1 Introduction and main result
q-series, which are also called basic hypergeometric series, play a very important role in many fields, such as affine root systems, Lie algebras and groups, number theory, orthogonal polynomials, and physics. The inequality technique is one of the useful tools in the study of special functions. There are many papers about the inequalities and the q-integral; see [1‐9]. In this paper, we derive an inequality for the q-integral of the bilateral basic hypergeometric function \({}_{r+1}\psi_{r+1}\). Some applications of the inequality are also given. The main result of this paper is the following inequality for q-integrals.
Theorem 1.1
Let
a, b
be any real numbers such that
\(0< q< b< a<1\), and let
\(a_{i}\), \(b_{i}\)
be any real numbers such that
\(|a_{i}|>q\), \(|b_{i}|<1\)
for
\(i=1,2,\ldots,r\)
with
\(r\geq1\)
and
\(|b_{1}b_{2}\cdots b_{r}|\leq|a_{1}a_{2}\cdots a_{r}|\). Then for any
\(c>0\), \(t>0\), such that
\(c>b/a\), \(c+t<1\), we have
where
$$ \biggl\vert \int_{0}^{t}{}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \leq \frac{Mt(q,b/a;q)_{\infty}}{(c+t,b/ac;q)_{\infty}}, $$
(1.1)
$$ M=\max \Biggl\{ \prod_{i=1}^{r} \frac{(-|a_{i}|; q)_{\infty}}{(|b_{i}|; q)_{\infty}},\prod_{i=1}^{r} \frac{(-q/|b_{i}|; q)_{\infty}}{(q/|a_{i}|; q)_{\infty}} \Biggr\} . $$
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Before we give the proof of the theorem, we recall some definitions, notation, and well-known results which will be used in this paper. Throughout the whole paper, it is supposed that \(0< q<1\). The q-shifted factorials are defined as
We also adopt the following compact notation for the multiple q-shifted factorial:
where n is an integer or ∞. We may extend the definition (1.2) of \((a; q)_{n}\) to
for any complex number α. In particular,
The following is the well-known Ramanujan \({}_{1}\psi_{1}\) summation formula [10, 11],
The bilateral basic hypergeometric series \({}_{r}\psi_{s}\) is defined by
Jackson defined the q-integral by [12]
and
In [13], the author uses Ramanujan’s \({}_{1}\psi_{1}\) summation formula to give the following inequality: Let a, b be any real numbers such that \(q< b< a<1\) or \(a< b<0\), and let \(a_{i}\), \(b_{i}\) be any real numbers such that \(|a_{i}|>q\), \(|b_{i}|<1\) for \(i=1,2,\ldots,r\) with \(r\geq1\) and \(|b_{1}b_{2}\cdots b_{r}|\leq|a_{1}a_{2}\cdots a_{r}|\). Then for any \(b/a<|z|<1\), we have
where
$$ (a; q)_{0}=1, \qquad (a; q)_{n}=\prod _{k=0}^{n-1}\bigl(1-aq^{k}\bigr), \qquad (a; q)_{\infty}=\prod_{k=0}^{\infty}\bigl(1-aq^{k}\bigr). $$
(1.2)
$$ (a_{1}, a_{2},\ldots,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n} \cdots (a_{m};q)_{n}, $$
(1.3)
$$ (a; q)_{\alpha}=\frac{(a; q)_{\infty}}{(aq^{\alpha}; q)_{\infty}}, $$
(1.4)
$$ (a; q)_{-n}=\frac{(a; q)_{\infty}}{(aq^{-n}; q)_{\infty}} =\frac{1}{(aq^{-n}; q)_{n}}= \frac{(-q/a)^{n}}{(q/a; q)_{n}}q^{n\choose 2}. $$
(1.5)
$$ \sum_{n=-\infty}^{\infty} \frac{(a;q)_{n}}{(b;q)_{n}}z^{n} =\frac{(q,b/a,az,q/az;q)_{\infty}}{(b,q/a,z,b/az;q)_{\infty}},\quad |b/a|< |z|< 1. $$
