An inequality for q-integral and its applications
© Wang; licensee Springer. 2014
Received: 20 April 2014
Accepted: 30 June 2014
Published: 22 July 2014
In this paper, we use the q-binomial theorem to establish an inequality for the q-integral. As applications of the inequality, we give some sufficient conditions for convergence of the q-integral.
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, physics, etc. The inequality technique is one of the useful tools in the study of special functions. There are many papers about the inequalities and q-integral; see [1–10]. Convergence is the key problem of a q-series. In order to give some new methods for convergence of a q-series, we derive an inequality for the q-integral with the basic hypergeometric series . Some applications of the inequality are also given. The main result of this paper is the following inequality.
where , for .
where n is an integer or ∞.
In , the author gives the following inequality.
where , for .
As an application of (1.8), the author give the following sufficient condition for convergence of q-series .
2 Proof of theorem
In this section, we use Theorems 1.2 and 1.3 to prove Theorem 1.1.
we see that the q-integral (2.1) converges absolutely.
where , for .
Substituting (2.5) into (2.4), we get (1.1). □
Thus, the inequality (2.6) holds. □
3 Some applications of the inequality
In this section, we use the inequality obtained in this paper to give a sufficient condition for convergence of a q-series. Convergence is an important problem in the study of a 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 .
where , for .
is absolutely convergent. From (3.5), it is sufficient to establish that (3.1) is absolutely convergent. □
from the theorem, we know that (3.6) is absolutely convergent. □
The author was supported by the National Natural Science Foundation (grant 11271057) of China.
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