Content uploaded by Malik Saad AlMuhja
Author content
All content in this area was uploaded by Malik Saad AlMuhja on Aug 17, 2020
Content may be subject to copyright.
On Extend the Domain of (Co)convex Polynomial
Malik Saad AlMuhja1,2,* Amer Himza Almyaly1
1 Department of Mathematics and Computer Application, College of Sciences, University of AlMuthanna, Samawa 66001, Iraq
2 Department of Mathematics and statistics, School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia, 06010 Sintok, Kedah,
Malaysia
Abstract. We will use different way (in this work) from the existing methods in the literature
which speaking in the separation of convex sets was carried out by hyperplanes. We are
examining the behavior of convex set which is the domain of convex and coconvex polynomial.
We simplify this term as (co)convex polynomial herein.
The main goal of the present work is:
What happens if
!
is a domain of (co)convex polynomial of
"
#
$
%#
&'
%,
()*
and
+,!
? Is
+
inflection point at
!
?
Keyword. (Co)convex polynomial, Convex set, Inflection point.
2010 Math. Sub. Classification. 41A10, 52A30, 52A27
I. INTRODUCTION
The domain of polynomial is beneficial to provide best approximation to a given function
through an approach which stipulates “
.
is convex function if
/01
#
.
% is a convex subset of
2345
”. Subsequently, the expansion of the domain of polynomials is necessary for the
assimilation of more than the characteristics of the convex sets such as supporting
hyperplane, strongly
6
hyperplane and separation optimization theorems (see [6, 8, 13]).
The last literature that discussed about best approximation in covex sets involving discrete
sets were found in 1979. Ref. [12, 15] introduced the characteristics of approximation to be
used in describing second separation theorem. All these characteristics were limited to the
approximation theory from an element to a convex set and best approximations by elements
of convex sets. But they have been yet to explore that approach in developing approximation
which stipulates “complex functions can be approximated by the simpler ones”. The new
results concept will offer a broader scope for discussion of results in approximation theory,
such as an extend the description of separation theorems.
Next, let
7
be a vector space that has topology
8
, then
7
is locally convex space if any point
has a neighborhood base consisting of convex sets (see, [11]). Assumption
.
is continuous
convex function from vector space
7
over field
9
onto that field
9
. Also, the element
:;,<
and
.
#
:;
%
=1>?
@
.
#
<
%A
B#C%
Now, we suppose
D
denote the set of all the functions
.
on
7
.
In 1979, Singer [15] proved for any convex subset
<
of
7
, (
<E7
) satisfying
1>?
@
.
#
7
%A
F1>?
@
.
#
<
%A
G#H%
* Corresponding author’s email: dr.almuhja@hotmail.com & malik@mu.edu.iq
where
7
is locally convex space. Furthermore, he found
1>?
@
.
#
<
%A
=IJ0
KL1>?
ML.
#
N
%

