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DEMONSTRATIO MATHEMATICA

Vol. 49 No 2 2016

Maja Andrić, Ana Barbir, Josip Pečarić, Gholam Roqia

CORRIGENDUM TO “GENERALIZATIONS OF

OPIAL-TYPE INEQUALITIES IN SEVERAL

INDEPENDENT VARIABLES” PUBLISHED IN

DEMONSTRATIO MATH. 4(47) (2014), 324–335

Communicated by A. Fryszkowski

Abstract.

The purpose of this corrigendum is to correct an error in the earlier paper

by the authors: Generalizations of Opial-type inequalities in several independent variables,

Demonstratio Math.

In the paper “Generalizations of Opial-type inequalities in several inde-

pendent variables” published in Demonstratio Mathematica ([

1

]), we have

considered certain multidimensional Opial-type inequalities, and for two of

them, inequalities obtained in Theorem 2.1 (page 841) and Theorem 2.3

(page 844), we give corrigendum. Namely, the error was made in a ﬁnal

step of the proof of Theorem 2.1, in the equality (2.7). Here we made neces-

sary corrections, which resulted from the need to observe the inequality on

Ω

“śm

j“1raj, bjs

with boundary conditions only in

a“ pa1, . . . , amq

. Since

applied Theorem 2.1 was used in Theorem 2.3, we made appropriate changes

in Theorem 2.3, also.

Following notation is used:

Let Ω

“śm

j“1raj, bjs

and

volp

Ω

q “ śm

j“1pbj´ajq

. Let

t“ pt1, . . . , tmq

be a general point in Ω,Ω

t“śm

j“1raj, tjs

and

dt “dt1. . . dtm

. Further, let

Dupxq “ d

dxupxq, Dkupt1, . . . , tmq “ B

Btk

upt1, . . . , tmq

and

Dkupt1, . . . , tmq “ D1¨ ¨ ¨ Dkupt1, . . . , tmq,

2010 Mathematics Subject Classiﬁcation: Primary 26D10; Secondary 26B25.

Key words and phrases: Opial-type inequalities, Willett’s inequality, Rozanova’s

inequality, several independent variables.

DOI: 10.1515/dema-2016-0013

c

Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology

150 M. Andrić, A. Barbir, J. Pečarić, G. Roqia

1≤k≤m. Let Ω1“śm

j“2raj, bjsand dt1“dt2. . . , dtm. Let

Djl upt1, . . . , tmq “ Bjl

Btl

j. . . Btl

1

upt1, . . . , tmq,

1≤j≤m,1≤l≤n.

Also, by

Cmnp

Ω

q

we denote the space of all functions

u

on Ωwhich have

continuous derivatives Djlufor j“1, . . . , m and l“1, . . . , n.

Proofs of corrected theorems follow the same step as in [

1

], but ﬁnish

with the inequality using boundary conditions only in

a

. First, we give

corrigendum to [

1

, Theorem 2.1]. Notice that the equation (2.2) from [

1

] is

explained in more detail here.

Theorem 1.

Let

m, n, p PN

. Let

f

be a nonnegative and diﬀerentiable

function on

r

0

,8qp

, with

fp

0

,...,

0

q “

0. Further, for

i“

1

, . . . , p

let

xiPCmnp

Ω

q

be such that

Djl xiptq|tj“aj“

0, where

j“

1

, . . . , m

and

l“

0

, . . . , n ´

1. Also, let

Dif

,

i“

1

, . . . , p

, be nonnegative, continuous and

nondecreasing on r0,8qp. Then the following inequality holds

(1) ż

Ωˆp

ÿ

i“1

Difp|x1ptq| ,...,|xpptq|q |Dmnxiptq|˙dt

≤pn´1q!m

pvolpΩqqn´1fˆpvolpΩqqn´1

pn´1q!mż

Ω

|Dmnx1ptq|dt, . . . ,

pvolpΩqqn´1

pn´1q!mż

Ω

|Dmnxpptq| dt˙.

Proof.

We extend technique used in [

2

, Theorem 2.1] on a multidimensional

case. For continuous function

g

: Ω

ÑR

, we should deﬁne

y

: Ω

ÑR

such

that

(2) Dmnypx1, . . . , xmq “ Bmny

Bxn

m¨ ¨ ¨ Bxn

1

“gpx1, . . . , xmq

and

(3) ypx1, . . . , xmq “ 1

pn´1q!mż

Ωx

m

ź

j“1

pxj´tjqn´1gpt1, . . . , tmqdt1¨ ¨ ¨ dtm,

where Ωx“śm

j“1raj, xjs.

