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Generalization of Ostrowski and Ostrowski–Grüss type inequalities on time scales
Computers and Mathematics with Applications 60 (2010) 803–811
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Generalization of Ostrowski and Ostrowski–Grüss type inequalities on time scales Adnan Tuna ∗ , Durmus Daghan Department of Mathematics, Faculty of Science and Arts, University of Niğde, Merkez, 51240, Niğde, Turkey
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Article history: Received 26 October 2009 Received in revised form 12 April 2010 Accepted 20 May 2010
In this paper, we obtain an Ostrowski and Ostrowski–Grüss type inequality on time scales, which not only provides a generalization of the known results on time scales, but also give some other interesting inequalities as special cases. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Grüss inequality Ostrowski inequality Ostrowski–Grüss inequality Time scales
1. Introduction In 1988, Hilger [1] introduced the time scales theory to unify continuous and discrete analysis. Since then, many authors have studied certain integral inequalities on time scales [2–6]. In 1938, Ostrowski derived a formula to estimate the absolute deviation of a differentiable function from its integral mean [7], the so-called Ostrowski inequality holds and can be shown by using the Montgomery identity [7]. These two properties was proved by Bohner and Matthews in [4] for general time scales, which unify discrete, continuous and many other cases. The Ostrowski inequality is generalized by using the Montgomery identity on time scales as following theorem are given in [4] with Theorem 3.5. Theorem 1. Let a, b, s, t ∈ T, a < b and if f : [a, b] → R be differentiable. Then
Z b M f (t ) − 1 (h2 (t , a) + h2 (t , b)), f (σ (s))1s ≤ b−a a b−a
(1.1)
where M = sup f ∆ (t ) .
a
This inequality is sharp in the sense that the right-hand side of (1.1) cannot be replaced by a smaller one. Additionally, Liu et al. [8] have extended a generalization of an Ostrowski type inequality with a parameter on time scales and a unified corresponding continuous and discrete case. Recently, Ngô and Liu in [9] gave the following sharp Grüss type inequality on time scales. Theorem 2. Let a, b, s, t ∈ T, a < b and if f : [a, b] → R be differentiable. If f ∆ is rd continuous and
γ ≤ f ∆ (t ) ≤ Γ , ∗
∀t ∈ [a, b].
Corresponding author. E-mail addresses: [email protected] (A. Tuna), [email protected] (D. Daghan).
0898-1221/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.05.027
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A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
Then we have
Z b f (b) − f (a) f (t ) − 1 f (σ ( s )) 1 s − h ( t , a ) − h ( t , b )) ( 2 2 2 b−a a ( b − a) Z b Γ −γ p(t , s) − h2 (t , a) − h2 (t , b) 1s, ≤ 2(b − a) a b−a n s − a, a ≤ s < t, for all t ∈ [a, b] and p(t , s) = s − b, t ≤ s ≤ b. Moreover, the constant 12 in (1.2) is sharp.
(1.2)
Shun-Feng Wang, Qiao-ling Xue and Wen-jun Liu in [10] obtained the following Ostrowski–Grüss type inequality for bounded differentiable mappings with a parameter. Besides, they also give some other interesting inequalities as special cases. Theorem 3. Let I ⊂ R be an open interval a, b ∈ I , a < b. If f : I → R is a differentiable function such that there exist constants γ , Γ ∈ R, with γ ≤ f 0 (t ) ≤ Γ , t ∈ [a, b]. Then for all t ∈ [a, b] and λ ∈ [0, 1], we have
Z b 1 − λ f (t ) − λ (t − b)f (b) − (t − a)f (a) − Γ + γ (1 − λ) t − a + b − 1 f (s)ds 2 2(b − a) 2 2 b−a a λ2 (t − a)2 + (b − t )2 ≤ 1−λ+ (Γ − γ ) . 2 4(b − a)
(1.3)
This work is organized as follows: in Section 2 we will briefly present the general definitions and theorems connected to the time scales calculus. Next, in Section 3 we will generalize the Montgomery identity, the Ostrowski inequality and Theorem 2 on time scales. Additionally, Theorem 3 will be proved for general time scales and we also give some other interesting inequalities as special cases. 2. General definitions Now we briefly introduce the time scales theory and refer the reader to Hilger [1] and the books [11–14] for further details. Definition 1. A time scale T is a nonempty closed subset of R. We assume throughout that T has the topology that is inherited from the standard topology on R. It also assumed throughout that in T the interval [a, b]T means the set {t ∈ T : s < t } for the points a < b in T. Since a time scale may not be connected, we need the following concept of jump operators. Definition 2. The mappings σ , ρ : T → T defined by σ (t ) = inf {s ∈ T : s > t } and ρ(t ) = sup {s ∈ T : s < t } are called the jump operators. The jump operators σ and ρ allow the classification of points in T in the following way: Definition 3. A nonmaximal element t ∈ T is said to be right-dense if σ (t ) = t, right-scattered if σ (t ) > t, left-dense if ρ(t ) = t, left-scattered if ρ(t ) < t. In the case T = R, we have σ (t ) = t, and if T = hZ, h > 0, then σ (t ) = t + h. Definition 4. The mapping µ : T → R+ defined by µ(t ) = σ (t ) − t is called the graininess function. The set Tk is defined as follows: if T has a left-scattered maximum m, then Tk = T − {m}; otherwise, Tk = T. If T = R, then µ(t ) = 0, and when T = Z, we have µ(t ) = 1. Definition 5. Let f : T → R. f is called differentiable at t ∈ Tk , with (delta) derivative f ∆ (t ) ∈ R if given ε > 0 there exists a neighborhood U of t such that, for all s ∈ U,
σ
f (t ) − f (s) − f ∆ (t )[σ (t ) − s] ≤ ε kσ (t ) − sk , where f σ = f ◦ σ . If T = R, then f ∆ (t ) =
df (t ) , dt
and if T = Z, then f ∆ (t ) = f (t + 1) − f (t ).
