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For the WSEAS: Summary and Importance of the Results

Involving the Deﬁnition of S−convexity

Dr. MARCIA PINHEIRO

WSEAS

PO Box 12396 A’Beckett St, Melbourne, VIC, 8006

AUSTRALIA

drmarciapinheiro@gmail.com

Abstract: In this note, we try to summarize the results we have so far in terms of the deﬁnition of the S-convexity

phenomenon, but we also try to explain in detail the relevance of those. For some of those results, we dare

presenting graphical illustrations to make our point clearer. S-convexity came to us through the work of Prof.

Dr. Dragomir (2001) and Prof. Dr. Dragomir claimed to have had contact with the concept through the hands of

Hudzik and Maligranda, who, in their turn, mention Breckner and Orlicz as an inspiration. We are working in a

professional way with the phenomenon since the year of 2001, and that was when we presented our ﬁrst talk on

the topic. In that talk, we introduced a conjecture about the shape of S-convexity. We have examined possible

examples, we have worked with the deﬁnition and examples, and we then concluded that we needed to reﬁne

the deﬁnition by much if we wanted to still call the phenomenon an extensional phenomenon in what regards

Convexity. We are now working on the fourth paper about the shape of S-convexity and trying to get both limiting

lines (negative and non-negative functions) to be as similar as possible. It is a delicate labour to the side of Real

Analysis, Vector Algebra, and even Calculus.

Key–Words: Analysis, Convexity, Deﬁnition, S-convexity, s−convexity, geometry, shape.

1 Introduction

In First Note on the Shape of S−convexity, we have

presented more evidence on our re-wording of the

piece of deﬁnition of the s−convexity phenomenon

that deals with non-negative real functions being

of fundamental importance for Mathematics and

we have proposed a geometric deﬁnition for the

phenomenon.

In Second Note on the Shape of S-convexity, we

have once more proved that the modiﬁcations to the

deﬁnition of the s−convexity phenomenon proposed

by us, this time for the negative share of the real

functions, constitute a major step towards making

the phenomenon be a proper extension of Convexity.

We have also proposed a geometric deﬁnition for the

negative case.

We hypothesized that our limiting curve for

s−convexity, when the real function is negative,

could be a bit bigger than the limiting curve for

s−convexity when the real function is not negative in

terms of length, what would mean that our lift could

not be the same for both cases.

Our perimeters seemed to be close enough in dimen-

sion, however, the difference being noticed by the ﬁrst

decimal digit only and being less than 0.5(consider-

ing our approximation for pi, our manual calculations,

and the approximation for the perimeter via elliptical

curve).

In Third Note on the Shape of s−convexity, we

studied the perimeter of our limiting curve for the

s−convexity phenomenon aiming equal perimeters

for geometrically equivalent situations.

The limiting curve originates in the application of the

deﬁnition of the phenomenon, so that, as a conse-

quence, we proposed new reﬁnements to the deﬁnition

of the phenomenon at the end of it.

The conclusions, for our third note, were:

1) There might be exponents that are nicer than our

chosen δ1and δ2and are still acceptable.

2) We have decided to deal with the s−convexity phe-

nomenon as if it were an exclusively extensional con-

cept for issues that have to do with practicality and

accuracy (we now forbid sto assume the value 1in

our deﬁnition).

3) We have extended the domain of the s−convex

functions to ℜbecause the deﬁnition should only

bring necessary limitations, and we have found prob-

lems only with the image of the functions so far in

what regards the current shape of the deﬁnition of the

phenomenon we study, not the domain.

4) We have decided to swap the coefﬁcients in our def-

inition because λ= 0 should bring f(x)to life, not

f(x+δ).

5) We have added the interval of deﬁnition of sto our

geometric deﬁnition to make it be independent from

the analytical deﬁnition.

6) Because of our new ﬁndings and decisions, we

have produced a new update for our deﬁnition of the

s−convexity phenomenon .