K5=
O
.5,DPIJ0
@
.5
#
<
%A
Q.5
#
+
%
G.5E*
R

M5=
O
N,7P.5
#
N
%
=IJ0
@
.5
#
<
%AR

and
1>?
@
.
#
<
%A
=IJ0
KS1>?
MS.
#
N
%

K$=
O
.$,DPIJ0
@
.$
#
<
%A
Q.$
#
+
%
G.$E*
R

M$=
O
N,7P.$
#
N
%
=IJ0
@
.$
#
<
%AR

where
+,7
and
.
#
+
%
F1>?
@
.
#
<
%A
B#T%
He further proposed some results, if
.
is finite, then (3) is valid.
Theorem A. [15] Let
7G.G<
be defined in above, and satisfying (2), and
+
be any element of
7
. If
+
is satisfying (3), and
:;,<
satisfying (1) iff there exists
.
U
,D
,
.
U
E*
, such that
.
U#
:;
%
=(VW
@
.
U#
<
%A
Q.
U#
+
%, and
.
#
:;
%
=XY.
Z,[.
#
N
%, where
.
U#
N
%
=.
U#
:;
%.
We will adopt the following concepts in this work.
Definition 1. [10] A subset
7
of
23
is convex set if \
+GN
]
^7
, whenever
+GN,7
.
Equivalently,
7
is convex if
#
C_`
%
+a`N,7
, for all
+GN,7
and
*F`FC
.
Definition 2. [5] The set b#
+Gc
%
,7d2P7^23Gc,2Gc).
#
+
%e is called the epigraph of
.
and denoted by
/01
#
.
%. We define the function
.
on
7
be a convex function on
7
if
/01
#
.
%
is convex subset of
2345
.
Remark 3. [14] A function
.P23f
\
_gGg
] is convex iff
.
h#
C_`
%
+a`N
i
Q
#
C_`
%
ja`k
,
*F`FC
,
whenever
.
#
+
%
Fj
and
.
#
N
%
Fk
.
Theorem 4. [6] Let
Y,
b
HGT
e and
.,l$3
\
_CGC
] be #
HY_C
%convex. Then
*Q
m
.
#
n
%
5
o5 _p3
#
.
%
Qq345
\
.
]
_
m
.
#
n
%
5
o5
.
The following result immediately of strongly
6
convex (see [3]). In 2016, Lara et al. [8,
Corollary 5] proposed a function called
r
strongly
6
convex such that
:
#
+
%
_rQs
#
+
%
Q:
#
+
%,
+,t
.
Let
u3
be the space of all algebraic polynomials of degree
QY_C
, and
"
#
$
%#
&'
% be the
collection of all functions
.,l
\
_CGC
] that change convexity at the points of the set
&'
, and
are convex in \
N'GC
]. The degree of best uniform coconvex polynomial approximation of
?
is
defined by
v3
#
$
%#
.G&'
%
=1>?
wx,yxz"
#
S
%#
{
%}
._W3
}
where
&'=
b
N~
e
~•5
'
such that
N€=_CFN5F•FN'FC=N'45
(see [7]).
If
&'=‚
, then
v3
#
$
%#
.G‚
%
=v3
#
$
%#
.
% which is usually referred to as the degree of best uniform
convex polynomial approximation (see [9]).
Definition 5. [4] The weighted DitzianTotik moduli of smoothness (DTMS) of
.,ƒw
\
_CGC
],
when
*FWQg
, is defined by
„…G†
‡
#
.Gn
%
w=IJ0
€ˆ‰Š‹
Œ
•
#
+
%
†"‰‡
…
#
.G+
%Œ
w
where
•
#
+
%
=
Ž
C_+$
. If
•=*
, then
„…
‡
#
.Gn
%
w=„…G€
‡
#
.Gn
%
w=IJ0
€ˆ‰Š‹
Œ
"‰‡
…
#
.G+
%Œ
w
is the usual DTMS. Also, note that
„€G†
‡
#
.Gn
%
w=
}
•†.
}
w
.
II. THE MAIN RESULTS
In this section, we will discuss the Domain of Convex Polynomial (DCP). Let
7^2
, then
Definition 1. A domain
!
of convex polynomial
W3
of
"
#
$
% is a subset of
7
and
7^2
,
satisfying the following properties:
1)
!,•‘
, where
•‘=
b
!P!1I’“”•0’“–IJ—I/–”?7
e
is the class of all domain of convex polynomial,
2) there is the point
n,7˜!
, such that
™
W3
#
n
%™
šIJ0
b™
W3
#
+
%™
P+,!
e, and
3) there is the function
.
of
"
#
$
%, such that
}
._W3
}
Q›
3S„$G$
‡
œ
.••G5
$
ž.
Let
!
and
7
be as in Definition 1.
Definition 2. If the compact set
Ÿ
is convex, so there is bounded neighborhood set
t=
b
,7P
™
™
$F¡
e for
¡
suitably near.
Theorem 3. If
!
is DCP of
W3
, and if
+;,!
. Then there is a compact neighborhood
¢
of the
point
+;
.
Proof. Suppose that
!
is DCP, from Definition 1, then
!
is compact subset (CS) of
7
, and
!,
•‘
.
Then
!
is compact and convex subset of
7
.
From Definition 2, there is bounded neighborhood
