DEMONSTRATIO MATHEMATICA
Vol. XLVI
No 4
2013
10.1515/dema-2013-0482
Li Yin and Changjian Zhao
SOME NEW GENERALIZATIONS OF MARONI INEQUALITY
ON TIME SCALES
Abstract. The aim of present paper is to establish some new integral inequalities
on time scales involving several functions and their derivatives which in the special cases
yield the well known Maroni inequality and some of its generalizations.
1. Introduction
Opial type inequality have many applications in the theory of differential
and difference equation, for instance some inequalities are used to prove
existence of solutions (see [1]). For some classical results on Opial inequality,
the readers are referred to the articles (see [2]-[9], [15]).
Now, we give some of earliest version of the Opial inequlality as follows:
Theorem 1.1. Let f (x) ∈ C 1 [0, a] with f (0) = f (a) = 0 and f (x) > 0,
(0 < x < a). Then
(1.1)
a f (x)f 0 (x) dx 6 a f 0 (x)2 dx,
40
0
a where the constant factor
if and only if
(1.2)
a
4
is the best possible. Equality holds true in (1.1)
f (x) =


cx, 0 6 x 6 a2 ,
 c(a − x),
a
2
6 x 6 a,
where c is a positive constant.
Beesack in [16] generalized Opial’s inequality as follows
2010 Mathematics Subject Classification: 26D15, 39A13.
Key words and phrases: Opial inequality; Maroni inequality; Hölder inequality; time
scales; delta differentiability.
This work is supported by the National Natural Science Foundation of China
(10971205) and by the Science Foundation of Binzhou University under grant BZXYL1303.
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L. Yin, C. J. Zhao
Theorem 1.2. Let the function p(x) be nonnegative and continuous on
[a, τ ]. If the function f (x) is absolutely continuous on [a, τ ] with f (a) = 0.
The following inequality holds true:
τ τ
τ
2 f (x)f 0 (x) dx 6 1 (p(x))−1 dx
p(x) f 0 (x) dx .
(1.3)
2 a
a
a
Subsequently, Maroni in [10] further generalized Beesack’s result in a
form given by Theorem 1.3.
Theorem 1.3. Let the function p(x) be nonnegative and continuous on
[a, τ ] with τa (p(x))1−u dx < ∞, (u ≥ 1). Also, let the function f (x) be
absolutely continuous on [a, τ ] and suppose that f (a) = f (τ ) = 0. The
following inequality holds true:
τ τ
2 τ
v v2
1 u
1−u
0
f (x)f (x) dx 6
(1.4)
p(x) f 0 (x) dx
(p(x)) dx
2 a
a
a
where
1
u
+
1
v
= 1.
Therefore, some very interesting generalizations are given by B. G. Pachpatte who works with several functions in Opial type inequalities. We give
the following case (see [8], [9], [11]).
Theorem 1.4. For f (x), g(x) ∈ C 1 [0, a] with f (a) = g(a) = 0, we have
(1.5)
b
b
b − a 0 2 0 2
(f (x)g 0 (x) + f 0 (x)g(x))dx 6
( f (x) + g (x) )dx
2 a
a
with equality f (x) = g(x) = c(x − a) for x ∈ [a, b] where c is constant.
Next, R. P. Agarwal extended Theorem A to hold true on time scales
in [12]. H. M. Srivastava etc. obtained some generalizations of Maroni
inequality on time scales in [13]. For more results on time scales see [14].
In this paper, we obtain some new Maroni type inequality on time scales
involving several functions. Our results are not only new for arbitrary time
scales, but also new for the continuous and the discrete cases. Our results
generalized some of the works done by B. G. Pachpathe.
2. Time scales essential
A time scale T is an arbitrary nonempty closed subset of the real numbers
R. The forward and backward jump operators σ, ρ : T → T are defined by
σ(t) = inf {s ∈ T : s > t} ,
ρ(t) = sup {s ∈ T : s < t} ,
where the supremum of the empty set is defined to be the infimum of T.
