Generalizations of Hölder inequality and some related results on time scales
© Chen et al.; licensee Springer. 2014
Received: 16 November 2013
Accepted: 12 May 2014
Published: 23 May 2014
In this paper, we establish some generalizations and refinements of the Hölder inequality on time scales via the diamond-α dynamic integral, which is defined as a linear combination of the delta and nabla integrals. Some related inequalities are also considered.
If and , then inequality (1.1) reduces to the famous Cauchy-Schwarz inequality (see ). Both the Cauchy-Schwarz inequality and the Hölder inequality play a significant role in different branches of modern mathematics. A great number of generalizations, refinements, variations, and applications of these inequalities have been studied in the literature (see [3–16] and the references therein).
The aim of this paper is to derive some new generalizations and refinements of the diamond-α integral Hölder inequality on time scales. Some related inequalities are also considered. The paper is organized as follows. In Section 2, we recall the basic definitions of time scale calculus, which can also be found in [13, 17–32], and of delta, nabla, and diamond-α dynamic derivatives. In Section 3, we will give the main results.
A time scale is an arbitrary nonempty closed subset of ℝ. The set of the real numbers, the integers, the natural numbers, and the Cantor set are examples of time scales. But the rational numbers, the irrational numbers, the complex numbers, and the open interval between 0 and 1 are not time scales. We first recall some basic concepts from the theory of time scales.
where and , ∅ denotes the empty set.
Definition 2.1 A point , , is said to be left-dense if , right-dense if and , left-scattered if , and right-scattered if .
Definition 2.2 A function is called rd-continuous if it is continuous at right-dense points and has finite left-sided limits at left-dense points. A function is called ld-continuous if it is continuous at left-dense points and has finite right-sided limits at right-dense points.
Definition 2.3 Assume that is a function, then we define the functions and .
We now introduce the basic notions of delta and nabla integrations.
Definition 2.5 An with is called a Δ-antiderivative of f, and then the Δ-integral of f is defined by for any . Also, with is called a ∇-antiderivative of f, and then the ∇-integral of f is defined by for any . It is known that rd-continuous functions have Δ-antiderivatives and ld-continuous functions have ∇-antiderivatives.
Proposition 2.1 (see )
Proposition 2.2 (see )
if for all , then ;
if for all , then ;
if for all , then if and only if .
3 Main results
In this section, we introduce the following lemma first before we give our results.
Lemma 3.1 (see )
- (1)for , we have(3.1)
- (2)for , (), we have(3.2)
- (1)for , we have(3.3)
- (2)for , (), we have(3.4)
- (2)Set , by (3.2), we obtain
Hence, we have the desired result. □
- (1)for , we have the following inequality:(3.5)
- (2), (), we have the following reverse inequality:(3.6)
- (2)This proof is similar to the proof of inequality (3.5), by (3.7), (3.8), and the reverse Hölder inequality (3.4), we have(3.10)
Substitution of in (3.10) leads to inequality (3.6) immediately. □
- (1)for , we have the following inequality:(3.11)
- (2), (), we have the following reverse inequality:(3.12)
- (1)for , we have the following inequality:(3.13)
- (2)for , (), we have the following reverse inequality:(3.14)
Since , () and , we have , by (3.6), we immediately have the inequality (3.14). This completes the proof. □
From Theorem 3.4, we obtain a Hölder type generalization of (3.15) as follows.
- (1)for , we have the following inequality:(3.16)
- (2)for , , we have the following reverse inequality:(3.17)
Now we present a refinement of inequality (3.13) and (3.14), respectively.
- (1)for , we have the following inequality:(3.18)
- (2)for , (), we have the following reverse inequality:(3.19)
is a nondecreasing function with .
This proof is similar to the proof of inequality (3.18), we have inequality (3.19). □
Remark 3.1 Taking , Theorem 3.5 presents refinement of (3.16) and (3.17). Moreover, letting and , then the results of this paper lead to the main results of .
The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper. This paper was partially supported by NNSFC (No. 11326161), the key projects of Science and Technology Research of the Henan Education Department (No. 14A110011), the key project of Guangxi Social Sciences (No. gxsk201424) and the Education Science fund of the Education Department of Guangxi (No. 2013JGB410).
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