Nabla discrete fractional Grüss type inequality
Journal of Inequalities and Applications volume 2014, Article number: 86 (2014)
Properties of the discrete fractional calculus in the sense of a backward difference are introduced and developed. Here, we prove a more general version of the Grüss type inequality for the nabla fractional case. An example of our main result is given.
MSC: 39A12, 34A25, 26A33, 26D15, 26D20.
The Grüss inequality is of great interest in differential and difference equations as well as many other areas of mathematics [1–5]. The classical inequality was proved by Grüss in 1935 : if f and g are continuous functions on satisfying and for all , then
The literature on Grüss type inequalities is extensive, and many extensions of the classical inequality (1) have been intensively studied by many authors in the 21st century [6–12]. Here we are interested in Grüss type inequalities in the nabla fractional calculus case.
We begin with basic definitions and notation from the nabla calculus that are used in this paper. The delta calculus analog has been studied in detail, and a general overview is given in . The domains used in this paper are denoted by where . This is a discrete time scale with a graininess of 1, so it is defined as
For , we use the terminology that a function’s domain, in the case studied here, is based at a. We use the ρ-function, or backward jump function, from the time scale as , given by . We define the backward difference operator, or the nabla operator , for a function by
In this paper, we use the convention that
We then define higher order differences recursively by
for where . We take as convention that is the identity operator.
Based on these preliminary definitions, we say F is an anti-nabla difference of f on if and only if for . We then define the definite nabla integral of by
where . We now give the fundamental theorem of nabla calculus.
Theorem 1.1 (Fundamental theorem of nabla calculus)
Let and let F be an anti-nabla difference of f on , then for any we have
The nabla product rule for two functions and is given by
This immediately leads to the summation by parts formula for the nabla calculus:
Now that we have established the basic definitions of the nabla calculus, we will move on to extending these definitions to the fractional case and establish definitions for the fractional sum and fractional difference which are analogues to the continuous fractional derivative.
In order to do this, we remind the reader of the rising factorial function. For , the rising factorial function is defined by
This definition can be extended for fractional values using the gamma function as follows.
Definition 1 (Rising function)
For , the rising function is defined by
To motivate the definition of a fractional sum, we look at the definition of the integral sum derived from the repeated summation rule.
Definition 2 For any given positive real number α, the (nabla) left fractional sum of order is defined by
where , and
Theorem 1.2 ()
Let f be a real-valued function defined on and . Then
Definition 3 (Fractional Caputo like nabla difference)
For , , (where is the ceiling function), , , we have the following:
We also will use the following discrete Taylor formula.
Theorem 1.3 ()
Let be a function and let . Then, for all with we have the representation
The following discrete backward fractional Taylor formula will be useful.
Theorem 1.4 ()
Let be a function and let . Here , , . Then for all we have the representation
Corollary 1.1 (To Theorem 1.4)
Additionally assume that , for . Then
Also, we have the following.
Theorem 1.5 ()
Let , , . Then
Now, we give the following discrete backward fractional extended Taylor formula.
Theorem 1.6 ()
Let be a function and let . Here , , . Consider . Then for all , , we have the representation
Corollary 1.2 (To Theorem 1.6)
Additionally assume that , for . Then
Remark 1.1 Let f be defined on , where is an integer. Then (3) and (6) are valid only for . Here we must assume that .
2 Main results
We present the following discrete nabla Grüss type inequality.
Theorem 2.1 Let , non-integer, ; with . Consider such that . Let f and g two real-valued functions defined on . Here . Assume that , for and
for , then
where , , , and are positive constants, and
Proof By Theorem 1.6, we have
If we take the sum from to b we get
On the other hand,
Multiplying the above two terms, we get
Using (8) and (9), we obtain
Next, in order to calculate the last two sums, we first observe that
Similarly, we get
which by use of Hölder’s inequality transforms into
Consequently, we get
Example Let , , , , in Theorem 2.1. Define
where . Here . Using Theorem 2.1, we obtain
where , , , and , , and .
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The authors declare that they have no competing interests.
AFG conceived the study, and participated in its design and coordination. BK carried out the mathematical studies and participated in the sequence alignment and drafted the manuscript. ACP participated in the design of the study. KT conceived the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Güvenilir, A.F., Kaymakçalan, B., Peterson, A.C. et al. Nabla discrete fractional Grüss type inequality. J Inequal Appl 2014, 86 (2014). https://doi.org/10.1186/1029-242X-2014-86