Abstract
In the article, we present several quadratic transformation inequalities for Gaussian hypergeometric function and find the analogs of duplication inequalities for the generalized Grötzsch ring function.
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1 Introduction
The Gaussian hypergeometric function \(_{2}F_{1}(a,b;c;x)\) with real parameters \(a,b\), and c \((c\neq0,-1,-2,\dots)\) is defined by [1, 4, 24, 41]
for \(x\in(-1,1)\), where \((a,n)=a(a+1)(a+2)\cdots(a+n-1)\) for \(n=1,2,\dots\), and \((a,0)=1\) for \(a\neq0\). The function \(F(a,b;c;x)\) is called zero-balanced if \(c=a+b\). The asymptotical behavior for \(F(a,b;c;x)\) as \(x\rightarrow1\) is as follows (see [4, Theorems 1.19 and 1.48])
where \(\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\,dt\) [10, 25, 43, 44, 47] and \(B(p,q)=[{\Gamma(p)\Gamma(q)}]/[{\Gamma(p+q)}]\) are the classical gamma and beta functions, respectively, and
\(\psi(z) =\Gamma'(z)/\Gamma(z)\)), and \(\gamma=\lim_{n\rightarrow \infty} (\sum_{k=1}^{n}{1}/{k}-\log n )=0.577\dots\) is the Euler–Mascheroni constant [21, 50].
As is well known, making use of the hypergeometric function, Branges proved the famous Bieberbach conjecture in 1984. Since then, \(F(a,b;c;x)\) and its special cases and generalizations have attracted attention of many researchers, and was studied deeply in various fields [2, 5, 9, 11–18, 20, 22, 23, 26, 30, 31, 35–37, 40, 45, 46, 48]. A lot of geometrical and analytic properties, and inequalities of the Gaussian hypergeometric function have been obtained [3, 6–8, 19, 29, 32, 34, 38, 49].
Recently, in order to investigate the Ramanujan’s generalized modular equation in number theory, Landen inequalities, Ramanujan cubic transformation inequalities, and several other quadratic transformation inequalities for zero-balanced hypergeometric function have been proved in [27, 28, 32, 39, 42]. For instance, using the quadratic transformation formula [24, (15.8.15), (15.8.21)]
Wang and Chu [32] found the maximal regions of the \((a,b)\)-plane in the first quadrant such that inequality
or its reversed inequality
holds for each \(r\in(0,1)\). Moreover, very recently in [33], some Landen-type inequalities for a class of Gaussian hypergeometric function \(_{2}F_{1} (a,b;(a+b+1)/2;x )\ (a,b>0)\), which can be viewed as a generalization of Landen identities of the complete elliptic integrals of the first kind
have also been proved. As an application, the analogs of duplication inequalities for the generalized Grötzsch ring function with two parameters [33]
have been derived. In fact, the authors have proved
Theorem 1.1
For \((a,b)\in\{(a,b)|a,b>0,ab\geq a+b-10/9, a+b\geq2\}\), let \(x=x(r)=2\sqrt{r}/(1+r)\), then the Landen-type inequality
holds for all \(r\in(0,1)\).
Theorem 1.2
For \((a,b)\in\{(a,b)|a,b>0,ab\geq a+b-10/9, a+b\geq2\}\), define the function g on \((0,1)\) by
Then g is strictly increasing from \((0,1)\) onto \((-\infty,0)\). In particular, the inequality
holds for each \(r\in(0,1)\) with \((a,b)\in\{(a,b)|a,b>0,ab\geq a+b-10/9, a+b\geq2\}\).
The purpose of this paper is to establish several quadratic transformation inequalities for Gaussian hypergeometric function \(_{2}F_{1}(a,b;(a+b+1)/2;x)\) \((a,b>0)\), such as inequalities (1.4), (1.5) and (1.7), and thereby prove the analogs of Theorem 1.2.