¢
of the point
+;
, such that
¢=
b
+,!P
™
+
™
$F¡
e, for
¡
suitably near,
and
¢^!
.
Since
+;,!
, and ™
+;
™
$F¡
, for
¡
suitably near. Then,
¢=!
. Therefore,
¢
is compact
neighborhood (CNE) of the point
+;
, and
¢
DCP of
W3
.
Corollary 4. If
+;,!
is DCP of
W3
. Then
!
is CNE of the point
+;
.
Proof. Clear.
Theorem 5. If
W3P!f!
is convex polynomial of
"
#
$
%, and
¢
is CS of
!
. Then
¢
is DCP of
W3
if and only if
W3
o5
#
¢
% is DCP of
W3
.
Proof. Suppose that
¢
is CS of
!
.
Case I. Suppose that
¢
is DCP of
W3
.
Since
W3P!f!
, and
¢^!^7
, then
¢
is CS of
7
, and
¢,•‘
.
Therefore,
W3
is continuous and
W3
o5
#
¢
%
=!
is CS of
7
.
Let
n£W3
o5
#
¢
%, then
n,7˜W3
o5
#
¢
%. From Definition 1, we have
™
W3
#
n
%™
š(VW
b™
W3
#
+
%™
P+,!
e, and
the function
.,"
#
$
%, such that
}
._W3
}
Q¤
3S„$G$
‡
œ
.••G5
$
ž.
Therefore,
W3
o5
#
¢
% is DCP of
W3
.
Case II. Suppose that
W3
o5
#
¢
% is DCP of
W3
.
Since
¢
is CS of
!
.
Let
N£!
, this is,
N,7˜!
, implies
N,7˜¢
. From Definition 1, we have
™
W3
#
N
%™
šIJ0
b™
W3
#
+
%™
P+,!
e,
then,
™
W3
#
N
%™
šIJ0
b™
W3
#
+
%™
P+,¢
e, and
the function
.,"
#
$
%, such that
}
._W3
}
Q›L
3S„$G$
‡
œ
.••G5
$
ž.
Therefore,
¢
is DCP of
W3
.
Theorem 6. If
!
is DCP of
W3
, and
¥^!
is DCP of
W3
. For every convex function
.
of
"
#
$
%,
defined on a neighborhood of
!
, then the set
¥¦.o5
#
*
% is DCP.
Proof. Suppose that
¥^!
is DCP of
W3
.
Let
+;,¥
, then from Theorem 3, there is CNE
¢
of the point
+;
.
If
.,"
#
$
%, such that
.
define on
¢
. From Theorem 5, then,
.o5
#
¢
% is DCP of
W3
.
Assume
+;=*
, then
¥¦.o5
#
*
% is DCP of
W3
.
Now, we will define the Domain of Coconvex Polynomial (DCCP).
Definition 7. A domain
!
of coconvex polynomial
W3
of
"
#
$
%#
&'
% is a subset of
7
and
7^2
,
satisfying the following properties:
1)
!,•‘
#
&'
%, where
•‘
#
&'
%
=
§
!P!1I’“”•0’“–IJ—I/–”?7G
’>¨W3“©’>ª/I“”>«/¬1–’–!
®
is the class of all domain of coconvex polynomial,
2)
N~
's are inflection points, such that
™
W3
#
N~
%™
Q5
$
,
X=CG¯G(
, and
3) there is the function
.
of
"
#
$
%#
&'
%, such that
}
._W3
}
Q›
3S„…G$
‡
œ
.••G5
3
ž.
Let
!
and
7
be as in Definition 7.
Theorem 8. If
W3P!f!
is coconvex polynomial of
"
#
$
%#
&'
% and
!
is DCCP of
W3
. Then
¢
is
DCCP of
W3
, if
¢
is CNE of the point
+;
, where
W3
#
+;
%
=5
$
.
Proof. Suppose that
W3P!f!
is coconvex polynomial of
"
#
$
%#
&'
%, such that
!
is CS of
7
,
and
W3
changes convexity at
!
.
Put
¢
is CNE of
+;
, implies
+;,¢
.
Since
W3
#
+;
%
=5
$
, and
!
is DCCP of
W3
. From Definition 7, then:
Case I. Either
+;
is inflection point at
!
.
Therefore,
+;,!
, and
¢^!
.
Since,
W3
#
+;
%
=5
$
. Then,
+;
is inflection point at
¢
.
Case II. Or
+;
is not inflection point at
!
.
Now, we must prove that
W3
changes convexity at
¢
. Let
CQ(Fg
,
N'o5GN',!
,
N'o5GN'
be
inflection points at
!
, such that
W3
#
N'o5
%
QW3
#
+;
%
QW3
#
N'
%. Since
W3
#
+;
%
=5
$
, and
N'
is
inflection points at
!
, implies
W3
#
+;
%
=W3
#
N'
%. This is contradiction.
Therefore,
+;
is inflection points at
¢
, and
¢^!
.
Thus,
W3
changes convexity at
¢
.
To prove
¢
have all inflection points
Q5
$
, let
N°
be inflection point at
¢
, such that
±F(
, and
²
W3
h
N°
i²
š5
$
. We get contradiction.
Since
¢^!
, then
.,"
#
$
%#
&'
%, such that
}
._W3
}
Q›L
3S„…G$
‡
œ
.••G5
3
ž.
Thus,
¢
is DCCP of
W3
.
Definition 9.
³

