1 Introduction
Let \(p>1\) and \(0\leq\theta\leq 1\). Then the function \(\sin_{p}^{-1}(\theta)\) and the number \(\pi_{p}\) are defined by
and
respectively, where B is the classical beta function. The inverse function of \(\sin_{p}^{-1}(\theta)\) defined on \([0, \pi_{p}/2]\) is said to be the generalized sine function and denoted by \(\sin_{p}\). From (1.1) and (1.2) we clearly see that \(\sin_{2}(\theta)=\sin(\theta)\) and \(\pi_{2}=\pi\). The generalized sine function \(\sin_{p}(\theta)\) and \(\pi_{p}\) appeared in the eigenvalue problem of one-dimensional p-Laplacian
$$ \sin_{p}^{-1}(\theta)= \int_{0}^{\theta}\frac{1}{(1-t^{p})^{1/p}}\,dt $$
(1.1)
$$ \frac{\pi_{p}}{2}=\sin_{p}^{-1}(1)= \int_{0}^{1}\frac{1}{(1-t^{p})^{1/p}}\,dt=\frac{\pi}{p\sin(\pi/p)}= \frac{1}{p} B(1/p,1-1/p), $$
(1.2)
$$ -\bigl( \bigl\vert u' \bigr\vert ^{p-2}u' \bigr)'=\lambda \vert u \vert ^{p-2} u,\quad u(0)=u(1)=0. $$
Indeed, the eigenvalues are given by \(\lambda_{n}=(p-1) (n\pi_{p})^{p}\) and the corresponding eigenfunction to \(\lambda_{n}\) is \(u(x)=\sin_{p}(n\pi_{n}x)\) for each \(n=1, 2, 3,\ldots \) . In the same way one can define the generalized cosine and tangent functions and their inverse functions [1‐3].
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Let \(x\in (-1,1)\), and a, b and c be the real numbers with \(c\neq 0,-1,-2,\ldots \) . Then the Gaussian hypergeometric function \(F(a, b; c; x)\) [4‐11] is defined by
where \((a,n)\) denotes the shifted factorial function \((a,n)=a(a+1)\cdots (a+n-1)\), \(n=1,2,\ldots \) , and \((a,0)=1\) for \(a\neq 0\). The well-known complete elliptic integrals \(\mathcal{K}(r)\) and \(\mathcal{E}(r)\) [12‐15] of the first and second kinds are respectively defined by
and
$$ F(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a,n)(b,n)}{(c,n)}\frac{x^{n}}{n!}, $$
(1.3)
$$ \mathcal{K}(r)=\frac{\pi}{2}F \biggl(\frac{1}{2},\frac{1}{2};1;r^{2} \biggr)= \int_{0}^{\pi/2}\frac{d\theta}{\sqrt{1-r^{2}\sin^{2}\theta}}= \int_{0}^{1}\frac{dt}{\sqrt{(1-t^{2})(1-r^{2}t^{2})}} $$
$$ \mathcal{E}(r)=\frac{\pi}{2}F \biggl(\frac{1}{2},- \frac{1}{2};1;r^{2} \biggr)= \int_{0}^{\pi/2}{\sqrt{1-r^{2} \sin^{2}\theta}}\,d\theta= \int_{0}^{1}{\sqrt{\frac{1-r^{2}t^{2}}{1-t^{2}}}}\,dt. $$
Let \(p\in (1,\infty)\) and \(r\in [0,1)\). Then the complete p-elliptic integrals \(\mathcal{K}_{p}(r)\) and \(\mathcal{E}_{p}(r)\) [16, 17] of the first and second kinds are respectively defined by
and
$$ \mathcal{K}_{p}(r)= \int_{0}^{\pi_{p}/2}\frac{d\theta}{(1-r^{p}\sin_{p}^{p}\theta)^{1-1/p}}= \int_{0}^{1}\frac{dt}{(1-t^{p})^{1/p}(1-r^{p}t^{p})^{1-1/p}} $$
(1.4)
$$ \mathcal{E}_{p}(r)= \int_{0}^{\pi_{p}/2}{\bigl(1-r^{p} \sin_{p}^{p}\theta\bigr)^{1/p}}\,d\theta= \int_{0}^{1}{ \biggl(\frac{1-r^{p}t^{p}}{1-t^{p}} \biggr)^{1/p}}\,dt. $$
(1.5)
From (1.4) and (1.5) we clearly see that the complete p-elliptic integrals \(\mathcal{K}_{p}(r)\) and \(\mathcal{E}_{p}(r)\) respectively reduce to the complete elliptic integrals \(\mathcal{K}(r)\) and \(\mathcal{E}(r)\) if \(p=2\). Recently, the complete p-elliptic integrals \(\mathcal{K}_{p}(r)\) and \(\mathcal{E}_{p}(r)\) and their special cases \(\mathcal{K}(r)\) and \(\mathcal{E}(r)\) have attracted the attention of many mathematicians [18‐30].
