+1,n)(3-a,n)(2-a)^{2}}{(3,n+1)}+ \frac {(a+1,n)(2-a,n)(2a-3)}{(3,n)} \biggr)\frac{(1-r^{2})^{n}}{n!} \\ &\quad =\sum_{n=0}^{\infty}\frac{(2-a,n)(n+2a^{2}-2a-1)(a+1,n)}{(3,n+1)} \frac {(1-r^{2})^{n}}{n!} \\ &\quad>\frac{2a^{2}-2a-1}{3}+\frac{a(2-a)(a^{2}-1)}{6}\bigl(1-r^{2}\bigr). \end{aligned}$$
(3.8)
Hence
$$\begin{aligned} \frac{2}{a^{2}(1-a)\pi}g''_{a}(r)>1+ \frac{2a^{2}-2a-1}{3}+\frac {a(2-a)(a^{2}-1)}{6}\bigl(1-r^{2}\bigr). \end{aligned}$$
(3.9)
Through direct calculation we have
$$\begin{aligned} &1+\frac{2a^{2}-2a-1}{3}+\frac{a(2-a)(a^{2}-1)}{6} \\ &\quad =\frac {-a^{4}+2a^{3}+5a^{2}-6a+4}{6}>0 ,\quad \forall a \in(0,1). \end{aligned}$$
(3.10)
Then we get \(g''_{a}(r)>0\). Thus \(g_{a}(r)\) is strictly convex on \((0,1)\). According to (3.3) and (2.3), we have
$$\begin{aligned} M_{a}(0)=\frac{a\pi}{2},\qquad M'_{a}(0)=0, \qquad\lim_{r\rightarrow1^{-}}M_{a}(r)=\frac{\sin(\pi a)}{2(1-a)}. \end{aligned}$$
(3.11)
Applying Lemma 2.3 and (2.6), we have
$$\begin{aligned} g'_{a}(0)=\lim_{r\rightarrow0} \frac{M_{a}(r)-M_{a}(0)}{r}-\frac {M_{a}(r')-M_{a}(1)}{r} =\lim_{x\rightarrow1} \frac{x'}{x^{4}} \bigl(\bigl(2-x^{2}\bigr)\mathscr {K}_{a}(x)-2\mathscr {E}_{a}(x) \bigr)=0 . \end{aligned}$$
Because of \(g''_{a}(r)>0\), \(g'_{a}(r)\) is increasing on \((0,1)\), and \(g'_{a}(0)=0\). Then the monotonicity of \(g_{a}(r)\) on \((0,1)\) is obtained. It follows from the convexity of \(g_{a}(r)\) that, for \(x\in(0, 1)\),
$$\begin{aligned} \frac{\pi a}{2}-\frac{\sin(\pi a)}{2(1-a)}< g_{a}(r)< \frac{\pi a}{2}- \frac{\sin(\pi a)}{2(1-a)}+ \biggl(\frac{\sin(\pi a)}{1-a}-\pi a \biggr)r. \end{aligned}$$
(3.12)
 □

Corollary 3.3

Let
$$\begin{aligned} L_{a}(p,q)=g_{a}(pq)-g_{a}(p)-g_{a}(q). \end{aligned}$$
(3.13)
Then we have
$$\begin{aligned} \frac{\pi a}{2}-\frac{\sin(\pi a)}{2(1-a)}< L_{a}(p,q)< \frac{\sin(\pi a)}{2(1-a)}- \frac{a\pi}{2} \end{aligned}$$
(3.14)
for all \(p,q\in(0,1)\).

Proof

By direct calculation we obtain
$$\begin{aligned} \frac{\partial}{\partial p}L_{a}(p,q)=sg'_{a}(pq)-g'_{a}(p), \qquad \frac{\partial^{2}}{\partial p\,\partial q}L_{a}(p,q)=g'_{a}(pq)+pqg''_{a}(pq). \end{aligned}$$
Considering the positivity of \(g'_{a}\) and \(g''_{a}\) on \((0,1)\), we have
$$\begin{aligned} \frac{\partial^{2}}{\partial p\,\partial q}L_{a}(p,q)>0, \end{aligned}$$
This means that \(\frac{\partial}{\partial p}L_{a}(p,q)\) is strictly increasing with respect to q. So we have
$$\begin{aligned} \frac{\partial}{\partial p}L_{a}(p,q)< \frac{\partial}{\partial p}L_{a}(p,q)\bigg|_{q=1}=0. \end{aligned}$$
(3.15)
Then the monotonicity of \(L_{a}(p,q)\) with respect to p is obtained, which leads to
$$\begin{aligned} \frac{\pi a}{2}-\frac{\sin(\pi a)}{2(1-a)}< L_{a}(p,q)< \frac{\sin(\pi a)}{2(1-a)}- \frac{a\pi}{2}. \end{aligned}$$
 □

