Abstract
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.
MSC:26D15.
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1 Introduction
Let . If and () is a continuous real-valued function on , then
-
(1)
for , we have the following Hölder inequality (see [1]):
(1.1) -
(2)
for , (), we have the following reverse Hölder inequality (see [2]):
(1.2)
If and , then inequality (1.1) reduces to the famous Cauchy-Schwarz inequality (see [3]). 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.
2 Preliminaries
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.
For , we define the forward jump operator by
and the backward jump operator by
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 .
Definition 2.4 Suppose that is a function, then for we define to be the number, if one exists, such that for all there is a neighborhood U of t such that for all
We say that f is delta differentiable on provided exists for all . Similarly, for we define to be the number value, if one exists, such that for all there is a neighborhood V of t such that for all
We say that f is nabla differentiable on provided exists for all .
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.
Recently, using the above derivatives and integrations, Sheng et al. [30] (see also [27–29]) have established the diamond-α derivative and the diamond-α integration on time scales.
Definition 2.6 If and is a function, then the -differentiation of f at a point is defined by
The -integral of f is defined by
Proposition 2.1 (see [30])
Let be a time scale with . Assume that f and g are continuous functions on . Let and . Then
-
(1)
;
-
(2)
;
-
(3)
;
-
(4)
;
-
(5)
.
Proposition 2.2 (see [30])
Let be a time scale with . Assume that f and g are continuous functions on ,
-
(1)
if for all , then ;
-
(2)
if for all , then ;
-
(3)
if for all , then if and only if .
Results about -derivatives and -integrals may be found in the papers [28–30, 32].
Throughout this work, we suppose that is a time scale, with and an interval means the intersection of a real interval with the given time scale.
3 Main results
In this section, we introduce the following lemma first before we give our results.
Lemma 3.1 (see [33])
Let , (). Then
-
(1)
for , we have
(3.1) -
(2)
for , (), we have
(3.2)
Theorem 3.2 Let be a time scale, with and . If , and () is continuous real-valued function on , then
-
(1)
for , we have
(3.3) -
(2)
for , (), we have
(3.4)
Proof (1) Let , by (3.1), we have
Therefore, we obtain the desired inequality.
-
(2)
Set , by (3.2), we obtain
Hence, we have the desired result. □
Theorem 3.3 Let be a time scale with and (, ), , . If , and () is a continuous real-valued function on , then
-
(1)
for , we have the following inequality:
(3.5) -
(2)
, (), we have the following reverse inequality:
(3.6)
Proof (1) Set
Applying the assumptions and , by computing, we can observe that
That is,
Hence, we obtain
By the Hölder inequality (3.3), we find
Substitution of in (3.9) conduce to inequality (3.5) immediately. This proves inequality (3.5).
-
(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. □
Corollary 3.1 Under the conditions of Theorem 3.3, let , for and with , then
-
(1)
for , we have the following inequality:
(3.11) -
(2)
, (), we have the following reverse inequality:
(3.12)
Theorem 3.4 Let be a time scale, with and , (, ), ,. If , and () is a continuous real-valued function on , then
-
(1)
for , we have the following inequality:
(3.13) -
(2)
for , (), we have the following reverse inequality:
(3.14)
Proof (1) Since and , we get . Then by (3.5), we immediately obtain the inequality (3.13).
-
(2)
Since , () and , we have , by (3.6), we immediately have the inequality (3.14). This completes the proof. □
Recently, Yang [11] established an extension of the Callebaut inequality, that is,
From Theorem 3.4, we obtain a Hölder type generalization of (3.15) as follows.
Corollary 3.2 Under the conditions of Theorem 3.4, and taking , , , , then
-
(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.
Theorem 3.5 Under the conditions of Theorem 3.4, we have
-
(1)
for , we have the following inequality:
(3.18)
where
is a nonincreasing function with ;
-
(2)
for , (), we have the following reverse inequality:
(3.19)
where
is a nondecreasing function with .
Proof (1) Let
By rearrangement, using the assumptions of Theorem 3.4, we have
Then by the Hölder inequality (3.3), we obtain
Therefore, we obtain the desired result.
-
(2)
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 [13].
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Acknowledgements
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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Chen, GS., Huang, FL. & Liao, LF. Generalizations of Hölder inequality and some related results on time scales. J Inequal Appl 2014, 207 (2014). https://doi.org/10.1186/1029-242X-2014-207
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DOI: https://doi.org/10.1186/1029-242X-2014-207