Ann. Funct. Anal. 4 (2013), no. 1, 149–162
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL:www.emis.de/journals/AFA/
GENERAL MULTIPLE OPIAL-TYPE INEQUALITIES FOR THE
CANAVATI FRACTIONAL DERIVATIVES
M. ANDRIĆ1 , J. PEČARIĆ2 AND I. PERIĆ3
∗
Communicated by S. Barza
Abstract. In this paper we establish some general multiple Opial-type inequalities involving the Canavati fractional derivatives. In some cases the best
possible constants are discussed.
1. Introduction and preliminaries
In 1960, Opial [7] proved the following inequality:
Let f ∈ C 1 [0, h] be such that f (0) = f (h) = 0 and f (x) > 0 for x ∈ (0, h). Then
Z h
Z
h h 0
2
0
[f (x)] dx ,
(1.1)
|f (x) f (x)| dx ≤
4 0
0
where h/4 is the best possible.
This inequality has been generalized and extended over the last 50 years in several
directions, and used in many applications in differential equations (for more details see [1], [9]). The aim of our research is an Opial-type inequality for fractional
derivatives, which has the general form
!p
Z b
pr+q
Z b
N
p+q
Y
β
ri
D i f (t)
w1 (t)
|Dα f (t)|q dt ≤ C
w2 (t) |Dα f (t)|p+q dt
,
a
a
i=1
PN
where w1 and w2 are weight functions, r = i=1 ri and Dγ f denotes the Canavati
fractional derivative of f of order γ.
First we survey some facts about the fractional integrals and derivatives needed
in this paper. For more details see the monographs [8, Chapter 1] and [3].
Date: Received: 17 July 2012; Accepted: 3 November 2012.
∗
Corresponding author.
2010 Mathematics Subject Classification. Primary 26A33; Secondary 26D15, 46N20.
Key words and phrases. Opial-type inequalities, Canavati fractional derivatives, weights.
149
150
M. ANDRIĆ, J. PEČARIĆ AND I. PERIĆ
By C n [a, b] we denote the space of all functions on [a, b] which have continuous
derivatives up to order n, and AC[a, b] is the space of all absolutely continuous
functions on [a, b]. By AC n [a, b] we denote the space of all functions f ∈ C n−1 [a, b]
with f (n−1) ∈ AC[a, b].
By Lp [a, b], 1 ≤ p < ∞, we denote the space of all Lebesgue measurable
functions f for which |f |p is Lebesgue integrable on [a, b], and by L∞ [a, b] the set
of all functions measurable and essentially bounded on [a, b]. Clearly, L∞ [a, b] ⊂
Lp [a, b] for all p ≥ 1.
Let x ∈ [a, b], α > 0, n =
R ∞[α] + 1, [α] denotes the integral part of α and Γ is the
gamma function Γ(α) = 0 e−t tα−1 dt. For f ∈ L1 [a, b] the Riemann-Liouville
α
α
f (right-sided) of order α are defined
f (left-sided) and Jb−
fractional integrals Ja+
by
Z x
1
α
Ja+ f (x) =
(x − t)α−1 f (t) dt ,
Γ(α) a
Z b
1
α
(t − x)α−1 f (t) dt .
Jb− f (x) =
Γ(α) x
α
α
The subspaces Ca+ [a, b] and Cb− [a, b] of C n−1 [a, b] are defined by
α
n−α (n−1)
Ca+
[a, b] = f ∈ C n−1 [a, b] : Ja+
f
∈ C 1 [a, b] ,
n−α (n−1)
α
Cb−
[a, b] = f ∈ C n−1 [a, b] : Jb−
f
∈ C 1 [a, b] .
α
α
α
f
[a, b] and g ∈ Cb−
[a, b] the Canavati fractional derivatives Da+
For f ∈ Ca+
α
(left-sided) and Db− g (right-sided) of order α are defined by
Z x
d n−α (n−1)
d
1
α
Da+ f (x) =
(x − t)n−α−1 f (n−1) (t) dt ,
J
f
(x) =
dx a+
Γ(n − α) dx a
Z b
(−1)n d
n−α (n−1)
α
n d
Db− g(x) = (−1)
(t − x)n−α−1 g (n−1) (t) dt .
J
g
(x) =
dx b−
Γ(n − α) dx x
In addition, we stipulate
0
0
Da+
f := f =: Ja+
f,
0
0
Db−
g := g =: Jb−
g.
α
α
If α ∈ N then Da+
f = f (α) and Db−
g = (−1)α g (α) , the ordinary α-order derivatives.
