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Second Note on the New Shape of S-Convexity

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Abstract In this note, we copy the work we presented in Second Note on the Shape of S-convexity [1], but apply the reasoning to one of the new limiting lines, limiting lines we presented in Summary and Importance of the Results Involving the Definition of S-Convexity [2]. That line was called New Positive System in Third Note on the Shape of S-convexity [3] because, on that instance, the images of the points of the domain had been replaced with a positive constant, which we called A . This is about Possibility 2 of Summary and Importance of the Results Involving the Definition of S-Convexity. We have called it S2 in Summary and Importance of the Results Involving the Definition of S-Convexity, and New Positive System in Third Note on the Shape of S-convexity. The second part has already been dealt with in Second Note on the Shape of S-convexity. In Second Note on the Shape of S-convexity, we have already performed the work we performed in First Note on the Shape of S-convexity [4] over the case in which the modulus does not equate the function in the system S2 from Summary and Importance of the Results Involving the Definition of S-Convexity. This paper is about progressing toward the main target: Choosing the best limiting lines amongst our candidates.
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Modern International Journal of Pure and Applied Mathematics 2017, 1(3): 46-49
DOI: 10.5923/j.mijpam.20170103.02
Second Note on the New Shape of S-Convexity
Marcia R. Pinheiro
IICSE University, USA
Abstract In this note, we copy the work we presented in Second Note on the Shape of S-convexity [1], but apply the
reasoning to one of the new limiting lines, limiting lines we presented in Summary and Importance of the Results Involving
the Definition of S-Convexity [2]. That line was called New Positive System in Third Note on the Shape of S-convexity [3]
because, on that instance, the images of the points of the domain had been replaced with a positive constant, which we
called
A
. This is about Possibility 2 of Summary and Importance of the Results Involving the Definition of S-Convexity.
We have called it
2
S
in Summary and Importance of the Results Involving the Definition of S-Convexity, and New
Positive System in Third Note on the Shape of S-convexity. The second part has already been dealt with in Second Note on
the Shape of S-convexity. In Second Note on the Shape of S-convexity, we have already performed the work we performed
in First Note on the Shape of S-convexity [4] over the case in which the modulus does not equate the function in the system
2
S
from Summary and Importance of the Results Involving the Definition of S-Convexity. This paper is about progressing
toward the main target: Choosing the best limiting lines amongst our candidates.
Keywords Analysis, Convexity, Definition, S-convexity, Geometry, Shape
1. Introduction
Figure 1. Third Note, page 736
* Corresponding author:
drmarciapinheiro@gmail.com (Marcia R. Pinheiro)
Published online at http://journal.sapub.org/mijpam
Copyright © 2017 Scientific & Academic Publishing. All Rights Reserved
Modern International Journal of Pure and Applied Mathematics 2017, 1(3): 46-49 47
Figure 2. Third Note, page 736
We here study the first inequality you see above, inside of the New Positive System, which is the first system we present in
this Introduction. We then study the inequality that is used when the modulus of the function is equal to the own function.
2. Continuity
We now prove that the function
22
11
11
log log
12 12
( ) (1 )
ss
f AA
λλ λ
−−
 
 
 
 
 
=−+
(1)
is continuous through a few theorems from Real Analysis.
We know that both the sum and the product of two continuous functions are continuous functions (see [5]). Notice that
21
1
log
12s
λ



is continuous, given that
01
λ
≤≤
and
01s<≤
.
A
is a constant, therefore could be seen as a constant
function, and the constant function is a continuous function.
21
1
log
12
(1 )
s
λ




is continuous due to the allowed values for
and
s
.
Making
1A=
and
0.5s=
in (1), we get the graph below:
Figure 3. Maple Plot
48 Marcia R. Pinheiro: Second Note on the New Shape of S-Convexity
Notice that
22
11
11
log log
12 12
( ) (1 ) ( ) ( )
ss
f fx fx
λλ λ δ
−−
−−
 
 
 
 
 
=− ++
is
C
,
that is, is smooth (see [6] and [7]).
Figure 4. Value of s
Because the coefficients that form the convexity limiting line use 100% split between the addends and form straight lines
and the coefficients that form the
S
convexity limiting line use more than 100% or 100% split between the addends,
given that
21
1
log
12
(1 ) (1 )
s
λλ