Update on our reﬁnements of

the deﬁnitions proposed by Hudzik and Maligranda

1) Analytical Deﬁnition

We have two possibilities that far for each piece of the

analytical deﬁnition of the phenomenon if we think of

including the deﬁnition of Hudzik and Maligranda in

Mathematics.

1.A) Possibility 1

Deﬁnition 1. A function f:X−>ℜ, where

|f(x)|=f(x), is told to belong to K2

sif the inequality

f((1 −λ)x+λ(x+δ))

≤(1 −λ)sf(x) + λsf(x+δ)

holds ∀λ/λ ∈[0,1]; ∀x/x ∈X;s=s2/0< s2<

1;X/X ⊆ ℜ ∧ X= [a, b];∀δ/0< δ ≤(b−x).

Deﬁnition 2. A function f:X−>ℜ, where

|f(x)|=−f(x), is told to belong to K2

sif the in-

equality

f((1 −λ)x+λ(x+δ))

≤(1−λ)

log2(1

1−2−s)f(x)+λ

log2(1

1−2−s)f(x+δ)

holds ∀λ/λ ∈[0,1]; ∀x/x ∈X;s=s2/0< s2<

1;X/X ⊆ ℜ ∧ X= [a, b];∀δ/0< δ ≤(b−x).

Remark 1. If the inequalities are obeyed in the re-

verse1situation by f, then fis said to be s2−concave.

1.B) Possibility 2

Deﬁnition 3. A function f:X−>ℜ, where

|f(x)|=f(x), is told to belong to K2

sif the inequality

f((1 −λ)x+λ(x+δ))

≤(1 −λ)

log2

1

1−2

−

1

sf(x) + λ

log2

1

1−2

−

1

sf(x+δ)

holds ∀λ/λ ∈[0,1]; ∀x/x ∈X;s=s2/0< s2<

1;X/X ⊆ ℜ ∧ X= [a, b];∀δ/0< δ ≤(b−x).

Deﬁnition 4. A function f:X−>ℜ, where

|f(x)|=−f(x), is told to belong to K2

sif the in-

equality

f((1 −λ)x+λ(x+δ))

≤(1 −λ)1

sf(x) + λ1

sf(x+δ)

holds ∀λ/λ ∈[0,1]; ∀x/x ∈X;s=s2/0< s2<

1;X/X ⊆ ℜ ∧ X= [a, b];∀δ/0< δ ≤(b−x).

Remark 2. If the inequalities are obeyed in the re-

verse2situation by f, then fis said to be s2−concave.

2) Geometric Deﬁnition

We then have two possibilities for each piece of the

geometric deﬁnition as well.

2.A) Possibility 1

Deﬁnition 5. A real function f:X−> Y , for which

|f(x)|=f(x), is called s−convex3if and only if, for

all choices (x1;y1)and (x2;y2), where {x1, x2} ⊂

X,{y1, y2} ⊂ Y,Y=Imf, and x1̸=x2, it happens

that the line drawn between (x1;y1)and (x2;y2)by

means of the expression (1 −λ)sy1+λsy2, where

λ∈[0,1], does not contain any point with height,

1Reverse here means ‘>’, not ‘≥’.

2See the previous footnote.

3smust be replaced, as needed, with a ﬁxed constant located

between 0and 1but different from 0and 1. For instance, if the

chosen constant is 0.5, then the function will be 0.5-convex or 1

2-

convex and swill be 0.5in the expression that deﬁnes the limiting

line.

measured against the vertical Cartesian axis, that is

inferior to the height of its horizontal equivalent in

the curve representing the ordered pairs of fin the

interval considered for the line in terms of distance

from the origin of the Cartesian axis.

Deﬁnition 6. A real function f:X−> Y , for

which |f(x)|=−f(x), is called s−convex4if and

only if, for all choices (x1;y1)and (x2;y2), where

{x1, x2} ⊂ X,{y1, y2} ⊂ Y,Y=Imf, and

x1̸=x2, it happens that the line drawn between

(x1;y1)and (x2;y2)by means of the expression (1 −

λ)

log2(1

1−2−s)y1+λ

log2(1

1−2−s)y2, where λ∈

[0,1], does not contain any point with height, mea-

sured against the vertical Cartesian axis, that is in-

ferior to the height of its horizontal equivalent in the

curve representing the ordered pairs of fin the inter-

val considered for the line in terms of distance from

the origin of the Cartesian axis.