We recall some basic facts about \(\mu_{a,b}(r)\) (see [33]). The limiting values of \(\mu_{a,b}(r)\) at 0 and 1 are
and the derivative formula of \(\mu_{a,b}(r)\) is
Here and in what follows,
2 Lemmas
In order to prove our main results, we need several lemmas, which we present in this section. Throughout this section, we denote
for \((a,b)\in(0,+\infty)\times(0,+\infty)\setminus\{p,q\}\) with \(p=(1/4, 3/4)\) and \(q=(3/4, 1/4)\), and
For the convenience of readers, we introduce some regions in \(\{(a,b)\in \mathbb{R}^{2}| a>0,b>0\}\) and refer to Fig. 1 for illustration:
Obviously, \(\bigcup_{i=1}^{4}D_{i}=(0,+\infty)\times(0,+\infty)\) and \(D_{i}\cap D_{j}=\emptyset\) for \(i\neq j\in\{1,2,3,4\}\) except that \(D_{1}\cap D_{2}=\{p,q\}\). Moreover, \(D_{1}\subset E_{1}\) and \(D_{2}\subset E_{2}\).
Lemma 2.1
([42, Theorem 2.1])
Suppose that the power series \(f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}\) and \(g(x)=\sum_{n=0}^{\infty}b_{n}x^{n}\) have the radius of convergence \(r>0\) with \(b_{n}>0\) for all \(n\in\{0,1,2,\dots\}\). Let \(h(x)=f(x)/g(x)\) and \(H_{f,g}=(f'/g')g-f\), then the following statements hold true:
-
1.
If the non-constant sequence \(\{a_{n}/b_{n}\}_{n=0}^{\infty}\) is increasing (decreasing) for all \(n>0\), then \(h(x)\) is strictly increasing (decreasing) on \((0,r)\);
-
2.
If the non-constant sequence \(\{a_{n}/b_{n}\}_{n=0}^{\infty}\) is increasing (decreasing) for \(0< n\leq n_{0}\) and decreasing (increasing) for \(n>n_{0}\), then \(h(x)\) is strictly increasing (decreasing) on \((0,r)\) if and only if \(H_{f,g}(r^{-})\geq(\leq) 0\). Moreover, if \(H_{f,g}(r^{-})<(>) 0\), then there exists an \(x_{0}\in(0,r)\) such that \(h(x)\) is strictly increasing (decreasing) on \((0,x_{0})\) and strictly decreasing (increasing) on \((x_{0},r)\).
Lemma 2.2
-
1.
The function \(\eta(x)=F(x)/\widehat{F}(x)\) is strictly decreasing on \((0,1)\) if \((a,b)\in D_{1}\setminus\{p,q\}\) and strictly increasing on \((0,1)\) if \((a,b)\in D_{2}\setminus\{p,q\}\). Moreover, if \((a,b)\in D_{3} (\textit{or }D_{4})\), then there exists \(\delta_{0}\in(0,1)\) such that \(\eta(x)\) is strictly increasing (decreasing) on \((0,\delta _{0})\) and strictly decreasing (increasing) on \((\delta_{0},1)\).
-
2.
The function \(\widetilde{\eta}(x)=G(x)/\widehat{G}(x)\) is strictly decreasing on \((0,1)\) if \((a,b)\in E_{1}\setminus\{p,q\}\) and strictly increasing on \((0,1)\) if \((a,b)\in E_{2}\setminus\{p,q\}\). In the remaining case, namely for \(x\in(0,+\infty)\times(0,+\infty )\setminus(E_{1}\cup E_{2})\), \(\widetilde{\eta}(x)\) is piecewise monotone on \((0,1)\).
Proof
Suppose that
then we have
It suffices to take into account the monotonicity of \(\{A_{n}/A^{*}_{n}\} _{n=0}^{\infty}\). By simple calculations, one has
where
We divide the proof into four cases.