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A point t ∈ T is said to be right-scattered if σ(t) > t and right-dense if
σ(t) = t, and t ∈ T with t > inf T is said to be left-scattered if ρ(t) < t
and left-dense if ρ(t) = t. A function g : T → R is said to be rd-continuous
provided g is continuous at right-dense points and has finite left-sided limits
at left-dense points in T. The graininess function µ for a time scales T is
defined by µ(t) = σ(t) − t, and for every function f : T → R the notation
f σ means the composition f ◦ σ.
For a function f : T → R, the (delta) derivative f ∆ (t) at t ∈ T is defined
to be the number (if it exists) such that for all ε > 0, there is a neighborhood
U of t with
(2.1)
|f (σ (t)) − f (s) − f ∆ (t)(σ (t) − s)| < ε|σ (t) − s|,
for all s ∈ U . If the (delta) derivative f ∆ (t) exits for all t ∈ T, then we
say that f is (delta) differentiable on T. We will make use of the following
product and rules for the derivatives of the product f g and the quotient f /g
(where gg σ 6= 0) of two (delta) differentiable functions f and g,
(f g)∆ = f ∆ g + f σ g ∆ = f g ∆ + f ∆ g σ ,
∆
f
f ∆g − f g∆
=
.
g
gg σ
(2.2)
(2.3)
A continuous function f : T → R is called pre-differentiable with D, provided
D ⊂ Tκ , Tκ \D is countable and contains no right-scattered elements of T,
and f is differentiable at each t ∈ D. Let f be rd-continuous. Then there
exists a function F which is pre-differentiable with region of differentiation
D, such that F ∆ (x) = f (t) holds for all t ∈ D. We define the Cauchy
integral by
c
(2.4)
f (t)∆t = F (c) − F (b),
b
where F is a pre-antiderivative of f and b, c ∈ T. The existence theorem [12,
p. 27, Theorem 1.74] reads as follows: Every rd-continuous function
t has an
antiderivative. In particular, if t0 ∈ T then F defined by F (t) = t0 f (τ )∆τ
is an antiderivative of f . An integration by parts formula reads
(2.5)
c
c
f (t)g ∆ (t)∆t = f (t)g(t)|cb − f ∆ (t)g(σ(t))∆t.
b
b
Note that in the case T = R, we have σ(t) = ρ(t) = t, µ(t) = 0, f ∆ (t) = f 0 (t),
(2.6)
c
b
c
f (t)∆t = f 0 (t)dt
∆
b
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L. Yin, C. J. Zhao
and in the case T = Z, we have σ(t) = t + 1, ρ(t) = t − 1, µ(t) ≡ 1,
f ∆ (t) = ∆f (t) := f (t + 1) − f (t)
(2.7)
and (if b < c )
c
(2.8)
f (t)∆t =
b
c−1
X
f (t).
t=b
Lemma 2.1. [12, p. 259, Theorem 6.13] Let a, b ∈ T. For f, g ∈ Crd (T, R),
we have
b
1
b
1 b
p
q
|g(x)|q ∆x ,
|f (x)g(x)| ∆x 6
|f (x)|p ∆x
(2.9)
a
a
a
where p > 1 and
1
p
+
1
q
= 1.
3. Main results
Theorem
3.1. For τ ∈ [a, b] ∩ T, let p(x) ∈ Crd ([a, τ ], R) be nonnegative
τ
with a (p(x))1−u ∆x < ∞ (u > 1). Let the function q(x) be positive, bounded
and non-increasing on [a, τ ] ∩ T. Suppose that the functions f (x) and g(x)
are delta differentiable on [a, τ ]∩T and f (a) = g(a) = 0. Then, the following
inequality holds true:
τ
(3.1)
q σ (x)(f ∆ (x)g(x) + f (x)g ∆ (x))∆x
a
τ
2
2 τ
v ∆ v
v 1 u
v
∆
1−u
σ
2
6
(p(x)) ∆x
p(x)(q (x)) ( f (x) + g (x) )∆x
2 a
a
where
1
u
+
1
v
= 1 and 1 < v ≤ 2.
