**R E S E A R C H**

**Open Access**

## Nabla discrete fractional Grüss type

## inequality

### A Feza Güvenilir

1_{, Billur Kaymakçalan}

2*_{, Allan C Peterson}

3_{and Kenan Ta¸s}

2
*_{Correspondence:}

billur@cankaya.edu.tr

2_{Department of Mathematics and}

Computer Science, Çankaya University, Ankara, 06810, Turkey Full list of author information is available at the end of the article

**Abstract**

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:** Primary 39A12; 34A25; 26A33; secondary 26D15; 26D20

**Keywords:** nabla discrete fractional calculus; nabla discrete Grüss inequality

**1 Introduction**

The Grüss inequality is of great interest in diﬀerential and diﬀerence equations as well
as many other areas of mathematics [–]. The classical inequality was proved by Grüss
in []: if*f* and*g*are continuous functions on [*a*,*b*] satisfying*ϕ*≤*f*(*t*)≤and*γ* ≤

*g*(*t*)≤for all*t*∈[*a*,*b*], then

_{b}_{–}_{a}*b*

*a*

*f*(*t*)*g*(*t*)*dt*–
(*b*–*a*)

*b*

*a*

*f*(*t*)*dt*

*b*

*a*

*g*(*t*)*dt*≤

(–*ϕ*)(–*γ*). ()

The literature on Grüss type inequalities is extensive, and many extensions of the clas-sical inequality () have been intensively studied by many authors in the st century [–]. 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 byN*a*where*a*∈R. This is a
discrete time scale with a graininess of , so it is deﬁned as

N*a*:={*a*,*a*+ ,*a*+ , . . .}.

ForN*a, 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*ρ*:N*a*→

N*a*, given by*ρ*(*t*) :=max{*a*,*t*– }. We deﬁne the backward diﬀerence operator, or the nabla
operator (∇), for a function*f* :N*a*→Rby

(∇*f*)(*t*) :=*f*(*t*) –*fρ*(*t*)=*f*(*t*) –*f*(*t*– ).

In this paper, we use the convention that

∇*f*(*t*) = (∇*f*)(*t*).

We then deﬁne higher order diﬀerences recursively by

∇*n _{f}*

_{(}

_{t}_{) :=}

_{∇}

_{∇}

*n*–

_{f}_{(}

_{t}_{)}

for*t*∈N*a*+*n*where*n*∈N. We take as convention that∇is the identity operator.

Based on these preliminary deﬁnitions, we say*F*is an anti-nabla diﬀerence of*f* onN*a*if
and only if∇*F*(*t*) =*f*(*t*) for*t*∈N*a*+. We then deﬁne the deﬁnite nabla integral of*f* :N*a*→R
by

*d*

*c*

*f*(*t*)∇*t*:=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

*d*

*t*=*c*+*f*(*t*), if*c*<*d*,

, if*c*=*d*,

–*c _{t}*

_{=}

_{d}_{+}

*f*(

*t*), if

*d*<

*c*,

where*c*,*d*∈N*a. We now give the fundamental theorem of nabla calculus.*

**Theorem .**(Fundamental theorem of nabla calculus) *Let f* :N*a*→R*and let F be an*

*anti-nabla diﬀerence of f on*N*a*,*then for any c*,*d*∈N*awe have*

*d*

*c*

*f*(*t*)∇*t*=*F*(*d*) –*F*(*c*).

The nabla product rule for two functions*u*,*ν*:N*a*→Rand*t*∈N*a*+is given by

∇*u*(*t*)*ν*(*t*)=*u*(*t*)∇*v*(*t*) +*vρ*(*t*)∇*u*(*t*).

This immediately leads to the summation by parts formula for the nabla calculus:

*c*

*s*=*b*+

*u*(*t*)∇*v*(*t*) =*u*(*t*)*v*(*t*)|*c _{b}*–

*c*

*s*=*b*+

*vρ*(*t*)∇*u*(*t*).

Discrete fractional initial value problems have been intensively studied by many authors. Some of the very recent results are in [–].