Case 1 \((a,b)\in D_{1}\setminus\{p,q\}\). Then it follows easily that \(a+b\leq1\), \(ab-\frac{3(a+b+1)}{32}\leq0\) and \(ab+\frac {a+b}{2}-\frac{11}{16}<0\). This, in conjunction with (2.4) and (2.5), implies that \(\{A_{n}/A^{*}_{n}\}_{n=0}^{\infty}\) is strictly decreasing for all \(n>0\). Therefore, (2.3) and Lemma 2.1(1) lead to the conclusion that \(\eta(x)\) is strictly decreasing on \((0,1)\).
Case 2 \((a,b)\in D_{2}\setminus\{p,q\}\). Then a similar argument as in Case 1 yields \(\Delta_{n}>0\) and this implies that \(\eta(x)\) is strictly increasing on \((0,1)\) from (2.3), (2.4) and Lemma 2.1(1).
Case 3 \((a,b)\in D_{3}\). It follows from (2.4) and (2.5) that the sequence \(\{A_{n}/A^{*}_{n}\}\) is increasing for \(0\leq n\leq n_{0}\) and decreasing for \(n\geq n_{0}\) for some integer \(n_{0}\). Furthermore, making use of the derivative formula for Gaussian hypergeometric function
and in conjunction with (1.1) and \(a+b<1\), we obtain
as \(x\rightarrow1^{-}\). Combing with (2.3), (2.6) and Lemma 2.1(2), we conclude that there exists an \(x_{1}\in(0,1)\) such that \(\eta(x)\) is strictly increasing on \((0,x_{1})\) and strictly decreasing on \((x_{1},1)\).
Case 4 \((a,b)\in D_{4}\). In this case, we follow a similar argument as in Case 3 and use the fact that
as \(x\rightarrow1^{-}\) since \(a+b>1\). Therefore, (2.3), (2.7) and Lemma 2.1(2) lead to the conclusion that there exists an \(x_{2}\in(0,1)\) such that \(\eta(x)\) is strictly decreasing on \((0,x_{2})\) and strictly increasing on \((x_{2},1)\).
Let
then we can write
Easy calculations lead to the conclusion that the monotonicity of \(\{ B_{n}/B^{*}_{n}\}_{n=0}^{\infty}\) depends on the sign of
Notice that
where
It follows easily from (1.1) and (2.11) that
Employing similar arguments mentioned in part (1), we obtain the desired assertions easily from (2.8)–(2.12). □
Lemma 2.3
Let \(D_{0}=\{(a,b)| a,b>0,a+b\geq7/4,ab\geq a+b-31/28\}\) and \(x'=\sqrt {1-x^{2}}\) for \(0< x<1\), then the function
is strictly increasing on \((0,1)\) if \((a,b)\in D_{0}\).
Proof
Taking the derivative of \(f(x)\) yields
where
We clearly see from (1.1) that
for \(0< x<1\). This implies, in conjunction with (2.15), that
where
It follows from the definition of hypergeometric function that
where
If \((a,b)\in D_{0}\), namely, \(a+b\geq7/4\) and \(ab\geq a+b-31/28\), we can verify
-
(i)
$$\begin{aligned} &4ab(a+b-1)-4(a-b)^{2}+1\\ &\quad\geq4 \biggl(a+b- \frac{31}{28} \biggr) (a+b-1)-4(a-b)^{2}+1 \\ &\quad=\frac{1}{7} \bigl[112ab-59(a+b)+38 \bigr]\geq\frac {53}{7} \biggl(a+b-\frac{86}{53} \biggr)\geq\frac{27}{28}, \end{aligned}$$
-
(ii)
$$\begin{aligned} 32ab+5(a+b)-29&\geq32 \biggl(a+b-\frac{31}{28} \biggr)+5(a+b)-29 \\ &=\frac{37}{7} \biggl[7(a+b)-\frac{451}{259} \biggr]\geq \frac {9}{28}, \end{aligned}$$
-
(iii)