Takeuchi [31] generalized several well-known theorems for the complete elliptic integrals \(\mathcal{K}(r)\) and \(\mathcal{E}(r)\), such as Legendre’s formula, Gaussian’s \(AGM\) approximation formulas for π, differential equations, and other similar results of the theory of complete elliptic integrals to the complete p-elliptic integrals \(\mathcal{K}_{p}(r)\) and \(\mathcal{E}_{p}(r)\), and proved that
$$\begin{aligned} &\mathcal{K}_{p}(r)=\frac{\pi_{p}}{2}F \biggl( \frac{1}{p},1-\frac{1}{p};1;r^{p} \biggr), \end{aligned}$$
(1.6)
$$\begin{aligned} &\mathcal{E}_{p}(r)=\frac{\pi_{p}}{2}F \biggl( \frac{1}{p},-\frac{1}{p};1;r^{p} \biggr). \end{aligned}$$
(1.7)
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Anderson, Qiu, and Vamanamurthy [32] discussed the monotonicity and convexity properties of the function
and proved that the double inequality
holds for all \(r\in (0,1)\). Both inequalities given in (1.8) are sharp as \(r\rightarrow 0\), while the second inequality is also sharp as \(r\rightarrow 1\). Here and in what follows, we denote \(r'=(1-r^{p})^{1/p}\), \(\mathcal{K}_{p}'(r)=\mathcal{K}_{p}(r')\), and \(\mathcal{E}_{p}'(r)=\mathcal{E}_{p}(r')\).
$$ r\mapsto f(r)=\frac{\mathcal{E}(r)-r^{\prime2}\mathcal{K}(r)}{r^{2}}\cdot\frac{r^{\prime2}}{\mathcal{E'}(r)-r^{2}\mathcal{K'}(r)} $$
$$ \frac{\pi}{4}< f(r)< \frac{\pi }{4}+ \biggl( \frac{4}{\pi}-\frac{\pi}{4} \biggr)r $$
(1.8)
Alzer and Richards [33] proved that the function
is strictly increasing and convex from \((0,1)\) onto \((\pi/4-1,1-\pi/4)\), and the double inequality
holds for all \(r\in (0,1)\) with the best constant \(\alpha=0\) and \(\beta=2-\pi/2\).
$$ r\mapsto \Delta(r)=\frac{\mathcal{E}(r)-r^{\prime2}\mathcal{K}(r)}{r^{2}}-\frac{\mathcal{E'}(r)-r^{2}\mathcal{K'}(r)}{r^{\prime2}} $$
$$ \frac{\pi}{4}-1+\alpha r< \Delta(r)< \frac{\pi }{4}-1+\beta r $$
(1.9)
Inequalities (1.8) and (1.9) have been generalized to the generalized elliptic integrals by Huang et al. in [34].
The main purpose of the article is to generalize inequalities (1.8) and (1.9) to the complete p-elliptic integrals. We discuss the monotonicity and convexity properties of the functions
and present their corresponding sharp inequalities.
$$\begin{aligned} &r\mapsto f_{p}(r)=\frac{\mathcal{E}_{p}(r)-r^{\prime p}\mathcal{K}_{p}(r)}{r^{p}}\cdot \frac{r^{\prime p}}{\mathcal{E'}_{p}(r)-r^{p}\mathcal{K'}_{p}(r)}, \end{aligned}$$
(1.10)
$$\begin{aligned} &r\mapsto g_{p}(r)=\frac{\mathcal{E}_{p}(r)-r^{\prime p}\mathcal{K}_{p}(r)}{r^{p}}- \frac{\mathcal{E'}_{p}(r)-r^{p}\mathcal{K'}_{p}(r)}{r^{\prime p}}, \end{aligned}$$
(1.11)
2 Lemmas
In order to prove our main results, we need several formulas and lemmas, which we present in this section.
The following formulas for the hypergeometric function and complete p-elliptic integrals can be found in the literature [5, 1.20(10), (1.16), 1.19(4), (1.48)]], [18], and [35, Equation (26)]:
where \(\Gamma(x)\) is the classical gamma function.
