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Volume 2007, Article ID 38347, 15 pagesdoi:10.1155/2007/38347 Research Article On Opial-Type Integral Inequalities Wing-Sum Cheung and Chang-Jian Zhao Received 22 January 2007; Accepted

Volume 2007, Article ID 38347, 15 pages

doi:10.1155/2007/38347

Research Article

On Opial-Type Integral Inequalities

Wing-Sum Cheung and Chang-Jian Zhao

Received 22 January 2007; Accepted 4 April 2007

Recommended by Peter Yu Hin Pang

We establish some new Opial-type inequalities involving functions of two and many in-dependent variables Our results in special cases yield some of the recent results on Opial’s inequality and also provide new estimates on inequalities of this type

Copyright © 2007 W.-S Cheung and C.-J Zhao This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

In the year 1960, Opial [1] established the following integral inequality

Theorem 1.1 Suppose f ∈ C1[0,h] satisfies f (0) = f (h) = 0 and f (x) > 0 for all x ∈

(0,h) Then the integral inequality holds

h 0

f (x) f (x)dx ≤ h

4

h 0



f (x)2

where this constant h/4 is best possible.

Opial’s inequality and its generalizations, extensions, and discretizations play a fun-damental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations [2–6] The inequality (1.1) has received considerable attention and a large number of pa-pers dealing with new proofs, extensions, generalizations, variants, and discrete analogs

of Opial’s inequality have appeared in some literature [7–22] For an extensive survey on these inequalities, see [2,6] The main purpose of the present paper is to establish some new Opial-type inequalities involving functions of two and many independent variables Our results in special cases yield some of the recent results on Opial’s inequality and pro-vide some new estimates on such types of inequalities

2 Main results

Our main results are given in the following theorems

Theorem 2.1 Let u i(s,t), v i(s,t), i =1, ,n, be real-valued absolutely continuous func-tions defined on [a,b] ×[c,d] and a,b,c,d ∈[0,∞ ) with u i(s,c) = u i(a,t) = u i(a,c) = 0,

v i(s,c) = v i(a,t) = v i(a,c) = 0, i =1, ,n Let F, G be real-valued nonnegative continuous and nondecreasing functions on [0, ∞)n with F(0, ,0) = 0, G(0, ,0) = 0 such that all their partial derivatives ∂2F/∂ | u i |2, ∂F/∂ | u i | , ∂2G/∂ | v i |2, ∂G/∂ | v i | , i =1, ,n are nonneg-ative continuous and nondecreasing functions on [0, ∞)n Let ∂ | u i | /∂s, ∂ | u i | /∂t, ∂2| u i | /∂s∂t,

∂ | v i | /∂s, ∂ | v i | /∂t, ∂2| v i | /∂s∂t, i =1, ,n, be nonnegative continuous and nondecreasing functions on [a,b] ×[c,d] Then

b

a

d

c



Fu1(s,t), ,u n(s,t) n

i =1



∂2G

∂v i 2· ∂v i

∂t · ∂v i

∂s +

∂G

∂v i  · ∂ | v i |

∂s∂t

+Gv1(s,t), ,v n(s,t) n

i =1



∂2F

∂u i 2· ∂u i

∂t · ∂u i

∂s +

∂F

∂u i  · ∂u i

∂s∂t

+S(s,t) dsdt

≤ F

b

a

d

c



∂2u1

∂s∂t



dsdt, ,

b

a

d

c



∂2u n

∂s∂t



dsdt

· G

b

a

d

c



∂2v1

∂s∂t



dsdt, ,

b

a

d

c



∂2v n

∂s∂t



dsdt

,

(2.1)

where

S(s,t) =

n



i =1

∂F

∂u i∂u i

∂s ·

n



i =1

∂G

∂v i∂v i

∂t +

n



i =1

∂F

∂u i∂u i

∂t ·

n



i =1

∂G

∂v i∂v i

Proof From the hypotheses on u i(s,t), v i(s,t), i =1, ,n, we have

u i(s,t)  ≤s

a

t

c



∂2u i

∂σ∂τ(σ,τ)



dσdτ,

v i(s,t)  ≤s

a

t

c



 ∂2v i

∂σ∂τ(σ,τ)



dσdτ,

(2.3)

fors ∈[a,b], t ∈[c,d].