$$\begin{aligned} &4ab(a+b+3) -4(a-b)^{2} -(3 a + 3 b + 5)\\ &\quad\geq4 \biggl(a+b-\frac{31}{28} \biggr) (a+b+3) \\ &\qquad{} -4(a-b)^{2}-(3a+3b+5)=\frac{16}{7} \bigl[7ab+2(a+b)-8 \bigr] \\ &\quad\geq \frac{16}{7} \biggl[7 \biggl(a+b-\frac{31}{28} \biggr)+2(a+b)-8 \biggr]=\frac{36}{7}\bigl[4(a+b)-7)\bigr]\geq0. \end{aligned}$$
This, in conjunction with (2.17) and (2.18), implies that \(f_{2}(x)>0\) for \(0< x<1\). Therefore, \(f(x)\) is strictly increasing on \((0,1)\), which follows from (2.14) and (2.16) if \((a,b)\in D_{0}\). □
Remark 2.4
The function \(f(x)\) defined in Lemma 2.3 is not monotone on \((0,1)\) if two positive numbers \(a,b\) satisfy \(a+b<1\), since \(\lim_{x\rightarrow 0^{+}}f(x)=\lim_{x\rightarrow1^{-}}f(x)=+\infty\) and Lemma 2.1(1) shows the monotonicity of \(f(x)\) on \((0,1)\) if \(a+b=1\). In the remaining case \(a+b>1\), it follows from (2.15) that \(f_{1}(0^{+})=(a+b-1)/2>0\). This, in conjunction with (2.14), implies that \(f(x)\) is strictly increasing on \((0,x^{*})\) for a sufficiently small \(x^{*}>0\). This enables us to find a sufficient condition for \(a,b\) with \(a+b>1\) such that \(f(x)\) is strictly increasing on \((0,1)\) in Lemma 2.3.
The following corollary can be derived immediately from the monotonicity of \(f(x)\) in Lemma 2.3 and the quadratic transformation equality (1.3).
Corollary 2.5
Let \(x=x(r)=\sqrt{8r(1+r)}/(1+3r)\), if \((a,b)\in D_{0}\), then the inequality
holds for all \(r\in(0,1)\).
3 Main results
Theorem 3.1
The quadratic transformation inequality
holds for all \(r\in(0,1)\) with \(a,b>0\) if and only if \((a,b)\in D_{1}\) and the reversed inequality
takes place for all \(r\in(0,1)\) if and only if \((a,b)\in D_{2}\), with equality only for \((a,b)=p\textit{ or }q\).
In the remaining case \((a,b)\in D_{3}\cup D_{4}\), neither of the above inequalities holds for all \(r\in(0,1)\).
Proof
Suppose that \(x(r)=[8r(1+r)]/(1+3r)^{2}\), then we clearly see that \(x(r)>r^{2}\) for \(0< r<1\). It follows from Lemma 2.1(1) that \(\eta (x(r))<\eta(r^{2})\) for \((a,b)\in D_{1}\setminus\{p,q\}\) and \(\eta (x(r))>\eta(r^{2})\) for \((a,b)\in D_{2}\setminus\{p,q\}\). This, in conjunction with the quadratic transformation formula (1.3), implies
for \((a,b)\in D_{1}\setminus\{p,q\}\), and it degenerates to the quadratic transformation equality for \((a,b)=p (\text{or} q)\). This completes the proof of (3.1).
Inequality (3.2) can be derived analogously, and the remaining case follows easily from Lemma 2.2(1). □
Theorem 3.2
We define the function
for \(r\in(0,1)\) with \(a,b>0\) and \((a,b)\neq p,q\). Let \(L_{1}=\{(a,b)| a+b=1, 0< a<\frac{1}{4}\textit{ or }\frac{3}{4}<a<1\}\) and \(L_{2}=\{ (a,b)| a+b=1, \frac{1}{4}< a<\frac{3}{4}\}\). Then the following statements hold true:
-
1.
If \((a,b)\in L_{1}(\textit{or }L_{2})\), then \(\varphi(r)\) is strictly increasing (resp., decreasing) from \((0,1)\) onto \((0,[R(a,b)-\log64]/B(a,b) )\) (resp., \(([R(a,b)-\log64]/B(a,b),0)\));
-
2.
If \((a,b)\in D_{1}\setminus L_{1}\), then \(\varphi(r)\) is strictly increasing from \((0,1)\) onto \((0,H(a,b))\);
-
3.