Remark 3. If all the points deﬁning the function are

located above the limiting line instead, then fis called

s−concave.

2.B) Possibility 2

Deﬁnition 7. A real function f:X−> Y , for which

|f(x)|=f(x), is called s−convex5if and only if, for

all choices (x1;y1)and (x2;y2), where {x1, x2} ⊂

X,{y1, y2} ⊂ Y,Y=Imf, and x1̸=x2, it hap-

pens that the line drawn between (x1;y1)and (x2;y2)

by means of the expression (1 −λ)

log2(1

1−2

−

1

s)y1+

λ

log2(1

1−2

−

1

s)y2, where λ∈[0,1], does not con-

tain any point with height, measured against the ver-

tical Cartesian axis, that is inferior to the height of its

horizontal equivalent in the curve representing the or-

dered pairs of fin the interval considered for the line

in terms of distance from the origin of the Cartesian

axis.

Deﬁnition 8. A real function f:X−> Y , for

which |f(x)|=−f(x), is called s−convex6if and

only if, for all choices (x1;y1)and (x2;y2), where

{x1, x2} ⊂ X,{y1, y2} ⊂ Y,Y=Imf, and

x1̸=x2, it happens that the line drawn between

(x1;y1)and (x2;y2)by means of the expression (1 −

λ)1

sy1+λ1

sy2, where λ∈[0,1], does not contain

4See the previous footnote.

5See the previous footnote.

6See the previous footnote.

any point with height, measured against the vertical

Cartesian axis, that is inferior to the height of its hor-

izontal equivalent in the curve representing the or-

dered pairs of fin the interval considered for the line

in terms of distance from the origin of the Cartesian

axis.

Remark 4. If all the points deﬁning the function are

located above the limiting line instead, then fis called

s−concave.

2 Development

We should probably talk about how important our

suggested modiﬁcations were for the concept and

deﬁnition of the S-convexity phenomenon .

We start with the dimensions involved in the deﬁni-

tion.

The original deﬁnition of the phenomenon was given

by Hudzik and Maligranda according to Dragomir

and it read [1]:

A function f: [0,∞)−>ℜis said to be

s−convex in the ﬁrst sense if f(ax +by)≤

asf(x) + bsf(y),∀x, y ∈[0,∞)and ∀a, b ≥0with

as+bs= 1,0< s ≤1.

A function f: [0,∞)−>ℜis said

to be s−convex in the second sense if

f(ax +by)≤asf(x) + bsf(y),∀x, y ∈[0,∞)and

∀a, b ≥0with a+b= 1,0< s ≤1.

We then observed that we were aiming at graphs

in ℜ2, and therefore we were drawing from ℜ, but we

were using two variables from the domain instead,

what would have to mean ℜ2in the domain.

Figure 1: Maple Plot I

Hudzik and Maligranda: x,y, and a.

This graph: xand a.

What we need is a(only).

Figure 2: Maple Fails

Maple fails because we have a 3Dgraph. Observe

that the previous example brings plot3d and this one

brings plot, which is the command for 2D graphs.

These plots are to show that, in the way it would

be, after the introduction of the spacer, we still have

three dimensions instead of two for our limiting line.

Notwithstanding, we will be changing the wording of

our deﬁnition once more in this paper to ﬁx also this

problem. We will say that for each xwe select from

our domain, the inequality should be veriﬁed, so that

we would be replacing xwith a constant on our next

plot, what would ﬁnally give us two dimensions.

To disappear with one dimension, we introduced

a constant, a spacer, as we said in the previous

paragraph, and we called it δ.

We also observed that we could not possibly be

willing to allow for the person to get a situation in

which x=ybecause we then would not have the

geometric extension of the concept Convexity, but

getting this extension could only be our main aim.