From (2.3) and in view of the hypotheses on all partial derivatives, and by letting

U i(s,t) =

s

a

t

c



∂2u i

∂σ∂τ(σ,τ)



dσdτ,

V i(s,t) =

s

a

t

c



∂2v i

∂σ∂τ(σ,τ)



dσdτ,

(2.4)

we obtain

b

a

d

c



Fu1(s,t), ,u n(s,t) n

i =1



∂2G

∂v i 2· ∂v i

∂t · ∂v i

∂s +

∂G

∂v i  · ∂2v i

∂s∂t

+Gv1(s,t), ,v n(s,t) n

i =1



∂2F

∂u i 2· ∂u i

∂t · ∂u i

∂s +

∂F

∂u i  · ∂2u i

∂s∂t

+

n



i =1

∂F

∂u i  · ∂u i

∂s ·n

i =1

∂G

∂v i  · ∂v i

∂t

+

n



i =1

∂F

∂u i  · ∂u i

∂t ·

n



i =1

∂G

∂v i  · ∂v i

∂s dsdt

≤

b

a

d

c



F

U1(s,t), ,U n(s,t)

·

n



i =1



∂2G

∂V i2· ∂V i

∂t · ∂V i

∂s +

∂G

∂V i · ∂2V i

∂s∂t

+G

V1(s,t), ,V n(s,t)

·n

i =1



∂2F

∂U i2· ∂U i

∂t · ∂U i

∂s +

∂F

∂U i · ∂2U i

∂s∂t

+

n



i =1

∂F

∂U i

∂U i

∂s ·n

i =1

∂G

∂V i

∂V i

∂t +

n



i =1

∂F

∂U i

∂U i

∂t ·n

i =1

∂G

∂V i

∂V i

∂s dsdt

=

b

a

d

c

∂2

∂s∂t

F

U1(s,t), ,U n(s,t)

· G

V1(s,t), ,V n(s,t)

dsdt

= F

U1(b,d), ,U n(b,d)

· G

V1(b,d), ,V n(b,d)

= F

b

a

d

c



∂2u1

∂s∂t



dsdt, ,

b

a

d

c



∂2u n

∂s∂t



dsdt

· G

b

a

d

c



∂2v1

∂s∂t



dsdt, ,

b

a

d

c



∂2v n

∂s∂t



dsdt

(2.5)

Remark 2.2 (i) Taking G =1 in inequality (2.1), and in view of

∂2G

∂v

i 2· ∂v

i

∂t · ∂v

i

∂s +

∂G

∂v

i∂v i

∂s∂t =0, S(s,t) =0, (2.6) fori =1, ,n, we have

b

a

d

c

n

i =1



∂2F

∂u i 2· ∂u i

∂t · ∂u i

∂s +

∂F

∂u i  · ∂2u i

∂s∂t dsdt

≤ F

b

a

d

c



∂2u1

∂s∂t



dsdt, ,

b

a

d

c



∂2u n

∂s∂t



dsdt

,

(2.7)

fori =1, ,n.

Letu i(s,t) reduce to u i(t), where i =1, ,n and with suitable modifications, then (2.7) becomes the following inequality:

b

a

n

i =1

F i u1(t), ,u

n(t)u 

i(t) dt ≤ Fb

a

u 

1(t)dt, ,b

a

u 

n(t)dt .

(2.8) This is a recent inequality which was given by Peˇcari´c and Brneti´c [18,19]

Takingn =1, inequality (2.7) reduces to

b

a

d

c



∂2F

∂ | u |2· ∂ | u |

∂t · ∂ | u |

∂s +

∂F

∂ | u | ·

∂2| u |

∂s∂t dsdt ≤ F

b

a

d

c





∂

2u

∂s∂t



dsdt

Letu(s,t) reduce to u(t) and with suitable modifications, then the above inequality

becomes the following inequality:

b

a F f (t)f (t)dt ≤ Fb

a

f (x)dt . (2.10)

This is an inequality which was given by Godunova and Levin [12]

(ii) TakingG = F and u i(s,t) = v i(s,t), i =1, ,n, in inequality (2.1), we have

b

a

d

c



Fu1(s,t), ,u

n(s,t) n

i =1



∂2G

∂u

i 2· ∂u

i

∂t · ∂u

i

∂s +

∂G

∂u

i∂u i

∂s∂t

+

n



i =1

∂F

∂u i

∂u i

∂s ·n

i =1

∂F

∂u i

∂u i

∂t dsdt

≤1

2· F2

b

a

d

c



∂2u1

∂s∂t



dsdt, ,

b

a

d

c



∂2u n

∂s∂t



dsdt

.