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*n*,*t*∈N, the
rising factorial function is deﬁned by

*tn*:=*t*(*t*+ )· · ·(*t*+*n*– ) = (*t*+*n*– )!
(*t*– )! .

**Deﬁnition **(Rising function) For*t*,*α*∈R, the rising function is deﬁned by

*tα*:=(*t*+*α*)
(*t*) .

To motivate the deﬁnition of a fractional sum, we look at the deﬁnition of the integral sum derived from the repeated summation rule.

**Deﬁnition ** For any given positive real number*α*, the (nabla) left fractional sum of order
*α*> is deﬁned by

∇–*α*

*a* *f*(*t*) =
(*α*)

*t*

*s*=*a*+

*t*–*ρ*(*s*)α–*f*(*s*),

where*t*∈ {*a*+ ,*a*+ , . . .}, and

∇–*α*

*a* *f*(*t*) =
(*α*)

*t*

*s*=*a*

*t*–*ρ*(*s*)α–*f*(*s*),

where*t*∈ {*a*,*a*+ ,*a*+ , . . .}.

**Theorem .**([]) *Let f be a real-valued function deﬁned on*N*aandα*,*β*> .*Then*

∇–*α*

*a* ∇*a*–*βf*(*t*) =∇*a*–(*α*+*β*)*f*(*t*) =∇*a*–*β*∇*a*–*αf*(*t*).

**Deﬁnition **(Fractional Caputo like nabla diﬀerence) For*μ*> ,*m*– <*μ*<*m*,*m*=*μ*
(where·is the ceiling function),*m*∈N,*v*=*m*–*μ*, we have the following:

∇*μ*

*a∗f*(*t*) =∇*a*–*v*

∇*m _{f}*

_{(}

_{t}_{)}

_{,}

_{t}_{∈}

_{N}

_{a.}We also will use the following discrete Taylor formula.

**Theorem .**([]) *Let f* :Z→R*be a function and let a*∈Z.*Then*,*for all t*∈Z*with*
*t*∈N*a*+*mwe have the representation*

*f*(*t*) =
*m*–

*k*=

(*t*–*a*)*k*

*k*! ∇

*k _{f}*

_{(}

_{a}_{) +} (

*m*– )!

*t*

*τ*=*a*+

(*t*–*τ*+ )*m*–∇*mf*(*τ*). ()

The following discrete backward fractional Taylor formula will be useful.

**Theorem .**([]) *Let f*:Z→R*be a function and let a*∈Z.*Here m*– <*μ*<*m*,*m*=*μ*,
*μ*> .*Then for all t*∈N*a*+*mwe have the representation*

*f*(*t*) =
*m*–

*k*=

(*t*–*a*)*k*

*k*! ∇

*k _{f}*

_{(}

_{a}_{) +} (

*μ*)

*t*

*τ*=*a*+

**Corollary .**(To Theorem .) *Additionally assume that*∇*k _{f}*

_{(}

_{a}_{) = ,}

_{for k}_{= , , , . . . ,}

*m*– .*Then*

*f*(*t*) =
(*μ*)

*t*

*τ*=*a*+

(*t*–*τ*+ )*μ*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*f*(

*τ*), ∀

*t*≥

*a*+

*m*. ()

Also, we have the following.

**Theorem .**([]) *Let p*∈N,*ν*>*p*,*a*∈N.*Then*

∇*p*_{∇}–*ν*

*a* *f*(*t*)

=∇* _{a}*–(

*ν*–

*p*)

*f*(

*t*) ()

*for t*∈N*a.*

Now, we give the following discrete backward fractional extended Taylor formula.