Definition 6. The function f : T → R is said to be rd-continuous (denote f ∈ Crd (T, R)) if, at all t ∈ T, (i) f is continuous at every right-dense point t ∈ T, (ii) lims→t − f (s) exists and is finite at every left-dense point t ∈ T. Theorem 4R (Existence of Antiderivatives). Every rd-continuous function has an antiderivative. In particular if t0 ∈ T, then F t defined by t f (s)1s for t ∈ T is an antiderivative of f . 0
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Definition 7. Let f ∈ Crd (T, R). Then g : T → R is called the antiderivative of f on T if it is differentiable on T and satisfies g ∆ (t ) = f (t ) for any t ∈ Tk . In this case, we defined t
Z
f (s)1s = g (t ) − g (a),
t ∈ T.
a
Theorem 5. Let f , g be rd-continuous, a, b, c ∈ T and α, β ∈ R. Then
Rb Rb Rb [f (t ) + g (t )] 1t = a f (t )1t + a g (t )1t, Rab Ra (2) a f (t )1t = − b f (t )1t, Rb Rc Rb (3) a f (t )1t = a f (t )1t + c f (t )1t, Rb Rb (4) a f (t )g ∆ (t )1t = (fg )(b) − (fg )(a) − a f ∆ (t )g (σ (t ))1t. (1)
Theorem 6. If f is ∆-integrable on [a, b], then so is |f |, and
Z
b
a
Z f (t )1t ≤
b
|f (t )| 1t . a
Definition 8. Let gk , hk : T2 → R, k ∈ N0 be defined by g 0 ( t , s ) = h0 ( t , s ) = 1
for all s, t ∈ T
and then recursively by gk+1 (t , s) =
t
Z
gk (σ (τ ), s)1s
for all s, t ∈ T
s
and hk+1 (t , s) =
t
Z
hk (τ , s)1s
for all s, t ∈ T.
s
3. Generalization of Ostrowski and Ostrowski–Grüss type inequalities Lemma 1. Let a, b, s, t ∈ T, a < b and if f : [a, b] → R be differentiable. Then for all s ∈ [a, b] and λ ∈ [0, 1], we have
λ
1−
f (t ) =
2
b
Z
1 b−a
f (σ (s))1s − λ a
(t − a)f (a) + (b − t )f (b) 1 + 2(b − a) b−a
b
Z
K (t , s)f ∆ (s)1s,
(3.1)
a
where
t −a , s − a + λ 2 K (t , s) = b−t s − b − λ ,
a ≤ s < t, (3.2) t ≤ s ≤ b.
2
Proof. Using Theorem 5(4), we have
Z t
s−
a+λ
a
t −a
f ∆ (s)1s = (t − a) 1 −
2
λ
2
f (t ) + λ
t −a 2
f (a) −
t
Z
f (σ (s))1s a
and b
Z
s−
b−λ
t
b−t
2
f ∆ (s)1s = (b − t ) 1 −
λ 2
f (t ) + λ
b−t 2
f (b) −
b
Z
f (σ (s))1s. t
Therefore b−a
=
b
Z
1
f (σ (s))1s + a
1
b−a
b
Z
b−a
1
f (σ (s))1s +
b
Z
K (t , s)f ∆ (s)1s
a
1
Z λ (t − a)f (a) + (b − t )f (b) (b − a) 1 − f (t ) + λ −
b−a 2 (t − a)f (a) + (b − t )f (b) = 1− f (t ) + λ , 2 2(b − a)
i.e., (3.1) holds.
a
λ
2
b
f (σ (s))1s a
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A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
Another generalized of Montgomery identity given by Liu et al. [8] on time scales for
b−a s − a + λ , 2 K ( t , s) = b−a s − b − λ ,
a ≤ s < t, t ≤ s ≤ b.
2
However, this result is different from ours, but is the same as in the case of λ = 0. Corollary 1. If we choose λ = 0 in Lemma 1, we get f (t ) =
Z
1 b−a
b
f (σ (s))1s +
a
1 b−a
b
Z
K (t , s)f ∆ (s)1s.
a
This result is the Montgomery identity on time scales, which can be found in Ref. [4] with Lemma 3.1 and it is also studied as continuous, discrete and quantum calculus cases in Ref. [4]. Theorem 7. Let a, b, s, t ∈ T, a < b and if f : [a, b] → R be differentiable. Then for all s ∈ [a, b] and λ ∈ [0, 1], we have
Z b 1 − λ f (t ) + λ (t − a)f (a) + (b − t )f (b) − 1 f (σ (s))1s 2 2(b − a) b−a a λ (t − a) f (a) + (b − t ) f (b) M ≤ (h2 (t , a) + h2 (t , b)) + f (t ) + λ b−a 2 2(b − a)
(3.3)
where M = sup f ∆ (t ) .
a
Proof. Using the definition of ∆-integrals, we have
Z t
s−
a+λ
a
t −a
2
f ∆ (s)1s =
t
Z
(s − a) f ∆ (s)1s − λ
t −a
a
2
(f (t ) − f (a))
and b
Z
s−
t
b−λ
b−t
2
f ∆ (s)1s =
b
Z t
(s − b) f ∆ (s)1s + λ
b−t 2
(f (b) − f (t )) .
Thus
Z b 1 1 K (t , s)f (s)1s b − a a Z t Z b 1 1 λ (t − a) f (a) + (b − t ) f (b) ∆ ∆ ≤ (s − a) f (s)1s + (s − b) f (s)1s − f (t ) + λ b−a a b−a t 2 2(b − a) Z t Z b ∆ 1 λ (t − a) f (a) + (b − t ) f (b) 1 1 |s − a| f (s) 1s + |s − b| f (s) 1s + f (t ) + λ ≤ b−a a b−a t 2 2(b − a) Z t Z b λ (t − a) f (a) + (b − t ) f (b) M M ≤ (s − a) 1s + (b − s) 1s + f (t ) + λ b−a a b−a t 2 2(b − a) (t − a) f (a) + (b − t ) f (b) λ M . ≤ (h2 (t , a) + h2 (t , b)) + f (t ) + λ b−a 2 2(b − a)
(3.4)
Using the Lemma 1, we have
Z b 1 − λ f (t ) + λ (t − a)f (a) + (b − t )f (b) − 1 f (σ ( s )) 1 s 2 2(b − a) b−a a Z b 1 = K (t , s)f 1 (s)1s b−a a λ (t − a) f (a) + (b − t ) f (b) M . ≤ (h2 (t , a) + h2 (t , b)) + f (t ) + λ b−a 2 2(b − a) Thus the proof is complete.