The composition identity for the Canavati left-sided fractional derivatives comes
from [5], and will be used in all presented Opial-type inequalities. Notice that we
relaxed some conditions on parameters and a function, comparing to the analogous identity given in [3].
Theorem 1.1. [5, Theorem 2.1] Let α > β ≥ 0, n = [α] + 1, m = [β] + 1. Let
β
α
f ∈ Ca+
[a, b] be such that f (i) (a) = 0 for i = m − 1, . . . , n − 2. Then f ∈ Ca+
[a, b]
and
Z x
1
β
α
Da+ f (x) =
(x − t)α−β−1 Da+
f (t) dt , x ∈ [a, b] .
(1.2)
Γ(α − β) a
GENERAL MULTIPLE OPIAL-TYPE INEQUALITIES
151
Our goal is to give general multiple Opial-type inequalities for the Canavati
fractional derivatives. The starting point is next Opial-type inequality for the
Riemann-Liouville1 left-sided fractional derivatives Dαa+ f that comes from [6].
Theorem 1.2. [6, Theorem 4.1] Let pi , q, βi , α, r (i = 1, . . . N ) be real numbers
P
such that pi ≥ 0, p := N
i=1 pi > 0, q > 0, α > βi + 1 ≥ 0 for all i = 1, . . . , N ,
and r > max {1, q, (α − βi )−1 : i = 1, . . . , N }. Suppose f ∈ L1 [a, b] has an integrable left-sided fractional derivative Dαa+ f ∈ L∞ [a, b] and Dα−j
a+ f (a) = 0 for
j = 1, . . . , [α] + 1. Then for any w1 , w2 ∈ C[a, b] with w1 ≥ 0 and w2 > 0,
x
Z
a
Z
N p i
α
q Y
βi
w1 (t) Da+ f (t)
Da+ f (t) dt ≤ A(x)
p+q
r
α
r
w2 (t) Da+ f (t) dt
,
x
a
i=1
where
A(x) =
q
p+q
rq "Z
x
w1 (t)
r
r−q
q
− r−q
w2 (t)
a
Z
Pi (t) =
t
N
Y
|Pi (t)|
pi (r−1)
r−q
# r−q
r
dt
,
i=1
1
r
w2 (τ )− r−1 Ki (t, τ ) r−1 dτ ,
a ≤ t ≤ x,
a
i −1
(t − τ )α−β
+
,
Ki (t, τ ) =
Γ(α − βi )
a ≤ t, τ ≤ x ,
(t − τ )+ = max{t − τ, 0} .
We will give two-weighted, one-weighted and non-weighted versions of this theorem involving the Canavati left-sided fractional derivatives. Also we will give
versions of those inequalities which include decreasing or bounded weight functions.
The right-sided versions of all inequalities in this paper can be established and
proven analogously.
2. Two-weighted case
First theorem is the Canavati fractional derivatives analogy of Theorem 1.2,
with relaxed conditions on the function (here the role of pi and r from Theorem
1.2 have ri p and p + q respectively).
Theorem 2.1. Let N ∈ N, α > βi ≥ 0, m = min{[βi ] + 1 : i = 1, . . . , N } and
α
n = [α]+1. Let f ∈ Ca+
[a, b] be such that f (i) (a) = 0 for i = m−1, . . . , n−2. Let
w1 and w2 be continuous weight functions on [a, x] with w1 ≥ 0 and w2 > 0. Let
P
PN
1
ri ≥ 0, r = N
i=1 ri > 0. Let p > 0, q ≥ 0, σ = p+q < 1, ρ =
i=1 ri (α − βi ) − rσ
1
Riemann-Liouville left-sided fractional derivative is defined by Dα
a+ f (x) =
dn n−α
dxn Ja+ f (x).
152
M. ANDRIĆ, J. PEČARIĆ AND I. PERIĆ
α
and let α > βi + σ for i = 1, . . . , N . Let also Da+
f ∈ Lp+q [a, b]. Then
x
Z
a
N ri
Y
βi
w1 (t)
Da+ f (t)
!p
α
Da+ f (t)q dt
i=1
x
Z
≤ C1

Z
·
rp+q
p+q
α
p+q
w2 (t) Da+ f (t) dt
a
x
q
1
[w1 (t)] σp [w2 (t)]− p
a
N Z t
Y
σ
[w2 (τ )]− 1−σ (t − τ )
α−βi −1
1−σ
σp
i
(1−σ)r
σ
dτ
dt ,
a
i=1
(2.1)
where
C1 =
N
Y
−ri p
[Γ(α − βi )]
i=1
q
rp + q
σq
.
