Combined dynamic Gruss inequalities on time scales

0.16 MB·English

by  Moulay Rchid Sidi Ammi

COMBINED DYNAMIC GRU¨SS INEQUALITIES
ON TIME SCALES
MOULAYRCHIDSIDIAMMIANDDELFIMF.M.TORRES
8
0 Abstract. WeproveamoregeneralversionoftheGru¨ssinequalitybyusing
0 the recent theory of combined dynamic derivatives on time scales and the
2 moregeneralnotionsofdiamond-αderivativeandintegral. Fortheparticular
n case when α = 1, one gets a delta-integral Gru¨ss inequality on time scales;
a for α = 0 a nabla-integral Gru¨ss inequality. If we further restrict ourselves
J by fixing the time scale to the real (or integer) numbers, then the standard
continuous (discrete)inequalitiesareobtained.
1
1
1. Introduction

]
A TheGru¨ssinequalityisofgreatinterestindifferentialanddifferenceequationsas
C well as many other areas of mathematics [8, 9, 16, 18, 19]. The classicalinequality
. was proved by G. Gru¨ss in 1935 [13]: if f and g are two continuous functions on
h
[a,b] satisfying ϕ≤f(t)≤Φ and γ ≤g(t)≤Γ for all t∈[a,b], then
t
a
m 1 b 1 b b 1
(1) f(t)g(t)dt− f(t)dt g(t)dt ≤ (Φ−ϕ)(Γ−γ).
[ (cid:12)(cid:12)b−aZa (b−a)2 Za Za (cid:12)(cid:12) 4
(cid:12) (cid:12)
1 The lit(cid:12)erature on Gru¨ss type inequalities is now vast, and(cid:12) many extensions of the
v classica(cid:12)l inequality (1) have been intensively studied by m(cid:12)any authors in the XXI
5
century[3,5,10,17,22,23,24,25]. HereweareinterestedinGru¨sstypeinequalities
6
on time scales.
8
1 The analysis on time scales is a relatively new area of mathematics that unify
. and generalize discrete and continuous theories. Moreover, it is a crucial tool in
1
0 many computational and numerical applications. The subject was initiated by
8 StefanHilger[14,15]andisbeingappliedtomanydifferentfieldsinwhichdynamic
0 processes can be described with discrete or continuous models: see [1, 4, 6, 7, 12]
:
v and references therein. One of the important subjects being developed within the
i theory of time scales is the study of inequalities [2, 11, 21, 29].
X
Recently, Bohner and Matthews [5] proved a time scales version of the Gru¨ss
r
a inequality (1) by using the delta integral. Roughly speaking, the main result in [5]
asserts that (1) continues to be valid on a general time scale by substituting the
Riemann integralby the ∆-integral. The main objective of this paper is to use the
more general diamond−α dynamic integral.
In2006,acombineddynamicderivative♦ wasintroducedas alinearcombina-
α
tionof∆and∇dynamicderivativesontimescales[28]. Thediamond–αderivative
reducestothestandard∆derivativeforα=1andtothestandard∇derivativefor
α=0. On the other hand, it represents a weighted dynamic derivative formula on
2000 Mathematics Subject Classification. 26D15,39A13.
Key words and phrases. Time scales, dynamic inequalities, combined dynamic derivatives,
Gru¨ssinequality.
ResearchpartiallysupportedbytheCentreforResearchonOptimizationandControl(CEOC)
fromthePortugueseFoundation forScienceandTechnology(FCT),cofinancedbytheEuropean
CommunityFundFEDER/POCI2010; andFCTpostdocfellowshipSFRH/BPD/20934/2004.
1
2 M.R.SIDIAMMIANDD.F.M.TORRES
anyuniformlydiscretetimescalewhenα= 1. Wereferthereaderto[20,26,27,28]
2
for an account of the calculus with diamond-α dynamic derivatives.
In Section 2 we briefly review the necessary definitions and calculus on time
scales. Our results are given in Section 3.