(3.5)
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Similarly, another generalized of Ostrowski inequality is proved by Liu et al. [8] on time scales by using the Montgomery identity. However, this result is also different from us, but is also same in the case of λ = 0. Corollary 2. In the case of λ = 0, we have
Z b ≤ M (h2 (t , a) + h2 (t , b)). f (t ) − 1 f (σ ( s )) 1 s b−a b−a a This is the Ostrowski inequality on time scales, which can be found in Ref. [4] with the Theorem 3.5 and it is also studied as continuous, discrete and quantum calculus cases in [4]. Generalized of the sharp Grüss type inequality is given with Theorem 2 in Section 1. Now, we will give another generalization of the Theorem 2 on time scales by using the Theorem 3.1 in Ref. [9]. In the case of λ = 0, we get Theorem 2 on time scales. Theorem 8. Let a, b, s, t ∈ T, a < b and f : [a, b] → R be differentiable. If f ∆ is rd-continuous and
γ ≤ f ∆ (t ) ≤ Γ ,
∀t ∈ [a, b]
then for all x ∈ [a, b] and λ ∈ [0, 1], we have
Z b (t − a)f (a) + (b − t )f (b) 1 − λ f (t ) − 1 f (σ (s))1s + λ 2 b−a a 2(b − a) 1 (t − a)2 − (b − t )2 f (b) − f (a) − (h2 (t , a) − h2 (t , b)) − λ b−a 2 b−a Z b 2 Γ −γ (t − a) − (b − t )2 K (t , s) − 1 ≤ h ( t , a ) − h ( t , b )) − λ ( 2 2 1s. 2(b − a) a b−a 2 Proof. Define the function
t −a , s − a + λ 2 K (t , s) = b−t s − b − λ , 2
a ≤ s < t, t ≤s≤b
then choosing f (t ) = K (t , s) and g (t ) = f ∆ (t ) in Theorem 3.1 in Ref. [9], we have b
Z
∆
K (t , s)f (s)1s −
a
≤
Γ −γ 2
b
Z a
1
Z
b
b−a a K (t , s) − 1 b−a
b
K (t , s)1s f (s)1s a Z b K (t , τ )1τ 1s. Z
∆
(3.6)
a
Integrating K (t , s) and f ∆ (t ) on [a, b], we get b
Z
1 b−a
K (t , s)1s = a
1 b−a
(h2 (t , a) − h2 (t , b)) − λ
(t − a)2 − (b − t )2
2
(3.7)
and b
Z
f ∆ (s)1s = f (b) − f (a).
(3.8)
a
Substituting Eq. (3.7) in the right hand side of Eq. (3.6), we arrive at the right hand side of Eq. (3.6) as given below.
Γ −γ
Z b K (t , s) − 1 1s K ( t , τ ) 1 τ 2 b−a a a Z Γ − γ b 1 (t − a)2 − (b − t )2 = K ( t , s ) − h ( t , a ) − h ( t , b )) − λ ( 2 2 1s. 2 b−a 2 a Z
b
(3.9)
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A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
By Eqs. (3.6) and (3.9), we obtain
Z
b
K (t , s)f ∆ (s)1s −
a
≤
Γ −γ 2
b
Z a
Z
1
b
K (t , s)1s
Z
b
f ∆ (s)1s
b−a a a (t − a)2 − (b − t )2 K (t , s) − 1 (h2 (t , a) − h2 (t , b)) − λ 1s. b−a 2
(3.10)
From Lemma 1, we have
1−
λ 2
f (t ) =
1 b−a
Z
b
f (σ (s))1s − λ
a
(t − a)f (a) + (b − t )f (b) 1 + 2(b − a) b−a
b
Z
K (t , s)f ∆ (s)1s.
(3.11)
a
Using Eqs. (3.7), (3.8) and (3.11) on the left hand side of Eq. (3.10), we get
Z b (t − a)f (a) + (b − t )f (b) 1 − λ (b − a)f (t ) − f (σ (s))1s + λ 2 2 a 2 (t − a) − (b − t )2 1 [f (b) − f (a)] − (h2 (t , a) − h2 (t , b)) − λ b−a 2 Z 1 Γ − γ b (t − a)2 − (b − t )2 ≤ K (t , s) − b − a (h2 (t , a) − h2 (t , b)) − λ 1s. 2 2 a Hence
Z b (t − a)f (a) + (b − t )f (b) 1 − λ f (t ) − 1 f (σ (s))1s + λ 2 b−a a 2(b − a) (t − a)2 − (b − t )2 f (b) − f (a) 1 − (h2 (t , a) − h2 (t , b)) − λ b−a 2 b−a Z b (t − a)2 − (b − t )2 Γ −γ K (t , s) − 1 h ( t , a ) − h ( t , b )) − λ ≤ ( 2 2 1s. 2(b − a) a b−a 2 This completes the proof.
Corollary 3. In the case of λ = 0, we have
Z b h2 (t , a) − h2 (t , b) f (b) − f (a) f (t ) − 1 f (σ ( s )) 1 s − b−a a b−a b−a Z b Γ −γ K (t , s) − h2 (t , a) − h2 (t , b) 1s. ≤ 2(b − a) a b−a
(3.12)
This result is given in Ref. [9] with Theorem 1.3. Corollary 4. We let T = R in Corollary 3. Then we have
Z b a + b f (b) − f (a) f (t ) − 1 ≤ 1 (Γ − γ ) (b − a) f (s)ds − t − 8 b−a a 2 b−a
(3.13)
for all t ∈ [a, b], where γ ≤ f 0 (t ) ≤ Γ . The constant 1/8 is the best possible, which is also given in Ref. [9] with Eq. (4.5). In addition to this, it is possible to derive the discrete and quantum calculus cases of Eq. (3.12). Theorem 9. Let a, b, s, t ∈ T, a < b and f : [a, b] → R be differentiable. If f ∆ is rd-continuous and
γ ≤ f ∆ (s) ≤ Γ ,
∀s ∈ [a, b].
Then for all x ∈ [a, b], we have
Z b 1 − λ f (t ) + λ (t − a)f (a) + (b − t )f (b) − 1 f (σ (s))1s 2 2(b − a) b−a a Γ +γ 1 (t − a)2 − (b − t )2 − (h2 (t , a) − h2 (t , b)) − λ 2 b−a 2
(3.14)
A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
≤
809
Γ −γ t −a t −a h2 a, a + λ + h2 t , a + λ 2(b − a) 2 2 ! b−t b−t + h2 t , b − λ + h2 b , b − λ 2
(3.15)
2
t for all λ ∈ [0, 1] such that a + λ t −2 a and b − λ b− are in T. 2
Proof. Let us define the function
t −a , s − a + λ 2 K (t , s) = b−t s − b − λ ,
a ≤ s < t, t ≤ s ≤ b.