Remark 3.4

Taking \(a= 1/2\) in Theorems 3.1 and 3.2, we get Theorems 1.1 and 1.2.

Declarations

Acknowledgements

This work was completed with the support of National Natural Science Foundation of China (No. 11401531, No. 11601485), the Natural Science Foundation of Zhejiang Province (No. Q17A010038), the Science Foundation of Zhejiang Sci-Tech University (ZSTU) (No. 14062093-Y), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 17KJD110004).

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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, Zhejiang Sci-Tech University
(2)
Taizhou Institute of Sci. & Tech., NUST

References

  1. Anderson, GD, Barnard, RW, Richards, KC, Vamanamurthy, MK, Vuorinen, M: Inequalities for zero-balanced hypergeometric functions. Trans. Am. Math. Soc. 347, 1713-1723 (1995) View ArticleMATHGoogle Scholar
  2. Anderson, GD, Qiu, SL, Vamanamurthy, MK, Vuorinen, M: Generalized elliptic integrals and modular equations. Pac. J. Math. 192, 1-37 (2000) View ArticleGoogle Scholar
  3. Baricz, Á: Turán type inequalities for generalized complete elliptic integrals. Math. Z. 256, 895-911 (2007) View ArticleMATHGoogle 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, 693-699 (2000) View ArticleMATHGoogle Scholar
  5. Neuman, E: Inequalities and bounds for generalized complete elliptic integrals. J. Math. Anal. Appl. 373, 203-213 (2011) View ArticleMATHGoogle Scholar
  6. Ponnusamy, S, Vuorinen, M: Asymptotic expansions and inequalities for hypergeometric functions. Mathematika 44, 278-301 (1997) View ArticleMATHGoogle Scholar
  7. Ponnusamy, S, Vuorinen, M: Univalence and convexity properties for Gaussian hypergeometric functions. Rocky Mt. J. Math. 31, 327-353 (2001) View ArticleMATHGoogle Scholar
  8. Yang, ZH, Chu, YM, Wang, M-K: Monotonicity criterion for the quotient of power series with applications. J. Math. Anal. Appl. 428, 587-604 (2015) View ArticleMATHGoogle Scholar
  9. Sun, MB, Yang, XP: Inequalities for the weighted mean of r-convex functions. Proc. Am. Math. Soc. 133(6), 1639-1646 (2005) View ArticleMATHGoogle Scholar
  10. Sun, MB, Yang, XP: Generalized Hadamard’s inequality and r-convex functions in Carnot groups. J. Math. Anal. Appl. 294(2), 387-398 (2004) View ArticleMATHGoogle Scholar
  11. Qiu, SL, Vuorinen, M: Duplication inequalities for the ratios of hypergeometric functions. Forum Math. 12, 109-133 (2000) MATHGoogle Scholar
  12. Heikkala, V, Linden, H, Vamanamurthy, MK, Vuorinen, M: Generalized elliptic integrals and the Legendre M-function. J. Math. Anal. Appl. 338(1), 223-243 (2008) View ArticleMATHGoogle Scholar
  13. Wang, MK, Chu, YM, Qiu, SL: Sharp bounds for generalized elliptic integrals of the first kind. J. Math. Anal. Appl. 429, 744-757 (2015) View ArticleMATHGoogle Scholar
  14. Chu, YM, Wang, MK, Qiu, YF: On Alzer and Qiu’s conjecture for complete elliptic integral and inverse hyperbolic tangent function. Abstr. Appl. Anal. 2011, Article ID 697547 (2011) MATHGoogle Scholar
  15. Chu, YM, Wang, MK, Qiu, SL, Jiang, YP: Bounds for complete elliptic integrals of the second kind with applications. Comput. Math. Appl. 63(7), 1177-1184 (2012) View ArticleMATHGoogle Scholar
  16. Chu, YM, Wang, MK, Qiu, YF: Hölder mean inequalities for the complete elliptic integrals. Integral Transforms Spec. Funct. 23(7), 521-527 (2012) View ArticleMATHGoogle Scholar
