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Exploring the Concept of s-Convexity

Authors:
  • IICSE University

Abstract

The purpose of this paper is to distinguish, as much as possible, the concept of s-convexity from the concept of convexity and the concept of s-convexity in the first sense from the concept of s-convexity in the second sense. In this respect, the present work further develops a previous study by Orlicz(1961, [3]), Hudzik and Maligranda (1994, [1]).
Exploring the concept of S-convexity
M.R. PINHEIRO
Matematica
UNESA
Rio de Janeiro, RJ
Brasil
e-mail address: inthenameofthegood@yahoo.com
Abstract: – The purpose of this paper is to distinguish, as much as possible, the
concept of s-convexity from the concept of convexity and the concept of s-convexity
in the first sense from the concept of s-convexity in the second sense. In this respect,
the present work further develops a previous study by Orlicz(1961, [3]), Hudzik and
Maligranda (1994, [1]).
Key-words 1:convex, s-convex, function
1 Introduction
Recently, Hudzik and Maligranda ([1]) stud-
ied some classes of functions introduced by
Orlicz ([3]), the classes of s-convex functions.
Although they claim, in their abstract, to be
providing several examples and to be clari-
fying the idea introduced by Orlicz further,
their work leaves plenty of room to build over
the concept.
The old conclusions presented here are:
1. theoretical definitions of convex/s-
convex functions;
2. a theorem which acts as a generator of
s-convex functions.
The new conclusions arisen from this paper
are:
1. a rephrasing of the theoretical defini-
tions of s-convex functions to look more
similar to the definition of convex func-
tion;
2. some new symbols to represent the
classes of s-convex functions;
1AMS: 26A51
1
3. an identity between the class of 1-
convex functions and the class of con-
vex functions;
4. a conjecture about the the looks of a
sconvex function;
5. some theorems on functions that are s-
convex in both senses;
6. a few other side results that might suit
future work or are, at least, useful to
clarify similarities and differences be-
tween functions that are s-convex in the
first sense and the ones which are s-
convex in the second sense.
The paper is organized as follows: First, in
section 2, we present the usual definition of
both convex and s-convex functions. In sec-
tion 3, we criticize the present presentation of
definitions of s-convex functions. In section
4, we introduce a few new ways of referring
to s-convex functions with views to have a
more mathematical jargon to deal with them.
In section 5, we re-write the definition of s-
convex functions based on our new symbol-
ogy, prove the equivalence between restric-
tions of convex functions and s-convex func-
tions, and present some consequences of the
definition of s-convex functions. In section
6, we recall one theorem on how to generate
s-convex functions, as presented by Hudzik
and Maligranda in [1]. Section 7 brings our
conjecture whilst section 8 presents our con-
clusions.
2 The usual definition
of convexity and s-
convexity
The concept of convexity that is mostly cited
in the bibliography is (as an example, [4]):
Definition 1. The function (f:X><f)2
is called convex if the inequality
f(λx +(1λ)y)λf(x)+(1λ)f(y)
holds λ[0,1], x, y Xsuch that the
right-hand side is well defined. It is called
strictly convex if the above inequality strictly
holds λ]0,1[ and for all pairs of dis-
tinct points x, y Xwith f(x)<and
f(y)<.
In some sources, such as [2], convexity is de-
fined only in geometrical terms as being the
property of a function whose graph bears
tangents only under it. In their words,
Citation 1.fis called convex if the graph
lies below the chord between any two points,
that is, for every compact interval JI,
with boundary ∂J, and every linear function
L, we have
supJ(fL)=sup∂J(fL)
One calls fconcave if fis convex.
The concept of s-convexity, on the other
hand, is split into two notions which are de-
scribed below with the basic condition that
0<s1. ([1])
Definition 2. A function f:[0,)><
is said to be s-convex in the first sense if
f(ax +by)asf(x)+bsf(y), x, y [0,)
and a, b 0 with as+bs=1.
2here, f means closure of <
2
Definition 3. A function f:[0,)><
is said to be s-convex in the second sense if
f(ax +by)asf(x)+bsf(y), x, y [0,)
and a, b 0 with a+b=1.