(2.2)
Proof. Let q 6= 0, δi = α − βi − 1, i = 1, . . . , N . Using composition identity (1.2),
1
the triangle inequality and Hölder’s inequality for 1−σ
and σ1 , for t ∈ [a, x] follows
βi
|Da+
f (t)|
t
1
≤
Γ (δi + 1)
Z
1
≤
Γ (δi + 1)
Z
α
[w2 (τ )]−σ [w2 (τ )]σ (t − τ )δi |Da+
f (τ )| dτ
a
t
σ
− 1−σ
[w2 (τ )]
(t − τ )
δi
1−σ
1−σ Z
dτ
a
t
α
1
w2 (τ ) Da+
f (τ ) σ dτ
σ
.
a
Now we have
x
Z
w1 (t)
a
≤
N
Y
βi
α
f (t)|ri p |Da+
f (t)|q dt
|Da+
i=1
1
N
Q
[Γ(δi + 1)]ri p
Z
x
α
w1 (t) [w2 (t)]−σq [w2 (t)]σq |Da+
f (t)|q
a
i=1
Z
·
a
t
1
α
w2 (τ ) |Da+
f (τ )| σ
dτ
σrp Y
N Z
i=1
t
σ
− 1−σ
[w2 (τ )]
(t − τ )
δi
1−σ
(1−σ)ri p
dτ
dt .
a
(2.3)
GENERAL MULTIPLE OPIAL-TYPE INEQUALITIES
1
, 1
σp σq
Applying Hölder’s inequality for
x
Z
w1 (t)
N
Y
a
≤
153
and simple integration, we get
βi
α
|Da+
f (t)|ri p |Da+
f (t)|q dt
i=1
1
N
Q
[Γ(δi + 1)]ri p
i=1

Z
·
x
q
1
[w1 (t)] σp [w2 (t)]− p
a
"Z
N Z t
Y
x
α
w2 (t) |Da+
f (t)|
·
1
σ
1
N
Q
dt
t
1
σ
α
w2 (τ ) |Da+
f (τ )| dτ
#σq
prq
dt
a
a
=
σp
a
i=1
Z
δi
σ
[w2 (τ )]− 1−σ (t − τ ) 1−σ dτ
i
(1−σ)r
σ
[Γ(δi + 1)]
ri p
q
rp + q
σq Z
x
1
α
w2 (t) |Da+
f (t)| σ
σ(rp+q)
dt
(2.4)
a
i=1

Z

·
x
[w1 (t)]
1
σp
− pq
[w2 (t)]
a
N Z
Y
t
σ
− 1−σ
[w2 (τ )]
(t − τ )
δi
1−σ
i
(1−σ)r
σ
σp
dt ,
dτ
a
i=1
which gives us inequality (2.1).
If q = 0 (σ = p1 ), then inequality (2.3) has the form
x
Z
w1 (t)
a
≤
N
Y
βi
|Da+
f (t)|ri p dt
i=1
Z
1
N
Q
[Γ(δi + 1)]
x
1
α
w2 (τ ) |Da+
f (τ )| σ
r
dτ
a
ri p
i=1
Z
·
x
w1 (t)
a
N Z
Y
i=1
t
σ
− 1−σ
[w2 (τ )]
(t − τ )
δi
1−σ
(1−σ)ri p
dτ
dt,
a
from which we get inequality (2.1) for q = 0.
Next results complement Theorem 2.1. To obtain inequality (2.5) we need a
monotonicity of w1 and w2 .
154
M. ANDRIĆ, J. PEČARIĆ AND I. PERIĆ
Theorem 2.2. Suppose that the assumptions of Theorem 2.1 hold. Suppose also
that w1 is an increasing and w2 is a decreasing functions. Then
!
Z x
N ri p Y
q
βi
α
Da+
w1 (t)
f (t) dt
Da+ f (t)
a
i=1
−σ(rp+q)
≤ C2 w1 (x) [w2 (x)]
(x − a)
(ρ+σ)p
x
Z
rp+q
p+q
α
p+q
,
w2 (t) Da+ f (t) dt
a
(2.5)
where
C1 σ σp (1 − σ)(1−σ)rp
C2 =
N
Q
(ρ + σ)σp
(α − βi − σ)ri (1−σ)p
(2.6)
i=1
and C1 is defined by (2.2).