Deﬁne

(4) ypxq “ żx

a

dt1żt1

a

dt2¨ ¨ ¨ żtn´2

a

dtn´1żtn´1

a

gptnqdtn

Corrigendum to “Generalizations of Opial-type inequalities” 151

or, in diﬀerent notations

(5) ypxq “ ż

Ωx

dt1ż

Ωt1

dt2¨ ¨ ¨ ż

Ωtn´2

dtn´1ż

Ωtn´1

gptnqdtn,

where

a“ pa1, . . . , amq, x “ px1, . . . , xmq, ti“ pti

1, . . . , ti

mq, dti“dti

1¨ ¨ ¨ dti

m,

i“1, . . . , n and

Ωti“

m

ź

j“1

raj, ti

js,ΩtiĎΩti´1, i “1, . . . , n ´1.

Since gis a continuous function, (2) obviously follows.

Obviously, integrals on the right-hand side of

(4)

or

(5)

, can be written

as iterations of the integrals of the form

żxj

aj

dt1

jżt1

j

aj

dt2

j¨ ¨ ¨ żtn´2

j

aj

dtn´1

jżtn´1

j

ajr

gptn

jqdtn

j,

which are known (and easy to deduce by interchanging the order of integration)

to be equal to 1

pn´1q!żxj

aj

pxj´tn

jqn´1r

gptn

jqdtn

j,

j“1, . . . , m, from which (3) easily follows.

Let

(6) yiptq “ 1

pn´1q!mż

Ωt

m

ź

j“1

ptj´sjqn´1|Dmnxipsq| ds,

for tPΩ,i“1, . . . , p. Hence

Dmnyiptq“|Dmnxiptq| and yiptq≥|xiptq|.

It is easy to conclude that for each

l“

0

, . . . , n ´

1we have

Djl yiptq≥

0and

nondecreasing on Ω(

i“

1

, . . . , p

and

j“

1

, . . . , m

). From

Djl yiptq|tj“aj“

0

follows

yiptq≤pvolpΩqqn´1

pn´1q!mDmpn´1qyiptq, t PΩ.

Deﬁne

uiptq “ pvolpΩqqn´1

pn´1q!mDmpn´1qyiptq

for

tP

Ωand

i“

1

, . . . , p

. Since

Dif

are nonnegative, continuous and

152 M. Andrić, A. Barbir, J. Pečarić, G. Roqia

nondecreasing on r0,8qp, it follows

(7) ż

Ω„p

ÿ

i“1

Difp|x1ptq| ,...,|xpptq|q |Dmnxiptq|dt

≤ż

Ω„p

ÿ

i“1

Difpy1ptq, . . . , ypptqq Dmnyiptqdt,

and

ż

Ω«p

ÿ

i“1

Difpy1ptq, . . . , ypptqq Dmnyiptqﬀdt

≤ż

Ω„p

ÿ

i“1

DifˆpvolpΩqqn´1

pn´1q!mDmpn´1qy1ptq,...,

pvolpΩqqn´1

pn´1q!mDmpn´1qypptq˙Dmnyiptqdt

≤żb1

a1„p

ÿ

i“1

Difpu1pt1, b2, . . . , bmq, . . . , uppt1, b2, . . . , bmqq ˆ ż

Ω1

Dmnyiptqdt1dt1

≤żb1

a1„p

ÿ

i“1

Difpu1pt1, b2, . . . , bmq, . . . , uppt1, b2, . . . , bmqq

ˆpn´1q!m

pvolpΩqqn´1D1uipt1, b2...,bmqdt1

“pn´1q!m

pvolpΩqqn´1żb1

a1

d

dt1

rfpu1pt1, b2, . . . , bmq, . . . , uppt1, b2, . . . , bmqqs dt1

“pn´1q!m

pvolpΩqqn´1fpu1pb1, b2, . . . , bmq, . . . , uppb1, b2, . . . , bmqq

“pn´1q!m

pvolpΩqqn´1f

ˆˆpvolpΩqqn´1

pn´1q!mż

Ω

|Dmnx1ptq| dt, . . . , pvolpΩqqn´1

pn´1q!mż

Ω

|Dmnxpptq| dt˙.