$$\begin{aligned} &F(a,b;a+b+1;x)=(1-x)F(a+1,b+1;a+b+1;x), \end{aligned}$$
(2.1)
$$\begin{aligned} &\frac{dF(a,b;c;x)}{dx}=\frac{ab}{c}F(a+1,b+1;c+1;x), \end{aligned}$$
(2.2)
$$\begin{aligned} &F(a,b;c;1)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)} \quad (c>a+b), \end{aligned}$$
(2.3)
$$\begin{aligned} &F(a,b;c;x)\sim -\frac{\log(1-x)}{B(a,b)} \quad (x\rightarrow1, c=a+b), \end{aligned}$$
(2.4)
$$\begin{aligned} &(\sigma-\rho)F(\alpha,\rho;\sigma+1;z)=\sigma F(\alpha,\rho; \sigma;z)-\rho F(\alpha,\rho+1;\sigma+1;z), \end{aligned}$$
(2.5)
$$\begin{aligned} &\frac{d\mathcal {K}_{p}(r)}{dr}=\frac{\mathcal {E}_{p}(r)-r^{\prime p}\mathcal {K}_{a}(r)}{rr^{\prime p}}, \qquad \frac{d\mathcal {E}_{a}(r)}{dr}=\frac{\mathcal {E}_{p}(r)-{\mathcal {K}_{p}}(r)}{r}, \end{aligned}$$
(2.6)
Lemma 2.1
Let
\(p \in(1,\infty)\). Then the function
is strictly increasing and convex from
\((0, 1)\)
onto
\(((p-1){\pi_{p}}/(2p), 1)\).
$$ r\mapsto f_{p}^{1}(r)=\frac{\mathcal{E}_{p}(r)-r^{\prime p} \mathcal{K}_{p}(r)}{r^{p}} $$
(2.7)
Proof
It follows from (1.3), (1.6), and (1.7) that
where \(a_{n}=(1/p,n)(1-1/p,n)(n!)^{-2}\). Therefore,
and \(f_{p}^{1}(r)\) is strictly increasing and convex on \((0, 1)\) due to \(1-1/p>0\).
$$\begin{aligned} &\mathcal{E}_{p}(r)-r^{\prime p} \mathcal{K}_{p}(r) \\ &\quad =\frac{\pi_{p}}{2} \biggl[F \biggl(\frac{1}{p},-\frac{1}{p}, \;1;r^{p} \biggr)-\bigl(1-r^{p}\bigr)F \biggl( \frac{1}{p},1-\frac{1}{p};1;r^{p} \biggr) \biggr] \\ &\quad =\frac{\pi_{p}}{2} \Biggl[\sum_{n=0}^{\infty}\frac{ (\frac{1}{p},n ) (-\frac{1}{p},n )}{(n!)^{2}}r^{pn}- \bigl(1-r^{p}\bigr)\sum _{n=0}^{\infty}\frac{ (\frac{1}{p},n ) (1-\frac{1}{p},n )}{(n!)^{2}}r^{pn} \Biggr] \\ &\quad =\frac{\pi_{p}}{2} \Biggl[\sum_{n=0}^{\infty}\frac{ (\frac{1}{p},n ) (-\frac{1}{p},n )- (\frac{1}{p},n ) (1-\frac{1}{p},n )}{(n!)^{2}}r^{pn}+ \sum_{n=0}^{\infty}\frac{ (\frac{1}{p},n ) (1-\frac{1}{p},n )}{(n!)^{2}}r^{p(n+1)} \Biggr] \\ &\quad =\frac{\pi_{p}}{2}\sum_{n=1}^{\infty}\frac{n^{2} (\frac{1}{p},n-1 ) (1-\frac{1}{p},n-1 ) -n (\frac{1}{p},n ) (1-\frac{1}{p},n-1 )}{(n!)^{2}}r^{pn} \\ &\quad =\frac{\pi_{p}}{2}\sum_{n=1}^{\infty}\frac{n (\frac{1}{p},n-1 ) (1-\frac{1}{p},n-1 ) (1-\frac{1}{p} )}{(n!)^{2}}r^{pn} \\ &\quad =\frac{\pi_{p} r^{p}}{2} \biggl(1-\frac{1}{p} \biggr)\sum _{n=0}^{\infty}\frac{1}{n+1} a_{n} r^{pn}, \end{aligned}$$
$$ f_{p}^{1}(r)=\frac{\mathcal{E}_{p}(r)-r^{\prime p} \mathcal{K}_{p}(r)}{r^{p}}= \frac{\pi_{p}}{2} \biggl(1-\frac{1}{p} \biggr)\sum _{n=0}^{\infty}\frac{1}{n+1} a_{n} r^{pn} $$
(2.8)
From (1.5), (1.6), (2.4), (2.7), and (2.8) we clearly see that
□
$$\begin{aligned} &\mathcal{E}_{p} \bigl(1^{-} \bigr)=1, \qquad \lim _{r\rightarrow 1^{-}}r^{\prime p} \mathcal{K}_{p}(r)=0, \\ &f_{p}^{1}\bigl(0^{+}\bigr)=\frac{(p-1){\pi_{p}}}{2p}, \qquad f_{p}^{1}\bigl(1^{-}\bigr)=1. \end{aligned}$$
Lemma 2.2
(see [18, Lemma 2.3])
Let
\(I\subset \mathbb{R}\)
be an interval and
\(f, g: I\rightarrow (0,\infty)\)
be two positive real-valued functions. Then the product
fg
is convex on
I
if both
f
and
g
are convex and increasing (decreasing) on
I.
