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For the WSEAS: Summary and Importance of the Results Involving the Definition of S􀀀convexity

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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.
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For the WSEAS: Summary and Importance of the Results
Involving the Definition of Sconvexity
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, sconvexity, geometry, shape.
1 Introduction
In First Note on the Shape of Sconvexity, we have
presented more evidence on our re-wording of the
piece of definition of the sconvexity 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 sconvexity 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
sconvexity, when the real function is negative,
could be a bit bigger than the limiting curve for
sconvexity 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 sconvexity, we
studied the perimeter of our limiting curve for the
sconvexity 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 sconvexity 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 sconvex
functions to because 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
sconvexity 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>, 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 (bx).
Definition 2. A function f:X>, where
|f(x)|=f(x), is told to belong to K2
sif the in-
equality
f((1 λ)x+λ(x+δ))
(1λ)
log2(1
12s)f(x)+λ
log2(1
12s)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 (bx).
Remark 1. If the inequalities are obeyed in the re-
verse1situation by f, then fis said to be s2concave.
1.B) Possibility 2
Definition 3. A function f:X>, 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
12
1
sf(x) + λ
log2
1
12
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 (bx).
Definition 4. A function f:X>, 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 (bx).
Remark 2. If the inequalities are obeyed in the re-
verse2situation by f, then fis said to be s2concave.
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 sconvex3if 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 sconvex4if 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
12s)y1+λ
log2(1
12s)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
sconcave.
2.B) Possibility 2
Definition 7. A real function f:X> Y , for which
|f(x)|=f(x), is called sconvex5if 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
12
1
s)y1+
λ
log2(1
12
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 sconvex6if 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
sconcave.
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,)>is said to be
sconvex 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,)>is said
to be sconvex 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 2, and therefore we were drawing from , but we
were using two variables from the domain instead,
what would have to mean 2in 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 Sconvexity and yet
were convex. The problem was that Sconvexity
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: sconvex in the first sense and sconvex
in the second sense. They were called S1and 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 Sconvexity, 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
Sconvexity 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
Sconvexity 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>, where
|f(x)|=f(x), is told to belong to K1
sif, for each
xXwe 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 (bx).
Definition 2. A function f:X>, where
|f(x)|=f(x), is told to belong to K1
sif, for each
xXwe select, and for all of them, the inequality
f((1 λ)x+λ(x+δ))
(1λ)
log2(1
12s)f(x)+λ
log2(1
12s)f(x+δ)
holds λ/λ [0,1]; s=s1/0< s1<1;X/X
ℜ ∧ X= [a, b];δ/0< δ (bx).
Remark 1. If the inequalities are obeyed in the re-
verse7situation by f, then fis said to be s1concave.
S2
Definition 3. A function f:X>, where
|f(x)|=f(x), is told to belong to K2
sif, for each
xXwe select, and for all of them, the inequality
f((1 λ)x+λ(x+δ))
(1 λ)
log2
1
12
1
sf(x) + λ
log2
1
12
1
sf(x+δ)
holds λ/λ [0,1]; s=s2/0< s2<1;X/X
ℜ ∧ X= [a, b];δ/0< δ (bx).
7Reverse here means ‘>’, not ‘’.
Definition 4. A function f:X>, where
|f(x)|=f(x), is told to belong to K2
sif, for each
xXwe 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 (bx).
Remark 2. If the inequalities are obeyed in the re-
verse8situation by f, then fis said to be s2concave.
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 sconvex9if 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 sconvex10 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 (1λ)
log2(1
12s)y1+λ
log2(1
12s)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
sconcave.
S2
Definition 7. A real function f:X> Y , for which
|f(x)|=f(x), is called sconvex11 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
12
1
s)y1+
λ
log2(1
12
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 sconvex12 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
sconcave.
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.
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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 Hudzik and Maligranda (1994, [1]).
See the previous footnote. 12 See the previous footnote. References: [1] Hudzik, H.& Maligranda, L. Some remarks on siconvex functions
See the previous footnote. 12 See the previous footnote. References: [1] Hudzik, H.& Maligranda, L. Some remarks on siconvex functions, Aequationes Mathematicae 48 (1994), 100–111.