**Theorem .**([]) *Let f* :Z→R*be a function and let a*∈Z+.*Here m*– <*μ*<*m*,*m*=

*μ*,*μ*> .*Consider p*∈N:*μ*>*p*.*Then for all t*≥*a*+*m*,*t*∈N,*we have the representation*

∇*p _{f}*

_{(}

_{t}_{) =}

*m*–

*k*=*p*

(*t*–*a*)*k*–*p*
(*k*–*p*)! ∇

*k _{f}*

_{(}

_{a}_{) +} (

*μ*–

*p*)

*t*

*τ*=*a*+

(*t*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*f*(

*τ*). ()

**Corollary .**(To Theorem .) *Additionally assume that*∇*kf*(*a*) = ,*for k*=*p*, . . . ,*m*– .

*Then*

∇*p _{f}*

_{(}

_{t}_{) =} (

*μ*–

*p*)

*t*

*τ*=*a*+

(*t*–*τ*+ )*μ*–*p*–_{∇}*μ*

(*a*+)∗*f*(*τ*), ∀*t*≥*a*+*m*,*t*∈N. ()

**Remark .** Let*f*be deﬁned on{*a*–*m*+ ,*a*–*m*+ , . . . ,*b*}, where*b*–*a*is an integer. Then
() and () are valid only for*t*∈ {*a*+*m*, . . . ,*b*}. Here we must assume that*a*+*m*<*b*.

**2 Main results**

We present the following discrete nabla Grüss type inequality.

**Theorem .** *Let m*– <*μ*<*m*,*m*=*μnon-integer*,*μ*> ;*p*,*a*∈Z+*withμ*>*p*.*Consider*

*b*∈N*such that a*+*m*<*b*.*Let f and g two real-valued functions deﬁned on*{*a*–*m*+ ,*a*–

*m*+ , . . . ,*b*}.*Here j*∈ {*a*+*m*, . . . ,*b*}.*Assume that*∇*k _{f}*

_{(}

_{a}_{) =}

_{∇}

*k*

_{g}_{(}

_{a}_{) = ,}

_{for k}_{=}

_{p}_{+ , . . . ,}

_{m}_{– }

*and*

*m*≤ ∇

*μ*

(*a*+)∗*f*(*s*)≤*M*, *m*≤ ∇

*μ*

(*a*+)∗*g*(*s*)≤*M*

*for s*∈ {*a*+ , . . . ,*b*},*then*

*b*–*a*–*m*

*b*

*j*=*a*+*m*+

∇*p _{f}*

_{(}

_{j}_{)}

_{∇}

*p*

_{g}_{(}

_{j}_{)}

_{–} (

*b*–

*a*–

*m*)

_{b}

*j*=*a*+*m*+

∇*p _{f}*

_{(}

_{j}_{)}

_{b}

*j*=*a*+*m*+

∇*p _{g}*

_{(}

_{j}_{)}

*where m*,*m*,*M*,*and M**are positive constants*,*and*

*C*:=

*Q*(*b*–*a*–*m*)(*b*–*a*–*m*+ ) _{+ (}_{b}_{–}_{a}_{–}_{m}_{)}_{}*μ*–*p*_{,}
*C*:=

(*b*–*a*+ )*μ*–*p*+_{– (}_{m}_{)}*μ*–*p*+

*μ*–*p*+ – (*b*–*a*–*m*)(*μ*–*p*+ )

,

*Q*:=

_{b}_{–}_{a}_{+}

*s*=*m*+

(*s*)*μ*–*p*

.

*Proof* By Theorem ., we have

∇*p _{f}*

_{(}

_{j}_{) =}

_{∇}

*p*

_{f}_{(}

_{a}_{) +} (

*μ*–

*p*)

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–_{∇}*μ*

(*a*+)∗*f*(*τ*)

,

∇*p _{g}*

_{(}

_{j}_{) =}

_{∇}

*p*

_{g}_{(}

_{a}_{) +} (

*μ*–

*p*)

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*g*(

*τ*)

for*j*∈ {*a*+*m*+ ,*a*+*m*+ , . . . ,*b*}.
We get

∇*p _{f}*

_{(}

_{j}_{)}

_{∇}

*p*

_{g}_{(}

_{j}_{)}

_{=} [(

*μ*–

*p*)]

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–_{∇}*μ*

(*a*+)∗*f*(*τ*)

·

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*g*(

*τ*)

+ ∇
*p _{f}*

_{(}

_{a}_{)}

(*μ*–*p*)