(1.6)
$$ {}_{r}\psi_{s} \biggl({{a_{1}, a_{2}, \ldots, a_{r}} \atop {b_{1}, b_{2}, \ldots, b_{s}}} ; q, z \biggr) = \sum_{n=-\infty}^{\infty}\frac{(a_{1}, a_{2}, \ldots, a_{r};q)_{n} }{(b_{1}, b_{2}, \ldots, b_{s} ;q)_{n}}(-1)^{(s-r)n}q^{(s-r){n\choose 2}}z^{n}. $$
(1.7)
$$ \int_{0}^{d}f(t)\, d_{q}t=d(1-q)\sum _{n=0}^{\infty}f\bigl(dq^{n} \bigr)q^{n} $$
(1.8)
$$ \int_{c}^{d}f(t)\, d_{q}t= \int_{0}^{d}f(t)\, d_{q}t- \int_{0}^{c}f(t)\, d_{q}t. $$
(1.9)
$$ \biggl\vert {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, z \biggr)\biggr\vert \leq M\frac{(q,b/a,a|z|,q/a|z|;q)_{\infty}}{(b,q/a,|z|,b/a|z|;q)_{\infty}}, $$
(1.10)
$$ M=\max \Biggl\{ \prod_{i=1}^{r} \frac{(-|a_{i}|; q)_{\infty}}{(|b_{i}|; q)_{\infty}},\prod_{i=1}^{r} \frac{(-q/|b_{i}|; q)_{\infty}}{(q/|a_{i}|; q)_{\infty}} \Biggr\} . $$
2 The proof of theorem
Proof
Under the conditions of the theorem 1.1, it is easy to see that
Letting \(z=c+tq^{n}\) in (1.10) gives
Since \(0< b< a(c+tq^{n})< a<1\), we have
and
Combining (2.2), (2.3), and (2.4), we get
By the definition of the q-integral (1.8), we get
Consequently,
Using (2.5) one gets
Thus, we complete the proof. □
$$ b/a< c+tq^{n}< 1. $$
(2.1)
$$\begin{aligned}& \biggl\vert {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+tq^{n} \biggr)\biggr\vert \\& \quad \leq M\frac{(q,b/a,a(c+tq^{n}),q/a(c+tq^{n});q)_{\infty }}{(b,q/a,c+tq^{n},b/a(c+tq^{n});q)_{\infty}},\quad n=0,1,2,\ldots. \end{aligned}$$
(2.2)
$$ \bigl(a\bigl(c+tq^{n}\bigr),q/a\bigl(c+tq^{n} \bigr);q\bigr)_{\infty}< (b,q/a;q)_{\infty} $$
(2.3)
$$ \bigl(c+tq^{n},b/a\bigl(c+tq^{n}\bigr);q \bigr)_{\infty}\geq(c+t,b/ac;q)_{\infty}. $$
(2.4)
$$ \biggl\vert {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+tq^{n} \biggr)\biggr\vert \leq M\frac{(q,b/a,;q)_{\infty}}{(c+t,b/ac;q)_{\infty}}, \quad n=0,1,2,\ldots. $$
(2.5)
$$\begin{aligned}& \int_{0}^{t}{}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z \\& \quad = t(1-q)\sum_{n=0}^{\infty}q^{n}{}_{r+1} \psi_{r+1} \biggl({{a_{1},a_{2}, \ldots,a_{r+1}} \atop {b_{1},b_{2}, \ldots,b_{r}}} ; q, tq^{n} \biggr). \end{aligned}$$
(2.6)
$$\begin{aligned}& \biggl\vert \int_{0}^{t}{}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \quad =\Biggl\vert t(1-q)\sum_{n=0}^{\infty}q^{n}{}_{r+1} \psi_{r+1} \biggl({{a_{1},a_{2}, \ldots,a_{r+1}} \atop {b_{1},b_{2}, \ldots,b_{r}}} ; q, tq^{n} \biggr)\Biggr\vert \\& \quad \leq t(1-q)\sum_{n=0}^{\infty}q^{n} \biggl\vert {}_{r+1}\psi_{r+1} \biggl({{a_{1},a_{2}, \ldots,a_{r+1}} \atop {b_{1},b_{2}, \ldots,b_{r}}} ; q, tq^{n} \biggr)\biggr\vert . \end{aligned}$$
(2.7)
$$\begin{aligned}& \biggl\vert \int_{0}^{t}{}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \quad \leq t(1-q)M\frac{(q,b/a,;q)_{\infty}}{(c+t,b/ac;q)_{\infty}}\sum_{n=0}^{\infty}q^{n} = \frac{Mt(q,b/a,;q)_{\infty}}{(c+t,b/ac;q)_{\infty}}. \end{aligned}$$
(2.8)
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Corollary 2.1
Under the conditions of the theorem, we have
where
\(s>0\)
and
\(c+s<1\).