Next comes a result for a convex function

f

. The proof follows the

same steps as in [

1

, Theorem 2.3], again with the diﬀerence of observing

the inequality on Ωwith boundary conditions only in

a

. We will use the

following lemma about convex function of several variables ([3, page 11]).

Lemma 1.

Suppose that

f

is deﬁned on the open convex set

UĂRn.

If

f

is

p

strictly

q

convex on

U

and the gradient vector

f1pxq

exists throughout

U

,

then f1is pstrictlyqincreasing on U.

Corrigendum to “Generalizations of Opial-type inequalities” 153

Theorem 2.

Let

m, n, p PN

. Let

f

be a convex and diﬀerentiable function

on r0,8qpwith fp0,...,0q “ 0. Further, for i“1, . . . , p let xiPCmnpΩqbe

such that

Djl xiptq|tj“aj“

0, where

j“

1

, . . . , m

and

l“

0

, . . . , n ´

1. Then

the following inequality holds

(8) ż

Ω´p

ÿ

i“1

Difp|x1ptq| ,...,|xpptq|q |Dmnxiptq|¯dt

≤pn´1q!m

pvolpΩqqnż

Ω

fˆpvolpΩqqn

pn´1q!m|Dmnx1ptq| ,...,

pvolpΩqqn

pn´1q!m|Dmnxpptq| ˙dt.

Proof.

As in the proof of the previous theorem, we obtain

p

1

q

with the

diﬀerence of applying Lemma 1in

p

7

q

since

f

is a convex function. Then,

from Jensen’s inequality for integrals (see for example [

3

, page 51]), we have

ż

Ω«p

ÿ

i“1

Difˆ|x1ptq| ,...,|xpptq|˙|Dmnxiptq|ﬀdt

≤pn´1q!m

pvolpΩqqn´1fˆpvolpΩqqn´1

pn´1q!mż

Ω

|Dmnx1ptq| dt, . . . ,

pvolpΩqqn´1

pn´1q!mż

Ω

|Dmnxpptq| dt˙

“pn´1q!m

pvolpΩqqn´1fˆ1

pvolpΩqq ż

Ω

pvolpΩqqn

pn´1q!m|Dmnx1ptq| dt, . . . ,

1

pvolpΩqq ż

Ω

pvolpΩqqn

pn´1q!m|Dmnxpptq| dt˙

≤pn´1q!m

pvolpΩqqnż

Ω

fˆpvolpΩqqn

pn´1q!m|Dmnx1ptq| ,..., pvolpΩqqn

pn´1q!m|Dmnxpptq|˙dt.

References

[1]

M. Andrić, A. Barbir, J. Pečarić, G. Roqia, Generalizations of Opial-type inequalities

in several independent variables, Demonstratio Math. 4(47) (2014), 324–335.

[2]

M. Andrić, A. Barbir, J. Pečarić, On Willett’s, Godunova-Levin’s and Rozanova’s

Opial-type inequalities with related Stolarsky type means, Math. Notes 96(6) (2014),

841–854.

154 M. Andrić, A. Barbir, J. Pečarić, G. Roqia

[3]

J. E. Pečarić, F. Proschan, Y. C. Tong, Convex Functions, Partial Orderings and

Statistical Applications, Academic Press, Inc., 1992.

M. Andrić, A. Barbir

FACULTY OF CIVIL ENGINEERING, ARCHITECTURE AND GEODESY

UNIVERSITY OF SPLIT

MATICE HRVATSKE 15

21000 SPLIT, CROATIA

E-mail: maja.andric@gradst.hr, ana.barbir@gradst.hr

J. Pečarić

FACULTY OF TEXTILE TECHNOLOGY

UNIVERSITY OF ZAGREB

PRILAZ BARUNA FILIPOVIĆA 28A

10000 ZAGREB, CROATIA

E-mail: pecaric@element.hr

G. Roqia

ABDUS SALAM SCHOOL OF MATHEMATICAL SCIENCES

68-B

NEW MUSLIM TOWN

LAHORE 54000, PAKISTAN

E-mail: rukiyya@gmail.com

Received July 20, 2015.

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