Nabla discrete fractional Grüss type inequality
© Güvenilir et al.; licensee Springer. 2014
Received: 20 February 2013
Accepted: 24 January 2014
Published: 20 February 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 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.
for where . We take as convention that is the identity operator.
where . We now give the fundamental theorem of nabla calculus.
Theorem 1.1 (Fundamental theorem of 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.
This definition can be extended for fractional values using the gamma function as follows.
Definition 1 (Rising function)
To motivate the definition of a fractional sum, we look at the definition of the integral sum derived from the repeated summation rule.
Theorem 1.2 ()
Definition 3 (Fractional Caputo like nabla difference)
We also will use the following discrete Taylor formula.
Theorem 1.3 ()
The following discrete backward fractional Taylor formula will be useful.
Theorem 1.4 ()
Corollary 1.1 (To Theorem 1.4)
Also, we have the following.
Theorem 1.5 ()
Now, we give the following discrete backward fractional extended Taylor formula.
Theorem 1.6 ()
Corollary 1.2 (To Theorem 1.6)
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.
where , , , and , , and .
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