Proof. Let
(3.2)
F (x) =
x
p
x
p
q σ (s) f ∆ (s) ∆s, G(x) =
q σ (s) g ∆ (s) ∆s.
a
a
Then
(3.3)
F ∆ (x) =
p
p
q σ (x) f ∆ (x) > 0, G∆ (x) = q σ (x) g ∆ (x) > 0
and
(3.4)
F (x) >
x
p
p
q σ (x) f ∆ (s) ∆s = q σ (x) |f (x)|,
a
G(x) >
p
q σ (x) |g(x)| .
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Therefore, we have
(3.5)
τ
q σ (x)(f ∆ (x)g(x) + f (x)g ∆ (x))∆x
a
τ
6 (F ∆ (x)G(x) + F (x)G∆ (x))∆x
a
τ
6 (F ∆ (x)G(x) + F σ (x)G∆ (x))∆x = F (x)G(x) |τa
=
a
τ p
q σ (x) f ∆ (x) ∆x
a
τ
p
q σ (x) g ∆ (x) ∆x.
a
In the case when u > 1, by using AGM inequality and Hölder’s inequality,
we have
(3.6)
τ
τ p
p
∆ σ
q (x) f (x) ∆x
q σ (x) g ∆ (x) ∆x
a
a
τ
2 τ p
2 i
1 h p σ
6
q (x) f ∆ (x) ∆x +
q σ (x) g ∆ (x) ∆x
2 a
a
τ
h
2
p
1
1
1
=
(p(x))− v (p(x)) v q σ (x) f ∆ (x) ∆x
2 a
2 i
τ
1
1p
+
(p(x))− v (p(x)) v q σ (x) g ∆ (x) ∆x
a
6
τ
1 2
+
(p(x))1−u dx
2 h τ
u
a
τ
2
v
v v
p(x)(q σ (x)) 2 f ∆ (x) ∆x
a
2 i
v
v v
∆
σ
2
p(x)(q (x)) g (x) ∆x
a
6
τ
1 2
a
(p(x))1−u ∆x
2 τ
u
2
v
v v v
p(x)(q σ (x)) 2 (f ∆ (x) + g ∆ (x) )∆x
a
where we use the elementary inequality
(3.7)
am + bm 6 (a + b)m ,
(m > 1, a, b > 0).
Thus, by means of (3.5) and (3.6), we complete the proof of Theorem 3.1.
Remark 3.1. Taking T = R, p(x) = q(x) = 1, τ = b and u = v = 2,
Theorem 3.1 reduces to Theorem 1.4 of B. G. Pachpatte.
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Theorem 3.2. For τ ∈ [a, b] ∩ T, let p(x) ∈ Crd ([τ, b], R) be nonnegative
with bτ (p(x))1−u ∆x < ∞ (u > 1). Let the function q(x) be positive, bounded
and non-decreasing on [τ, b] ∩ T. Suppose that the functions f (x) and g(x)
are delta differentiable on [τ, b] ∩ T and f (b) = g(b) = 0. Then
(3.8)
b
q σ (x)(f ∆ (x)g(x) + f (x)g ∆ (x))∆x
τ
2 2
v v
v 1 u
v
(p(x))1−u ∆x
p(x)(q σ (x)) 2 (f ∆ (x) + g ∆ (x) )∆x .
2 τ
τ
b
6
b
Proof. Let
F (x) =
(3.9)
b p
q σ (s) f ∆ (s) ∆s, G(x) =
q σ (s) g ∆ (s) ∆s.
b p
x
x
Then
(3.10)
p
p
F ∆ (x) = − q σ (x) f ∆ (x) 6 0, G(x) = − q σ (x) g ∆ (x) 6 0
and
(3.11)
F (x) >
p
b p
q σ (x) f ∆ (s) ∆s = q σ (x) |f (x)|, G(x)
>
p
q σ (x) |g(x)| .
x
Therefore, we have
(3.12)
b
q σ (x)(f ∆ (x)g(x) + f (x)g ∆ (x))∆x
τ
b
6 − (F ∆ (x)G(x) + F (x)G∆ (x))∆x
τ
τ
6 (F ∆ (x)G(x) + F σ (x)G∆ (x))∆x = F (x)G(x) |τb .
b
The rest of the proof of Theorem 3.2 follows by suitable modifications of the
proof of Theorem 3.1. We omit the further details.