Proof. We start the proof with obtained inequality (2.1) form Theorem 2.1. By
monotonicity of w1 and w2 follows

σp
i
(1−σ)r
Z x
N Z t
σ
Y
δi
q
1
σ

[w1 (t)] pσ [w2 (t)]− p
dt
[w2 (τ )]− 1−σ (t − τ ) 1−σ dτ
a
a
i=1

Z
−σ(rp+q) 
≤ w1 (x) [w2 (x)]
N Z t
xY
a
= w1 (x) [w2 (x)]−σ(rp+q)
i=1
δi
(t − τ ) 1−σ dτ
dt
a
(1 − σ)(1−σ)rp
N
Q
σp
i
(1−σ)r
σ
(1−σ)ri p
(δi + 1 − σ)
(x − a)(ρ+σ)p
σ σp
.
(ρ + σ)σp
(2.7)
i=1
Inequality (2.5) now follows from (2.4) and (2.7).
For q = 0, we proceed the same as in the proof of Theorem 2.1.
For the next theorem we suppose that weight functions are bounded.
Theorem 2.3. Suppose that the assumptions of Theorem 2.1 hold. Suppose also
w1 (t) ≤ B and A ≤ w2 (t) for t ∈ [a, x]. Then
!
Z x
N ri p Y
q
βi
α
Da+
w1 (t)
f (t) dt
Da+ f (t)
a
i=1
≤ C2 B A
−σ(rp+q)
(ρ+σ)p
Z
(x − a)
x
rp+q
p+q
α
p+q
,
w2 (t) Da+ f (t) dt
(2.8)
a
where C2 is defined by (2.6).
Proof. The proof of (2.8) is the same as the one for (2.5), except one change:
instead of inequalities w1 (t) ≤ w1 (x), w2 (t) ≥ w2 (x) we use w1 (t) ≤ B, w2 (t) ≥ A
respectively.
GENERAL MULTIPLE OPIAL-TYPE INEQUALITIES
155
With extra parameters s1 , s2 and s3 we can extract expressions containing just
weight functions to get inequality (2.9).
Theorem 2.4. Suppose that the assumptions of Theorem 2.1 hold. Suppose also
that sk > 1 and s1k + s10 = 1 for k = 1, 2, 3. Then
k
x
Z
a
!p
N ri
Y
βi
w1 (t)
Da+ f (t)
α
Da+ f (t)q dt
i=1
≤ C3 P (x) Q(x) R(x) (x − a)
ρp+ sσp
−
s
2 3
(1−σ)rp
s01
x
Z
rp+q
p+q
α
p+q
dt
w2 (t) Da+ f (t)
,
a
(2.9)
where
(1−σ)rp
s1
C3 =
N
Q
[s1 (α − βi − 1) + 1 − σ]
C1 (1 − σ)
N
(1−σ)ri p
P
s
sσps
[s1 (α − βi − 1) + 1 − σ] ri s2 s3 + σs1
1
i=1
σp
(σs1 ) s2 s3
2 3
i=1
and
x
Z
P (x) =
(1−σ)rp
s01
0
σ
,
[w2 (t)]− 1−σ s1 dt
a
x
Z
Q(x) =
[w1 (t)]
s02
σp
σp0
dt
s2
,
(2.10)
a
x
Z
R(x) =
[w2 (t)]
q
−p
s2 s03
σp0
s 2 s3
.
dt
a
Proof. We start the proof with obtained inequality (2.1) form Theorem 2.1. By
Hölder’s inequality we have
Z t
δi
σ
[w2 (τ )]− 1−σ (t − τ ) 1−σ dτ
a
Z
t
σ
− 1−σ
s01
≤
[w2 (τ )]
s10 Z
dτ
1
(t − τ )
a
Z
t
δi
s
1−σ 1
s1
1
dτ
a
x
σ
− 1−σ
s01
≤
[w2 (τ )]
10 dτ
s1
a
1−σ
δi s 1 + 1 − σ
s1
1
δi
(t − a) 1−σ
+ s1
1
.
Now follows
i
(1−σ)r
N Z t
σ
Y
δi
σ
− 1−σ
(t − τ ) 1−σ dτ
[w2 (τ )]
i=1
a
Z
≤
x
σ
− 1−σ
s01
[w2 (τ )]
a
(1−σ)r
0
dτ
(1 − σ)
σs1
N
Q
(1−σ)r
σs1
(δi s1 + 1 − σ)
i=1
(1−σ)ri
σs1
(t − a)
PN
i=1 (δi s1 +1−σ)ri
σs1
.