(2.11)

Takingn =1, (2.11) reduces to

b

a

d

c



Fu(s,t) ∂2G

∂ | u |2· ∂ | u |

∂t · ∂ | u |

∂s +

∂G

∂ | u |

∂ | u |

∂s∂t +

∂F

∂u

∂u

∂s

∂F

∂u

∂u

∂t dsdt

≤1

2· F2

b

a

d

c



∂s∂t ∂2udsdt

.

(2.12)

Let u(s,t) reduce to u(t) and with suitable modifications, then (2.12) becomes the following inequality:

b

a

Fu(t) · F u(t) · u (t)dt ≤1

2F2

b

a

u (t)dt . (2.13)

This is an inequality given by Pachpatte in [15]

Inequality (2.12) is also a similar form of the following inequality which was given by Yang [22]:

b1

a1

b2

a2



f

t1,t2  ∂2f

∂t1∂t2



dt1dt2≤



b1− a1 

b2− a2 

8

b1

a1

b2

a2



∂2f

∂t1∂t2



t1,t2

2

dt1dt2.

(2.14) (iii) Letu i(s,t) and v i(s,t) reduce to u i(s) and v i(s), respectively, and with suitable

mod-ifications (wherei =1, ,n), then inequality (2.1) changes to the following inequality:

b

a



Fu1(t), ,u n(t) n

i =1

G  iv1(t), ,v n(t)v 

i(t)

+Gv1(t), ,v n(t) n

i =1

F i u1(t), ,u n(t)u 

i(t) dt

≤ F

b

a

u 

1(t)dt, ,b

a

u 

n(t)dt · Gb

a

u 

1(t)dt, ,b

a

u 

n(t)dt .

(2.15) This is an inequality given by Agarwal and Pang in [2]

Takingn =1,G =1,F(u) = u2, (2.15) changes to

b

a

u(t)u (t)dt ≤1

2(b − a)

b

a

u (t) 2

This is another version of the Opial’s inequality, (see [13])

(iv) TakingG =1,F =(| u1|, , | u n |)=n

i =1f i(| u i |),i =1, ,n, in (2.1), (2.1) changes

to a general form of the inequality which was given by Pachpatte [16], where the functions

f imust satisfy some suitable conditions, (see [16])

Theorem 2.3 Let u i(s,t), v i(s,t), F, G, ∂2F/∂ | u i |2, ∂F/∂ | u i | , ∂ | u i | /∂s, ∂ | u i | /∂t, ∂2| u i | /∂s∂t,

∂2G/∂ | v i |2, ∂G/∂ | v i | , ∂ | v i | /∂s, ∂ | v i | /∂t, ∂2| v i | /∂s∂t, i =1, ,n, be as in Theorem 2.1 Let

p i(s,t), q i(s,t), i =1, ,n, be real-valued positive functions defined on [a,b] ×[c,d] satis-fying

b

a

d

c p i(s,t)dsdt =1,

b

a

d

c q i(s,t)dsdt =1 (i =1, ,n). (2.17)

Let h i , w i , i =1, ,n, be real-valued positive convex and increasing functions on (0, ∞)2 Then the following integral inequality holds:

b

a

d

c



Fu1(s,t), ,u n(s,t) n

i =1



∂2G

∂v i 2· ∂v i

∂t · ∂v i

∂s +

∂G

∂v i∂2v i

∂s∂t

+Gv1(s,t), ,v n(s,t) n

i =1



∂2F

∂u i 2· ∂u i

∂t · ∂u i

∂s +

∂F

∂u i∂2u i

∂s∂t

+S(s,t) dsdt

≤ F



h −1

b

a

d

c p1(s,t)h1

 ∂2u1/∂s∂t

p1(s,t) dsdt , ,

h −1

n

b

a

d

c p n(s,t)h n ∂2u n /∂s∂t

p n(s,t) dsdt

· G



w −1

b

a

d

c q1(s,t)w1

 ∂2v1/∂s∂t

q1(s,t) dsdt

, ,

w n −1

b

a

d

c q n(s,t)w n

 ∂2v n /∂s∂t

q n(s,t) dsdt ,

(2.18)

where

S(s,t) =

n



i =1

∂F

∂u i∂u i

∂s ·

n



i =1

∂G

∂v i∂v i

∂t +

n



i =1

∂F

∂u i∂u i

∂t ·

n



i =1

∂G

∂v i∂v i

∂s .