2. Preliminaries
A time scale T is an arbitrarynonempty closedsubset of the realnumbers. The
time scales calculus was initiated by S. Hilger in his PhD thesis in order to unify
discrete and continuous analysis [14, 15]. Let T be a time scale with the topology
that it inherits from the real numbers. For t ∈ T, we define the forward jump
operator σ :T→T by
σ(t)=inf{s∈T:s>t} ,
and the backward jump operator ρ:T→T by
ρ(t)=sup{s∈T:s<t} .
If σ(t) > t we say that t is right-scattered, while if ρ(t) < t we say that t is
left-scattered. Pointsthat are simultaneously right-scatteredandleft-scattered are
said to be isolated. If σ(t) = t, then t is called right-dense; if ρ(t) = t, then t
is called left-dense. Points that are right-dense and left-dense at the same time
are called dense. The mappings µ,ν : T → [0,+∞) defined by µ(t) := σ(t)−t
and ν(t) := t−ρ(t) are called, respectively, the forward and backward graininess
function.
Given a time scale T, we introduce the sets Tk, T , and Tk as follows. If T
k k
has a left-scattered maximum t , then Tk = T−{t }, otherwise Tk = T. If T has
1 1
a right-scattered minimum t , then T = T−{t }, otherwise T = T. Finally,
2 k 2 k
Tk =Tk T .
k k
Let f : T → R be a real valued function on a time scale T. Then, for t ∈ Tk,
T
we define f∆(t) to be the number, if one exists, such that for all ǫ > 0, there is a
neighborhood U of t such that for all s∈U,
f(σ(t))−f(s)−f∆(t)(σ(t)−s) ≤ǫ|σ(t)−s|.
We say that f is(cid:12)delta differentiable on Tk provid(cid:12)ed f∆(t) exists for all t ∈ Tk.
(cid:12) (cid:12)
Similarly, for t∈T we define f∇(t) to be the number, if one exists, such that for
k
all ǫ>0, there is a neighborhood V of t such that for all s∈V
f(ρ(t))−f(s)−f∇(t)(ρ(t)−s) ≤ǫ|ρ(t)−s|.
Wesaythatf isna(cid:12)bladifferentiableonT ,provided(cid:12) thatf∇(t)existsforallt∈T .
(cid:12) k (cid:12) k
Forf :T→Rwedefinethefunctionfσ :T→Rbyfσ(t)=f(σ(t))forallt∈T,
thatis, fσ =f◦σ. Similarly, we define the function fρ :T→R by fρ(t)=f(ρ(t))
for all t∈T, that is, fρ =f ◦ρ.
A function f : T → R is called rd-continuous, provided it is continuous at all
right-densepointsinTanditsleft-sidedlimitsfiniteatallleft-densepointsinT. A
functionf :T→Riscalledld-continuous,provideditiscontinuousatallleft-dense
points in T and its right-sided limits finite at all right-dense points in T.
A function F : T → R is called a delta antiderivative of f : T → R, provided
F∆(t)=f(t) holds for all t∈Tk. Then the delta integral of f is defined by
b
f(t)∆t=F(b)−F(a).
Za
A function G : T → R is called a nabla antiderivative of g : T → R, provided
G∇(t) = g(t) holds for all t ∈ T . Then the nabla integral of g is defined by
k
b
g(t)∇t=G(b)−G(a). For more details on time scales one can see [1, 6, 7].
a
R
COMBINED DYNAMIC GRU¨SS INEQUALITIES ON TIME SCALES 3
Nowweintroduce the diamond-αderivativeandintegral,referringthe readerto
[20, 26, 27, 28] for more on the associated calculus.
LetTbeatimescaleandf differentiableonTinthe∆and∇senses. Fort∈T,
we define the diamond-α dynamic derivative f♦α(t) by
f♦α(t)=αf∆(t)+(1−α)f∇(t), 0≤α≤1.
Thus, f is diamond-α differentiable if and only if f is ∆ and ∇ differentiable.
The diamond-α derivative reduces to the standard ∆ derivative for α = 1, or the
standard ∇ derivative for α = 0. On the other hand, it represents a “weighted
derivative” for α ∈ (0,1). Diamond-α derivatives have shown in computational
experiments to provide efficient and balanced approximation formulas, leading to
the design of more reliable numerical methods [27, 28].