156
M. ANDRIĆ, J. PEČARIĆ AND I. PERIĆ
Let ε =
N
P
(δi s1 + 1 − σ)ri . Applying Hölder’s inequalities we get
i=1

Z

x
[w1 (t)]
1
σp
− pq
[w2 (t)]
a
N Z
Y
≤ P (x)
N
Q
σ
− 1−σ
(1−σ)rp
s1
(δi s1 + 1 − σ)
(t − τ )
[w2 (τ )]
σp
i
(1−σ)r
σ
δi
1−σ
dτ
dt
a
i=1
(1 − σ)
t
x
Z
[w1 (t)]
(1−σ)ri p
s1
1
σp
− pq
[w2 (t)]
(t − a)
ε
σs1
σp
dt
a
i=1
(1 − σ)
≤ P (x)
N
Q
(1−σ)rp
s1
(δi s1 + 1 − σ)
Z
x
[w1 (t)]
(1−σ)ri p
s1
s02
σp
σp0
s2
dt
a
i=1
x
Z
·
−
[w2 (t)]
qs2
p
(t − a)
ε
s
σs1 2
σp
s2
dt
a
≤ P (x) Q(x)
(1 − σ)
N
Q
(1−σ)rp
s1
(δi s1 + 1 − σ)
(1−σ)ri p
s1
i=1
x
Z
·
− pq s2 s03
[w2 (t)]
σp0 Z
s2 s3
dt
x
(t − a)
ε
s s
σs1 2 3
sσps
2 3
dt
a
a
= P (x) Q(x) R(x)
(1 − σ)
N
Q
(1−σ)rp
s1
(δi s1 + 1 − σ)
(1−σ)ri p
s1
σs1
εs2 s3 + σs1
sσps
2 3
εp
(x − a) s1
+ sσp
s
2 3
.
i=1
(2.11)
Inequality (2.9) now follows from (2.4) and (2.11).
For q = 0, we proceed the same as in the proof of Theorem 2.1.
If we choose a convenient parameter s3 , then we get next corollary.
Corollary 2.5. Suppose that the assumptions of Theorem 2.4 hold. Suppose also
σps01
that s3 = σps0 −q(1−σ)s
> 1. Then
2
1
!
Z x
N ri p Y
q
βi
α
Da+
w1 (t)
f (t) dt
Da+ f (t)
a
i=1
e3 Pe(x) Q(x) (x − a)
≤ C
ρp+ σp
−
s
2
(1−σ)(rp+q)
s01
Z
x
rp+q
p+q
α
p+q
w2 (t) Da+ f (t) dt
,
a
where Q is defined by (2.10) and
e3 =
C
C1 (1 − σ)
N
Q
(1−σ)rp
s1
[(α − βi − 1)s1 + 1 − σ]
i=1
(1−σ)ri
s1
σps01 − q(1 − σ)s2
(ρs2 + σ)ps01 − (1 − σ)(rp + q)s2
σp − q(1−σ)
0
s2
s1
,
GENERAL MULTIPLE OPIAL-TYPE INEQUALITIES
x
Z
Pe(x) =
0
σ
w2 (t)− 1−σ s1 dt
157
(1−σ)(rp+q)
0
s1
.
a
3. One-weighted case
First result is a direct consequence of Theorem 2.1.
Theorem 3.1. Let N ∈ N, α > βi ≥ 0, m = min{[βi ] + 1 : i = 1, . . . , N } and
α
n = [α]+1. Let f ∈ Ca+
[a, b] be such that f (i) (a) = 0 for i = m−1, . . . , n−2. Let
P
w be continuous positive weight function on [a, x]. Let ri ≥ 0, r = N
i=1 ri > 0.
P
N
1
Let p > 0, q ≥ 0, σ = p+q < 1, ρ = i=1 ri (α − βi ) − rσ and let α > βi + σ for
α
f ∈ Lp+q [a, b]. Then
i = 1, . . . , N . Let also Da+
x
Z
a
N ri
Y
βi
w(t)
Da+ f (t)
!p
α
Da+ f (t)q dt
i=1
x
Z
≤ C1

Z

·
rp+q
p+q
α
p+q
w(t) Da+ f (t) dt
a
x
w(t)
a
N Z
Y
t
σ
− 1−σ
[w(τ )]
(t − τ )
α−βi −1
1−σ
i
(1−σ)r
σ
dτ
σp
dt ,
a
i=1
where C1 is defined by (2.2) .
If we have a decreasing weight function, then we need the assumption r ≥ 1.