Theorem 3.3. For τ ∈ [a, b] ∩ T, let p(x) ∈ Crd ([a, b], R) be nonnegative
with τa (p(x))1−u ∆x < ∞ (u > 1) and bτ (p(x))1−u ∆x < ∞ (u > 1). Suppose
that the function q(x) is positive, bounded and non-increasing on [a, τ ] ∩ T
and non-decreasing on [τ, b] ∩ T. Suppose that the functions f (x) and g(x)
are delta differentiable on [a, b] ∩ T and f (a) = f (b) = g(a) = g(b) = 0.
Then
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b
(3.13)
651
q σ (x)(f ∆ (x)g(x) + f (x)g ∆ (x))∆x
a
b
v
v 2
v A
p(x)(q σ (x)) 2 (f ∆ (x) + g ∆ (x) )∆x) v
2 a
6
where
1
u
+
1
v
= 1, 1 < v 6 2 and A, τ are so constrained that
τ
(3.14)
1−u
(p(x))
2
b
2
u
u
1−u
∆x
= (p(x)) ∆x
= A.
a
τ
Proof. It is easily observed from the hypothesis of Theorem 3.3 that
(3.15)
b
q σ (x)(f ∆ (x)g(x) + f (x)g ∆ (x))∆x
a
τ
= q σ (x)(f ∆ (x)g(x) + f (x)g ∆ (x))∆x
a
b
+ q σ (x)(f ∆ (x)g(x) + f (x)g ∆ (x))∆x
τ
6
τ
2
v
v v A h v
p(x)(q σ (x)) 2 (f ∆ (x) + g ∆ (x) )∆x
2 a
+
b
2 i
v
v v v
p(x)(q σ (x)) 2 (f ∆ (x) + g ∆ (x) )∆x
τ
6
b
v
v 2
v A
p(x)(q σ (x)) 2 (f ∆ (x) + g ∆ (x) )∆x) v .
2 a
This completes the proof.
Theorem 3.4. For τ ∈ [a, b] ∩ T, p(x) ∈ Crd ([a, b], R) nonnegative with
τ
(3.16)
1
(p(x))− 2m+1 ∆x < ∞
a
and
b
1
(p(x))− 2m+1 ∆x < ∞.
τ
Let the functions f (x), g(x) and h(x) be delta differentiable on τ ∈ [a, b] ∩ T
and f (a) = f (b) = g(a) = g(b) = h(a) = h(b) = 0. Then
(3.17)
b
Ξ(f (x), g(x))∆x
a
6
b
2m+2 ∆ 2m+2 ∆ 2m+2
A p(x)(f ∆ (x)
+ g (x)
+ h (x)
)∆x
m+1a
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where m ≥ 0,
Ξ(f (x), g(x)) = |f (x)g(x)|m (f ∆ (x)g(x) + f (x)g ∆ (x))
+ |g(x)h(x)|m (g ∆ (x)h(x) + g(x)h∆ (x))
+ |h(x)f (x)|m (h∆ (x)f (x) + h(x)f ∆ (x))
(3.18)
and A, τ are so constrained that
τ
(3.19)
1
(p(x))− 2m+1 ∆x
2m+1
=
b
1
(p(x))− 2m+1 ∆x
2m+1
= A.
τ
a
Proof. Let
(3.20)
x
x
x
F (x) = f ∆ (s) ∆s, G(x) = g ∆ (s) ∆s, H(x) = h∆ (s) ∆s.
a
a
a
Then
(3.21) F ∆ (x) = f ∆ (x) > 0, G∆ (x) = g ∆ (x) > 0, H ∆ (x) = h∆ (x) > 0
and
F (x) > |f (x)| , G(x) > |g(x)| , H(x) > |h(x)| .