Let f,g :T→R be diamond-α differentiable at t∈T. Then,
(i) f +g :T→R is diamond-α differentiable at t∈T with
(f +g)♦α(t)=(f)♦α(t)+(g)♦α(t).
(ii) For any constant c, cf :T→R is diamond-α differentiable at t∈T with
(cf)♦α(t)=c(f)♦α(t).
(ii) fg :T→R is diamond-α differentiable at t∈T with
(fg)♦α(t)=(f)♦α(t)g(t)+αfσ(t)(g)∆(t)+(1−α)fρ(t)(g)∇(t).
Let a,t ∈ T, and h : T → R. Then, the diamond-α integral from a to t of h is
defined by
t t t
h(τ)♦ τ =α h(τ)∆τ +(1−α) h(τ)∇τ, 0≤α≤1.
α
Za Za Za
We may notice the absence of an anti-derivative for the ♦ combined derivative.
α
For t∈T, in general
t ♦α
h(τ)♦ τ 6=h(t).
α
(cid:18)Za (cid:19)
Although the fundamental theorem of calculus does not hold for the ♦ -integral,
α
other properties do. Let a,b,t∈T, c∈R. Then,
t t t
(i) {f(τ)+g(τ)}♦ τ = f(τ)♦ τ + g(τ)♦ τ;
a α a α a α
t t
(ii) cf(τ)♦ τ =c f(τ)♦ τ;
Ra α a R α R
t b t
(iii) f(τ)♦ τ = f(τ)♦ τ + f(τ)♦ τ;
Ra α aR α b α
(iv) If f(t)≥0 for all t, then bf(t)♦ t≥0;
R R a R α
(v) If f(t)≤g(t) for all t, then bf(t)♦ t≤ bg(t)♦ t;
R a α a α
(vi) If f(t)≥0 for all t, then f(t)≡0 if and only if bf(t)♦ t=0.
R R a α
In [11] a diamond-α Jensen’s inequality on time scaRles is proved: let c and d
be real numbers, u : [a,b] → (c,d) be continuous and h : (c,d) → R be a convex
function. Then,
(2) h abu(t)♦αt ≤ 1 bh(u(t))♦ t.
α
b−a b−a
R ! Za
We willuse the diamond-αJensen’sinequality (2)inthe proofofourTheorem3.4.
4 M.R.SIDIAMMIANDD.F.M.TORRES
3. Main Results
In the sequel we assume that T is a time scale, a, b ∈ T with a < b, and [a,b]
denote [a,b]∩T. We begin by proving some auxiliary lemmas.
Lemma 3.1. If f ∈ C([a,b],R) satisfy m ≤ f(t) ≤ M for all t ∈ [a,b] and
b
f(t)♦ t=0, then
a α
R 1 b 1
f2(t)♦ t≤ (M −m)2.
α
b−a 4
Za
Proof. Define ϕ(t) = f(t)−m. It follows that f(t) = m+(M −m)ϕ(t). One can
M−m
easily check that 0≤ϕ(t)≤1. Then,
1 b
f2(t)♦ t
α
b−a
Za
1 b
= (m+(M −m)ϕ(t))2♦ t
α
b−a
Za
1 b
= m2+2m(M −m)ϕ(t)+(M −m)2ϕ2(t) ♦ t
α
b−a
Za
2m((cid:0)M −m) b (M −m)2 b (cid:1)
=m2+ ϕ(t)♦ t+ ϕ2(t)♦ t
α α
b−a b−a
Za Za
2m(M −m) b (M −m)2 b
≤m2+ ϕ(t)♦ t+ ϕ(t)♦ t
α α
b−a b−a
Za Za
2m(M −m) b f(t)−m (M −m)2 b f(t)−m
=m2+ ♦ t+ ♦ t
α α
b−a M −m b−a M −m
Za Za
−m −m
=m2+2m(M −m) +(M −m)2
M −m M −m
(cid:18) (cid:19) (cid:18) (cid:19)
1
=−mM = (M −m)2−(M +m)2
4
1 (cid:0) (cid:1)
≤ (M −m)2.