Theorem 3.2. Suppose that the assumptions of Theorem 3.1 hold. Suppose also
that r ≥ 1 and w is a decreasing function. Then
x
Z
a
N ri
Y
βi
w(t)
Da+ f (t)
!p
α
Da+ f (t)q dt
i=1
(1−r)σp
≤ C2 [w(x)]
(x − a)
(ρ+σ)p
Z
x
pr+q
p+q
α
p+q
w(t) Da+ f (t) dt
,
(3.1)
a
where C2 is defined by (2.6) .
Proof. Let q 6= 0, δi = α − βi − 1, i = 1, . . . , N . Since w is decreasing, then
w(τ )
1≤
w(t)
σ
,
τ ≤ t.
(3.2)
158
M. ANDRIĆ, J. PEČARIĆ AND I. PERIĆ
Using composition identity (1.2), the triangle inequality and Hölder’s inequality
1
for 1−σ
and σ1 , for t ∈ [a, x] follows
ri p
N
N Z t
Y
Y
1
βi
δi
α
ri p
(t − τ ) |Da+ f (τ )| dτ
|Da+ f (t)|
≤ N
Q
a
ri p i=1
i=1
[Γ(δi + 1)]
i=1
N Z
Y
[w(t)]−σrp
≤
N
Q
[Γ(δi +
t
σ
δi
(t − τ ) [w(τ )]
α
f (τ )| dτ
|Da+
ri p
a
1)]ri p i=1
i=1
[w(t)]−σrp (1 − σ)(1−σ)rp
≤
N
Q
[Γ(δi + 1)(δi + 1 − σ)1−σ ]ri p
i=1
Z t
σrp
PN
1
(δi +1−σ)ri p
α
i=1
σ
· (t − a)
w(τ ) |Da+ f (τ )| dτ
.
a
Therefore
x
Z
w(t)
N
Y
a
βi
α
|Da+
f (t)|ri p |Da+
f (t)|q dt
i=1
≤
N
Y
"
i=1
Z
1−σ
δi + 1 − σ
1−σ
1
Γ(δi + 1)
#ri p
x
α
[w(t)]1−σrp |Da+
f (t)|q (t − a)
σrp
aZ t
1
α
dt.
w(τ ) |Da+ f (τ )| σ dτ
·
·
(3.3)
PN
i=1 (δi +1−σ)ri p
(3.4)
a
Applying Hölder’s inequality for
1
σp
x
Z
1−σrp
α
|Da+
f (t)|q
ρp
[w(t)]
(t − a)
Z x
σp
ρ
σ
≤
(t − a) dt
a
and
Z
1
σq
t
with ρp =
N
P
(δi + 1 − σ)ri p, we obtain
i=1
1
α
w(τ ) |Da+
f (τ )| σ
σrp
dτ
dt
a
a
Z
·
x
[w(t)]
1−σrp−σq
σq
a
α
w(t) |Da+
f (t)|
1
σ
Z
t
1
σ
α
w(τ ) |Da+
f (τ )| dτ
rpq
!σq
dt
a
σp
p(1−r)
σ
≤ (x − a)
[w(x)] p+q
ρ+σ
σq Z x
σ(rp+q)
1
q
α
·
w(t)|Da+ f (t)| σ dt
.
rp + q
a
(ρ+σ)p
The inequality (3.1) now follows from (3.3) and (3.5).
For q = 0, we proceed the same as in the proof of Theorem 2.1.
(3.5)
GENERAL MULTIPLE OPIAL-TYPE INEQUALITIES
159
If r = 1 then we have Alzer’s inequality [2, Theorem 1] for the Canavati leftsided fractional derivatives:
Corollary 3.3. Suppose that the assumptions of Theorem 3.2 hold and let r = 1.
Then
!
Z x
N ri p Y
q
βi
α
Da+
w(t)
f (t) dt
Da+ f (t)
a
i=1
N
P
e2 (x − a)i=1
≤ C
x
Z
ri (α−βi )p
α
p+q
w(t) Da+
f (t) dt,
a
where
"
e2 = σ q σq
C
N
X
#−σp
ri (α − βi )
"
N
Y
i=1
i=1
1−σ
α − βi − σ
1−σ
1
Γ(α − βi )
#ri p
.
For the next theorem we suppose that weight function is bounded.
Theorem 3.4. Suppose that the assumptions of Theorem 3.1 hold. Suppose also
that r ≥ 1 and A ≤ w(t) ≤ B for t ∈ [a, x]. Then
!