  17. Song, YQ, Zhou, PG, Chu, YM: Inequalities for the Gaussian hypergeometric function. Sci. China Math. 57(11), 2369-2380 (2014) View ArticleMATHGoogle Scholar
  18. Wang, GD, Zhang, XH, Chu, YM: Inequalities for the generalized elliptic integrals and modular functions. J. Math. Anal. Appl. 331(2), 1275-1283 (2007) View ArticleMATHGoogle Scholar
  19. Wang, GD, Zhang, XH, Chu, YM: A power mean inequality involving the complete elliptic integrals. Rocky Mt. J. Math. 44(5), 1661-1667 (2014) View ArticleMATHGoogle Scholar
  20. Wang, MK, Li, YM, Chu, YM: Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J. (2017). doi:10.1007/s11139-017-9888-3 Google Scholar
  21. Wang, MK, Chu, YM: Asymptotical bounds for complete elliptic integrals of the second kind. J. Math. Anal. Appl. 402(1), 119-126 (2013) View ArticleMATHGoogle Scholar
  22. Wang, MK, Chu, YM, Qiu, YF, Qiu, SL: An optimal power mean inequality for the complete elliptic integrals. Appl. Math. Lett. 24(6), 887-890 (2011) View ArticleMATHGoogle Scholar
  23. Wang, MK, Chu, YM, Jiang, YP: Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions. Rocky Mt. J. Math. 46(2), 679-691 (2016) View ArticleMATHGoogle Scholar
  24. Wang, MK, Chu, YM: Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. 37B(3), 607-622 (2017) View ArticleGoogle Scholar
  25. Wang, MK, Chu, YM, Song, YQ: Asymptotical formulas for Gaussian and generalized hypergeometric functions. Appl. Math. Comput. 276, 44-60 (2016) Google Scholar
  26. Wang, MK, Qiu, SL, Chu, YM, Jiang, YP: Generalized Hersch-Pfluger distortion function and complete elliptic integrals. J. Math. Anal. Appl. 385(1), 221-229 (2012) View ArticleMATHGoogle Scholar
  27. Wang, MK, Qiu, SL, Chu, YM, Jiang, YP: Convexity of the complete elliptic integrals of the first kind with respect to Hölder means. J. Math. Anal. Appl. 388(2), 1141-1146 (2012) View ArticleMATHGoogle Scholar
  28. Yang, ZH, Chu, YM: A monotonicity property involving the generalized elliptic integral of the first kind. Math. Inequal. Appl. 20(3), 729-735 (2017) MATHGoogle Scholar
  29. Zhang, XH, Wang, DG, Chu, YM: Convexity with respect to Hölder mean involving zero-balanced hypergeometric functions. J. Math. Anal. Appl. 353(1), 256-259 (2009) View ArticleMATHGoogle Scholar
  30. Zhang, XH, Wang, DG, Chu, YM: Remarks on generalized elliptic integrals. Proc. R. Soc. Edinb. A 139(2), 417-426 (2009) View ArticleMATHGoogle Scholar
  31. Anderson, GD, Qiu, SL, Vamanamurthy, MK: Elliptic integral inequalities, with applications. Constr. Approx. 14, 195-207 (1998) View ArticleMATHGoogle Scholar
  32. Alzer, H, Richards, K: A note on a function involving complete elliptic integrals: monotonicity, convexity, inequalities. Anal. Math. 41(3), 133-139 (2015) View ArticleMATHGoogle Scholar
  33. Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York (1966) MATHGoogle Scholar
  34. Prudnikov, AP, Brychkov, YA, Marichev, OI: Integrals and Series, vol. 3. Gordon & Breach, Amsterdam (1990) MATHGoogle Scholar
  35. Qiu, SL, Vuorinen, M: Special functions in geometric function theory. In: Kühnau, R (ed.) Handbook of Complex Analysis: Geometric Function Theory, vol. 2, pp. 621-659. Elsevier, Amsterdam (2005) View ArticleGoogle Scholar

Copyright

© The Author(s) 2017