4
(cid:3)
Lemma 3.2. If f ∈ C([a,b],R) satisfy m ≤ f(t) ≤ M for all t ∈ [a,b] and
b
f(t)♦ t6=0, then
a α
2
R 1 b 1 b 1
f2(t)♦ t− f(t)♦ t ≤ (M −m)2.
α α
b−a b−a 4
Za Za !
Proof. Setting
1 b
f(t)♦ t=I(b−a),
α
b−a
Za
I ∈R, and F(t)=f(t)−I(b−a), we then have
m−I(b−a)≤F(t)≤M −I(b−a).
Therefore,
1 b 1 b
F(t)♦ t= (f(t)−I(b−a))♦ t
α α
b−a b−a
Za Za
(3) 1 b
= f(t)♦ t−I(b−a)
α
b−a
Za
=0.
COMBINED DYNAMIC GRU¨SS INEQUALITIES ON TIME SCALES 5
It follows from Lemma 3.1 that
1 b 1
F2(t)♦ t≤ ((M −I(b−a))−(m−I(b−a)))2
α
b−a 4
(4) Za
1
= (M −m)2.
4
On the other hand, using (3) we have
2
1 b 1 b
F2(t)♦ t− F(t)♦ t
α α
b−a b−a
Za Za !
1 b
= F2(t)♦ t
α
b−a
Za
1 b
= f2(t)−2I(b−a)f(t)+I2(b−a) ♦ t
α
b−a
(5) Za
1 b(cid:0) (cid:1)
= f2(t)♦ t−2I2(b−a)+I2(b−a)
α
b−a
Za
1 b
= f2(t)♦ t−I2(b−a)
α
b−a
Za
2
1 b 1 b
= f2(t)♦ t− f(t)♦ t .
α α
b−a b−a
Za Za !
Then, using (4) and (5), we conclude with
2
1 b 1 b 1
f2(t)♦ t− f(t)♦ t ≤ (M −m)2.
α α
b−a b−a 4
Za Za !
(cid:3)
From Lemma 3.1 and Lemma 3.2 we have the following corollary.
Corollary 3.3. If f ∈C([a,b],R) satisfy m≤f(t)≤M for all t∈[a,b], then
2
1 b 1 b 1
f2(t)♦ t− f(t)♦ t ≤ (M −m)2.
α α
b−a b−a 4
Za Za !
We are now in conditions to prove the intended Gru¨ss inequality.
Theorem 3.4. (The diamond-α Gru¨ss inequality on time scales, α ∈ [0,1]). Let
T be a time scale, a,b ∈ T with a < b. If f,g ∈ C([a,b],R) satisfy ϕ ≤ f(t) ≤ Φ
and γ ≤g(t)≤Γ for all t∈[a,b]∩T, then
1 b 1 b b 1
(6) f(t)g(t)♦ t− f(t)♦ t g(t)♦ t ≤ (Φ−ϕ)(Γ−γ).
(cid:12)(cid:12)b−aZa α (b−a)2 Za α Za α (cid:12)(cid:12) 4
(cid:12) (cid:12)
Proo(cid:12)f. A straightforwardcomputation leads to (cid:12)
(cid:12) (cid:12)
1 b 1 b b
f(t)g(t)♦ t− f(t)♦ t g(t)♦ t
b−a α (b−a)2 α α
Za Za Za
1 1 b
(7) = (f(t)+g(t))2−(f(t)−g(t))2 ♦αt
4 (b−a)
Za
(cid:0) (cid:1)
4 b b
− f(t)♦ t g(t)♦ t .
(b−a)2 α α
Za Za !
6 M.R.SIDIAMMIANDD.F.M.TORRES
If we consider the function h(x) = x2, which is obviously convex, then using the
diamond-α Jensen’s inequality on time scales (2), we obtain
(8) ab(f(t)−g(t))♦αt 2 ≤ 1 b(f(t)−g(t))2♦ t.
α
b−a b−a
R ! Za
Then we have by (7) and (8) that
(9)
1 b 1 b b
f(t)g(t)♦ t− f(t)♦ t g(t)♦ t
b−a α (b−a)2 α α
Za Za Za
2
1 1 b 1 b
≤ ((f(t)+g(t))2♦ t− (f(t)−g(t))♦ t
4 (b−a) α (b−a)2 α
( Za Za !