(2.19)

Proof From the hypotheses, we have

b

a

d

c



∂u2i

∂s∂t



dsdt =

b

a

d

c p i(s,t)∂u2

i /∂s∂tp i(s,t)dsdt

b

a

d

c p i(s,t)dsdt ,

b

a

d

c



∂v2i

∂s∂t



dsdt =

b

a

d

c q i(s,t)∂v2

i /∂s∂t |q i(s,t)dsdt

b

a

d

c q i(s,t)dsdt ,

(2.20)

fori =1, ,n.

From (2.20), the hypotheses onh i,w i,i =1, ,n, and in view of Jensen’s inequality,

we obtain

h i

b

a

d

c



∂u2i

∂s∂t



dsdt

≤

b

a

d

c p i(s,t) · h i ∂2u i /∂s∂t

p i(s,t) dsdt,

w i

b

a

d

c



∂v2i

∂s∂t



dsdt

≤

b

a

d

c q i(s,t) · w i

 ∂2v i /∂s∂t

q i(s,t) dsdt,

(2.21)

fori =1, ,n.

From (2.21), we observe that

b

a

d

c



∂u2i

∂s∂t



dsdt ≤ h(i) −1

b

a

d

c p i(s,t) · h i ∂2u i /∂s∂t

p i(s,t) dsdt

,

b

a

d

c



∂v2i

∂s∂t



dsdt ≤ w(i) −1

b

a

d

c q i(s,t) · w i

 ∂2v i /∂s∂t

q i(s,t) dsdt

,

(2.22)

fori =1, ,n.

From (2.22) and in view of inequality (2.1), we get inequality (2.18) and the proof is

Remark 2.4 (i) Taking G =1 in inequality (2.18), and in view of

∂2G

∂v i 2· ∂v i

∂t · ∂v i

∂s +

∂G

∂v i∂v i

fori =1, ,n, and

(2.18) becomes

b

a

d

c

n

i =1



∂2F

∂u i 2· ∂u i

∂t · ∂u i

∂s +

∂F

∂u i  · ∂2u i

∂s∂t dsdt

≤ F



h −1

b

a

d

c p1(s,t)h1

 ∂2u1/∂s∂t

p1(s,t) dsdt

, ,

h −1

n

b

a

d

c p n(s,t)h n

 ∂2u n /∂s∂t

p n(s,t) dsdt ,

(2.25)

fori =1, ,n.

Letu i(s,t), h i(s,t), and p i(s,t) change to f i(t), h i(t), and p i(t), respectively, where i =

1, ,n, then (2.25) reduces to the following inequality:

b

a

n

i =1

D i Ff1(t), ,f n(t)f 

i(t) dt

≤ F



h −11

b

a p1(t)h1

 f 

1(t)

p1(t) dt

, ,h − n1

b

a p n(t)h n

 f 

n(t)

p n(t) dt

, (2.26)

whereD i F is as in [18] This is an inequality given by Peˇcari´c in [18]

TakingF(x1, ,x n)=n

i =1F i(x i),i =1, ,n, (2.25) changes to a general form of the inequality which was given by Pachpatte [16] Takingn =1, (2.25) reduces to a general form of the inequality which was given by Godunova and Levin [12]

On the other hand, inequality (2.18) is also a general form of another inequality in Peˇcari´c and Brneti´c [20, Theorem 1]

(ii) TakingG = F and u i(s,t) = v i(s,t), i =1, ,n, in inequality (2.18), we have

b

a

d

c



Fu1(s,t), ,u

n(s,t) n

i =1



∂2G

∂u

i 2· ∂u

i

∂t · ∂u

i

∂s +

∂G

∂u

i∂u i

∂s∂t

+

n



i =1

∂F

∂u i

∂u i

∂s ·n

i =1

∂F

∂u i

∂u i

∂t dsdt

≤1

2· F2



h −1

b

a

d

c p1(s,t)h1

 ∂2u1/∂s∂t

p1(s,t) dsdt

, ,

h − n1

b

a

d

c p n(s,t)h n

 ∂2u n /∂s∂t

p n(s,t) dsdt .