2
Using Lemma 1, we have b
Z
1 b−a
K (t , s)f ∆ (s)1s =
1−
a
λ
2
f (t ) −
b
Z
1 b−a
f (σ (s))1s + λ a
(t − a)f (a) + (b − t )f (b) . 2(b − a)
(3.16)
We have also
b−a Let C =
b
Z
1
K (t , s)1s = a
Γ +γ
1
2
b−a
b−a
(h2 (t , a) − h2 (t , b)) − λ
(t − a)2 − (b − t )2
2
.
(3.17)
. Using the Eqs. (3.16) and (3.17), we get
b
Z
1
a
λ
Z b (t − a)f (a) + (b − t )f (b) 1 K (t , s) f (s) − C 1s = 1 − f (σ (s))1s f (t ) + λ − 2 2(b − a) b−a a Γ +γ 1 (t − a)2 − (b − t )2 . − (h2 (t , a) − h2 (t , b)) − λ 2 b−a 2
∆
(3.18)
On the other hand, we have
Z 1 b − a
b a
1 K (t , s) f ∆ (s) − C 1s ≤ max f ∆ (s) − C s∈[a,b] b−a
b
Z a
|K (t , s)| 1s .
(3.19)
We have also max f ∆ (s) − C ≤
Γ −γ
s∈[a,b]
(3.20)
2
and 1 b−a
b
Z
|K (t , s)| 1s = a
"
1
h2
b−a
a, a + λ
+ h2 t , b − λ
b−t 2
t −a 2
+ h2 t , a + λ
t −a
# .
+ h2 b , b − λ
b−t 2
2 (3.21)
From the Eqs. (3.19)–(3.21), it follows that
Z 1 b − a
b
∆
K (t , s) f (s) − a
Γ +γ 2
Γ −γ t −a ≤ h2 a, a + λ 2(b − a) 2
1s
+ h2 t , a + λ
t −a
2
Finally, using the Eqs. (3.18) and (3.22), we get Eq. (3.15).
+ h2 t , b − λ
b−t 2
+ h2 b, b − λ
b−t 2
!
.
(3.22)
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Corollary 5. In the case of T = R in Theorem 9, we have
Z b 1 − λ f (t ) + λ (t − a)f (a) + (b − t )f (b) − Γ + γ (1 − λ) t − a + b − 1 f ( s ) ds 2 2(b − a) 2 2 b−a a λ2 (t − a)2 + (b − t )2 ≤ 1−λ+ (Γ − γ ) . 2 4(b − a)
(3.23)
This result was given in Ref. [10] with Theorem 3. It is also shown that, for the case of λ = 1, λ = 0 are given with Theorem 2 and Corollary 5 in [10], respectively. b Corollary 6. We assume that t = a+ ∈ T in Theorem 9, we have 2
1 − λ f a + b + λ f (a) + f (b) 2 2 4 Z b a+b a+b 1 Γ +γ 1 h2 , a − h2 ,b − f (σ (s))1s − 2 b−a 2 2 b−a a Γ −γ b−a a+b b−a ≤ h 2 a, a + λ + h2 ,a + λ 2(b − a) 4 2 4 ! a+b b−a b−a + h2 ,b − λ + h2 b , b − λ 2
4
4
(3.24)
a a for all λ ∈ [0, 1] such that a + λ b− and b − λ b− in T. This is the new result for a special case of Theorem 9. 4 4
Corollary 7. In the case of T = R in Corollary 6, we have
Z b 1 − λ f a + b + λ f (a) + f (b) − 1 f ( s ) ds 2 2 4 b−a a λ2 b − a ≤ 1−λ+ (Γ − γ ) . 2
8
(3.25)
This result is the same as Corollary 6 which was given in Ref. [10]. Corollary 8. Choosing t = a in Theorem 9, we have
Z b 1 − λ f (a) + λ f (b) − Γ + γ λ (b − a)2 − h2 (a, b) − 1 f (σ (s))1s 2 2 2(b − a) 2 b−a a b−a b−a Γ −γ h 2 a, b − λ + h2 b, b − λ ≤ 2(b − a) 2 2
(3.26)
a for all λ ∈ [0, 1] such that b − λ b− in T. This is another new result for a special case of Theorem 9. 4
Corollary 9. We let T = R in Corollary 8, we have
Z b 1 − λ f (a) + λ f (b) + Γ + γ (1 − λ)(b − a) − 1 f (s)ds 2 2 4 b−a a λ2 Γ − γ b − a ≤ 1−λ+ . 2
2
2
(3.27)
This result is the same as Corollary 7 which was given in Ref. [10]. Corollary 10. We let t = b in Theorem 9, we have
Z b 1 − λ f (b) + λ f (a) − Γ + γ h2 (b, a) − λ (b − a)2 − 1 f (σ (s))1s 2 2 2(b − a) 2 b−a a b−a b−a Γ −γ ≤ h 2 a, a + λ + h2 b , a + λ 2(b − a) 2 2 a for all λ ∈ [0, 1] such that a + λ b− in T. This is also a new result for a special case of Theorem 9. 4
(3.28)
A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
811
Corollary 11. We let T = R in Corollary 10, we have
Z b 1 − λ f (b) + λ f (a) − Γ + γ (1 − λ) (b − a) − 1 f ( s ) ds 2 2 4 b−a a λ2 Γ − γ b − a ≤ 1−λ+ . 2
2
2
(3.29)
This result is the same as Corollary 8 which was given in Ref. [10].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Univarsität Würzburg, 1988. R. Agarwal, M. Bohner, A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl. 4 (4) (2001) 535–557. M. Bohner, T. Matthews, The Grüss inequality on time scales, Commun. Math. Anal. 3 (1) (2007) 1–8. M. Bohner, T. Matthews, Ostrowski inequalities on time scales, JIPAM. J. Inequal. Pure Appl. Math. 9 (1) (2008) Art. 6, 8 pp. R. Hilscher, A time scales version of a Wirtinger-type inequality and applications, J. Comput. Appl. Math. 141 (1–2) (2002) 219–226. W.J. Liu, Q.A. Ngô, An Ostrowski–Grüss type inequality on time scales, Comput. Math. Appl. 58 (2009) 1207–1210. A. Ostrowski, Über die Absolutabweichung einer differenzierbaren Funktion von ihrem integralmittelwert, Comment. Math. Helv. 10 (1) (1937) 226–227. W.J. Liu, Q.A. Ngô, W.B. Chen, A new generalization of Ostrowski type inequality on time scales, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 17 (2009) 101–114. Q.A. Ngô, W.J. Liu, A sharp Grüss type inequality on time scales and application to the sharp Ostrowski–Grüss inequality, Commun. Math. Anal. 6 (2) (2009) 33–41. Shun-Feng Wang, Qiao-Ling Xue, Wen-Jun Liu, Further generalization of Ostrowski–Grüss type inequalities, Adv. Appl. Math. Anal. 3 (1) (2008) 17–20. R.P. Agarwal, M. Bohner, D. O’Regan, A. Peterson, Dynamic equations on time scales: a survey, J. Comput. Appl. Math. 141 (1–2) (2002) 1–26. M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser, Boston, 2001. M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003. V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, 1996.
