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Third Note on the Shape of S-convexity

Authors:
  • IICSE University

Abstract

As promised in Second Note on the Shape of S−convexity, we now discuss the exponent of the piece of definition for S−convexity that deals with negative images of real functions. We also present a severely improved definition for the phenomenon.
International Journal of Pure and Applied Mathematics
Volume 93 No. 5 2014, 729-739
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
doi: http://dx.doi.org/10.12732/ijpam.v93i5.11
P
A
ijpam.eu
THIRD NOTE ON THE SHAPE OF SCONVEXITY
I.M.R. Pinheiro
P.O. Box 12396, A’Beckett St
Melbourne, Victoria, AUSTRALIA, 8006
Abstract: As promised in Second Note on the Shape of Sconvexity, we
now discuss the exponent of the piece of definition for Sconvexity that deals
with negative images of real functions. We also present a severely improved
definition for the phenomenon.
AMS Subject Classification: 26A51
Key Words: analysis, convexity, definition, S-convexity, geometry, shape
1. Introduction
In First Note on the Shape of Sconvexity, we confirmed the value of our re-
wording of the piece of definition for the phenomenon Sconvexity that deals
with non-negative real functions and we proposed a geometric definition for the
phenomenon.
In Second Note on the Shape of S-convexity, we confirmed that our added
piece of definition for the phenomenon Sconvexity, that for negative real func-
tions, is a proper extension of Convexity and we proposed a geometric definition
for that case.
Still in our second note, we observed that our limiting curve for Sconvexity,
when the real function is negative, is a bit bigger than the limiting curve for
Sconvexity when the real function is not negative in terms of length, what
means that our lift is not the same for both cases.
Received: April 14, 2014 c
2014 Academic Publications, Ltd.
url: www.acadpubl.eu
730 I.M.R. Pinheiro
Our perimeters are still close enough: The discrepancy appears by the first
decimal digit only and is less than 0.5 in dimension (considering our approxi-
mation for pi, our manual calculation, and the approximation for the perimeter
via elliptical curve or circumference).
In Third Note on the Shape of Sconvexity, we study the perimeter of our
limiting curve for the phenomenon Sconvexity aiming at equal perimeters for
situations that could be seen as similar, geometrically speaking.
As a consequence, we here propose new refinements to the definition of the
phenomenon.
Pieces of the analytical definition of the phenomenon Sconvexity that we
here deal with ([1] and [2])
Definition 2. A function f:X→ ℜ, where |f(x)|=f(x), is told to
belong to K2
sif the inequality
f(λx + (1 λ)(x+δ)) λsf(x) + (1 λ)sf(x+δ)
holds λ/λ [0,1]; x/x X;s=s2/0< s21;X/X ⊆ ℜ+X= [a, b];
δ/0< δ (bx).
Definition 3. A function f:X→ ℜ, where |f(x)|=f(x), is told to
belong to K2
sif the inequality
f(λx + (1 λ)(x+δ)) (1 λ)1
sf(x) + λ1
sf(x+δ)
holds λ/λ [0,1]; x/x X;s=s2/0< s21;X/X ⊆ ℜ+X= [a, b];
δ/0< δ (bx).
Remark 1. If the inequality is obeyed in the reverse1situation by f, then
fis told to be s2concave.
Pieces of the geometrical definition that we here deal with ([3] and [4])
Definition 4. A real function f:XY, for which |f(x)|=f(x),
is called Sconvex 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)sy1+λsy2,
where λ[0,1], does not contain any point with height, measured against the
vertical Cartesian axis, that be 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.
1Reverse here means >, not .
THIRD NOTE ON THE SHAPE OF SCONVEXITY 731
Definition 5. A real function f:XY, for which |f(x)|=f(x),
is called Sconvex 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 be 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.
6. Chasing Equal Lengths for Both Cases
In [3], we have reached the following expression for the Arc Length of our
limiting curve (non-negative real functions):
pZ1
0p1 + [s(1 λ)s1y1+s1y2]2dλ.
In Second Note on the Shape of Sconvexity, the expression for the Arc Length
was (negative real functions):
pZ1
0r1 + [1
s(1 λ)1s
sy1+1
sλ1s
sy2]2dλ.
Our tables, containing samples of values for sand their respective Arc Lengths,
had been put together through approximating values to no decimals in the third
listed value for s, 0.25, and through approximating the value of pi to 3.141516
plus the result of the calculation to two decimal digits in the second listed value
for s, 0.5.
The negative case is giving more rope than the positive case, since we have
1.2 (non-negative) for one case and 1.0 (negative) for the other.
That should mean that we get more functions in each s-group if we choose
to equate the limiting line for the non-negative functions to the limiting line
for the negative ones.