(2.27) Takingn =1, (2.27) reduces to

b

a

d

c



Fu(s,t) ∂2G

∂ | u |2· ∂ | u |

∂t · ∂ | u |

∂s +

∂G

∂ | u |

∂ | u |

∂s∂t +

∂F

∂u

∂u

∂s

∂F

∂u

∂u

∂t dsdt

≤1

2· F2



h −1

b

a

d

c p(s,t)h ∂2u/∂s∂t

p(s,t) dsdt

.

(2.28)

This is a general form of the inequality which was given by Pachpatte [14]

(iii) Letu i(s,t), v i(s,t), h i(s,t), w i(s,t), p i(s,t), and q i(s,t) reduce to u i(t), v i(t), h i(t),

w i(t), p i(t), and q i(t), respectively, and with suitable modifications (where i =1, ,n),

then inequality (2.18) changes to the following inequality:

b

a



Fu1(t), ,u

n(t) n

i =1

G  iv1(t), ,v

n(t)v 

i(t)

+Gv1(t), ,v

n(t) n

i =1

F i u1(t), ,u

n(t)u 

i(t) dt

≤ F



h −1

b

a p1(t)h1

 u 

1(t)

p1(t) dt

, ,h −1

n

b

a p n(t)h n

 u 

n(t)

p n(t) dt

· G



w −1

b

a q1(t)w1

 v 

1(t)

q1(t) dt

, ,w n −1

b

a q n(t)w n

 v 

n(t)

q n(t) dt

.

(2.29) This is just an inequality given by Agarwal and Pang in [2]

Theorem 2.5 Let u i(s,t), v i(s,t), F, G, be as in Theorem 2.1 Let φ i , ψ i , i =1, ,n, be real-valued positive convex and increasing functions on (0, ∞)2 Let r i(s,t) ≥ 0, ∂2r i /∂s∂t > 0,

r i(s,c) = r i(a,t) = r i(a,c) = 0, ∂2e i /∂s∂t > 0, e i(s,c) = e i(a,t) = e i(a,c) = 0, i =1, ,n Let

∂2F/∂M2i , ∂F/∂M i , ∂2G/∂N2i , ∂G/∂N i , i =1, ,n, be nonnegative continuous and nonde-creasing functions on [0, ∞)n Let ∂M i /∂s, ∂M i /∂t, ∂2M i /∂s∂t, ∂N i ∂s, ∂N i /∂t, ∂2N i /∂s∂t,

i =1, ,n, be nonnegative continuous and nondecreasing functions on [a,b] ×[c,d] Then the following inequality holds:

b

a

d

c



F

M1(s,t), ,M n(s,t)

·

n



i =1



∂2G

∂N2i · ∂N i

∂t · ∂N i

∂s +

∂G

∂N i · ∂2N i

∂s∂t

+G

N1(s,t), ,N n(s,t)

·

n



i =1



∂2F

∂M2i · ∂M i

∂t · ∂M i

∂s +

∂F

∂M i · ∂2M i

∂s∂t

+S(s,t) dsdt

≤ F

b

a

d

c

∂2r1

∂s∂t · φ1

 ∂2u1/∂s∂t

∂2r1/∂s∂t dsdt, ,

b

a

d

c

∂2r n

∂s∂t · φ n∂2u n /∂s∂t

∂2r n /∂s∂t dsdt

· G

b

a

d

c

∂2e1

∂s∂t · ψ1

 ∂2v1/∂s∂t

∂2e1/∂s∂t dsdt, ,

b

a

d

c

∂2e n

∂s∂t · ψ n

 ∂2v n /∂s∂t

∂2e n /∂s∂t dsdt

, (2.30)

where

M i(s,t) = r i(s,t) · φ i u

i(s,t)

r i(s,t) ,

N i(s,t) = e i(s,t) · ψ i v

i(s,t)

e i(s,t) ,

(2.31)

for i =1, ,n, and

S(s,t) =

n



i =1

∂F

∂M i

∂M i

∂s ·

n



i =1

∂G

∂N i

∂N i

∂t +

n



i =1

∂F

∂M i

∂M i

∂t ·

n



i =1

∂G

∂N i

∂N i

for i =1, ,n.

Proof From the hypotheses on u i(s,t), v i(s,t), r i(s,t), e i(s,t), i =1, ,n, we have

u i(s,t)  ≤s

a

t

c



∂2u i

∂σ∂τ(σ,τ)



dσdτ,

v

i(s,t)  ≤s

a

t

c



 ∂2v i

∂σ∂τ(σ,τ)



dσdτ,

r i(s,t) =

s

a

t

c

∂2r i

∂σ∂τ(σ,τ)dσdτ,

e i(s,t) =

s

a

t

c

∂2e i

∂σ∂τ(σ,τ)dσdτ,

(2.33)

fors ∈[a,b], t ∈[c,d].