3 What are the criticisms
to the present defini-
tion of s-convexity?
It seems that there is lack of objectivity
in the present definition of s-convexity
for there are some redundant things;
It takes us a long time, the way the
definition is written now, to work out
the true difference between convex and
s-convex functions;
So far, we did not find references, in
the bibliography, to the geometry of
an s-convex function, what, once more,
makes it less clear to understand the
difference between an s-convex and a
convex function whilst there are clear
references to the geometry of the con-
vex functions.
4 New Symbology
In this paper, we mean that fis an
s-convex function in the first sense by
saying that fK1
s;
We use the same reasoning for a func-
tion g, s-convex in the second sense and
say then that gK2
s;
We name s1the generic class constant
for those functions that are s-convex in
the first sense;
We name s2the generic class constant
for those functions that are s-convex in
the second sense.
5 The first few new re-
sults
5.1 Re-writting the definition
of s-convex function
It is trivial to prove that a, b [0,1] is a
consequence of the present definition of s-
convexity.
Lemma 5.1. If fK1
sor fK2
sthen
f(au +bv)asf(u)+bsf(v)
with a, b [0,1], exclusively.
Proof. We present the proof for K1
sonly,
since the proof for K2
sis analogous.
For K1
s: We first prove that it is not the case
that a>1 and b>1. Supposing that it is
the case that a>1 and b>1, that implies
having
a=1+
b=1+δ
as+bs=1,0<s1
Therefore,
(1 + )1
n+(1+δ)1
n=1,1n<+
As x1
nis a decreasing function of n, for x>1,
and, as n>+, the above result is not
verified, being as+bs>1, k(a>1b>1).
Secondly, we prove that it is not the case
3
that a>1 and b<1, or vice-versa, just by
re-analyzing the previous case again. There-
fore, k(a<1b>1) k(a>1b<1).
Thirdly, we conclude that it must be the case
that (a1b1). But since the definition
of s-convexity uses a, b 0, we have that
a, b [0,1]
With this, we may re-write the definitions of
s-convexity in each of the senses as being:
Definition 4. A function f:X><
is said to be s-convex in the first sense if
f(λx +(1λs)1
sy)λsf(x)+(1λs)f(y),
x, y Xand λ[0,1] where X⊂<
+.
Definition 5. A function f:X><is
said to be s-convex in the second sense if
f(λx +(1λ)y)λsf(x)+(1λ)sf(y),
x, y Xand λ[0,1] where X⊂<
+.
6 The classes K1
1,K2
1, and
convex coincide when
the domains are re-
stricted to <+
Theorem 6.1. The classes K1
1,K2
1, and
convex are equivalent when the domain is re-
stricted to <+.
Proof. Just a matter of applying the defini-
tions.
Natural implication: All 1convex functions
are convex.
7 Some natural conse-
quences of the defini-
tion of s-convex func-
tions
Theorem 7.1.
fK1
s=fu+v
21
sf(u)+f(v)
2
Proof. Simply consider the case where as=
bs=1
2.
Theorem 7.2.
fK2
s=fu+v
2f(u)+f(v)
2s
Proof. Simply consider the case where a=
b=1
2.
Theorem 7.3. For a function that is both
s1and s2-convex, there is a perfect bijection
between the set of (a’s,b’s) used in s1and the
set of (a’s, b’s) used in s2.
Proof. Each amay be written as an as
1and
each bas a bs
1and vice-versa. This happens
because a, b [0,1], s[0,1](each 1
s-root in
(0,1) will give us a number in (0,1)).
Theorem 7.4. If a function belongs to both
K1
sand K2
s, then
f(a1u+b1v)as
1f(u)+bs
1f(v)as
2f(u)+bs
2f(v)
for some {a1,b
1,a
2,b
2}⊂[0,1] and such that
it occurs to each and all of them.
Proof. It follows from the bijection proved
before. For each a2,b
2such that a2+b2=1,
it corresponds a1,b
1such that as
1+bs
1=1
and a2a1,b2b1since {a, b}⊂[0,1].
4
Theorem 7.5. If a function belongs to both
K1
sand K2
sand its domain coincides with its
counter-domain then the composition f(f)is
s2
1-convex.