Z x
N ri p Y
q
βi
α
Da+
f (t) dt
w(t)
Da+ f (t)
a
i=1
≤ C2
B
Ar
σp
(x − a)
(ρ+σ)p
x
Z
pr+q
p+q
α
p+q
w(t) Da+ f (t)
dt
,
(3.6)
a
where C2 is given by (2.6).
Proof. The proof of (3.6) is the same as the one for (3.1), except two changes.
Instead of inequality (3.2) we use 1 ≤ (w(τ )/A)σ . Moreover, in (3.4) we apply
the inequality w(t) = [w(t)]σp [w(t)]σq ≤ B σp [w(t)]σq . These two changes lead to
the inequality (3.6).
4. Non-weighted case
The last result is a non-weighted case of previous theorems. Here we also give
a case with a best possible solution.
Proposition 4.1. Let N ∈ N, α > βi ≥ 0, m = min{[βi ] + 1 : i = 1, . . . , N } and
α
n = [α]+1. Let f ∈ Ca+
[a, b] be such that f (i) (a) = 0 for i = m−1, . . . , n−2. Let
PN
P
1
ri ≥ 0, r = i=1 ri > 0. Let p > 0, q ≥ 0, σ = p+q
< 1, ρ = N
i=1 ri (α − βi ) − rσ
α
and let α > βi + σ for i = 1, . . . , N . Let also Da+ f ∈ Lp+q [a, b]. Then
!
Z x Y
N ri p q
βi
α
Da+
f (t) dt
Da+ f (t)
a
i=1
(ρ+σ)p
Z
≤ C2 (x − a)
a
where C2 is defined by (2.6).
x
pr+q
p+q
α
p+q
Da+ f (t) dt
,
(4.1)
160
M. ANDRIĆ, J. PEČARIĆ AND I. PERIĆ
Inequality (4.1) is sharp if and only if α = βi + 1, i = 1, . . . , N and q = 1. The
α
f (t) = 1, t ∈ [a, x].
equality in this case is attained for a function f such that Da+
Proof. Let q 6= 0, δi = α − βi − 1, i = 1, . . . , N . As in the proof of Theorem 2.1,
using composition identity (1.2), the triangle inequality and Hölder’s inequality
1
for 1−σ
and σ1 , for t ∈ [a, x] follows
N
Y
βi
|Da+
f (t)|ri p
i=1
≤
"Z
N
Y
1
N
Q
(t − τ )
δi
1−σ
1−σ Z
t
1
σ
α
|Da+
f (τ )| dτ
dτ
a
1)]ri p i=1
[Γ(δi +
t
σ #ri p
(4.2)
a
i=1
PN
(1 − σ)(1−σ)rp (t − a) i=1 (δi +1−σ)ri p
=
N ri p
Q
Γ(δi + 1) (δi + 1 − σ)1−σ
Z
t
1
α
|Da+
f (τ )| σ
σrp
dτ
.
a
i=1
Now we have
Z xY
N
βi
α
|Da+
f (t)|ri p |Da+
f (t)|q dt
a
≤
i=1
(1 − σ)(1−σ)rp
N ri p
Q
Γ(δi + 1) (δi + 1 − σ)1−σ
i=1
Z t
σrp
Z x
PN
1
(δ
+1−σ)r
p
α
α
i
i
·
(t − a) i=1
|Da+ f (τ )| σ dτ
|Da+
f (t)|q dt .
a
a
1
1
and σq
and a simple integration inequality (4.1)
By Hölder’s inequality for σp
follows.
For q = 0, we proceed as in the proof of Theorem 2.1.
Using the equality condition in Hölder’s inequality we have equality in (4.2) if
δi
1
α
α
and only if |Da+
f (τ )| σ = λ(t − τ ) 1−σ , i = 1, . . . , N , which implies (since Da+
f (τ )
depends only on τ ) that δi = 0, that is α = βi + 1 for i = 1, . . . , N . Due to
α
homogeneous property of inequality (4.1) we can take Da+
f (τ ) = 1, which gives
βi
α−1
Da+ f (τ ) = Da+ (τ ) = τ − a, i = 1, . . . , N . Substituting this in equality (4.1) for
the left side we get
Z
N
xY
(t − a)
a
ri p
x
Z
(t − a)rp dt =
dt =
a
i=1
(x − a)rp+1
.
rp + 1
For the right side, with ρ = r − rσ, follows
(ρ+σ)p
Z
C2 (x − a)
a
x
pr+q
p+q
dt
=
σ σp q σq
(x − a)rp+1 .