4 b b
− f(t)♦ t g(t)♦ t
(b−a)2 α α
Za Za )
2
1 1 b 1 b
≤ ((f(t)+g(t))2♦ t− (f(t)+g(t))♦ t
4 (b−a) α (b−a)2 α
( Za Za !
2 2
1 b 1 b
+ (f(t)+g(t))♦ t − (f(t)−g(t))♦ t
(b−a)2 α (b−a)2 α
Za ! Za !
4 b b
− f(t)♦ t g(t)♦ t
(b−a)2 α α
Za Za )
2
1 1 b 1 b
≤ ((f(t)+g(t))2♦ t− (f(t)+g(t))♦ t ,
4(b−a) α (b−a)2 α 
 Za Za ! 
since  
2 2
1 b 1 b
(f(t)+g(t))♦ t − (f(t)−g(t))♦ t
(b−a)2 α (b−a)2 α
Za ! Za !
4 b b
− f(t)♦ t g(t)♦ t=0.
(b−a)2 α α
Za Za
Onthe other hand, we haveϕ+Φ≤(f+g)(t)≤γ+Γ. Applying Corollary3.3
to the function f +g, we get
2
1 b 1 b
(10) (f(t)+g(t))2♦ t− (f(t)+g(t))♦ t ♦ t
b−a α (b−a)2 α α
Za Za !
1 1
≤ ((Φ+Γ)−(ϕ+γ))2 = (Φ+Γ−ϕ−γ)2.
4 4
Gathering (8), (9) and (10), we obtain:
1 b 1 b b
f(t)g(t)♦ t− f(t)♦ t f(t)♦ t
b−a α (b−a)2 α α
Za Za Za
1
≤ ((Φ+Γ)−(ϕ+γ))2
16
1 1
≤ (Φ−ϕ)(Γ−γ)+ (Φ−ϕ−(Γ−γ))2 .
4 4
(cid:18) (cid:19)
COMBINED DYNAMIC GRU¨SS INEQUALITIES ON TIME SCALES 7
We consider now two cases: (i) if Φ−ϕ=Γ−γ, then
1 b 1 b b 1
(11) f(t)g(t)♦ t− f(t)♦ t g(t)♦ t≤ (Φ−ϕ)(Γ−γ);
b−a α (b−a)2 α α 4
Za Za Za
(ii) if Φ−ϕ6=Γ−γ, let us define
Γ−γ Φ−ϕ
β = , µ= ,
sΦ−ϕ sΓ−γ
and
f (t)=βf(t), g (t)=µg(t).
1 1
Note that βµ=1. Then, we have
m =βϕ≤f (t)≤βΦ=M , m =µγ ≤g (t)≤µΓ=M ,
1 1 1 2 1 2
and it follows that
M −m =βΦ−βϕ=β(Φ−ϕ)
1 1
(12)
= (Φ−ϕ)(Γ−γ)=µ(Γ−γ)=M −m .
2 2
Using the fact that βµ=1p, (11) and (12) (with f , g ), we get
1 1
1 b 1 b b
(13) f(t)g(t)♦ t− f(t)♦ t g(t)♦ t
b−a α (b−a)2 α α
Za Za Za
1 b 1 b b
= βµf(t)g(t)♦ t− βf(t)♦ t µg(t)♦ t
b−a α (b−a)2 α α
Za Za Za
1 b 1 b b
= f (t)g (t)♦ t− f (t)♦ t g (t)♦ t
b−a 1 1 α (b−a)2 1 α 1 α
Za Za Za
1 1 1
≤ (M −m )(M −m )= βµ(Φ−ϕ)(Γ−γ)= (Φ−ϕ)(Γ−γ).
1 1 2 2
4 4 4
If we now consider the case of −f we have −Φ≤−f(t)≤−ϕ. Using (13),
1 b 1 b b
(14) (−f)(t)g(t)♦ t− (−f)(t)♦ t g(t)♦ t
b−a α (b−a)2 α α
Za Za Za
1 b 1 b b
=− f(t)g(t)♦ t− f(t)♦ t g(t)♦ t
b−a α (b−a)2 α α
( Za Za Za )
1 1
≤ (−ϕ−(−Φ))(Γ−γ)= (Φ−ϕ)(Γ−γ).