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Generalization of Ostrowski and Ostrowski–Grüss type inequalities on time scales Adnan Tuna ∗ , Durmus Daghan Department of Mathematics, Faculty of Science and Arts, University of Niğde, Merkez, 51240, Niğde, Turkey
article
abstract
info
Article history: Received 26 October 2009 Received in revised form 12 April 2010 Accepted 20 May 2010
In this paper, we obtain an Ostrowski and Ostrowski–Grüss type inequality on time scales, which not only provides a generalization of the known results on time scales, but also give some other interesting inequalities as special cases. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Grüss inequality Ostrowski inequality Ostrowski–Grüss inequality Time scales
1. Introduction In 1988, Hilger [1] introduced the time scales theory to unify continuous and discrete analysis. Since then, many authors have studied certain integral inequalities on time scales [2–6]. In 1938, Ostrowski derived a formula to estimate the absolute deviation of a differentiable function from its integral mean [7], the so-called Ostrowski inequality holds and can be shown by using the Montgomery identity [7]. These two properties was proved by Bohner and Matthews in [4] for general time scales, which unify discrete, continuous and many other cases. The Ostrowski inequality is generalized by using the Montgomery identity on time scales as following theorem are given in [4] with Theorem 3.5. Theorem 1. Let a, b, s, t ∈ T, a < b and if f : [a, b] → R be differentiable. Then
Z b M f (t ) − 1 (h2 (t , a) + h2 (t , b)), f (σ (s))1s ≤ b−a a b−a
(1.1)
where M = sup f ∆ (t ) .
a
This inequality is sharp in the sense that the right-hand side of (1.1) cannot be replaced by a smaller one. Additionally, Liu et al. [8] have extended a generalization of an Ostrowski type inequality with a parameter on time scales and a unified corresponding continuous and discrete case. Recently, Ngô and Liu in [9] gave the following sharp Grüss type inequality on time scales. Theorem 2. Let a, b, s, t ∈ T, a < b and if f : [a, b] → R be differentiable. If f ∆ is rd continuous and
γ ≤ f ∆ (t ) ≤ Γ , ∗
∀t ∈ [a, b].
Corresponding author. E-mail addresses: [email protected] (A. Tuna), [email protected] (D. Daghan).
0898-1221/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.05.027
804
A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
Then we have
Z b f (b) − f (a) f (t ) − 1 f (σ ( s )) 1 s − h ( t , a ) − h ( t , b )) ( 2 2 2 b−a a ( b − a) Z b Γ −γ p(t , s) − h2 (t , a) − h2 (t , b) 1s, ≤ 2(b − a) a b−a n s − a, a ≤ s < t, for all t ∈ [a, b] and p(t , s) = s − b, t ≤ s ≤ b. Moreover, the constant 12 in (1.2) is sharp.
(1.2)
Shun-Feng Wang, Qiao-ling Xue and Wen-jun Liu in [10] obtained the following Ostrowski–Grüss type inequality for bounded differentiable mappings with a parameter. Besides, they also give some other interesting inequalities as special cases. Theorem 3. Let I ⊂ R be an open interval a, b ∈ I , a < b. If f : I → R is a differentiable function such that there exist constants γ , Γ ∈ R, with γ ≤ f 0 (t ) ≤ Γ , t ∈ [a, b]. Then for all t ∈ [a, b] and λ ∈ [0, 1], we have
Z b 1 − λ f (t ) − λ (t − b)f (b) − (t − a)f (a) − Γ + γ (1 − λ) t − a + b − 1 f (s)ds 2 2(b − a) 2 2 b−a a λ2 (t − a)2 + (b − t )2 ≤ 1−λ+ (Γ − γ ) . 2 4(b − a)
(1.3)
This work is organized as follows: in Section 2 we will briefly present the general definitions and theorems connected to the time scales calculus. Next, in Section 3 we will generalize the Montgomery identity, the Ostrowski inequality and Theorem 2 on time scales. Additionally, Theorem 3 will be proved for general time scales and we also give some other interesting inequalities as special cases. 2. General definitions Now we briefly introduce the time scales theory and refer the reader to Hilger [1] and the books [11–14] for further details. Definition 1. A time scale T is a nonempty closed subset of R. We assume throughout that T has the topology that is inherited from the standard topology on R. It also assumed throughout that in T the interval [a, b]T means the set {t ∈ T : s < t } for the points a < b in T. Since a time scale may not be connected, we need the following concept of jump operators. Definition 2. The mappings σ , ρ : T → T defined by σ (t ) = inf {s ∈ T : s > t } and ρ(t ) = sup {s ∈ T : s < t } are called the jump operators. The jump operators σ and ρ allow the classification of points in T in the following way: Definition 3. A nonmaximal element t ∈ T is said to be right-dense if σ (t ) = t, right-scattered if σ (t ) > t, left-dense if ρ(t ) = t, left-scattered if ρ(t ) < t. In the case T = R, we have σ (t ) = t, and if T = hZ, h > 0, then σ (t ) = t + h. Definition 4. The mapping µ : T → R+ defined by µ(t ) = σ (t ) − t is called the graininess function. The set Tk is defined as follows: if T has a left-scattered maximum m, then Tk = T − {m}; otherwise, Tk = T. If T = R, then µ(t ) = 0, and when T = Z, we have µ(t ) = 1. Definition 5. Let f : T → R. f is called differentiable at t ∈ Tk , with (delta) derivative f ∆ (t ) ∈ R if given ε > 0 there exists a neighborhood U of t such that, for all s ∈ U,
σ
f (t ) − f (s) − f ∆ (t )[σ (t ) − s] ≤ ε kσ (t ) − sk , where f σ = f ◦ σ . If T = R, then f ∆ (t ) =
df (t ) , dt
and if T = Z, then f ∆ (t ) = f (t + 1) − f (t ).