Proof. f(a1u+(1as
1)1
sv)as
1f(u)+(1
as
1)f(v)=f(as
1f(u)+(1as
1)f(v))
(as
1)sf(f(u))+(1as
1)sf(f(v)) = as
2f(f(u))+
bs
2f(f(v))
Theorem 7.6. f:I><,I[0,),f
being a convex, non-negative function, then
s(0,1],fis s2-convex.
Proof.
a+b=1
f(ax +by)af (x)+bf(y)asf(x)+
bsf(y)
8 A new conjecture
Taking into account the relationship between
asand a, we may wonder whether the follow-
ing is true or not:
Conjecture 1.fis called sconvex if the
graph lies below the ‘bent chord‘ between any
two points, that is, for every compact interval
JI, with boundary ∂J, and every linear
function L, we have
G(s)supJ(fL)sup∂J(fL)
9 Conclusions
In this paper, we proved that s-convexity
may be stated in a very similar way to con-
vexity, as written below:
Definition 6. the function (f:X><f)3
is called convex if the inequality
f(λx +(1λ)y)λf(x)+(1λ)f(y)
holds λ[0,1], x, y X.
For 0 <s
1,s
21,
Definition 7. A function f:X><is
said to be s1-convex if the inequality
f(λx +(1λs)1
sy)λsf(x)+(1λs)f(y)
holds λ[0,1], x, y Xsuch that
X⊂<
+.
Definition 8. A function f:X><is
said to be s2convex if the inequality
f(λx +(1λ)y)λsf(x)+(1λ)sf(y)
holds λ[0,1], x, y Xsuch that
X⊂<
+.
The own re-definition of s-convexity included
our new way of referring to s-convex func-
tions by creating class-like symbology for
them:
K1
sfor the class of s-convex functions
in the first sense, some s;
K2
sfor the class of s-convex functions
in the second sense, some s;
K0for the class of convex functions;
s1for the constant s,0<s1, used
in the first definition of s-convexity;
3here, f means closure of <
5
s2for the constant s,0<s1, used
in the second definition of s-convexity.
thirdly, we pointed out that the class of
1-convex functions is just a restriction
of the class of convex functions, that is,
when X=<+,
K1
1K2
1K0
In fourth, we introduced the following side-
theorems:
Theorem 9.1. For a function that is both
s1and s2-convex, there is a perfect bijection
between the set of (a’s,b’s) used in s1and the
set of (a’s, b’s) used in s2.
Theorem 9.2. If a function belongs to both
K1
sand K2
s, then
f(a1u+b1v)as
1f(u)+bs
1f(v)as
2f(u)+bs
2f(v)
for some {a1,b
1,a
2,b
2}⊂[0,1] obeying K1
s
and K2
srules, and such that it occurs to each
and all of them.
Theorem 9.3. If a function belongs to both
K1
sand K2
sand its domain coincides with its
counter-domain then the composition f(f)is
s2
1-convex.
Theorem 9.4. f:I><,I[0,),f
being a convex, non-negative function, then
s(0,1],fis s2-convex.
In fifth we bring our conjecture as a prospec-
tive future work:
Conjecture 2.fis called sconvex if the
graph lies below the ‘bent chord‘ between any
two points, that is, for every compact interval
JI, with boundary ∂J, and every linear
function L, we have
G(s)supJ(fL)sup∂J(fL)
10 Acknowledgements
We wish to thank Dr. Carlos Gustavo Mor-
eira, from IMPA (RJ, Brasil) for his precious
revision and contribution to this paper.
References
[1] H. Hudzik and L. Maligranda. Some remarks on si-convex functions. Aequationes Math.,
48: 100–111, 1994.
[2] Lars Hormander. Notions of convexity. Birkhauser , 1994.
[3] W. Orlicz. A note on modular spaces I. Bull. Acad. Polon. Sci. Ser. Math. Astronom.
Phys., 9: 157–162, 1961.
[4] Th. Precupanu, V. Barbu. Convexity and Optimization in Banach Spaces. Editura acad-
emiei/D. Reidel Publishing Company, 1986.
6
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