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*g*(

*τ*)

+ ∇
*p _{g}*

_{(}

_{a}_{)}

(*μ*–*p*)

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*f*(

*τ*)

+∇*p _{f}*

_{(}

_{a}_{)}

_{∇}

*p*

_{g}_{(}

_{a}_{)}

_{.}

If we take the sum from*a*+*m*+ to*b*we get

*b*

*j*=*a*+*m*+

∇*p _{f}*

_{(}

_{j}_{)}

_{∇}

*p*

_{g}_{(}

_{j}_{)}

_{=}

_{∇}

*p*

_{f}_{(}

_{a}_{)}

_{∇}

*p*

_{g}_{(}

_{a}_{)}

+ ∇
*p _{f}*

_{(}

_{a}_{)}

(*μ*–*p*)

_{b}

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*g*(

*τ*)

+ ∇
*p _{g}*

_{(}

_{a}_{)}

(*μ*–*p*)

_{b}

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–_{∇}*μ*

(*a*+)∗*f*(*τ*)

+

[(*μ*–*p*)]

*b*

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*f*(

*τ*)

·

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–_{∇}*μ*

(*a*+)∗*g*(*τ*)

Then

*b*–*a*–*m*

*b*

*j*=*a*+*m*+

∇*p _{f}*

_{(}

_{j}_{)}

_{∇}

*p*

_{g}_{(}

_{j}_{)}

=

*b*–*a*–*m*

∇*p _{f}*

_{(}

_{a}_{)}

_{∇}

*p*

_{g}_{(}

_{a}_{)}

+

(*b*–*a*–*m*)[(*μ*–*p*)]

*b*

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*f*(

*τ*)

·

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*g*(

*τ*)

+ ∇

*p _{f}*

_{(}

_{a}_{)}

(*b*–*a*–*m*)(*μ*–*p*)

_{b}

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*g*(

*τ*)

+ ∇

*p _{g}*

_{(}

_{a}_{)}

(*b*–*a*–*m*)(*μ*–*p*)

_{b}

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*f*(

*τ*)

. ()

On the other hand,

*b*–*a*–*m*

_{b}

*j*=*a*+*m*+

∇*p _{f}*

_{(}

_{j}_{)}

=∇*pf*(*a*)

+

(*b*–*a*–*m*)(*μ*–*p*)
*b*

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–_{∇}*μ*

(*a*+)∗*f*(*τ*)

and

*b*–*a*–*m*

_{b}

*j*=*a*+*m*+

∇*p _{g}*

_{(}

_{j}_{)}

=∇*pg*(*a*)

+

(*b*–*a*–*m*)(*μ*–*p*)
*b*

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*g*(

*τ*)

.

Multiplying the above two terms, we get

(*b*–*a*–*m*)

_{b}

*j*=*a*+*m*+

∇*p _{f}*

_{(}

_{j}_{)}

_{b}

*j*=*a*+*m*+

∇*p _{g}*

_{(}

_{j}_{)}

=∇*pf*(*a*)∇*pg*(*a*)

+

(*b*–*a*–*m*)_{[}_{}_{(}_{μ}_{–}_{p}_{)]}

_{b}

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*f*(

*τ*)

·

_{b}

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*g*(

*τ*)

+ (∇

*p _{f}*

_{(}

_{a}_{))}

(*b*–*a*–*m*)(*μ*–*p*)

_{b}

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*g*(

*τ*)

+ (∇

*p _{g}*

_{(}

_{a}_{))}

(*b*–*a*–*m*)(*μ*–*p*)

_{b}

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–∇_{(}*μ _{a}*

_{+)}

_{∗}

*f*(

*τ*)

. ()

Using () and (), we obtain

(*b*–*a*–*m*)

*b*

*j*=*a*+*m*+

∇*p _{f}*

_{(}

_{j}_{)}

_{∇}

*p*

_{g}_{(}

_{j}_{)}

–

(*b*–*a*–*m*)