As a consequence of taking notice of the just-

mentioned facts, we have proposed a reﬁnement to

the deﬁnition of the convexity phenomenon as well.

Once we chose our x, we would naturally choose a y

after it and a ythat is inside of the interval, and let’s

call it I, so that we can say that we have xand x+δ

where δ̸= 0 and δis between xand the end of I.

We observed that bdepended on our choice of value

for a, since as+bs= 1 or a+b= 1. In this case,

one of them could be called dependant variable, for it

would be given in function of the other.

We also observed that convexity could use any do-

main in the reals, but S-convexity, that far, was using

only the non-negative part, what then eliminated the

possibility of it being a proper extension of Convexity.

Upon examining the restriction from closer, we did

not ﬁnd any reason not to have the domain in the reals

instead.

As a consequence of the just-mentioned ﬁndings,

which we communicated to Academia also through

[2], [3], and [4], we reﬁned the deﬁnition of the

phenomenon a bit more.

Upon studying the deﬁnitions for longer, we found

counter-examples: Families of functions that would

not ﬁt inside of the concept S−convexity and yet

were convex. The problem was that S−convexity

lifted and bent the convexity line, so that the compar-

ison signs and expressions could not ﬁt both the case

of the negative and the case of the positive convex

functions. We then had to split the deﬁnition into

two subdeﬁnitions: one for when the modulus of

the function equated the function and one for when

the modulus of the function were equal to minus the

function.

The original deﬁnition of S-convexity was, as we see

on the paragraphs above this one, split into two other

deﬁnitions: s−convex in the ﬁrst sense and s−convex

in the second sense. They were called S1−and S2−

convex functions, respectively.

One of the senses could not be considered an exten-

sion of convexity, as we proved on [5], so that we had

to drop it.

We were then seen talking about splitting only the

second sense of S−convexity, the one that remained,

S2, into two subdeﬁnitions. That is seen on [6].

Finally, we found out that, even with all the just-

mentioned ﬁxings, our split deﬁnition presented a

mistake, which was a bit more rope to one of the

subdeﬁnitions than to the other one in what had to do

with the limiting line: We needed to make sure that

equivalent situations would return equivalent limiting

lines, regardless of one being to the negative and the

other being to the positive side.

That is when we came up with another exponent, as

we see in [4].

We also reﬁned the only geometric deﬁnition we

found for the convexity phenomenon as we pro-

posed the ﬁrst ever seen geometric deﬁnition of the

S−convexity phenomenon. These results also appear

on [4].

As the reader can see, aand bare interchangeable in

the deﬁnition attributed to Hudzik and Maligranda.

We, however, tied the position of the coefﬁcients. We

did this because otherwise the limiting line would not

be acceptable. See the graphs:

Figure 3: Coefﬁcients, importance of the order

These are probably the most meaningful modiﬁca-

tions that we have proposed to the deﬁnition of this

phenomenon and we now need to tell the reader about

our next step.

Having reached clarity and coherence in our deﬁni-

tion, we now wonder about whether this is the best ex-

tension for the convexity phenomenon: Could we ﬁnd

something better, perhaps more analytical or more

perfect?

The ﬁrst idea that occurred to us was that we could

have a piece of a circumference as a limiting line,

so that we would achieve maximum analyticity. We

could then choose to have the midpoint of the limit-

ing line for convexity involved somehow in the deter-

mination of the center and the distance between that

midpoint and the extremes as a radius. One of the

problems with that is that we lose the variation of the s

and we then have something totally different. Another

problem is that the expression will look totally differ-

ent from that of the limiting line for convexity, and

therefore will not allow us to easily compare them.

There is also an issue with the concept extension: That

would be perhaps a geometric extension, but not nec-

essarily an analytical one. Besides, we then get an

expression that seems to be independent of the func-

tion itself, for it depends solely on the extreme val-

ues picked from any chosen interval. With convex-

ity, however, and also with the current concept of S-

convexity, things depend on the value of the function

at each stage of the comparison path. It perhaps stops

being an extension and becomes simply an inequality

if we introduce this change.