$$\begin{aligned}& \biggl\vert \int_{s}^{t} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \quad \leq \frac{M(q,b/a,;q)_{\infty}}{(c+t,c+s,b/ac;q)_{\infty}}\bigl[t(c+s;q)_{\infty }+s(c+t;q)_{\infty} \bigr], \end{aligned}$$
(2.9)
Proof
By the definition of the q-integral (1.9), we get
Thus, the inequality (2.9) holds. □
$$\begin{aligned}& \biggl\vert \int_{s}^{t} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \quad = \biggl\vert \int_{0}^{t} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z \\& \qquad {}- \int_{0}^{s} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \quad \leq \biggl\vert \int_{0}^{t} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \qquad {}+\biggl\vert \int_{0}^{s} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \quad \leq \frac{Mt(q,b/a,;q)_{\infty}}{(c+t,b/ac;q)_{\infty}} +\frac{Ms(q,b/a,;q)_{\infty}}{(c+s,b/ac;q)_{\infty}} \\& \quad = \frac{M(q,b/a,;q)_{\infty}}{(c+t,c+s,b/ac;q)_{\infty }}\bigl[t(c+s;q)_{\infty}+s(c+t;q)_{\infty} \bigr]. \end{aligned}$$
(2.10)
3 Some applications of the inequality
In this section, we use the inequality obtained in this paper to give some sufficient conditions for the convergence of the q-series. Convergence is an important problem in the study of q-series. There are some results about it. For example, Ito used an inequality technique to give a sufficient condition for the convergence of a special q-series called the Jackson integral [14].
Theorem 3.1
Suppose that
(1)
a, b, c
be any positive real numbers such that
\(0< q< b< a<1\), \(c>b/a\);
(2)
\(a_{i}\), \(b_{i}\)
be any real numbers such that
\(|a_{i}|>q\), \(|b_{i}|<1\)
for
\(i=1,2,\ldots,r\)
with
\(r\geq1\)
and
\(|b_{1}b_{2}\cdots b_{r}|\leq|a_{1}a_{2}\cdots a_{r}|\);
(3)
\(\{t_{n}\}\)
be any positive number series, such that
\(c+t_{n}<1\)
and
\(\sum_{n=1}^{\infty}t_{n}\)
converges.
Then the
q-series
converges absolutely.
$$ \sum_{n=0}^{\infty} \int_{0}^{t_{n}}{}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z $$
(3.1)
Proof
Since \(\sum_{n=1}^{\infty}t_{n}\) converges, we get
So, there exists an integer \(N_{0}\) such that, when \(n>N_{0}\),
When \(n>N_{0}\), letting \(t=t_{n}\) in (1.1) gives
From (3.4) and the convergence of \(\sum_{n=1}^{\infty}t_{n}\), it is sufficient to establish that (3.1) is absolutely convergent. □
$$ \lim_{n\rightarrow\infty}t_{n}=0. $$
(3.2)
$$ c+t_{n}\leq d< 1. $$
(3.3)
$$\begin{aligned}& \biggl\vert \int_{0}^{t_{n}}{}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \quad \leq \frac{Mt_{n}(q,b/a;q)_{\infty}}{(c+t_{n},b/ac;q)_{\infty}}\leq \frac{M(q,b/a;q)_{\infty}}{(d,b/ac;q)_{\infty}}t_{n}. \end{aligned}$$
(3.4)
Corollary 3.2
Let
\(\{s_{n}\}\)
be any positive number series such that
\(c+s_{n}<1\)
and
\(\sum_{n=1}^{\infty}s_{n}\)
converges. Under the conditions of Theorem
3.1, then the
q-series
converges absolutely.