´
is said to be supporting hyperplane to domain of (co)convex polynomial
W3
if at least one point
+
µ
;
of
!
lies in
³

´
, and
W3
#
N
%
)j
µ, for all
N,!_
b
+
µ
;
e, and
j
µ
,2
.
Definition 10. If
!
is domain of (co)convex polynomial of
W3
,
+£!
.
³

´
is said to be strictly
separates
!
, if we choose
¶,2
such that
IJ0
b
W3
#
N
%
PN,!
e
F¶FW3
#
+
%.
Definition 11. If
6P
\
*GC
]
f2
is given function,
!5
and
!$
are domains of (co)convex
polynomials of
W3
and
·3
respectively.
³

´
and
³6

´
are said to be strongly hyperplane and
strongly
6
hyperplane respectively, if and only if
1>?
b
W3
#
¸
%
P¸,!5
e
šIJ0
b
·3
#
¹
%
P¹,!$
e
and
1>?
b
6
#
n
%
W3
#
¸
%
P¸,!5
e
šIJ0
b
6
#
n
%
·3
#
¹
%
P¹,!$
e,
where
n,
\
*GC
].
II. CONCLUSIONS
Example 1. Let
Y=T
,
WºP!f
#
_gGg
% be polynomial of degree
QY_C
, such that
!=
\
_TGT
] and
Wº
#
+
%
=*B»+$_+
.
1) Suppose that
+=T
,
N=_T
and
`=*B¼
(
*F`FC
). Then,
Wº
#
+=T
%
=CB»
and
Wº
#
N=_T
%
=½B»
.
Now,
Wº
h#
C_`
%
+a`N
i
=Wº
#
_*B¼
%
=_*B½¾
,
also, #
C_`
%
Wº
#
T
%
a`Wº
#
_T
%
=
#
*B¿
%
d
#
CB»
%
a
#
*B¼
%
d
#
½B»
%
=»BC
.
Therefore,
Wº
h#
C_`
%
d
#
T
%
a`d
#
_T
%i
Q
#
C_`
%
Wº
#
T
%
a`Wº
#
_T
%.
Then,
Wº
is convex polynomial, and
!,•‘
.
2) Let
n=¼,2˜!
, then
Wº
#
n=¼
%
=CH
,
and
IJ0
b™
Wº
#
+
%™
P+,!
e
=
™
Wº
#
+=_T
%™
=½B»
.
3) Let
.P!f
#
_gGg
% such that
.
#
+
%
=
À
5
$+Á_
#
+_C
%
º_H+$Â1?*Q+QT
+Â1?_TQ+F*
,
.•
#
+
%
=
§
H+º_T
#
+_C
%
$_¿+Â1?*Q+QT
CÂ1?_TQ+F*
and
.••
#
+
%
=
§
¼+$_¼+aHÂ1?*Q+QT
*Â1?_TQ+F*
.
Let
+;=C
,
N;=H
and
`=*B»
(
*F`FC
). Then,
.
#
+;=C
%
=_CB»
and
.
#
N;=H
%
=_C
.
So,
.
h#
C_`
%
d
#
C
%
a`
#
H
%i
=.
#
CB»
%
=_HB*ÃT
,
also, #
C_`
%
.
#
C
%
a`.
#
H
%
=_CBH»
.
Therefore,
.
h#
C_`
%
d
#
C
%
a`
#
H
%i
Q
#
C_`
%
.
#
C
%
a`.
#
H
%.
Hence,
.
is convex function, and it has
.••
.
Now, }
.
#
T
%
_Wº
#
T
%}
=
Äœ
5
$+Á_
#
+_C
%
º_H+$
ž
_
#
*B»+$_+
%Ä
=CT
,
and
"
#
€BÁ
%
‡
$
#
.ÅÅG+
%
=
Æ œ
H
X
ž#
_C
%
$o~.ÅÅ
@
+_$d
#
€BÁ
%
$aXd
#
*B¿
%A
$
~•€ =CBÃH
Therefore,
„$G$
‡
@
.ÅÅ
#
+
%
=¼+$_¼+aHGC
H
A
=IJ0
€ˆ‰Š5
$
}#
C_+$
%
d"‰
$
#
.ÅÅG+
%}
=
™#
_¾
%
d
#
CBÃH
%™
=C»BT¼
.
Thus, }
._Wº
}
Q›L
Ç„$G$
‡
œ
.ÅÅ
#
+
%
=¼+$_¼+aHG5
$
ž,
where
È5=½B¼H
.
Example 2. Let
Y=»
,
WÉP!f
#
_gGg
% be polynomial of degree
QY_C
, such that
!=
\
_TGT
] and
WÉ
#
+
%
=
#
+aH