Lemma 2.3
Let
\(p>1\). Then the function
is strictly increasing from
\((0,1)\)
onto
\((-\infty,0)\).
$$\begin{aligned} r\mapsto J(r)=\frac{(1-r^{p}) (\mathcal{E}_{p}(r)-(1+r^{p})\mathcal{K}_{p}(r) )}{r^{2p-1}} \end{aligned}$$
(2.9)
Proof
Let
Then it follows from (1.3), (1.6), (1.7), (2.9), and (2.10) that
Equations (2.11) and (2.12) lead to
It is easy to verify that
for \(p>1\) and \(n\geq 2\).
$$ f_{1}(r)=\bigl(1-r^{p}\bigr) \biggl[F \biggl( \frac{1}{p},-\frac{1}{p};1;r^{p} \biggr)- \bigl(1+r^{p}\bigr)F \biggl(\frac{1}{p},1-\frac{1}{p};1;r^{p} \biggr) \biggr]. $$
(2.10)
$$\begin{aligned} &J(r)=\frac{\pi_{p}}{2r^{2p-1}}f_{1}(r), \end{aligned}$$
(2.11)
$$\begin{aligned} &f_{1}(r)=\bigl(1-r^{p}\bigr) \Biggl[\sum _{n=0}^{\infty}\frac{ (\frac{1}{p},n ) (-\frac{1}{p},n )}{(n!)^{2}}r^{pn}- \bigl(1+r^{p}\bigr)\sum_{n=0}^{\infty}\frac{ (\frac{1}{p},n ) (1-\frac{1}{p},n )}{(n!)^{2}}r^{pn} \Biggr] \\ &\hphantom{f_{1}(r)}=\bigl(1-r^{p}\bigr) \Biggl[\sum_{n=0}^{\infty}\frac{ (\frac{1}{p},n ) (-\frac{1}{p},n )- (\frac{1}{p},n ) (1-\frac{1}{p},n )}{(n!)^{2}}r^{pn}- \sum_{n=0}^{\infty}\frac{ (\frac{1}{p},n ) (1-\frac{1}{p},n )}{(n!)^{2}}r^{p(n+1)} \Biggr] \\ &\hphantom{f_{1}(r)}=\bigl(1-r^{p}\bigr)\sum_{n=1}^{\infty}\frac{-n^{2} (\frac{1}{p},n-1 ) (1-\frac{1}{p},n-1 ) -n (\frac{1}{p},n ) (1-\frac{1}{p},n-1 )}{(n!)^{2}}r^{pn} \\ &\hphantom{f_{1}(r)}=\bigl(1-r^{p}\bigr)\sum_{n=1}^{\infty}\frac{ (\frac{1}{p},n-1 ) (1-\frac{1}{p},n-1 ) ( (1-\frac{1}{p} )n-2n^{2} )}{(n!)^{2}}r^{pn} \\ &\hphantom{f_{1}(r)}=\sum_{n=1}^{\infty}\frac{ (\frac{1}{p},n-1 ) (1-\frac{1}{p},n-1 ) ( (1-\frac{1}{p} )n-2n^{2} )}{(n!)^{2}}r^{pn} \\ &\hphantom{f_{1}(r)=}{}-\sum_{n=2}^{\infty}\frac{ (\frac{1}{p},n-2 ) (1-\frac{1}{p},n-2 ) ( (1-\frac{1}{p} )(n-1)-2(n-1)^{2} )}{((n-1)!)^{2}}r^{pn} \\ &\hphantom{f_{1}(r)}= \biggl(-1-\frac{1}{p} \biggr)r^{p} \\ &\hphantom{f_{1}(r)=}{}+\sum _{n=2}^{\infty}\frac{ (\frac{1}{p},n-2 ) (1-\frac{1}{p},n-2 ) [2n (n+\frac{1}{p^{3}}-2 )+\frac{1}{p^{3}}-\frac{2}{p^{2}}-\frac{1}{p}+2 ]}{(n!)^{2}}r^{pn}. \end{aligned}$$
(2.12)
$$ \begin{aligned}[b] J(r)={}&\frac{\pi_{p}}{2} \Biggl\{ -\frac{1+p}{pr^{p-1}}\\ &{}+ \sum _{n=2}^{\infty}\frac{ (\frac{1}{p},n-2 ) (1-\frac{1}{p},n-2 ) [2n (n+\frac{1}{p^{3}}-2 ) +\frac{1}{p^{3}}-\frac{2}{p^{2}}-\frac{1}{p}+2 ]}{(n!)^{2}}r^{p(n-2)+1} \Biggr\} . \end{aligned} $$
(2.13)
$$ 2n \biggl(n+\frac{1}{p^{3}}-2 \biggr)+\frac{1}{p^{3}}- \frac{2}{p^{2}}-\frac{1}{p}+2>0 $$
(2.14)
Therefore, the monotonicity of \(J(r)\) on the interval \((0, 1)\) follows easily from (2.13) and (2.14).