If \((a,b)\in D_{2}\setminus L_{2}\), then \(\varphi(r)\) is strictly decreasing from \((0,1)\) onto \((-\infty,0)\).
As a consequence, the inequality
holds for all \(r\in(0,1)\) if \((a,b)\in D_{1}\setminus L_{1}\), and the following inequality is valid for all \(r\in(0,1)\):
if \((a,b)\in L_{1} (\textit{resp., }L_{2})\).
Proof
Let \(z=z(r)=[8\sqrt{r}(1+\sqrt{r})]/(1+3\sqrt{r})^{2}\), then we clearly see that
Taking the derivative of \(\varphi(r)\) with respect to r and using (3.5) yields
We substitute \(\sqrt{r}\) for r in the quadratic transformation equality (1.3), then differentiate it with respect to r to obtain
in other words,
If \((a,b)\in D_{1}\setminus\{p,q\}\), then it follows from Lemma 2.2(2) that \(G(x)/\widehat{G}(x)\) is strictly decreasing on \((0,1)\). This, in conjunction with \(z>r\), implies that \(G(z)/\widehat{G}(z)< G(r)/\widehat {G}(r)\), that is,
Combing (3.6), (3.7) with the inequality (3.8), we clearly see that
It follows from Lemma 2.2(1) that \(\widehat{F}(r)/F(r)\) is strictly increasing on \((0,1)\) if \((a,b)\in D_{1}\setminus\{p,q\}\). This, in conjunction with (3.9), implies that \(\varphi(r)\) is strictly increasing on \((0,1)\) if \((a,b)\in D_{1}\).
Analogously, if \((a,b)\in D_{2}\setminus\{p,q\}\), then we obtain the following inequality:
By using a similar argument as above, we have
since \(F(r)/\widehat{F}(r)\) is strictly increasing on (0,1) if \((a,b)\in D_{2}\setminus\{p,q\}\) by Lemma 2.2(1). Hence, \(\varphi(r)\) is strictly decreasing on \((0,1)\) if \((a,b)\in D_{2}\).
Notice that \(\varphi(0^{+})=0\) and
Therefore, we obtain the desired assertion from (3.10). □
Theorem 3.3
If we define the function
for \((a,b)\in D_{0}\), then \(\phi(r)\) is strictly increasing from \((0,1)\) onto \((-\infty,0)\). As a consequence, the inequality
holds for all \(r\in(0,1)\) if \((a,b)\in D_{0}\).
Proof
Remark 2.4 enables us to consider the case for \(a+b>1\). Note that \(\phi (1^{-})=0\) and
Let \(x=x(r)=\sqrt{8r(1+r)}/(1+3r)\) and \(x'=\sqrt{1-x^{2}}\). Then
Taking the derivative of \(\phi(r)\) and using (3.12) leads to
Therefore, the monotonicity of \(\phi(r)\) follows immediately from (2.19) and (3.13). This, in conjunction with (3.11), gives rise to the desired result. □
4 Results and discussion
In the article, we establish several quadratic transformation inequalities for Gaussian hypergeometric function \(_{2}F_{1}(a,b;(a+b+1)/2;x)\) \((0< x<1)\). As applications, we provide the analogs of duplication inequalities for the generalized Grötzsch ring function
introduced in [33].
5 Conclusion
We find several quadratic transformation inequalities for the Gaussian hypergeometric function and Grötzsch ring function. Our approach may have further applications in the theory of special functions.
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This work was supported by the Natural Science Foundation of China (Grant Nos. 11701176, 11626101, 11601485), the Science and Technology Research Program of Zhejiang Educational Committee (Grant no. Y201635325).
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Zhao, TH., Wang, MK., Zhang, W. et al. Quadratic transformation inequalities for Gaussian hypergeometric function. J Inequal Appl 2018, 251 (2018). https://doi.org/10.1186/s13660-018-1848-y
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DOI: https://doi.org/10.1186/s13660-018-1848-y