%#
+aC
%#
+_C
%#
+_H
% .
1) Suppose that
+=CB»
,
N=C
and
`=*B»
(
*F`FC
). Then,
WÉ
#
+=CB»
%
=_HBC¾½»
and
WÉ
#
N=C
%
=*
.
Now,
WÉ
h#
C_`
%
+a`N
i
=WÉ
#
CBH»
%
=_CBT½
,
also, #
C_`
%
WÉ
#
CB»
%
a`WÉ
#
C
%
=
#
*B»
%
d
#
_HBC¾½»
%
a
#
*B»
%
d
#
*
%
=_CB*Ã
.
Therefore,
WÉ
h#
C_`
%
d
#
CB»
%
a`
#
C
%i
Q
#
C_`
%
WÉ
#
CB»
%
a`WÉ
#
C
%.
Then,
WÉ
is changes convexity at
!,•‘
.
2) Let
&'=
b
N~
e
~•5
'•Á
such that
N€=_TFN5=_HFN$=_CFNº=CFNÁ=HFN'45 =
T
and are convex in \
NÁGT
]. Then,
™
WÉ
#
N~
%™
=*Q5
$
,
X=CG¯G¿
,
3) Let
.P!f
#
_gGg
% such that
.
#
+
%
=
§™
+$_¿
™
a+Â1?_TQ+Q*
™
H+_¿
™
_+Â1?*F+QT
,
.•
#
+
%
=
Ê
Ë
Ì
H+º_¾+
™
+$_¿
™
aCÂ1?_TQ+Q*
¿+_¾
™
H+_¿
™
_CÂ1?*F+QT
and
.••
#
+
%
=
Íh²
ÎSoÁ
²i
Sd
#
ÏÎoÐ
%
o
h
$ÎÑoÐÎ
i
S
#™
ÎSoÁ
™%
ÑÂ1?_TQ+Q*
*Â1?*F+QT
.
Let
+;=*
,
N;=*B»
and
`=*B»
(
*F`FC
). Then,
.
#
+;=*
%
=¿
and
.
#
N;=*B»
%
=HB»
.
So,
.
h#
C_`
%
d
#
*
%
a`
#
HB»
%i
=.
#
CBH»
%
=*BH»
,
also, #
C_`
%
.
#
*
%
a`.
#
*B»
%
=TBH»
.
Therefore,
.
h#
C_`
%
d
#
*
%
a`
#
*B»
%i
Q
#
C_`
%
.
#
*
%
a`.
#
*B»
%.
Hence,
.
is changes convexity at
!
, and it has
.••
.
Now, }
.
#
_T
%
_WÉ
#
_T
%}
=
}#™
+$_¿
™
a+
%
_
#
+Á_»+$a¿
%}
=T¾
,
and
"
#
€B5
%
‡
Á
#
.ÅÅG+
%
=
Æ œ
¿
X
ž#
_C
%
Áo~.ÅÅ
@
+_Ád
#
€B5
%
$aXd
#
*BC
%A
Á
~•€ =CH¿B¼½¾
Therefore,
„ÁG$
‡
Ò
.ÅÅ
#
+
%
=
#™
+$_¿
™%
$d
#
¼+_¾
%
_
#
H+º_¾+
%
$
#™
+$_¿
™%
ºGC
»
Ó
=IJ0
€ˆ‰ŠL
S
Œ#
C_+$
%
d"
#
€B5
%
Á
#
.ÅÅG+
%Œ
=
™#
_¾
%
d
#
CH¿B¼½¾
%™
=ÃÃ½B¿H¿
.
Thus, }
._WÉ
}
Q›S
$É„ÁG$
‡
@
.ÅÅ
#
+
%
=
h²
ÎSoÁ
²i
Sd
#
ÏÎoÐ
%
o
h
$ÎÑoÐÎ
i
S
#™
ÎSoÁ
™%
ÑG5
É
A,
where
È$=*BÃ»T
.
These results (Definitions 1, 7) are able to answer the question above. Also, it's a possibility
of supporting the separation hyperplane theorem later by using DCP (see [1], [2]) like:
If
W3
and
·3
are two convex polynomials of
"
#
$
%. If
!wx
is a nonempty compact (and
!Ôx
is a
nonempty closed), such that
!wx
and
!Ôx
are disjoint. Are
W3
and
·3
strongly separated by
a hyperplane?
Funding
The research did not receive specific funding yet. The research was performed as part of the
employment by the authors.
Acknowledgements
The authors are indebted to administrative and technical support by University of Al
Muthanna.
Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this
paper.
REFERENCES