Any attempt to have the same geometric shape over

all convexity limiting lines as a limiting line for

S−convexity is then a vain attempt.

Even though we could probably have different ex-

ponents, it looks like our extension is now an ex-

cellent choice. When we studied the exponents,

we concluded that only one of them could be nice

(negative/non-negative) if we wanted to have equiv-

alent situations returning equivalent shapes. That has

been done on [4].

Our extension, as it is now, extends the concept of

convexity both analytically and geometrically, so that

it does look like an ideal choice.

To illustrate things graphically, we have:

Figure 4: S1

Figure 5: S2

We notice that our limiting lines are not yet perfect,

since, ideally, they would both have the same shape

and dimensions (non-negative and negative case).

Even though we put some effort into equating what

we called the size of the rope, we have only equated

the middle point match on the limiting line in terms of

height, as one can tell by simply studying our devel-

opments in [4].

3 Conclusion

We have decided to keep the name S1and replace

the previous class K1

swith a new version of it, which

would be one of our possible deﬁnitions, as for [4].

So far, we have:

1) Analytical Deﬁnition

S1

Deﬁnition 1. A function f:X−>ℜ, where

|f(x)|=f(x), is told to belong to K1

sif, for each

x∈Xwe select, and for all of them, the inequality

f((1 −λ)x+λ(x+δ))

≤(1 −λ)sf(x) + λsf(x+δ)

holds ∀λ/λ ∈[0,1]; s=s1/0< s1<1;X/X ⊆

ℜ ∧ X= [a, b];∀δ/0< δ ≤(b−x).

Deﬁnition 2. A function f:X−>ℜ, where

|f(x)|=−f(x), is told to belong to K1

sif, for each

x∈Xwe select, and for all of them, the inequality

f((1 −λ)x+λ(x+δ))

≤(1−λ)

log2(1

1−2−s)f(x)+λ

log2(1

1−2−s)f(x+δ)

holds ∀λ/λ ∈[0,1]; s=s1/0< s1<1;X/X ⊆

ℜ ∧ X= [a, b];∀δ/0< δ ≤(b−x).

Remark 1. If the inequalities are obeyed in the re-

verse7situation by f, then fis said to be s1−concave.

S2

Deﬁnition 3. A function f:X−>ℜ, where

|f(x)|=f(x), is told to belong to K2

sif, for each

x∈Xwe select, and for all of them, the inequality

f((1 −λ)x+λ(x+δ))

≤(1 −λ)

log2

1

1−2

−

1

sf(x) + λ

log2

1

1−2

−

1

sf(x+δ)

holds ∀λ/λ ∈[0,1]; s=s2/0< s2<1;X/X ⊆

ℜ ∧ X= [a, b];∀δ/0< δ ≤(b−x).

7Reverse here means ‘>’, not ‘≥’.

Deﬁnition 4. A function f:X−>ℜ, where

|f(x)|=−f(x), is told to belong to K2

sif, for each

x∈Xwe select, and for all of them, the inequality

f((1 −λ)x+λ(x+δ))

≤(1 −λ)1

sf(x) + λ1

sf(x+δ)

holds ∀λ/λ ∈[0,1]; s=s2/0< s2<1;X/X ⊆

ℜ ∧ X= [a, b];∀δ/0< δ ≤(b−x).

Remark 2. If the inequalities are obeyed in the re-

verse8situation by f, then fis said to be s2−concave.

2) Geometric Deﬁnition

We then have two possibilities for each piece of the

geometric deﬁnition as well.

S1

Deﬁnition 5. A real function f:X−> Y , for which

|f(x)|=f(x), is called s−convex9if and only if, for

all choices (x1;y1)and (x2;y2), where {x1, x2} ⊂

X,{y1, y2} ⊂ Y,Y=Imf, and x1̸=x2, it happens

that the line drawn between (x1;y1)and (x2;y2)by

means of the expression (1 −λ)sy1+λsy2, where

λ∈[0,1], does not contain any point with height,

measured against the vertical Cartesian axis, that is

inferior to the height of its horizontal equivalent in

the curve representing the ordered pairs of fin the

interval considered for the line in terms of distance

from the origin of the Cartesian axis.

