Corrigendum to “Generalizations of Opial-Type Inequalities in Several Independent Variables” Published in Demonstratio Math. 4(47) (2014), 324–335

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.

Full text

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 Dmnyiptqdt,
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˙Dmnyiptqdt
≤żb1
a1„p
ÿ
i“1
Difpu1pt1, b2, . . . , bmq, . . . , uppt1, b2, . . . , bmqq ˆ ż
Ω1
Dmnyiptqdt1dt1
≤żb1
a1„p
ÿ
i“1
Difpu1pt1, b2, . . . , bmq, . . . , uppt1, b2, . . . , bmqq
ˆpn´1q!m
pvolpΩqqn´1D1uipt1, b2...,bmqdt1
“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.