[1] M. AlMuhja, M. Misiran, and Z. Omar, “The development of approximation theory and some proposed
applications”, International Journal of Engineering and Technology (UAE), vol. 8, pp. 90–94, 2019.
[2] M. AlMuhja, “On hyperplane type convex polynomial and its applications: Approximation”, (preprint).
[3] H. Angulo, J. Gimenez, A. Moros, and K. Nikodem, “On strongly hconvex functions”, Annals of Functional
Analysis, vol. 2, pp. 85–91, 2011.
[4] R. DeVore and G. Lorentz, Constructive approximation. New York: A Series of Comprehensive Studies in
Mathematics 303, SpringerVerlag, 1993.
[5] T. Hoang, Convex analysis and global optimization: Springer optimization and its applications: 110. New
York: Springer International Publishing Switzerland, 2016.
[6] A. Komisarski and S. Wąsowicz, “Inequalities between remainders of quadratures”, Aequationes
mathematicae, vol. 6, pp. 1103–1114, 2017.
[7] K. Kopotun, D. Leviatan, and I. Shevchuk, “The degree of coconvex polynomial approximation”, Proceedings
of the American Mathematical Society, vol. 127, pp. 409–415, 1999.
[8] T. Lara, N. Merentes, and K. Nikodem, “Strong hconvexity and separation theorems”, International Journal
of Analysis, vol. 2016(Article ID 7160348), pp. 1–5, 2016.
[9] D. Leviatan, “Pointwise estimates for convex polynomial approximation”, Proceedings of the American
Mathematical Society, vol. 98, pp. 471–474, 1986.
[10] B. Mordukhovich and N. Nam, An easy path to convex analysis and applications. Wiliston: Morgan and
Claypool Publishers, 2014.
[11] Osborne, M. (2014). Locally convex spaces. New York, NY: Springer International Publishing Switzerland.
[12] P. Papini and I. Singer, “Best coapproximation in normed linear spaces”, Monatshefte Mathematik, vol. 88,
pp. 27–44, 1979.
[13] L. Pellegrini, “On generalized constrained optimization and separation theorems”, Taiwanese Journal of
Mathematics, vol. 15, pp. 659–671, 2011.
[14] R. Rockafellar, Convex analysis. New Jersey: Princeton Univ. Press, Princeton, 1970.
[15] I. Singer, “Generalizations of methods of best approximation to convex optimization in locally convex
spaces. II: Hyperplane theorems”, Journal of Mahtematical Analysis and Applications, vol. 69, pp. 571–584,
1979.