_{b}

*j*=*a*+*m*+

∇*p _{f}*

_{(}

_{j}_{)}

_{b}

*j*=*a*+*m*+

∇*p _{g}*

_{(}

_{j}_{)}

≤ *M**M*

(*b*–*a*–*m*)[(*μ*–*p*)]

*b*

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–

– *m**m*

(*b*–*a*–*m*)_{[}_{}_{(}_{μ}_{–}_{p}_{)]}

_{b}

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–

·

_{b}

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–

= *M**M*
(*b*–*a*–*m*)[(*μ*–*p*)]

*b*

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–

– *m**m*

(*b*–*a*–*m*)_{[}_{}_{(}_{μ}_{–}_{p}_{)]}

_{b}

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–

.

Next, in order to calculate the last two sums, we ﬁrst observe that

*j*

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–=

*j*

*a*

(*j*–*τ*+ )*μ*–*p*–∇*τ*

=
*μ*–*p*

(*j*–*a*+ +*μ*–*p*)

(*j*–*a*+ ) –(*μ*–*p*+ )

.

Therefore,

*b*

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–

=
*μ*–*p*

*b*

*j*=*a*+*m*+

(*j*–*a*+ +*μ*–*p*)

(*j*–*a*+ ) –(*μ*–*p*+ )

=
*μ*–*p*

(*m*+ +*μ*–*p*)
(*m*+ ) +· · ·+

(*b*–*a*+ +*μ*–*p*)
(*b*–*a*+ )

–*b*–*a*–*m*

=
*μ*–*p*

(*m*+ )*μ*–*p*+ (*m*+ )*μ*–*p*+· · ·+ (*b*–*a*+ )*μ*–*p*

–*b*–*a*–*m*

*μ*–*p* (*μ*–*p*+ )

=
*μ*–*p*

*b*_{}–*a*+

*s*=*m*+

(*s*)*μ*–*p*–*b*–*a*–*m*

*μ*–*p* (*μ*–*p*+ )

=
*μ*–*p*

*b*–*a*+

*m*+

*τμ*–*p*∇*τ*–*b*–*a*–*m*

*μ*–*p* (*μ*–*p*+ )

=
*μ*–*p*

(*b*–*a*+ )*μ*–*p*+_{– (}_{m}_{+ )}*μ*–*p*+

*μ*–*p*+

–*b*–*a*–*m*

*μ*–*p* (*μ*–*p*+ ).

Similarly, we get

*b*

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–

≤

(*μ*–*p*)

_{b}

*j*=*a*+*m*+

(*j*–*τ*+ )*μ*–*p*

+

(*μ*–*p*)

*μ*–*p*(*b*–*a*–*m*),

which by use of Hölder’s inequality transforms into

*b*

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–

≤(*b*–*a*–*m*+ )/

(*μ*–*p*)

_{b}_{–}_{a}_{+}

*s*=*m*+

(*s*)*μ*–*p*

/

+*b*–*a*–*m*
(*μ*–*p*)

*μ*–*p*

.

By hypothesis,

*b*

*j*=*a*+*m*+

_{j}

*τ*=*a*+

(*j*–*τ*+ )*μ*–*p*–

≤*Q*(*b*–*a*–*m*+ )/

(*μ*–*p*) +

(*b*–*a*–*m*)[*μ*–*p*_{]}

(*μ*–*p*) .

Consequently, we get

*b*–*a*–*m*

*b*

*j*=*a*+*m*+

∇*p _{f}*

_{(}

_{j}_{)}

_{∇}

*p*

_{g}_{(}

_{j}_{)}

–

(*b*–*a*–*m*)

_{b}

*j*=*a*+*m*+

∇*p _{f}*

_{(}

_{j}_{)}

_{b}

*j*=*a*+*m*+

∇*p _{g}*

_{(}

_{j}_{)}

≤ *M**M**C*–*m**m**C*

(*b*–*a*–*m*)_{[}_{}_{(}_{μ}_{–}_{p}_{+ )]}.

