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International Journal of Pure and Applied Mathematics

Volume 106 No. 3 2016, 699-713

ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)

url: http://www.ijpam.eu

doi: 10.12732/ijpam.v106i3.1

P

A

ijpam.eu

SUMMARY AND IMPORTANCE OF THE RESULTS

INVOLVING THE DEFINITION OF S−CONVEXITY

Marcia Pinheiro

RGMIA

P.O. Box 12396 A’Beckett St

Melbourne, VIC, 8006, AUSTRALIA

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, S-convexity, s−convexity, geometry, shape

1. Introduction

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

Received: January 30, 2016

Published: March 2, 2016

c

2016 Academic Publications, Ltd.

url: www.acadpubl.eu

700 M. Pinheiro

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 dimension, however, the dif-

ference being noticed by the ﬁrst decimal digit only and being less than 0.5

(considering our approximation for pi, our manual calculations, and the ap-

proximation 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 consequence, 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 phenomenon as if it were

an exclusively extensional concept for issues that have to do with practicality

and accuracy (we now forbid sto assume the value 1 in 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 problems

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 deﬁnition 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.

SUMMARY AND IMPORTANCE OF THE RESULTS... 701

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 deﬁni-

tion 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 inequality

f((1 −λ)x+λ(x+δ)) ≤(1 −λ)

log21

1−2−sf(x) + λ

log21

1−2−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 1. If the inequalities are obeyed in the reverse1situation by f,

then fis said to be s2−concave.

1Reverse here means ‘>’, not ‘≥’.

702 M. Pinheiro

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) + λlog2

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 inequality

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 reverse2situation 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 x16=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.

2See the previous footnote.

3smust be replaced, as needed, with a ﬁxed constant located between 0 and 1 but diﬀerent

from 0 and 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 0.5 in the expression that deﬁnes the limiting line.

SUMMARY AND IMPORTANCE OF THE RESULTS... 703

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 x16=x2, it happens that the line

drawn between (x1;y1) and (x2;y2) by means of the expression

(1 −λ)

log21

1−2−sy1+λ

log21

1−2−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 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.

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 x16=x2, it happens that the line

drawn between (x1;y1) and (x2;y2) by means of the expression

(1 −λ)

log21

1−2

−

1

sy1+λ

log21

1−2

−

1

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 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 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 x16=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

4See the previous footnote.

5See the previous footnote.

6See the previous footnote.

704 M. Pinheiro

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.

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 deﬁnition.

The original deﬁnition of the phenomenon was given by Hudzik and Ma-

ligranda according to Dragomir and it read [1]:

A function f: [0,∞)→ ℜ 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, 0 < s ≤1.

A function f: [0,∞)→ ℜ 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, 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.

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 x

with 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

SUMMARY AND IMPORTANCE OF THE RESULTS... 705

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

geometric extension of the concept Convexity, but getting this extension could

706 M. Pinheiro

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 yafter 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+δwhere δ6= 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 domain 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: Fam-

ilies 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 comparison 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 extension 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.

SUMMARY AND IMPORTANCE OF THE RESULTS... 707

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 proposed 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 at-

tributed to Hudzik and Maligranda. We, however, tied the position of the

coeﬃcients. We did this because otherwise the limiting line would not be ac-

ceptable. See the graphs:

Figure 3: Coeﬃcients, importance of the order

These are probably the most meaningful modiﬁcations that we have pro-

posed 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 deﬁnition, we now wonder

about whether this is the best extension 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.

708 M. Pinheiro

We could then choose to have the midpoint of the limiting line for convexity

involved somehow in the determination 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 sand we then have something totally diﬀerent.

Another problem is that the expression will look totally diﬀerent 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 necessarily an analytical one. Besides, we then

get an expression that seems to be independent of the function itself, for it

depends solely on the extreme values picked from any chosen interval. With

convexity, 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 exponents, it looks like our

extension is now an excellent choice. When we studied the exponents, we

concluded that only one of them could be nice (negative/non-negative) if we

wanted to have equivalent situations returning equivalent shapes. That has

been done on [4].

Our extension, as it is now, extends the concept of convexity both analyti-

cally and geometrically, so that it does look like an ideal choice.

We notice upon examining Figure 4 and Figure 5 (they both appear after

this paragraph), 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 developments in [4].

SUMMARY AND IMPORTANCE OF THE RESULTS... 709

Figure 4: S1

Figure 5: S2

710 M. Pinheiro

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)

log21

1−2−sf(x) + λ

log21

1−2−sf(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 reverse7situation 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) + λlog2

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 ‘≥’.

SUMMARY AND IMPORTANCE OF THE RESULTS... 711

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 reverse8situation 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 x16=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 x16=x2, it happens that the line

drawn between (x1;y1) and (x2;y2) by means of the expression

(1 −λ)

log21

1−2−sy1+λ

log21

1−2−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 horizontal equivalent

8See the previous footnote.

9smust be replaced, as needed, with a ﬁxed constant located between 0 and 1 but diﬀerent

from 0 and 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 0.5 in the expression that deﬁnes the limiting line.

10See the previous footnote.

712 M. Pinheiro

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.

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 x16=x2, it happens that the line

drawn between (x1;y1) and (x2;y2) by means of the expression

(1 −λ)

log21

1−2

−

1

sy1+λ

log21

1−2

−

1

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 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 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 x16=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 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.

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.

SUMMARY AND IMPORTANCE OF THE RESULTS... 713

References

[1] Hudzik, H., Maligranda, L. (1994). Some remarks on si-convex functions, Aequationes

Mathematicae 48, 100–111.

[2] Pinheiro, M. R. (2012). Minima Domain Intervals and the S-Convexity, as well as the

Convexity, Phenomenon, Advances in Pure Mathematics 2, 457–458.

[3] Pinheiro, M. R. (2004). Exploring the concept of S-Convexity, Proceedings of the WSEAS

International Conference on Mathematics and Computers in Physics (MCF ’04).

[4] Pinheiro, M. R. (2014). Third Note on the Shape of S-Convexity, International Journal

of Pure and Applied Mathematics 93(5), 729–739.

[5] Pinheiro, M. R. (2015). Second Note on the Deﬁnition of S1-convexity, Advances in Pure

Mathematics 5, 127–130.

[6] Pinheiro, M. R. (2011). First Note on the Deﬁnition of S2-convexity, Advances in Pure

Mathematics 1, 1–2.

714