From (2.3), (2.4), (2.10), (2.11), and (2.13) we clearly see that \(J(0^{+})=-\infty\) and
□
$$\begin{aligned} &\lim_{r\rightarrow 1^{-}}F \biggl(\frac{1}{p}, -\frac{1}{p}; 1; r^{p} \biggr)=\frac{1}{\Gamma(1-1/p)\Gamma(1+1/p)}, \qquad \lim_{r\rightarrow 1^{-}} \bigl(1-r^{p} \bigr)F \biggl(\frac{1}{p}, 1-\frac{1}{p}; 1; r^{p} \biggr)=0, \\ &J\bigl(1^{-}\bigr)=0. \end{aligned}$$
Lemma 2.4
(see [5, Theorem 1.25])
Let
\(a, b\in \mathbb{R}\)
with
\(a< b\), \(f,g: [a,b]\rightarrow \mathbb{R}\)
be continuous on
\([a, b]\)
and be differentiable on
\((a, b)\)
such that
\(g'(x) \neq 0\)
on
\((a, b)\). If
\(f'(x)/g'(x)\)
is increasing (decreasing) on
\((a, b)\), then so are the functions
If
\(f'(x)/g'(x)\)
is strictly monotone, then the monotonicity in the conclusion is also strict.
$$ \frac{f(x)-f(a)}{g(x)-g(a)}, \quad\quad \frac{f(x)-f(b)}{g(x)-g(b)}. $$
3 Main results
Theorem 3.1
Let
\(p>1\)
and
\(f_{p}(r)\)
be defined by (1.10). Then
\(f_{p}(r)\)
is strictly increasing and convex from
\((0,1)\)
onto
\(((p-1)\pi_{p}/(2p), 2p/[(p-1)\pi_{p}])\), and the double inequality
holds for all
\(r\in(0,1)\)
if and only if
\(\alpha\leq 0\)
and
\(\beta\geq 2p/[(p-1)\pi_{p}]-(p-1)\pi_{p}/(2p)\). Moreover, both inequalities in (3.1) are sharp as
\(r\rightarrow 0\), while the second inequality is sharp as
\(r\rightarrow 1\).
$$ \frac{(p-1) \pi_{p} }{2p}+\alpha r< f_{p}(r)< \frac{(p-1) \pi_{p} }{2p}+ \beta r $$
(3.1)
Proof
Let \(f^{1}_{p}(r)\) be defined by (2.7). Then \(f_{p}(r)\) can be rewritten as
$$ f_{p}(r)=\frac{\mathcal{E}_{p}(r)-r^{\prime p}\mathcal{K}_{p}(r)}{r^{p}}\cdot\frac{r^{\prime p}}{\mathcal{E'}_{p}(r)-r^{p}\mathcal{K'}_{p}(r)}=f^{1}_{p}(r) \cdot \frac{1}{f^{1}_{p}(r')}. $$
(3.2)
From Lemma 2.1 we know that both the functions \(f_{p}^{1}(r)\) and \(1/f^{1}_{p}(r')\) are positive and strictly increasing on \((0,1)\), hence \(f_{p}(r)\) is also strictly increasing on \((0,1)\).
Next, we prove that \(1/f^{1}_{a}(r')\) is convex on \((0,1)\). It follows from (2.6) and (2.7) that
where
where \(J(r)\) is defined by (2.9).
$$\begin{aligned} \frac{d}{dr} \biggl(\frac{1}{f_{p}^{1}(r)} \biggr)&= \frac{pr^{p-1}(\mathcal{E}_{p}(r)-r^{\prime p}\mathcal{K}_{p}(r))-(p-1)r^{2p-1}\mathcal{K}_{p}(r) }{(\mathcal{E}_{p}(r)-r^{\prime p}\mathcal{K}_{p}(r))^{2}} \\ &=\frac{p(\mathcal{E}_{p}(r)-r^{\prime p}\mathcal{K}_{p}(r))-(p-1)r^{p}\mathcal{K}_{p}(r) }{\frac{(\mathcal{E}_{p}(r)-r^{\prime p}\mathcal{K}_{p}(r))^{2}}{r^{p-1}}}=\frac{g_{1}(r)}{g_{2}(r)}, \end{aligned}$$
(3.3)
$$ \begin{aligned} &g_{1}(r)=p \bigl(\mathcal{E}_{p}(r)-r^{\prime p} \mathcal{K}_{p}(r) \bigr)+(1-p)r^{p}\mathcal{K}_{p}(r), \qquad g_{2}(r)=\frac{ (\mathcal{E}_{p}(r)-r^{\prime p}\mathcal{K}_{p}(r) )^{2}}{r^{p-1}}, \\ &\frac{g'_{1}(r)}{g'_{2}(r)}=\frac{r^{2p-1}}{(1-r^{p}) (\mathcal{E}_{p}(r)-(1+r^{p})\mathcal{K}_{p}(r) )}=\frac{1}{J(r)}, \end{aligned} $$
(3.4)
From (1.3), (1.6), (1.7), and Lemma 2.1 we clearly see that
$$ g_{1}\bigl(0^{+}\bigr)=g_{2} \bigl(0^{+}\bigr)=0. $$
(3.5)
Equations (3.3)–(3.5) and Lemmas 2.3 and 2.4 lead to the conclusion that the function \(\frac{d}{dr} (1/f_{p}^{1}(r) )\) is strictly decreasing on \((0, 1)\), which implies that the function \(\frac{d}{dr} (1/f_{p}^{1}(r') )\) is strictly increasing on \((0, 1)\) and \(1/f^{1}_{a}(r')\) is convex on \((0,1)\).