(3.22)
Therefore, we have
(3.23)
τ
Ξ(f (x), g(x))∆x 6
τ
[(F (x)G(x))m (F (x)G∆ (x) + F ∆ (x)G(x))
a
a
+ (G(x)H(x))m (G∆ (x)H(x) + G(x)H ∆ (x))
+ (H(x)F (x))m (H ∆ (x)F (x) + H(x)F ∆ (x))]∆x
τ
6 [(F (x)G(x))m (F (x)G(x))∆ + (G(x)H(x))m (G(x)H(x))∆
a
+ (H(x)F (x))m (F (x)H(x))∆ ]∆x
(F (τ )G(τ ))m+1 + (G(τ )H(τ ))m+1 + (H(τ )F (τ ))m+1
m+1
(F (τ ))2m+2 + (G(τ ))2m+2 + (H(τ ))2m+2
6
m+1
τ h
2m+2 τ 2m+2 τ 2m+2 i
1
∆
∆
=
f (x) ∆x
+
g (x) ∆x
+ h∆ (x) ∆x
m+1 a
a
a
6
where we apply the inequality
(3.24)
cd + de + ec 6 c2 + d2 + e2 .
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Some new generalizations of Maroni inequality on time scales
Let u =
2m+2
2m+1
653
and v = 2m + 2. By using Hölder inequality, we have
(3.25)
2m+2
2m+2 τ τ 2m+2 τ h∆ (x) ∆x
g ∆ (x) ∆x
f ∆ (x) ∆x
+
+
a
a
a
=
τ
1
1
(p(x))− 2m+1 (p(x)) 2m+1
2m+2
∆ f (x) ∆x
a
+
τ
2m+2
1
1
(p(x))− 2m+1 (p(x)) 2m+1 g ∆ (x) ∆x
a
+
τ
2m+2
1
1
(p(x))− 2m+1 (p(x)) 2m+1 h∆ (x) ∆x
a
6
τ
a
τ
1
(p(x))− 2m+1 ∆x
2m+1
2m+2 ∆ 2m+2 ∆ 2m+2
p(x)(f ∆ (x)
+ g (x)
+ h (x)
)∆x.
a
Considering (3.23) and (3.25), we have
(3.26)
τ
2m+1
1
1 (p(x))− 2m+1 ∆x
m+1 a
τ
Ξ(f (x), g(x))∆x 6
a
τ
2m+2 ∆ 2m+2 ∆ 2m+2
)∆x.
+ h (x)
+ g (x)
p(x)(f ∆ (x)
a
By following steps very similar to those of the proof above, we have
(3.27)
b
2m+1
1
1 (p(x))− 2m+1 ∆x
m+1 τ
b
Ξ(f (x), g(x))∆x 6
τ
b
2m+2 ∆ 2m+2 ∆ 2m+2
)∆x.
p(x)(f ∆ (x)
+ g (x)
+ h (x)
τ
Associating (3.26) with (3.27), we complete the proof.
Remark 3.2. Result of Theorem 3.4 is not only new for arbitrary time
scales, but also new for the continuous and the discrete cases.
Remark 3.3. If T = R, p(x) = 1, τ = a+b
2 and m = 0, Theorem 3.4 is a
particular case of Theorem 1 of B. G. Pachpatte [15].
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L. Yin
DEPARTMENT OF MATHEMATICS AND INFORMATION SCIENCE
BINZHOU UNIVERSITY
BINZHOU CITY, SHANDONG PROVINCE, 256603, CHINA
E-mail: [email protected]
C. J. Zhao
DEPARTMENT OF MATHEMATICS
CHINA JILIANG UNIVERSITY
HANGZHOU, 310000, P.R. CHINA
E-mail: [email protected]
Received January 19, 2012.
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