**Example** Let*μ*= .,*m*= ,*p*= ,*b*= ,*a*= in Theorem .. Deﬁne

where*t*∈ {–, –, , , . . . , }. Here*j*∈ {, , }. Using Theorem ., we obtain

*t*=

∇_{(}_{t}_{– )}_{∇}_{(}_{t}_{– )}_{–}

_{}

*t*=

∇_{(}_{t}_{– )}

_{}

*t*=

∇_{(}_{t}_{– )}

≤*M**M**C*–*m**m**C*

[(.)] ,

where*M*= ,*M*= ,*m*= ,*m*= and*C*= ,,*C*= ,, and*Q*= ..

**Competing interests**

The authors declare that they have no competing interests.

**Authors’ contributions**

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.

**Author details**

1_{Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey.}2_{Department of Mathematics and}

Computer Science, Çankaya University, Ankara, 06810, Turkey. 3_{Department of Mathematics, University of}
Nebraska-Lincoln, Lincoln, NE 68588-0130, USA.

Received: 20 February 2013 Accepted: 24 January 2014 Published:20 Feb 2014

**References**

1. Boros, G, Moll, V: Irresistible Integrals: Symbols, Analysis and Experiments in the Evaluation of Integrals. Cambridge University Press, Cambridge (2004)

2. Cerone, P, Dragomir, SS: A reﬁnement of the Grüss inequality and applications. Tamkang J. Math.38, 37-49 (2007) 3. Grüss, G: Über das Maximum des absoluten Betrages von 1

*b*–*a*

*b*

*af*(*x*)*g*(*x*)*dx*–

1
(*b*–*a*)2

*b*
*af*(*x*)*dx*

*b*

*ag*(*x*)*dx*. Math. Z.39,

215-226 (1935)

4. Mercer, AMD: An improvement of the Grüss inequality. JIPAM. J. Inequal. Pure Appl. Math.6(4), Article ID 93 (2005) (electronic)

5. Mitrinovi´c, DS, Pe˘cari´c, JE, Fink, AM: Classical and New Inequalities in Analysis. Kluwer Acadademic, Dordrecht (1993) 6. Anastassiou, GA: Multivariate Fink type identity and multivariate Ostrowski, comparison of means and Grüss type

inequalities. Math. Comput. Model.46, 351-374 (2007)

7. Bohner, M, Matthews, T: The Grüss inequality on time scales. Commun. Math. Anal.3, 1-8 (2007) (electronic) 8. Dragomir, SS: A Grüss type discrete inequality in inner product spaces and applications. J. Math. Anal. Appl.250,

494-511 (2000)

9. Liu, Z: Notes on a Grüss type inequality and its applications. Vietnam J. Math.35, 121-127 (2007) 10. Peri´c, I, Raji´c, R: Grüss inequality for completely bounded maps. Linear Algebra Appl.390, 292-298 (2004)

11. Pe˘cari´c, JE, Tepe˘s, B: On the Grüss type inequalities of Dragomir and Fedotov. J. Inequal. Pure Appl. Math.4(5), Article ID 91 (2003) (electronic)

12. Pe˘cari´c, JE, Tepe˘s, B: A note on Grüss type inequality in terms of-seminorms. Pril. - Maked. Akad. Nauk. Umet., Odd. Mat.-Teh. Nauki23/24, 29-35 (2004)

13. Goodrich, CS: Continuity of solutions to discrete fractional initial value problems. Comput. Math. Appl.59, 3489-3499 (2010)

14. Goodrich, CS: Existence and uniqueness of solutions to a fractional diﬀerence equation with nonlocal conditions. Comput. Math. Appl.61, 191-202 (2011)

15. Goodrich, CS: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl.385, 111-124 (2012) 16. Abdeljawad, T, Atici, FM: On the deﬁnitions of nabla fractional operators. Abstr. Appl. Anal.2012, Article ID 406757,

Special Issue (2012)

17. Anderson, DR: Taylor polynomials for nabla dynamic equations on time scales. Panam. Math. J.12, 17-27 (2002) 18. Anastassiou, GA: Nabla discrete fractional calculus and nabla inequalities. Math. Comput. Model.51, 562-571 (2010)

10.1186/1029-242X-2014-86