4 4
Then, using (13) and (14), we arrive to
1 b 1 b b 1
f(t)g(t)♦ t− f(t)♦ t g(t)♦ t ≤ (Φ−ϕ)(Γ−γ).
(cid:12)(cid:12)b−aZa α (b−a)2 Za α Za α (cid:12)(cid:12) 4
Th(cid:12)(cid:12)is completes the proof of our Theorem 3.4. (cid:12)(cid:12) (cid:3)
(cid:12) (cid:12)
Remark 3.1. When α=1, we have the following Gru¨ss inequality on time scales:
1 b 1 b b 1
f(t)g(t)∆t− f(t)∆t g(t)∆t ≤ (Φ−ϕ)(Γ−γ).
(cid:12)(cid:12)b−aZa (b−a)2 Za Za (cid:12)(cid:12) 4
(cid:12) (cid:12)
Rem(cid:12)ark 3.2. For T=R Theorem 3.4 gives the classical in(cid:12)equality (1).
(cid:12) (cid:12)
Applying the Gru¨ss inequality in Theorem 3.4 to the time scale T = Z with
a=0, b=n, andf(i)=x , i=1,...,n, we arriveto the following corollarywhich
i
improves [10].
8 M.R.SIDIAMMIANDD.F.M.TORRES
Corollary 3.5. If ϕ≤x ≤Φ and γ ≤y ≤Γ for all 1≤i≤n, then the following
i i
discrete Gru¨ss inequality holds:
n n n
1 1 1
x y − x y ≤ (Φ−ϕ)(Γ−γ).
(cid:12)n i i n2 i i(cid:12) 4
(cid:12) Xi=1 Xi=1 Xi=1 (cid:12)
(cid:12) (cid:12)
Theorem 3.4(cid:12)is valid for an arbitrary tim(cid:12)e scale. For example, let q > 1 and
(cid:12) (cid:12)
T={qN0}.
Corollary 3.6. (Quantum Gru¨ss inequality) If f and g satisfy ϕ≤f(qi)≤Φ and
γ ≤g(qi)≤Γ for all qi, i=m,...,n, then the following inequality holds:
n−1qif(qi+1)g(qi+1) 1 n−1 n−1
i=m − qif(qi+1) qig(qi+1)
(cid:12)(cid:12)(cid:12)P in=−m1 qi in=−m1 qi 2 iX=m ! iX=m !(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) P (cid:16)P (cid:17) ≤ 1(Φ−ϕ)(Γ−(cid:12)(cid:12)γ).
(cid:12) (cid:12)
4
References
[1] R.Agarwal,M.Bohner,D.O’ReganandA.Peterson, Dynamicequations ontimescales: a
survey,J.Comput.Appl.Math.141(2002), no.1-2,1–26.
[2] R.Agarwal,M.BohnerandA.Peterson,Inequalitiesontimescales: asurvey,Math.Inequal.
Appl.4(2001), no.4,535–557.
[3] G. A. Anastassiou, MultivariateFinktype identity and multivariate Ostrowski,comparison
ofmeansandGru¨sstypeinequalities,Math.Comput.Modelling46(2007),no.3-4,351–374.
[4] Z.BartosiewiczandD.F.M.Torres,Noether’stheoremontimescales,J.Math.Anal.Appl.,
inpress.
[5] M.BohnerandT.Matthews,TheGru¨ssinequalityontimescales,Commun.Math.Anal.3
(2007), no.1,1–8(electronic).
[6] M.BohnerandA.Peterson,Dynamic equations on time scales,Birkha¨userBoston,Boston,
MA,2001.
[7] M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Birkha¨user
Boston,Boston,MA,2003.
[8] P.CeroneandS.S.Dragomir,ArefinementoftheGru¨ssinequalityandapplications,Tamkang
J.Math.38(2007), no.1,37–49.
[9] S. S. Dragomir, A generalization of Gru¨ss’s inequality in inner product spaces and applica-
tions,J.Math.Anal.Appl.237(1999), no.1,74–82.