From (2.33) and using the hypotheses onφ i,ψ i,i =1, ,n, and Jensen’s inequality, we

have

M i(s,t) ≤

s

a

t

c

∂2r i

∂σ∂τ · φ i

 ∂2u i /∂σ∂τ

(σ,τ)



∂2r i /∂σ∂τ

(σ,τ) dσdτ,

N i(s,t) ≤

s

a

t

c

∂2e i

∂σ∂τ · ψ i ∂2v i /∂σ∂τ

(σ,τ)



∂2e i /∂σ∂τ

(σ,τ) dσdτ,

(2.34)

fors ∈[a,b], t ∈[c,d].

From (2.34), using the hopytheses on all partial derivatives and in view of

M i(s,t) =

s

a

t

c

∂2r i

∂σ∂τ · φ i

 ∂2u i /∂σ∂τ

∂2r i /∂σ∂τ dσdτ,

N i(s,t) =

s

a

t

c

∂2e i

∂σ∂τ · ψ i ∂2v i /∂σ∂τ

∂2e i /∂σ∂τ dσdτ,

(2.35)

we have

b

a

d

c



F

M1(s,t), ,M n(s,t)

·n

i =1



∂2G

∂N2i · ∂N i

∂t · ∂N i

∂s +

∂G

∂N i · ∂2N i

∂s∂t

+G

N1(s,t), ,N n(s,t)

·n

i =1



∂2F

∂M2i · ∂M i

∂t · ∂M i

∂s +

∂F

∂M i · ∂2M i

∂s∂t

+S(s,t) dsdt

≤

b

a

d

c



F

M1(s,t), ,M n(s,t)

·n

i =1



∂2G

∂N2

i

· ∂N i

∂t · ∂N i

∂s +

∂G

∂N i · ∂2N i

∂s∂t

+G

N1(s,t), ,N n(s,t)

·

n



i =1



∂2F

∂M2

i

· ∂M i

∂t · ∂M i

∂s +

∂F

∂M i · ∂2M i

∂s∂t

+

n



i =1

∂F

∂M i

∂M i

∂s ·

n



i =1

∂G

∂N i

∂N i

∂t +

n



i =1

∂F

∂M i

∂M i

∂t ·

n



i =1

∂G

∂N i

∂N i

∂s dsdt

=

b

a

d

c

∂2

∂s∂t



F

M1(s,t), ,M n(s,t)

· G

N1(s,t), ,N n(s,t)

dsdt

= F

M1(b,d), ,M n(b,d)

· G

N1(b,d), ,N n(b,d)

= F

b

a

d

c

∂2r1

∂s∂t · φ1

 ∂2u1/∂s∂t

∂2r1/∂s∂t dsdt, ,

b

a

d

c

∂2r n

∂s∂t · φ n

 ∂2u n /∂s∂t

∂2r n /∂s∂t dsdt

· G

b

a

d

c

∂2e1

∂s∂t · ψ1

 ∂2v1/∂s∂t

∂2e1/∂s∂t dsdt, ,

b

a

d

c

∂2e n

∂s∂t · ψ n ∂2v n /∂s∂t

∂2e n /∂s∂t dsdt

.

(2.36)

Remark 2.6 (i) Taking n =1, (2.30) changes to a general form of the inequality which was given by Pachpatte [17]

(ii) TakingG =1, (2.30) changes to a general form of the inequality which was given

by Peˇcari´c and Brneti´c [19]

(iii) Takingn =1,G =1, (2.30) changes to the following inequality:

b

a

d

c



∂2F

∂M2· ∂M

∂t · ∂M

∂s +

∂F

∂M · ∂2M

∂s∂t dsdt ≤ F

b

a

d

c

∂2r

∂s∂t · φ ∂2u/∂s∂t

∂2r/∂s∂t dsdt

, (2.37) which is a general form of the follwing inequality established by Rozanova [21]:

b

a F 



r(t)φ f (t)

r(t) r

(t)φ f (t)

r (t) dt ≤ F

b

a r (t)φ f (x)

r (t) dt . (2.38)

f imust satisfy some suitable conditions, (see [16])

Theorem...

∂2G... =1, ,n.

From (2.20), the hypotheses on< i>h i,w i,i