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## 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.
On Extend the Domain of (Co)convex Polynomial
Malik Saad Al-Muhja1,2,* Amer Himza Almyaly1
1 Department of Mathematics and Computer Application, College of Sciences, University of Al-Muthanna, 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, ). 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  proved for any convex subset < of 7 , ( <E7 ) satisfying 1>? @ . # 7 %A F1>? @ . # < %A G---------------------------------------------------------#H% * Corresponding author’s e-mail: dr.al-muhja@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.  Let 7G.G< be defined in above, and satisf-ying (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.  A subset 7 of 23 is convex set if \ +GN ] ^7 , whenever +GN,7 . Equivalently, 7 is convex if # C_ % +aN,7 , for all +GN,7 and *FFC . Definition 2.  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.  A function .P23f \ _gGg ] is convex iff . h# C_ % +aN i Q # C_ % jak , *FFC , whenever . # + % Fj and . # N % Fk . Theorem 4.  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 ). 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=_CFN5FFN'FC=N'45
(see ).
If
&'=
, then
v3
#
$%# .G % =v3 #$
%#
.
% which is usually referred to as the degree of best uniform
convex polynomial approximation (see ).
Definition 5.  The weighted Ditzian-Totik 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
.,"
#
$%, such that } ._W3 } QL 3S$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'
Therefore,
+;
is inflection points at
¢
, and
¢^!
.
Thus,
W3
changes convexity at
¢
.
To prove
¢
have all inflection points
Q5
$, let N° be inflection point at ¢ , such that ±F( , and ² W3 h N° š5$
Since
¢^!
, then
.,"
#
$%# &' %, such that } ._W3 } QL 3S…G$
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
FFW3
#
+
%.
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
(
*FFC
). Then,
Wº
#
+=T
%
=CB»
and
Wº
#
N=_T
%
=½B»
.
Now,
Wº
h#
C_
%
+aN
i
=Wº
#
_*B¼
%
=_*B½¾
,
also, #
C_
%
Wº
#
T
%
aWº
#
_T
%
=
#
*B¿
%
d
#
CB»
%
a
#
*B¼
%
d
#
½B»
%
=»BC
.
Therefore,
Wº
h#
C_
%
d
#
T
%
#
_T
%i
Q
#
C_
%
Wº
#
T
%
aWº
#
_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
%
$_¿+--Â1?-*Q+QT C---------------------------Â1?_TQ+F*-- and .•• # + % = § ¼+$_¼+aH--Â1?-*Q+QT
*---------------Â1?_TQ+F*-
.
Let
+;=C
,
N;=H
and
=*B»
(
*FFC
). Then,
.
#
+;=C
%
=_CB»
and
.
#
N;=H
%
=_C
.
So,
.
h#
C_
%
d
#
C
%
a
#
H
%i
=.
#
CB»
%
=_HB*ÃT
,
also, #
C_
%
.
#
C
%
a.
#
H
%
=_CB
.
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+
%}
=
™#
%
d
#
CBÃH
%™
=B
.
Thus, }
._Wº
}
QL
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»
(
*FFC
). Then,
WÉ
#
+=CB»
%
=_HBC¾½»
and
WÉ
#
N=C
%
=*
.
Now,
WÉ
h#
C_
%
+aN
i
=WÉ
#
CB
%
=_CB
,
also, #
C_
%
WÉ
#
CB»
%
aWÉ
#
C
%
=
#
*B»
%
d
#
_HBC¾½»
%
a
#
*B»
%
d
#
*
%
=_CB
.
Therefore,
WÉ
h#
C_
%
d
#
CB»
%
a
#
C
%i
Q
#
C_
%
WÉ
#
CB»
%
aWÉ
#
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
.
#
+
%
=
§
+$_¿ a+--------Â1?_TQ+Q* H+_¿ _+-------Â1?-*F+QT---- , .• # + % = Ê Ë Ì H+º_¾+ +$_¿
aC--------Â1?_TQ+Q*
¿+_¾
H+_¿
_C-----------Â1?-*F+QT
and
.••
#
+
%
=
Í
ÎS
²i
Sd
#
ÏÎoÐ
%
o
h
ÑoÐÎ
i
S
#
ÎS
%
Ñ--------Â1?_TQ+Q*
*---------------------------------------------Â1?-*F+QT
.
Let
+;=*
,
N;=*B»
and
=*B»
(
*FFC
). Then,
.
#
+;=*
%
=¿
and
.
#
N;=*B»
%
=HB»
.
So,
.
h#
C_
%
d
#
*
%
a
#
HB»
%i
=.
#
CB
%
=*B
,
also, #
C_
%
.
#
*
%
a.
#
*B»
%
=TB
.
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¿
%}
=
,
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_+$% d" # €B5 % Á # .ÅÅG+ %Œ = ™# % d # CH¿B¼½¾ %™ =ÃÃ½B¿H¿ . Thus, } ._WÉ } QS ÁG$
@
.ÅÅ
#
+
%
=
ÎS
²i
Sd
#
ÏÎoÐ
%
o
h
ÑoÐÎ
i
S
#
ÎS
%
ÑG5
É
A,
where
È$=*BÃ»T . These results (Definitions 1, 7) are able to answer the question above. Also, it's a possibility of supporting the separation hyperplane theorem later by using DCP (see , ) like: If W3 and ·3 are two convex polynomials of " #$
%. 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.
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