Therefore, \(f_{p}(r)\) is convex on \((0, 1)\) follows from Lemmas 2.1 and 2.2 together with (3.2) and the convexity of \(1/f^{1}_{a}(r')\).
The limit values
follow easily from Lemma 2.1 and (3.2).
$$ f_{p}\bigl(0^{+}\bigr)=\frac{(p-1)\pi_{p}}{2p}, \qquad f_{p}\bigl(1^{-}\bigr)=\frac{2p}{(p-1)\pi_{p}} $$
(3.6)
It follows from (2.8) and (3.2) that
$$\begin{aligned} &\lim_{r\rightarrow 0^{+}}\frac{df_{p}(r)}{dr} \\ &\quad =\lim_{r\rightarrow 0^{+}}\frac{\sum_{n=0}^{\infty}\frac{a_{n}}{n+1}(1-r^{p})^{n}\sum_{n=1}^{\infty}\frac{pna_{n}}{n+1}r^{pn-1} +r^{p-1}\sum_{n=0}^{\infty}\frac{a_{n}}{n+1}r^{pn}\sum_{n=1}^{\infty}\frac{pna_{n}}{n+1}(1-r^{p})^{n-1}}{ [\sum_{n=0}^{\infty}\frac{a_{n}}{n+1}(1-r^{p})^{n} ]^{2}} \\ &\quad=0. \end{aligned}$$
(3.7)
Therefore, inequality (3.1) holds for all \(r\in(0,1)\) if and only if \(\alpha\leq 0\), and \(\beta\geq 2p/[(p-1)\pi_{p}]-((p-1)\pi_{p}/(2p)\) follows easily from (3.6) and (3.7) together with the monotonicity and convexity of \(f_{p}(r)\) on \((0, 1)\). From (3.6) we clearly see that both inequalities in (3.1) are sharp as \(r\rightarrow 0\) and the second inequality is sharp as \(r\rightarrow 1\). □
Theorem 3.2
Let
\(p\geq 2\)
and
\(g_{p}(r)\)
be defined in (1.11). Then
\(g_{p}(r)\)
is strictly increasing and convex from
\((0,1)\)
onto
\(((p-1)\pi_{p}/(2p)-1, 1-(p-1)\pi_{p}/(2p))\), and the double inequality
holds for all
\(r \in (0, 1)\)
if and only if
\(\alpha\leq 0\)
and
\(\beta\geq 2-(p-1)\pi_{p}/p\). Moreover, both inequalities in (3.8) are sharp as
\(r\rightarrow 0\), while the second inequality is sharp as
\(r\rightarrow 1\).
$$ \frac{(p-1)\pi_{p}}{2p}-1+\alpha r< g_{p}(r)< \frac{(p-1)\pi_{p}}{2p}-1+\beta r $$
(3.8)
Proof
Let
Then from (1.3), (1.6), (1.7), and (1.11) we get
It follows from (2.1), (2.2), (2.5), and (3.10) that
Equations (1.3) and (3.12)–(3.14) lead to
for \(p\geq 2\).