Definition 6. The function f : T → R is said to be rd-continuous (denote f ∈ Crd (T, R)) if, at all t ∈ T, (i) f is continuous at every right-dense point t ∈ T, (ii) lims→t − f (s) exists and is finite at every left-dense point t ∈ T. Theorem 4R (Existence of Antiderivatives). Every rd-continuous function has an antiderivative. In particular if t0 ∈ T, then F t defined by t f (s)1s for t ∈ T is an antiderivative of f . 0
A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
805
Definition 7. Let f ∈ Crd (T, R). Then g : T → R is called the antiderivative of f on T if it is differentiable on T and satisfies g ∆ (t ) = f (t ) for any t ∈ Tk . In this case, we defined t
Z
f (s)1s = g (t ) − g (a),
t ∈ T.
a
Theorem 5. Let f , g be rd-continuous, a, b, c ∈ T and α, β ∈ R. Then
Rb Rb Rb [f (t ) + g (t )] 1t = a f (t )1t + a g (t )1t, Rab Ra (2) a f (t )1t = − b f (t )1t, Rb Rc Rb (3) a f (t )1t = a f (t )1t + c f (t )1t, Rb Rb (4) a f (t )g ∆ (t )1t = (fg )(b) − (fg )(a) − a f ∆ (t )g (σ (t ))1t. (1)
Theorem 6. If f is ∆-integrable on [a, b], then so is |f |, and
Z
b
a
Z f (t )1t ≤
b
|f (t )| 1t . a
Definition 8. Let gk , hk : T2 → R, k ∈ N0 be defined by g 0 ( t , s ) = h0 ( t , s ) = 1
for all s, t ∈ T
and then recursively by gk+1 (t , s) =
t
Z
gk (σ (τ ), s)1s
for all s, t ∈ T
s
and hk+1 (t , s) =
t
Z
hk (τ , s)1s
for all s, t ∈ T.
s
3. Generalization of Ostrowski and Ostrowski–Grüss type inequalities Lemma 1. Let a, b, s, t ∈ T, a < b and if f : [a, b] → R be differentiable. Then for all s ∈ [a, b] and λ ∈ [0, 1], we have
λ
1−
f (t ) =
2
b
Z
1 b−a
f (σ (s))1s − λ a
(t − a)f (a) + (b − t )f (b) 1 + 2(b − a) b−a
b
Z
K (t , s)f ∆ (s)1s,
(3.1)
a
where
t −a , s − a + λ 2 K (t , s) = b−t s − b − λ ,
a ≤ s < t, (3.2) t ≤ s ≤ b.
2
Proof. Using Theorem 5(4), we have
Z t
s−
a+λ
a
t −a
f ∆ (s)1s = (t − a) 1 −
2
λ
2
f (t ) + λ
t −a 2
f (a) −
t
Z
f (σ (s))1s a
and b
Z
s−
b−λ
t
b−t
2
f ∆ (s)1s = (b − t ) 1 −
λ 2
f (t ) + λ
b−t 2
f (b) −
b
Z
f (σ (s))1s. t
Therefore b−a
=
b
Z
1
f (σ (s))1s + a
1
b−a
b
Z
b−a
1
f (σ (s))1s +
b
Z
K (t , s)f ∆ (s)1s
a
1
Z λ (t − a)f (a) + (b − t )f (b) (b − a) 1 − f (t ) + λ −
b−a 2 (t − a)f (a) + (b − t )f (b) = 1− f (t ) + λ , 2 2(b − a)
i.e., (3.1) holds.
a
λ
2
b
f (σ (s))1s a
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A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
Another generalized of Montgomery identity given by Liu et al. [8] on time scales for
b−a s − a + λ , 2 K ( t , s) = b−a s − b − λ ,
a ≤ s < t, t ≤ s ≤ b.
2
However, this result is different from ours, but is the same as in the case of λ = 0. Corollary 1. If we choose λ = 0 in Lemma 1, we get f (t ) =
Z
1 b−a
b
f (σ (s))1s +
a
1 b−a
b
Z
K (t , s)f ∆ (s)1s.
a
This result is the Montgomery identity on time scales, which can be found in Ref. [4] with Lemma 3.1 and it is also studied as continuous, discrete and quantum calculus cases in Ref. [4]. Theorem 7. Let a, b, s, t ∈ T, a < b and if f : [a, b] → R be differentiable. Then for all s ∈ [a, b] and λ ∈ [0, 1], we have
Z b 1 − λ f (t ) + λ (t − a)f (a) + (b − t )f (b) − 1 f (σ (s))1s 2 2(b − a) b−a a λ (t − a) f (a) + (b − t ) f (b) M ≤ (h2 (t , a) + h2 (t , b)) + f (t ) + λ b−a 2 2(b − a)
(3.3)
where M = sup f ∆ (t ) .
a
Proof. Using the definition of ∆-integrals, we have
Z t
s−
a+λ
a
t −a
2
f ∆ (s)1s =
t
Z
(s − a) f ∆ (s)1s − λ
t −a
a
2
(f (t ) − f (a))
and b
Z
s−
t
b−λ
b−t
2
f ∆ (s)1s =
b
Z t
(s − b) f ∆ (s)1s + λ
b−t 2
(f (b) − f (t )) .
Thus
Z b 1 1 K (t , s)f (s)1s b − a a Z t Z b 1 1 λ (t − a) f (a) + (b − t ) f (b) ∆ ∆ ≤ (s − a) f (s)1s + (s − b) f (s)1s − f (t ) + λ b−a a b−a t 2 2(b − a) Z t Z b ∆ 1 λ (t − a) f (a) + (b − t ) f (b) 1 1 |s − a| f (s) 1s + |s − b| f (s) 1s + f (t ) + λ ≤ b−a a b−a t 2 2(b − a) Z t Z b λ (t − a) f (a) + (b − t ) f (b) M M ≤ (s − a) 1s + (b − s) 1s + f (t ) + λ b−a a b−a t 2 2(b − a) (t − a) f (a) + (b − t ) f (b) λ M . ≤ (h2 (t , a) + h2 (t , b)) + f (t ) + λ b−a 2 2(b − a)
(3.4)
Using the Lemma 1, we have
Z b 1 − λ f (t ) + λ (t − a)f (a) + (b − t )f (b) − 1 f (σ ( s )) 1 s 2 2(b − a) b−a a Z b 1 = K (t , s)f 1 (s)1s b−a a λ (t − a) f (a) + (b − t ) f (b) M . ≤ (h2 (t , a) + h2 (t , b)) + f (t ) + λ b−a 2 2(b − a) Thus the proof is complete.