$$ \sum_{n=0}^{\infty} \int_{s_{n}}^{t_{n}}{}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z $$
(3.5)
Proof
By the definition of the q-integral (1.9), we get
Since both
and
are absolutely convergent, so (3.5) is absolutely convergent. □
$$\begin{aligned}& \int_{s_{n}}^{t_{n}} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z \\& \quad = \int_{0}^{t_{n}} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z \\& \qquad {}- \int_{0}^{s_{n}} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z. \end{aligned}$$
(3.6)
$$ \int_{0}^{t_{n}} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z $$
(3.7)
$$ \int_{0}^{s_{n}} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z $$
(3.8)
Theorem 3.3
Suppose that
(1)
a, b, c, d
be any positive real numbers such that
\(0< q< b< a<1\), \(c>b/a\), \(c+d<1\);
(2)
\(a_{i}\), \(b_{i}\)
be any real numbers such that
\(|a_{i}|>q\), \(|b_{i}|<1\)
for
\(i=1,2,\ldots,r\)
with
\(r\geq1\)
and
\(|b_{1}b_{2}\cdots b_{r}|\leq|a_{1}a_{2}\cdots a_{r}|\);
(3)
\(\{t_{n}\}\)
be any positive number series, such that
\(t_{n}\leq d\)
and
\(c+d<1\);
(4)
\(\sum_{n=1}^{\infty}e_{n}\)
converges absolutely.
Then the
q-series
converges absolutely.
$$ \sum_{n=0}^{\infty}e_{n} \int_{0}^{t_{n}}{}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z $$
(3.9)
Proof
Using (1.1) gives
Because \(\sum_{n=1}^{\infty}e_{n}\) converges absolutely, (3.10) is sufficient to establish that (3.9) is absolutely convergent. □
$$\begin{aligned}& \biggl\vert e_{n} \int_{0}^{t_{n}}{}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \quad \leq \frac{Mt_{n}(q,b/a;q)_{\infty}}{(c+t_{n},b/ac;q)_{\infty}}|e_{n}| \leq\frac{Md(q,b/a;q)_{\infty}}{(c+d,b/ac;q)_{\infty}}|e_{n}|. \end{aligned}$$
(3.10)
Corollary 3.4
Suppose that
(1)
a, b, c, d
be any positive real numbers such that
\(0< q< b< a<1\), \(c>b/a\), \(c+d<1\);
(2)
\(a_{i}\), \(b_{i}\)
be any real numbers such that
\(|a_{i}|>q\), \(|b_{i}|<1\)
for
\(i=1,2,\ldots,r\)
with
\(r\geq1\)
and
\(|b_{1}b_{2}\cdots b_{r}|\leq|a_{1}a_{2}\cdots a_{r}|\);
(3)
\(\{t_{n}\}\), \(\{s_{n}\}\)
be any positive number series, such that
\(t_{n}\leq d\), \(s_{n}\leq d\), and
\(c+d<1\);
(4)
\(\sum_{n=1}^{\infty}e_{n}\)
converges absolutely.
Then the
q-series
converges absolutely.
$$ \sum_{n=0}^{\infty}e_{n} \int_{s_{n}}^{t_{n}}{}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z $$
(3.11)
Proof
By the definition of the q-integral (1.9), we get
Since \(\sum_{n=1}^{\infty}e_{n}\) converges absolutely, (3.11) converges absolutely. □
$$\begin{aligned}& \biggl\vert e_{n} \int_{s_{n}}^{t_{n}} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \quad = \biggl\vert e_{n} \int_{0}^{t_{n}} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z \\& \qquad {}-e_{n} \int_{0}^{s_{n}} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \quad \leq \biggl\vert e_{n} \int_{0}^{t_{n}} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \qquad {}+\biggl\vert e_{n} \int_{0}^{s_{n}} {}_{r+1}\psi_{r+1} \biggl({{a,a_{1},\ldots,a_{r}} \atop {b,b_{1}, \ldots,b_{r}}} ; q, c+z \biggr)\, d_{q}z\biggr\vert \\& \quad \leq \frac{Mt_{n}(q,b/a;q)_{\infty}}{(c+t_{n},b/ac;q)_{\infty}}|e_{n}| +\frac{Ms_{n}(q,b/a;q)_{\infty}}{(c+s_{n},b/ac;q)_{\infty}}|e_{n}| \\& \quad \leq \frac{2Md(q,b/a;q)_{\infty}}{(c+d,b/ac;q)_{\infty}}|e_{n}|. \end{aligned}$$
(3.12)
Acknowledgements
This work was supported by the National Natural Science Foundation (grant No. 11271057) of China.
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Authors’ contributions
All authors equally have made contributions. All authors read and approved the final manuscript.