Deﬁnition 6. A real function f:X−> Y , for

which |f(x)|=−f(x), is called s−convex10 if

and only if, for all choices (x1;y1)and (x2;y2),

where {x1, x2} ⊂ X,{y1, y2} ⊂ Y,Y=Imf,

and x1̸=x2, it happens that the line drawn be-

tween (x1;y1)and (x2;y2)by means of the expres-

sion (1−λ)

log2(1

1−2−s)y1+λ

log2(1

1−2−s)y2, where

λ∈[0,1], does not contain any point with height,

measured against the vertical Cartesian axis, that is

inferior to the height of its horizontal equivalent in the

8See the previous footnote.

9smust be replaced, as needed, with a ﬁxed constant located

between 0and 1but different from 0and 1. For instance, if the

chosen constant is 0.5, then the function will be 0.5-convex or 1

2-

convex and swill be 0.5in the expression that deﬁnes the limiting

line.

10See the previous footnote.

curve representing the ordered pairs of fin the inter-

val considered for the line in terms of distance from

the origin of the Cartesian axis.

Remark 3. If all the points deﬁning the function are

located above the limiting line instead, then fis called

s−concave.

S2

Deﬁnition 7. A real function f:X−> Y , for which

|f(x)|=f(x), is called s−convex11 if and only if, for

all choices (x1;y1)and (x2;y2), where {x1, x2} ⊂

X,{y1, y2} ⊂ Y,Y=Imf, and x1̸=x2, it hap-

pens that the line drawn between (x1;y1)and (x2;y2)

by means of the expression (1 −λ)

log2(1

1−2

−

1

s)y1+

λ

log2(1

1−2

−

1

s)y2, where λ∈[0,1], does not con-

tain any point with height, measured against the ver-

tical Cartesian axis, that is inferior to the height of its

horizontal equivalent in the curve representing the or-

dered pairs of fin the interval considered for the line

in terms of distance from the origin of the Cartesian

axis.

Deﬁnition 8. A real function f:X−> Y , for

which |f(x)|=−f(x), is called s−convex12 if and

only if, for all choices (x1;y1)and (x2;y2), where

{x1, x2} ⊂ X,{y1, y2} ⊂ Y,Y=Imf, and

x1̸=x2, it happens that the line drawn between

(x1;y1)and (x2;y2)by means of the expression (1 −

λ)1

sy1+λ1

sy2, where λ∈[0,1], does not contain

any point with height, measured against the vertical

Cartesian axis, that is inferior to the height of its hor-

izontal equivalent in the curve representing the or-

dered pairs of fin the interval considered for the line

in terms of distance from the origin of the Cartesian

axis.

Remark 4. If all the points deﬁning the function are

located above the limiting line instead, then fis called

s−concave.

We may come back to the subject shape in the near

future to see if we can get even closer to our goal.

11See the previous footnote.

12See the previous footnote.

References:

[1] Hudzik, H.& Maligranda, L. Some remarks on si-

convex functions, Aequationes Mathematicae 48

(1994), 100–111.

[2] Pinheiro, M. R., Minima Domain Intervals and

the S-Convexity, as well as the Convexity, Phe-

nomenon, Advances in Pure Mathematics 2

(2012), 457–458.

[3] Pinheiro, M. R. Exploring the concept of S-

Convexity, Proceedings of the WSEAS Interna-

tional Conference on Mathematics and Comput-

ers in Physics (MCF ’04) (2004).

[4] Pinheiro, M. R. Third Note on the Shape of S-

Convexity, International Journal of Pure and Ap-

plied Mathematics 93(5) (2014), 729–739.

[5] Pinheiro, M. R. Second Note on the Deﬁnition

of S1-convexity, Advances in Pure Mathematics

5(2015), 127–130.

[6] Pinheiro, M. R. First Note on the Deﬁnition of

S2-convexity, Advances in Pure Mathematics 1

(2011), 1–2.