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Summary and importance of the results involving the definition of S-Convexity

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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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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 SCONVEXITY
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, sconvexity, geometry, shape
1. Introduction
In First Note on the Shape of Sconvexity, 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 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 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 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 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 sconvexity 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 sconvex functions to because 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 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 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 (bx).
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
12sf(x) + λ
log21
12sf(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 reverse1situation by f,
then fis said to be s2concave.
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
12
1
sf(x) + λlog2
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→ 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 (bx).
Remark 2. If the inequalities are obeyed in the reverse2situation 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:XY, 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 x16=x2, it happens that the line
drawn between (x1;y1) and (x2;y2) by means of the expression (1λ)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:XY, 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 x16=x2, it happens that the line
drawn between (x1;y1) and (x2;y2) by means of the expression
(1 λ)
log21
12sy1+λ
log21
12sy2,
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 sconcave.
2.B) Possibility 2
Definition 7. A real function f:XY, 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 x16=x2, it happens that the line
drawn between (x1;y1) and (x2;y2) by means of the expression
(1 λ)
log21
12
1
sy1+λ
log21
12
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:XY, 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 x16=x2, it happens that the line
drawn between (x1;y1) and (x2;y2) by means of the expression (1λ)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 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 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 sconvex 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 sconvex 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 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.
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 Sconvexity and yet
were convex. The problem was that Sconvexity 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: sconvex in the first sense and
sconvex in the second sense. They were called S1and S2convex 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 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.
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
Sconvexity 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 Sconvexity 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 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→ ℜ, 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)
log21
12sf(x) + λ
log21
12sf(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 reverse7situation by f,
then fis said to be s1concave.
S2
Definition 3. A function f:X→ ℜ, 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) + λlog2
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 ‘’.
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 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 reverse8situation 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:XY, 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 x16=x2, it happens that the line
drawn between (x1;y1) and (x2;y2) by means of the expression (1λ)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:XY, 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 x16=x2, it happens that the line
drawn between (x1;y1) and (x2;y2) by means of the expression
(1 λ)
log21
12sy1+λ
log21
12sy2,
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 sconcave.
S2
Definition 7. A real function f:XY, 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 x16=x2, it happens that the line
drawn between (x1;y1) and (x2;y2) by means of the expression
(1 λ)
log21
12
1
sy1+λ
log21
12
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:XY, 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 x16=x2, it happens that the line
drawn between (x1;y1) and (x2;y2) by means of the expression (1λ)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 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.
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.
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... In [8], we have decided to keep the name S 1 and replace the previous class K 1 s with a new version of it, which would be one of our possible definitions, as for [9]. So far, we have: ...
... We have discarded the original definition of K 1 s [11]. In [8], we decided to use its name and symbols to designate another class of functions because the intentions involved in the creation of this other class of functions are the same. ...
... All the conditions for us to have a proper extension seem to have been satisfied, so that this definition seems to be as good as the one we had before it,which is what we wanted to verify with this paper. The replacement has been proposed because ideally we would have the same height and shape for the limiting curve in both the negative and the nonnegative case, but such fact was not being verified with the previous definition, as seen in [8]. More studies are necessary to reach a final conclusion. ...
Data
Full-text available
... In [8], we have decided to keep the name S 1 and replace the previous class K 1 s with a new version of it, which would be one of our possible definitions, as for [9]. So far, we have: ...
... We have discarded the original definition of K 1 s [11]. In [8], we decided to use its name and symbols to designate another class of functions because the intentions involved in the creation of this other class of functions are the same. ...
... All the conditions for us to have a proper extension seem to have been satisfied, so that this definition seems to be as good as the one we had before it,which is what we wanted to verify with this paper. The replacement has been proposed because ideally we would have the same height and shape for the limiting curve in both the negative and the nonnegative case, but such fact was not being verified with the previous definition, as seen in [8]. More studies are necessary to reach a final conclusion. ...
Article
Full-text available
In this note we copy the work we presented on Second Note on the Shape of S-convexity [1], but apply the reasoning to one of the new limiting lines, limiting lines we presented on Summary and Importance of the Results Involving the Definition of S-Convexity [2]. This is about Possibility 1, second part of the definition, that is, the part that deals with negative real functions. We have called it S1 in Summary [2]. The first part has already been dealt with in First Note on the Shape of S-convexity [3]. This paper is about progressing toward the main target: Choosing the best limiting lines amongst our candidates.
... We now prove that the function 2 2 1 1 1 1 log log ...
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We are now in the final stages of publication. You will find the last version close to this paper online. It is a supplementary resource. The Research Gate system now does not allow for us to delete files or update. Shame.
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