To chase equal lengths for both cases, we first observe that if the percentage
we select inside of the brackets when the definition of the Sconvexity limiting
line is used is the same, what commands the result is the value of the function,
first of all, and then the value of the exponent we raise the percentage to.
This way, to compare both cases and chase the same result, we should start
by finding functions that hold the same value in modulus in the interval we
choose to consider.
732 I.M.R. Pinheiro
Whilst, in the non-negative case, we increase the value of the percentage
when raising it to a fractionary exponent, given that it is always at most 100%
and at least 0%; in the negative case, we decrease it if doing the same because
increasing the portion we take from a negative value is making the result smaller,
not bigger, as it happens with the positive value.
That tells the reader why we guessed 1
sfor the second part of our definition.
To chase equal lengths, we use the convexity limiting line as a reference line
for both cases and equate the moduli of the differences between the limiting
lines for the extension of convexity and the limiting lines for convexity.
We choose the constant functions to work with because they are the nicest
functions in the group of functions available to us inside of the phenomenon.
With this, suppose that g(x) = A, A > 0 (notice that A= 0 implies that
the limiting line for convexity is equal to the limiting line for Sconvexity),
and h(x) = A.
If we look for an exponent that produce the same effect we get with 1
swhen
we have a negative function, we may call this exponent δ1and think in the
following way:
Suppose that f(λcp) = (1 λ)A+λA,f(λscp ) = (1λ)δ1A+λδ1A,f(λcn) =
(λ1)AλA, and f(λscn) = (1 λ)1
sAλ1
sA.
We then have |f(λscp)f(λcp)|=|f(λscn)f(λcn )|and, with this, |(1
λ)δ1A+λδ1A[(1 λ)A+λA]|=| − (1 λ)1
sAλ1
sA[(λ1)AλA]|.
What follows is:
|(1 λ)δ1+λδ11|=| − (1 λ)1
sλ1
s+ 1|.
Making λ=1
2for practical purposes, we have:
|2δ1+ 2δ11|=| − 21
s21
s+ 1|.
What follows is:
|22δ11|=| − 221
s+ 1|.
Then:
(A) 2 2δ11 = 221
s+ 1 or
(B) 2 2δ11 = 2 21
s1.
From A, we get:
21δ11 = 211
s+ 1,
THIRD NOTE ON THE SHAPE OF SCONVEXITY 733
21δ1= 2 211
s,
2δ1=2
2211
s
,
2δ1=1
121
s
.
A then leads to our sbeing replaced with log21
121
sand B leads to our
replacement being 1
s.
Because both differences should be non-negative (we are always doing the
limiting line of the extension minus the limiting line of convexity), only A is a
valid result.
It is then the case that δ1= log21
121
s.
With this, our function is smooth of class C([5], p. 511).
To calculate our perimeters, we now have to use the formula we had for the
negative case and the new formula, which will be a result of our calculations.
Assume that A= 1.
f(λscp) =(1 λ)
log21
121
s+λ
log21
121
s,
f(λscp) = log21
121
s(1 λ)
log21
121
s1
+ log21
121
sλ
log21
12
1
s1
,
(f(λscp))2=
log21
121
s(1 λ)
log21
12
1
s1
+ log21
121
sλ
log21
12
1
s1
2
,
pZ1
0
1 + log21
121
s(1 λ)
log21
12
1
s1
734 I.M.R. Pinheiro
+ log21
121
sλ
log21
12
1
s12
1
2
dλ.
Let’s call this new set of functions New Positive.
Now we try to use the exponent we have for the non-negative case as a
model and see what we get ([4]).
If we look for an exponent that produce the same effect we get with swhen
we have a non-negative function, we may call this exponent δ2and think in the
following way:
Suppose that f(λcp ) = (1 λ)A+λA,f(λscp) = (1 λ)sA+λsA,f(λcn) =
(λ1)AλA, and f(λscn) = (1 λ)δ2Aλδ2A.
We then have |f(λscp)f(λcp)|=|f(λscn)f(λcn )|and, with this, |(1
λ)sA+λsA[(1 λ)A+λA]|=| − (1 λ)δ2Aλδ2A[(λ1)AλA]|.
What follows is:
|(1 λ)s+λs1|=| − (1 λ)δ2λδ2+ 1|.
Making λ=1
2for practical purposes, we have:
|2s+ 2s1|=| − 2δ22δ2+ 1|.
What follows is:
|22s1|=| − 22δ2+ 1|.
Then:
(A) 2 2s1 = 22δ2+ 1 or
(B) 2 2s1 = 2 2δ21.
From A, we get:
21s1 = 21δ2+ 1,
21δ2=21s+ 2,
2δ2=2
221s,
2δ2=1
12s.