− ≥−
and
21
1
log
12
s
λλ





(we are using the non-negativity of the function here, so
0A
, but also [9]), we know that the limiting line for
S
convexity lies always above or over the limiting line for
convexity, with two points that always belong to both the convexity and the
s
convexity limiting lines (first and last or
(
11
;xy
) and (
22
;xy
)).
We now have then proved, in a definite manner, that our limiting line for
2
S
, first part, is smooth, continuous, and
located above or over the limiting line for the convexity phenomenon. Our
S
convexity limiting line should also be
concave when seenfrom the limiting convexity line for the same points ((
11
;xy
) and (
22
;xy
)) (taking away the cases in
which
12
0yy= =
or
1s=
).
3. Conclusions
1
S
and
2
S
are equally acceptable as extensions of the phenomenon Convexity. Notwithstanding, one must remember
that the names remain the same (
1
s
and
2
s
), but we have in much modified the original definitions that appear in
association with them. Throughout the years we have even completely discarded the original definition associated with
1
s
([8],[9]). We are then left with three possibilities to choose from: Not only our
1
S
and
2
S
, but also the more
geometrically unpleasant, but more analytically welcome, couple formed from the second part of the New Positive and the
first part of the New Negative systems [10].
We must say that we did not find an ideal match, a match that would bear both analytical and geometrical perfection.
The main issue is that we are looking for a single analytical shape that will satisfy both negative and non-negative functions,
if possible, something that somehow connects to both the analytical and the geometrical definition [11] of convexity, and
will still represent a geometrical distance that is equal in terms of both the
S
convexity and the convexity line for both
parts of the definition (non-negative and negative). If we were looking into increasing the scope of the line to the upper part
on the non-negative side and to the lower part on the negative side, we would not have the same difficulties...
Considering the simplest case, of a straight line on a 45-degree angle with the horizontal axis, we would have to move
the line that describes the function and the limiting line to the second quadrant, and therefore we would have to mess up
with our domain variable, horizontal change, and we then would have to move the set to the third quadrant, and therefore
Modern International Journal of Pure and Applied Mathematics 2017, 1(3): 46-49 49
we would have to mess up with our image variable, vertical change. On the last move, it is simply going down the
Cartesian Plane, so that we would have to subtract something from the function itself. That would lead to an undesirable
algebraic shape straight away, since we would lose in terms of the meaning of the factors that mark our percentage in
Convexity.
That was thinking as in Vector Algebra, but notice that the fact that the function changes sign will make the inequality
change direction, and that is a lot of trouble.
It looks like preserving that meaning will cost us somewhere: geometrical aspect, algebraic aspect, complexity of the
calculation, and so on. Most of us would privilege calculation, easiness, over perfect geometric match between
non-negative and negative parts of the function.
REFERENCES
[1] Pinheiro, M. R. (2014). Second Note on the Shape of S-convexity, International Journal of Pure and Applied Mathematics 92(2),
297-303.
[2] Pinheiro, M. R. (2016). Summary and importance of the results involving the definition of S-Convexity, International Journal of
Pure and Applied Mathematics 106(3), 699-713.
[3] Pinheiro, M. R. (2014). Third Note on the Shape of S-convexity, International Journal of Pure and Applied Mathematics 93(5),
729-739.
[4] Pinheiro, M. R. (2014). First Note on the Shape of S-convexity, International Journal of Pure and Applied Mathematics 90(1),
101-107.
[5] L. S. Hush. (1995). Continuous Functions. Visual Calculus. http://archives.math.utk.edu/visual.calculus/1/continuous.5/index.html.
Accessed on the 8th of December of 2011.
[6] Planetmath authors. (2017). http://planetmath.org/encyclopedia/ContinuouslyDifferentiable.html. Accessed on the 9th of December
of 2011.
[7] Mueller, W. (2017). Exponential Functions. http://wmueller.com/precalculus/families/1{\_}21.html. Accessed on the 9th of
November of 2017.
[8] Pinheiro, M. R. (2015). Second Note on the Definition of S1-Convexity. Advances in Pure Mathematics 5, 127-130.
[9] Pinheiro, M. R. and Maleki, Y. (2016). Master Yaser and Dr. Pinheiro: S1-Convexity.
http://mathematicalcircle.blogspot.com.au/2016/12/master-yaser-and-dr-pinheiro-s1.html. Accessed on the 9th of November of
2017.
[10] Pinheiro, M. R. (2011). First Note on the Definition of S2-convexity. Advances in Pure Mathematics 1. 1-2.
[11] Pinheiro, M. R. (2012). Minima Domain Intervals and the S-Convexity, as well as the Convexity, Phenomenon. Advances in Pure
Mathematics 2(6).
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Continuous Functions. Visual Calculus
  • L S Hush
L. S. Hush. (1995). Continuous Functions. Visual Calculus. http://archives.math.utk.edu/visual.calculus/1/continuous.5/index.html. Accessed on the 8th of December of 2011.
Exponential Functions
  • W Mueller
Mueller, W. (2017). Exponential Functions. http://wmueller.com/precalculus/families/1{\_}21.html. Accessed on the 9th of November of 2017.