$$ M_{p}(r)=\frac{\pi_{p}}{2r^{p}} \biggl[F \biggl(\frac{1}{p},- \frac{1}{p};1;r^{p} \biggr)-r^{\prime p}F \biggl(\frac{1}{p},1-\frac{1}{p};1;r^{p} \biggr) \biggr]. $$
$$\begin{aligned} & M_{p}(r)=\frac{\pi_{p}(p-1)}{2p}F \biggl(1- \frac{1}{p},\frac{1}{p};2;r^{p} \biggr), \end{aligned}$$
(3.9)
$$\begin{aligned} &\begin{aligned}[b] g_{p}(r)&=M_{p}(r)-M_{p} \bigl(r'\bigr)\\ &=\frac{\pi_{p}(p-1)}{2p} \biggl[F \biggl(1- \frac{1}{p},\frac{1}{p};2;r^{p} \biggr)-F \biggl(1- \frac{1}{p},\frac{1}{p};2;1-r^{p} \biggr) \biggr]. \end{aligned} \end{aligned}$$
(3.10)
$$\begin{aligned} &\frac{dg_{p}(r)}{dr}=\frac{(p-1)^{2}\pi_{p}}{4p^{2}}r^{p-1} \biggl[F \biggl(2- \frac{1}{p}, 1+\frac{1}{p};3;r^{p} \biggr)+F \biggl(2- \frac{1}{p},1+\frac{1}{p};3;1-r^{p} \biggr) \biggr], \\ &\frac{dg_{p}(r)}{dr}|_{r=0}=0, \end{aligned}$$
(3.11)
$$\begin{aligned} &\begin{aligned}[b] \frac{d^{2}g_{p}(r)}{dr^{2}}={}&\frac{(p-1)^{3}\pi_{p}}{4p^{2}}r^{p-2} \biggl[F \biggl(2- \frac{1}{p},1+\frac{1}{p};3;r^{p} \biggr)+F \biggl(2- \frac{1}{p},1+\frac{1}{p};3;1-r^{p} \biggr) \\ &{}+\frac{\pi_{p}(p-1)^{2}(1+p)(2p-1)}{12p^{3}}r^{2p-2} \\ &{}\times\biggl(F \biggl(3-\frac{1}{p},2+ \frac{1}{p};4;r^{p} \biggr) -F \biggl(3-\frac{1}{p},2+ \frac{1}{p};4;1-r^{p} \biggr) \biggr) \biggr], \end{aligned} \end{aligned}$$
(3.12)
$$\begin{aligned} &F \biggl(3-\frac{1}{p},2+\frac{1}{p};4;1-r^{p} \biggr)=\frac{1}{r^{p}}F \biggl(2-\frac{1}{p},1+\frac{1}{p};4;1-r^{p} \biggr), \end{aligned}$$
(3.13)
$$\begin{aligned} &\begin{aligned}[b] &\biggl(2-\frac{1}{p} \biggr)F \biggl(2- \frac{1}{p},1+\frac{1}{p};4;1-r^{p} \biggr) \\ &\quad =3F \biggl(2-\frac{1}{p},1+\frac{1}{p};3;1-r^{p} \biggr) - \biggl(1+\frac{1}{p} \biggr)F \biggl(2-\frac{1}{p},2+ \frac{1}{p};4;1-r^{p} \biggr). \end{aligned} \end{aligned}$$
(3.14)
$$\begin{aligned} &\frac{4p^{2}}{(p-1)^{2}\pi_{p}}r^{p-2}g''_{p}(r) \\ &\quad = (p-1)F \biggl(2-\frac{1}{p},1+\frac{1}{p};3;r^{p} \biggr) +(p-1)F \biggl(2-\frac{1}{p},1+\frac{1}{p};3;1-r^{p} \biggr) \\ &\qquad {}+\frac{(2p-1)(p+1)}{3p}r^{p}F \biggl(3-\frac{1}{p},2+ \frac{1}{p};4;r^{p} \biggr) \\ &\qquad {}-r^{p}\frac{(2p-1)(p+1)}{3p}F \biggl(3-\frac{1}{p},2+\frac{1}{p};4;1-r^{p} \biggr) \\ &\quad = (p-1)F \biggl(2-\frac{1}{p},1+\frac{1}{p};3;r^{p} \biggr)+(p-1)F \biggl(2-\frac{1}{p},1+\frac{1}{p};3;1-r^{p} \biggr) \\ &\qquad {}+\frac{(2p-1)(p+1)}{3p}r^{p}F \biggl(3-\frac{1}{p},2+ \frac{1}{p};4;r^{p} \biggr) \\ &\qquad {} -\frac{(2p-1)(p+1)}{3p}F \biggl(2- \frac{1}{p},1+\frac{1}{p};4;1-r^{p} \biggr) \\ &\quad = (p-1)F \biggl(2-\frac{1}{p},1+\frac{1}{p};3;r^{p} \biggr)+\frac{(2p-1)(p+1)}{3p}r^{p}F \biggl(3-\frac{1}{p},2+ \frac{1}{p};4;r^{p} \biggr) \\ &\qquad {}+(p-1)F \biggl(2-\frac{1}{p},1+\frac{1}{p};3;1-r^{p} \biggr)-(2p-1)F \biggl(2-\frac{1}{p},1+\frac{1}{p};3;1-r^{p} \biggr) \\ &\qquad {}+\frac{(2p-1)^{2}}{3p}F \biggl(3-\frac{1}{p},1+\frac{1}{p};4;1-r^{p} \biggr) \\ &\quad = (p-1)F \biggl(2-\frac{1}{p},1+\frac{1}{p};3;r^{p} \biggr)+\frac{(2p-1)(p+1)}{3p}r^{p}F \biggl(3-\frac{1}{p},2+ \frac{1}{p};4;r^{p} \biggr) \\ &\qquad {}-pF \biggl(2-\frac{1}{p},1+\frac{1}{p};3;1-r^{p} \biggr)+\frac{(2p-1)^{2}}{3p}F \biggl(1+\frac{1}{p},3-\frac{1}{p};4;1-r^{p} \biggr) \\ &\quad > p-1-pF \biggl(2-\frac{1}{p},1+\frac{1}{p};3;1-r^{p} \biggr)+\frac{(2p-1)^{2}}{3p}F \biggl(1+\frac{1}{p},3-\frac{1}{p};4;1-r^{p} \biggr) \\ &\quad = p-1-p\sum_{n=0}^{\infty}\frac{(1+\frac{1}{p},n)(2-\frac{1}{p},n)}{(3,n)} \frac{(1-r^{p})^{n}}{n!} \\ &\qquad {}+\frac{(2p-1)^{2}}{3p}\sum_{n=0}^{\infty} \frac{(1+\frac{1}{p},n)(3-\frac{1}{p},n)}{(4,n)}\frac{(1-r^{p})^{n}}{n!} \\ &\quad = \sum_{n=0}^{\infty}\frac{(1+\frac{1}{p},n)(2-\frac{1}{p},n)}{(3,n+1)} \biggl[(p-1)n+p+\frac{1}{p}-4 \biggr]\frac{(1-r^{p})^{n}}{n!} \\ &\quad \geq \sum_{n=0}^{1}\frac{(1+\frac{1}{p},n)(2-\frac{1}{p},n)}{(3,n+1)} \biggl[(p-1)n+p+\frac{1}{p}-4 \biggr]\frac{(1-r^{p})^{n}}{n!} \\ &\quad = \frac{20p^{4}-36p^{3}-p^{2}+6p-1}{12p^{3}}>0 \end{aligned}$$
(3.15)
Therefore, the monotonicity and convexity for \(g_{p}(r)\) on the interval \((0, 1)\) follow from (3.11) and (3.15).
Remark 3.3
Corollary 3.4
Let
\(p\geq 2\), \(g_{p}(r)\)
be defined by (1.11) and
Then the double inequality
holds for all
\(x,y\in (0,1)\).
$$ L_{p}(x,y)=g_{p}(xy)-g_{p}(x)-g_{p}(y). $$
(3.17)
$$ \frac{(p-1)\pi_{p}}{2p}-1< L_{p}(x,y)< 1-\frac{(p-1)\pi_{p}}{2p} $$
Proof
It follows from (3.17) and the proof of Theorem 3.2 that
for all \(x, y\in (0, 1)\).
$$\begin{aligned} &\frac{\partial}{\partial x}L_{p}(x,y)=yg'_{p}(xy)-g'_{p}(x), \end{aligned}$$
(3.18)
$$\begin{aligned} &\frac{\partial^{2}}{\partial x\partial y}L_{p}(x,y)=g'_{p}(xy)+xyg''_{p}(xy)>0 \end{aligned}$$
(3.19)
4 Methods
The main purpose of the article is to generalize inequalities (1.8) and (1.9) for the complete elliptic integrals to the complete p-elliptic integrals. To achieve this goal we discuss the monotonicity and convexity properties for the functions given by (1.10) and (1.11) by use of the analytical properties of the Gaussian hypergeometric function and the well-known monotone form of l’Hôpital’s rule given in [5, Theorem 1.25].
5 Results and discussion
In the article, we present the monotonicity and convexity properties and provide the sharp bounds for the functions
and
on the interval \((0, 1)\).
$$ r\mapsto f_{p}(r)=\frac{\mathcal{E}_{p}(r)-r^{\prime p}\mathcal{K}_{p}(r)}{r^{p}}\cdot\frac{r^{\prime p}}{\mathcal{E'}_{p}(r)-r^{p}\mathcal{K'}_{p}(r)} $$
$$ r\mapsto g_{p}(r)=\frac{\mathcal{E}_{p}(r)-r^{\prime p}\mathcal{K}_{p}(r)}{r^{p}}-\frac{\mathcal{E'}_{p}(r)-r^{p}\mathcal{K'}_{p}(r)}{r^{\prime p}} $$
6 Conclusion
In this paper, we generalize the monotonicity, convexity, and bounds for the functions involving the complete elliptic integrals to the complete p-elliptic integrals. The given idea may stimulate further research in the theory of generalized elliptic integrals.
Competing interests
The authors declare that they have no competing interests.
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