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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

s−convex 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 +(1−λ)y)≤λf(x)+(1−λ)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 J⊂I,

with boundary ∂J, and every linear function

L, we have

supJ(f−L)=sup∂J(f−L)

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<s≤1. ([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 f∈K1

s;

•We use the same reasoning for a func-

tion g, s-convex in the second sense and

say then that g∈K2

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 f∈K1

sor f∈K2

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<s≤1

Therefore,

(1 + )1

n+(1+δ)1

n=1,1≤n<+∞

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>1∧b>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<1∧b>1) ∧k(a>1∧b<1).

Thirdly, we conclude that it must be the case

that (a≤1∧b≤1). 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 +(1−λs)1

sy)≤λsf(x)+(1−λs)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 +(1−λ)y)≤λsf(x)+(1−λ)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 1−convex functions

are convex.

7 Some natural conse-

quences of the deﬁni-

tion of s-convex func-

tions

Theorem 7.1.

f∈K1

s=⇒fu+v

21

s≤f(u)+f(v)

2

Proof. Simply consider the case where as=

bs=1

2.

Theorem 7.2.

f∈K2

s=⇒fu+v

2≤f(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 a2≥a1,b2≥b1since {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+(1−as

1)1

sv)≤as

1f(u)+(1−

as

1)f(v)=⇒f(as

1f(u)+(1−as

1)f(v)) ≤

(as

1)sf(f(u))+(1−as

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 s−convex if the

graph lies below the ‘bent chord‘ between any

two points, that is, for every compact interval

J⊂I, with boundary ∂J, and every linear

function L, we have

G(s)≥supJ(f−L)≥sup∂J(f−L)

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 +(1−λ)y)≤λf(x)+(1−λ)f(y)

holds ∀λ∈[0,1], ∀x, y ∈X.

For 0 <s

1,s

2≤1,

Deﬁnition 7. A function f:X−><is

said to be s1-convex if the inequality

f(λx +(1−λs)1

sy)≤λsf(x)+(1−λs)f(y)

holds ∀λ∈[0,1], ∀x, y ∈Xsuch that

X⊂<

+.

Deﬁnition 8. A function f:X−><is

said to be s2−convex if the inequality

f(λx +(1−λ)y)≤λsf(x)+(1−λ)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<s≤1, used

in the ﬁrst deﬁnition of s-convexity;

3here, f means closure of <

5

•s2for the constant s,0<s≤1, 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

1≡K2

1≡K0

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 s−convex if the

graph lies below the ‘bent chord‘ between any

two points, that is, for every compact interval

J⊂I, with boundary ∂J, and every linear

function L, we have

G(s)≥supJ(f−L)≥sup∂J(f−L)

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