A then leads to our sbeing replaced with log21
12sand B leads to our
replacement being s.
THIRD NOTE ON THE SHAPE OF SCONVEXITY 735
Because both differences should be non-negative (we are always doing the
limiting line of the extension minus the limiting line of convexity), only A is a
valid result.
It is then the case that δ2= log21
12s.
With this, our function is smooth of class C([5], p. 511).
Once more, to calculate our perimeters, we have to use the formula we
had for the negative case and the new formula, which will be a result of our
calculations.
Assume that A= 1.
f(λscn) = (1 λ)
log21
12sλ
log21
12s,
f(λscn) = log21
12s(1 λ)
log21
12s1
log21
12sλ
log21
12s1
,
(f(λscn))2=log21
12s(1 λ)
log21
12s1
log21
12sλ
log21
12s12
,
pZ1
0
1 + log21
12s(1 λ)
log21
12s1
log21
12sλ
log21
12s12
1
2
dλ.
Let’s call this new set of functions New Negative.
Our new tables2, for the cases in which our functions are f(x) = 1 and
f(x) = 1, just to exemplify, are:
2The first value for Arc Length in the table has been attained through simple substitution in the formula. The
second and the third values have been attained by means of the formula for circumference length (average between
vertical and horizontal values).
736 I.M.R. Pinheiro
New Positive System
f(λscp) = (1 λ)
log21
12
1
sA+λ
log21
12
1
sA
for when |f|=fand
f(λscn) = (1 λ)1
sAλ1
sA
for when |f|=f.
Table 1: f(x)=1
sArc Length
1 1
0.5 1.57
0.25 2.16
New Negative System
f(λscp) = (1 λ)sA+λsA
for when |f|=fand
f(λscn) = (1 λ)
log21
12sAλ
log21
12sA
for when |f|=f.
Table 2: f(x)=-1
sArc Length
1 1
0.5 1.43
0.25 1.85
THIRD NOTE ON THE SHAPE OF SCONVEXITY 737
7. Interval of Interest
For the sin Sconvexity to make sense, we should probably stick to the New
Negative System. In this case, no better exponent should be found.
We then have 1/s providing us with a nicer shape, and perhaps making us
have less work with the formula by the time of applying it, and the δ2we have
found here, which gives us the so expected equal distance for both pieces of the
definition of the phenomenon (non-negative and negative).
If we stick to what we had before, we are farther from what we expect
in Mathematics, so that we here suggest that we now use the New Negative
System as a definition for Sconvexity.
8. Conclusion
We have refined the definition of the phenomenon Sconvexity once more.
This time, it is about the distance between the limiting line of the extension
of convexity and the limiting line of convexity being the same in both pieces
of our definition (the non-negative functions form one piece and the negative
functions form another).
We have decided to deal with the phenomenon Sconvexity 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).
We have extended the domain of the Sconvex functions to because the
definition should only impose limitations to the image of the functions, not to
the domain.
We have decided to swap the coefficients in our definition because λ= 0
should bring f(x)to life, not f(x+δ).
We have added the interval of definition of sto our geometric definition to
make it be independent from the analytical definition.
Based on our new findings and decisions, we produce a new update in our
definition for the phenomenon Sconvexity, update that we present below.
Analytical Definition
Definition 9. 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).
738 I.M.R. Pinheiro
Definition 10. 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 2. If the inequalities are obeyed in the reverse situation by f,
then fis said to be s2concave.
Geometric Definition
Definition 11. 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,x16=x2, and y16=y2, 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 12. 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,x16=x2, and y16=y2, 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.
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.
4smust 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.
THIRD NOTE ON THE SHAPE OF SCONVEXITY 739
References
[1] M.R. Pinheiro, First note on the definition of S2convexity, Advances in
Pure Mathematics,1(2011), 1-2, doi: 10.4236/apm.2011.11001.
[2] M.R. Pinheiro, Minima domain intervals and the S-convexity, as well as the
convexity, phenomenon, Advances in Pure Mathematics,2, No. 6 (2012),
457-458, doi: 10.4236/apm.2012.26069.
[3] M.R. Pinheiro, First note on the shape of S-convexity, International Jour-
nal of Pure and Applied Mathematics,90, No. 1 (2014), 101-107, doi:
10.12732/ijpam.v90i1.12.
[4] M.R. Pinheiro, Second note on the shape of S-convexity, International
Journal of Pure and Applied Mathematics,92, No. 2 (2014), 297-303, doi:
10.12732/ijpam.v92i2.11.
[5] E.V. Shikin, Handbook and Atlas of Curves, CRC Press (1995), ISBN:
9780849389634.
740
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