(3.5)
A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
807
Similarly, another generalized of Ostrowski inequality is proved by Liu et al. [8] on time scales by using the Montgomery identity. However, this result is also different from us, but is also same in the case of λ = 0. Corollary 2. In the case of λ = 0, we have
Z b ≤ M (h2 (t , a) + h2 (t , b)). f (t ) − 1 f (σ ( s )) 1 s b−a b−a a This is the Ostrowski inequality on time scales, which can be found in Ref. [4] with the Theorem 3.5 and it is also studied as continuous, discrete and quantum calculus cases in [4]. Generalized of the sharp Grüss type inequality is given with Theorem 2 in Section 1. Now, we will give another generalization of the Theorem 2 on time scales by using the Theorem 3.1 in Ref. [9]. In the case of λ = 0, we get Theorem 2 on time scales. Theorem 8. Let a, b, s, t ∈ T, a < b and f : [a, b] → R be differentiable. If f ∆ is rd-continuous and
γ ≤ f ∆ (t ) ≤ Γ ,
∀t ∈ [a, b]
then for all x ∈ [a, b] and λ ∈ [0, 1], we have
Z b (t − a)f (a) + (b − t )f (b) 1 − λ f (t ) − 1 f (σ (s))1s + λ 2 b−a a 2(b − a) 1 (t − a)2 − (b − t )2 f (b) − f (a) − (h2 (t , a) − h2 (t , b)) − λ b−a 2 b−a Z b 2 Γ −γ (t − a) − (b − t )2 K (t , s) − 1 ≤ h ( t , a ) − h ( t , b )) − λ ( 2 2 1s. 2(b − a) a b−a 2 Proof. Define the function
t −a , s − a + λ 2 K (t , s) = b−t s − b − λ , 2
a ≤ s < t, t ≤s≤b
then choosing f (t ) = K (t , s) and g (t ) = f ∆ (t ) in Theorem 3.1 in Ref. [9], we have b
Z
∆
K (t , s)f (s)1s −
a
≤
Γ −γ 2
b
Z a
1
Z
b
b−a a K (t , s) − 1 b−a
b
K (t , s)1s f (s)1s a Z b K (t , τ )1τ 1s. Z
∆
(3.6)
a
Integrating K (t , s) and f ∆ (t ) on [a, b], we get b
Z
1 b−a
K (t , s)1s = a
1 b−a
(h2 (t , a) − h2 (t , b)) − λ
(t − a)2 − (b − t )2
2
(3.7)
and b
Z
f ∆ (s)1s = f (b) − f (a).
(3.8)
a
Substituting Eq. (3.7) in the right hand side of Eq. (3.6), we arrive at the right hand side of Eq. (3.6) as given below.
Γ −γ
Z b K (t , s) − 1 1s K ( t , τ ) 1 τ 2 b−a a a Z Γ − γ b 1 (t − a)2 − (b − t )2 = K ( t , s ) − h ( t , a ) − h ( t , b )) − λ ( 2 2 1s. 2 b−a 2 a Z
b
(3.9)
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A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
By Eqs. (3.6) and (3.9), we obtain
Z
b
K (t , s)f ∆ (s)1s −
a
≤
Γ −γ 2
b
Z a
Z
1
b
K (t , s)1s
Z
b
f ∆ (s)1s
b−a a a (t − a)2 − (b − t )2 K (t , s) − 1 (h2 (t , a) − h2 (t , b)) − λ 1s. b−a 2
(3.10)
From Lemma 1, we have
1−
λ 2
f (t ) =
1 b−a
Z
b
f (σ (s))1s − λ
a
(t − a)f (a) + (b − t )f (b) 1 + 2(b − a) b−a
b
Z
K (t , s)f ∆ (s)1s.
(3.11)
a
Using Eqs. (3.7), (3.8) and (3.11) on the left hand side of Eq. (3.10), we get
Z b (t − a)f (a) + (b − t )f (b) 1 − λ (b − a)f (t ) − f (σ (s))1s + λ 2 2 a 2 (t − a) − (b − t )2 1 [f (b) − f (a)] − (h2 (t , a) − h2 (t , b)) − λ b−a 2 Z 1 Γ − γ b (t − a)2 − (b − t )2 ≤ K (t , s) − b − a (h2 (t , a) − h2 (t , b)) − λ 1s. 2 2 a Hence
Z b (t − a)f (a) + (b − t )f (b) 1 − λ f (t ) − 1 f (σ (s))1s + λ 2 b−a a 2(b − a) (t − a)2 − (b − t )2 f (b) − f (a) 1 − (h2 (t , a) − h2 (t , b)) − λ b−a 2 b−a Z b (t − a)2 − (b − t )2 Γ −γ K (t , s) − 1 h ( t , a ) − h ( t , b )) − λ ≤ ( 2 2 1s. 2(b − a) a b−a 2 This completes the proof.
Corollary 3. In the case of λ = 0, we have
Z b h2 (t , a) − h2 (t , b) f (b) − f (a) f (t ) − 1 f (σ ( s )) 1 s − b−a a b−a b−a Z b Γ −γ K (t , s) − h2 (t , a) − h2 (t , b) 1s. ≤ 2(b − a) a b−a
(3.12)
This result is given in Ref. [9] with Theorem 1.3. Corollary 4. We let T = R in Corollary 3. Then we have
Z b a + b f (b) − f (a) f (t ) − 1 ≤ 1 (Γ − γ ) (b − a) f (s)ds − t − 8 b−a a 2 b−a
(3.13)
for all t ∈ [a, b], where γ ≤ f 0 (t ) ≤ Γ . The constant 1/8 is the best possible, which is also given in Ref. [9] with Eq. (4.5). In addition to this, it is possible to derive the discrete and quantum calculus cases of Eq. (3.12). Theorem 9. Let a, b, s, t ∈ T, a < b and f : [a, b] → R be differentiable. If f ∆ is rd-continuous and
γ ≤ f ∆ (s) ≤ Γ ,
∀s ∈ [a, b].