(r − rσ + σ)σp (rp + q)σq
GENERAL MULTIPLE OPIAL-TYPE INEQUALITIES
161
Hence,
q
1
q p+q
=
p
q
rp + 1
[r(p + q) − r + 1] p+q (rp + q) p+q
which is equivalent with
[r(p + q) − r + 1]p [rp + q]q = q q (rp + 1)p+q .
(4.3)
For q = 1 equality (4.3) obviously holds. For q = 0 equality (4.3) implies r = 0,
which gives trivial identity in (4.1). By simple rearrangements equation (4.3)
becomes
p q
q−1
p 1−q
1+r
= 1.
(4.4)
1+r
rp + 1
q rp + 1
2 p
1−p
For p = q the left-hand side of equation (4.4) is equal to 1 − r rp+1
, which
is strictly less then 1, except in trivial cases. For 0 < p < q, q 6= 1, r > 0, using
the Bernoulli inequality, we have
p q−1 q
p 1−q
p q−1
p 1−q
1+r
1+r
< 1+r
1+r
,
rp + 1
q rp + 1
q rp + 1
q rp + 1
which is obviously strictly less then 1. For 0 < q < p, q 6= 1, r > 0, using the
Bernoulli inequality, we have
p q−1
q−1
p 1−q q
1−q
1+r
< 1+r
1+r
1+r
,
rp + 1
q rp + 1
rp + 1
rp + 1
which is again obviously strictly less then 1. It follows that (4.3) holds if and
only if q = 1.
Remark 4.2. Let N = 1, α = 1, β1 = 0, r1 = r = 1, p = q = 1, a = 0 and x = h.
Then inequality (4.1) becomes Beesack’s inequality [4]
Z h
Z
h h 0 2
0
|f (t) f (t)| dt ≤
[f (t)] dt .
(4.5)
2 0
0
He proved that inequality (4.5) is valid for any function f absolutely continuous
on [0, h] satisfying single boundary condition f (0) = 0.
In order to get classical Opial’s inequality (1.1) we need right-sided version of
inequality (4.1) for N = 1, r1 = r = 1 and p = q = 1:
Z b
Z b
β1
α
α
|Db−
f (t)|2 dt
(4.6)
|Db− f (t)| |Db− f (t)| dt ≤ C2 (b − x)
x
x
satisfying f (b) = 0. Observe the inequality
Z a+b
Z b
2
β1
β1
α
α
f (t)| dt
|Da+ f (t)| |Da+ f (t)| dt +
|Db−
f (t)| |Db−
a+b
2
a
≤ C2
b−a
2
Z
a
a+b
2
α
Da+ f (t)2 dt +
Z
b
a+b
2
!
α
2
Db− f (t) dt .
(4.7)
162
M. ANDRIĆ, J. PEČARIĆ AND I. PERIĆ
If we put α = 1, β1 = 0, a = 0 and b = h, then inequality (4.7) becomes Opial’s
inequality (1.1) (having boundary conditions f (0) = f (h) = 0).
References
1. R.P. Agarwal and P.Y.H. Pang, Opial Inequalities with Applications in Differential and
Difference Equations, Kluwer Academic Publishers, Dordrecht, Boston, London, 1995.
2. H. Alzer, On some inequalities of Opial-type, Arch. Math. 63 (1994), 431–436.
3. G.A. Anastassiou, Fractional Differentiation Inequalities, Springer, 2009.
4. P.R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc. 104 (1962),
470–475.
5. M. Andrić, J. Pečarić and I. Perić, Improvements of composition rule for the Canavati
fractional derivatives and applications to Opial-type inequalities, Dynam. Systems Appl. 20
(2011), 383–394.
6. W.-S. Cheung, Z. Dandan and J. Pečarić, Opial-type inequalities for differential operators,
Nonlinear Analysis 66 (2007), no. 9, 2028–2039.
7. Z. Opial, Sur une inégalité, Ann. Polon. Math. 8 (1960), 29–32.
8. S.G. Samko, A.A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory
and Applications, Gordon and Breach, Reading, 1993.
9. D. Willet, The existence-uniqueness theorem for an nth order linear ordinary differential
equation, Amer. Math. Monthly 75 (1968), 174–178.
1
Faculty of Civil Engineering, Architecture and Geodesy, University of
Split, Split, Croatia.
E-mail address: maja.andric@gradst.hr
2
Faculty of Textile Technology, University of Zagreb, Zagreb, Croatia.
E-mail address: pecaric@element.hr
3
Faculty of Food Technology and Biotechnology, University of Zagreb, Zagreb, Croatia.
E-mail address: iperic@pbf.hr