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 \(a=1/2\).
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1 Introduction
For real numbers a, b, and c with \(c\neq0,-1,-2,\ldots \) , the Gaussian hypergeometric function is defined by
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 [1–10].
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
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, 11–30].
Anderson, Qiu, and Vamanamurthy [31] considered the monotonicity and convexity of the function
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,
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
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
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
and
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 [33–35]:
the differential formula
the asymptotic limit
and the contiguous relation
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]:
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
is a strictly decreasing automorphism of \((0,1)\) if and only if \(4ab\leq a+b \).
Lemma 2.4
The function
is increasing from \((0,1)\) onto \((-\infty,0)\).
Proof
Let
By the series expansion for \(F(a,b;c;x)\) we have
By the definition of the generalized elliptic integrals of the first and second kinds (1.2) we have
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
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,
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
Then
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
where
Obviously, \(g_{1}(0^{+})=0\). By Lemma 2.1 we get \(g_{2}(0^{+})=0\). Moreover,
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
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
By the series expansion for \(F(a,b;c;x)\) we obtain
Then
Using the differentiation formula (2.2), we have
By formula (2.1),we get
Using the contiguous relation (2.4), we take \(\alpha=a+1\), \(\rho=2-a\), \(\sigma=3\), and \(z=1-r^{2}\) and obtain
Hence, it follows from (3.6), (3.7), and the last formula that
By the series expansion for \(F(a,b;c;x)\) we have
Hence
Through direct calculation we have
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
Applying Lemma 2.3 and (2.6), we have
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 \(x\in(0, 1)\),
□
Corollary 3.3
Let
Then we have
for all \(p,q\in(0,1)\).
Proof
By direct calculation we obtain
Considering the positivity of \(g'_{a}\) and \(g''_{a}\) on \((0,1)\), we have
This means that \(\frac{\partial}{\partial p}L_{a}(p,q)\) is strictly increasing with respect to q. So we have
Then the monotonicity of \(L_{a}(p,q)\) with respect to p is obtained, which leads to
□
Remark 3.4
Taking \(a= 1/2\) in Theorems 3.1 and 3.2, we get Theorems 1.1 and 1.2.
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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).
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Huang, T., Tan, S. & Zhang, X. Monotonicity, convexity, and inequalities for the generalized elliptic integrals. J Inequal Appl 2017, 278 (2017). https://doi.org/10.1186/s13660-017-1556-z
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DOI: https://doi.org/10.1186/s13660-017-1556-z