Then for all x ∈ [a, b], we have
Z b 1 − λ f (t ) + λ (t − a)f (a) + (b − t )f (b) − 1 f (σ (s))1s 2 2(b − a) b−a a Γ +γ 1 (t − a)2 − (b − t )2 − (h2 (t , a) − h2 (t , b)) − λ 2 b−a 2
(3.14)
A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
≤
809
Γ −γ t −a t −a h2 a, a + λ + h2 t , a + λ 2(b − a) 2 2 ! b−t b−t + h2 t , b − λ + h2 b , b − λ 2
(3.15)
2
t for all λ ∈ [0, 1] such that a + λ t −2 a and b − λ b− are in T. 2
Proof. Let us define the function
t −a , s − a + λ 2 K (t , s) = b−t s − b − λ ,
a ≤ s < t, t ≤ s ≤ b.
2
Using Lemma 1, we have b
Z
1 b−a
K (t , s)f ∆ (s)1s =
1−
a
λ
2
f (t ) −
b
Z
1 b−a
f (σ (s))1s + λ a
(t − a)f (a) + (b − t )f (b) . 2(b − a)
(3.16)
We have also
b−a Let C =
b
Z
1
K (t , s)1s = a
Γ +γ
1
2
b−a
b−a
(h2 (t , a) − h2 (t , b)) − λ
(t − a)2 − (b − t )2
2
.
(3.17)
. Using the Eqs. (3.16) and (3.17), we get
b
Z
1
a
λ
Z b (t − a)f (a) + (b − t )f (b) 1 K (t , s) f (s) − C 1s = 1 − f (σ (s))1s f (t ) + λ − 2 2(b − a) b−a a Γ +γ 1 (t − a)2 − (b − t )2 . − (h2 (t , a) − h2 (t , b)) − λ 2 b−a 2
∆
(3.18)
On the other hand, we have
Z 1 b − a
b a
1 K (t , s) f ∆ (s) − C 1s ≤ max f ∆ (s) − C s∈[a,b] b−a
b
Z a
|K (t , s)| 1s .
(3.19)
We have also max f ∆ (s) − C ≤
Γ −γ
s∈[a,b]
(3.20)
2
and 1 b−a
b
Z
|K (t , s)| 1s = a
"
1
h2
b−a
a, a + λ
+ h2 t , b − λ
b−t 2
t −a 2
+ h2 t , a + λ
t −a
# .
+ h2 b , b − λ
b−t 2
2 (3.21)
From the Eqs. (3.19)–(3.21), it follows that
Z 1 b − a
b
∆
K (t , s) f (s) − a
Γ +γ 2
Γ −γ t −a ≤ h2 a, a + λ 2(b − a) 2
1s
+ h2 t , a + λ
t −a
2
Finally, using the Eqs. (3.18) and (3.22), we get Eq. (3.15).
+ h2 t , b − λ
b−t 2
+ h2 b, b − λ
b−t 2
!
.
(3.22)
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A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
Corollary 5. In the case of T = R in Theorem 9, we have
Z b 1 − λ f (t ) + λ (t − a)f (a) + (b − t )f (b) − Γ + γ (1 − λ) t − a + b − 1 f ( s ) ds 2 2(b − a) 2 2 b−a a λ2 (t − a)2 + (b − t )2 ≤ 1−λ+ (Γ − γ ) . 2 4(b − a)
(3.23)
This result was given in Ref. [10] with Theorem 3. It is also shown that, for the case of λ = 1, λ = 0 are given with Theorem 2 and Corollary 5 in [10], respectively. b Corollary 6. We assume that t = a+ ∈ T in Theorem 9, we have 2
1 − λ f a + b + λ f (a) + f (b) 2 2 4 Z b a+b a+b 1 Γ +γ 1 h2 , a − h2 ,b − f (σ (s))1s − 2 b−a 2 2 b−a a Γ −γ b−a a+b b−a ≤ h 2 a, a + λ + h2 ,a + λ 2(b − a) 4 2 4 ! a+b b−a b−a + h2 ,b − λ + h2 b , b − λ 2
4
4
(3.24)
a a for all λ ∈ [0, 1] such that a + λ b− and b − λ b− in T. This is the new result for a special case of Theorem 9. 4 4
Corollary 7. In the case of T = R in Corollary 6, we have
Z b 1 − λ f a + b + λ f (a) + f (b) − 1 f ( s ) ds 2 2 4 b−a a λ2 b − a ≤ 1−λ+ (Γ − γ ) . 2
8
(3.25)
This result is the same as Corollary 6 which was given in Ref. [10]. Corollary 8. Choosing t = a in Theorem 9, we have
Z b 1 − λ f (a) + λ f (b) − Γ + γ λ (b − a)2 − h2 (a, b) − 1 f (σ (s))1s 2 2 2(b − a) 2 b−a a b−a b−a Γ −γ h 2 a, b − λ + h2 b, b − λ ≤ 2(b − a) 2 2
(3.26)
a for all λ ∈ [0, 1] such that b − λ b− in T. This is another new result for a special case of Theorem 9. 4
Corollary 9. We let T = R in Corollary 8, we have
Z b 1 − λ f (a) + λ f (b) + Γ + γ (1 − λ)(b − a) − 1 f (s)ds 2 2 4 b−a a λ2 Γ − γ b − a ≤ 1−λ+ . 2
2
2
(3.27)
This result is the same as Corollary 7 which was given in Ref. [10]. Corollary 10. We let t = b in Theorem 9, we have
Z b 1 − λ f (b) + λ f (a) − Γ + γ h2 (b, a) − λ (b − a)2 − 1 f (σ (s))1s 2 2 2(b − a) 2 b−a a b−a b−a Γ −γ ≤ h 2 a, a + λ + h2 b , a + λ 2(b − a) 2 2 a for all λ ∈ [0, 1] such that a + λ b− in T. This is also a new result for a special case of Theorem 9. 4
(3.28)
A. Tuna, D. Daghan / Computers and Mathematics with Applications 60 (2010) 803–811
811
Corollary 11. We let T = R in Corollary 10, we have
Z b 1 − λ f (b) + λ f (a) − Γ + γ (1 − λ) (b − a) − 1 f ( s ) ds 2 2 4 b−a a λ2 Γ − γ b − a ≤ 1−λ+ . 2
2
2
(3.29)
This result is the same as Corollary 8 which was given in Ref. [10].
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