[10] S.S.Dragomir,AGru¨sstypediscreteinequalityininnerproductspacesandapplications,J.
Math.Anal.Appl.250(2000), no.2,494–511.
[11] R.A.C.Ferreira,M.R.SidiAmmiandD.F.M.Torres,Diamond-alphaintegralinequalities
ontimescales,Proc.8thPortugueseConferenceonAutomaticControl–CONTROLO’2008,
2008,inpress.
[12] R.A.C.FerreiraandD.F.M.Torres,Remarksonthecalculusofvariationsontimescales,
Int.J.Ecol.Econ.Stat.9(2007), no.F07,65–73.
b
[13] G. Gru¨ss, U¨ber das Maximum des absoluten Betrages von b−1aR f(x)g(x)dx −
a
b b
(b−1a)2 Ra f(x)dxRa g(x)dx,Math.Z.39(1935), no.1,215–226.
[14] S.Hilger,Analysisonmeasurechains—aunifiedapproachtocontinuousanddiscretecalculus,
ResultsMath.18(1990), no.1-2,18–56.
[15] S. Hilger, Differential and difference calculus—unified!, Nonlinear Anal. 30 (1997), no. 5,
2683–2694.
[16] Z. Liu, Notes on a Gru¨ss type inequality and its application, Vietnam J. Math. 35 (2007),
no.2,121–127.
[17] A. McD. Mercer, An improvement of the Gru¨ss inequality, JIPAM. J. Inequal. Pure Appl.
Math.6(2005), no.4,Article93,4pp.(electronic).
[18] D.S.Mitrinovi´c,Analytic inequalities,Springer,NewYork,1970.
[19] D. S. Mitrinovi´c, J. E. Peˇcari´c and A. M. Fink, Classical and new inequalities in analysis,
KluwerAcad.Publ.,Dordrecht,1993.
COMBINED DYNAMIC GRU¨SS INEQUALITIES ON TIME SCALES 9
[20] D. Mozyrska and D. F. M. Torres, Diamond-alpha polynomial series on time scales, Proc.
8thPortugueseConferenceonAutomaticControl–CONTROLO’2008,2008,inpress.
[21] U.M.OzkanandH.Yildirim,Steffensen’sintegralinequalityontimescales,J.Inequal.Appl.
2007,Art.ID46524,10pp.
[22] B. G. Pachpatte, Some new Ostrowski and Gru¨ss type inequalities, Tamkang J. Math. 38
(2007), no.2,111–120.
[23] J.E.Peˇcari´candB.Tepeˇs, OnGru¨sstypeinequalitiesofDragomirandFedotov,JIPAM.J.
Inequal.PureAppl.Math.4(2003), no.5,Article91,6pp.(electronic).
[24] J.Peˇcari´candB.Tepeˇs,AnoteonGru¨sstypeinequalityintermsof∆-seminorms,Makedon.
Akad.Nauk.Umet.Oddel.Mat.-Tehn.Nauk.Prilozi23/24(2002/03),no.1-2,29–35(2004).
[25] I. Peri´cand R.Raji´c, Gru¨ssinequality forcompletely bounded maps,Linear AlgebraAppl.
390(2004), 287–292.
[26] J.W.Rogers,Jr.andQ.Sheng,Notesonthediamond-αdynamicderivativeontimescales,
J.Math.Anal.Appl.326(2007), no.1,228–241.
[27] Q. Sheng, Hybrid approximations via second order combined dynamic derivatives on time
scales,Electron.J.Qual.TheoryDiffer.Equ.2007,No.17,13pp.(electronic).
[28] Q. Sheng, M. Fadag, J. Henderson and J. M. Davis, An exploration of combined dynamic
derivativesontimescalesandtheirapplications,NonlinearAnal.RealWorldAppl.7(2006),
no.3,395–413.
[29] M.R.SidiAmmiandD.F.M.Torres,Existenceofpositivesolutionsfornonlocalp-Laplacian
thermistor problems on time scales, JIPAM. J. Inequal. Pure Appl. Math. 8 (2007), no. 3,
Article69,10pp.
DepartmentofMathematics,University of Aveiro,3810-193Aveiro,Portugal
E-mail address: [email protected]
E-mail address: [email protected]