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This note supplements First Note on the Shape of S-convexity. We here deal with the negative pieces of the real functions.
International Journal of Pure and Applied Mathematics
Volume 92 No. 2 2014, 297-303
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
doi: http://dx.doi.org/10.12732/ijpam.v92i2.11
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ijpam.eu
SECOND NOTE ON THE SHAPE OF SCONVEXITY
I.M.R. Pinheiro
P.O. Box 12396 A’Beckett St.
Melbourne, Victoria, 8006, AUSTRALIA
Abstract: This note supplements First Note on the Shape of Sconvexity.
We here deal with the negative pieces of the real functions.
AMS Subject Classification: 26A51
Key Words: analysis, convexity, definition, S-convexity, geometry, shape
1. Introduction
We seem to have progressed quite a lot with the wording of the analytical
definitions for the Sconvex real functions (see, for instance, [1], paper that
comes straight after [2] in our series on improvements of the wording of the
definition of the Sconvexity phenomenon). One of the resulting analytical
definitions in [1] is:
Definition 1. A function f:X>, where |f(x)|=f(x), is told to
belong to K2
sif the inequality
f(λx + (1 λ)(x+δ)) λ1
sf(x) + (1 λ)1
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).
Received: February 12, 2014 c
2014 Academic Publications, Ltd.
url: www.acadpubl.eu
298 I.M.R. Pinheiro
Remark 1. If the inequality is obeyed in the supplementary1situation by
f, then fis said to be s2concave.
We now need to have a geometric definition for this group of Sconvex
functions.
The geometric definition of the convex functions has been worded by us in
[1] in the following way:
Definition 2. A real function f:X> Y is called convex 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 chord drawn between (x1;y1) and
(x2;y2) 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 chord
in terms of distance from the origin of the Cartesian axis.
We would like to word the geometric definition of the Sconvex functions
in a similar way:
Definition 3. A real function f:X> Y , 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.
To prove that the geometric rule for pertinence to the Sconvex class of
functions is the one that we here present, we remind the reader that (1λ)1
sy1+
λ1
sy2(1 λ)y1+λy2due to the allowed (by the definition) values for λ,s,
y1, and y2.
We prove that the geometric limiting line for Sconvexity is continuous in
Section 2.
In Section 3, we prove that, as sdecreases in value (as the distance from the
convexity limiting line is increased), the length of the limiting line increases,
therefore we prove that we have more functions in the 1/4-convex class than
in the 1/2-convex class, for instance, what provides us with certainty that
Sconvexity is a proper extension of convexity, also geometrically speaking.
1Supplementary here means ’>’, not ’’.
SECOND NOTE ON THE SHAPE OF SCONVEXITY 299
2. Continuity
We now prove that the function f(λ) = (1λ)1
sy1+λ1
sy2is continuous through
a few theorems from Real Analysis.
We know, for instance, that both the sum and the product of two continuous
functions are continuous functions (see, for instance, [3]). Notice that λ1
sis
continuous, given that 0 λ1 and 0 < s 1. y1and y2are constants,
therefore could be seen as constant functions, which are continuous functions.
(1 λ)1
sis continuous due to the allowed values for λand s. As a consequence,
(1 λ)1
sy1+λ1
sy2is continuous.
Notice that f(λ) = (1 λ)1
sy1+λ1
sy2is C, that is, is smooth (see [4], for
instance).
We do notice that we will have problems, for instance, with the first deriva-
tive of f(λ) of the sort n(see [5], for instance) or 00, when λ∈ {0,1}and
s= 0 or s>, but, in excluding 0 from our set of possible values for s, and
limiting it adequately, what is done in the definition of S-convex functions, we
disappear with those problems.
We say that s(0,1) when f(λ) = 1
s(1 λ)1
s
1y1+1
sλ1
s
1y2and s= 1
when f(λ) = y2y1.
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 Sconvexity limiting line use more than 100% or 100% split between the
addends, given that (1 λ)1
s≥ −(1 λ) and λ1
s≥ −λ(we are using the
negativity of the function here), we know that the limiting line for Sconvexity
lies always above or over the limiting line for convexity, and contains two points
that always belong to both the convexity and the sconvexity limiting lines
(first and last or (x1;y1) and (x2;y2)).
We now have then proved, in a definite manner, also in the shape of a paper,
that our limiting line for the Sconvexity phenomenon is smooth, continuous,
and located above or over the limiting line for the convexity phenomenon. Our
Sconvexity limiting line should also be concave when seen from the limiting
convexity line for the same points ((x1;y1) and (x2;y2)) (taking away the cases
in which y1=y2= 0 or s= 1), as we asserted to be the case in our talks at
the Victoria University of Technology (2001) and at the Adelaide University
(2005).
300 I.M.R. Pinheiro
3. Arc Length
Arc length is defined as the length along a curve,
sZγ
|dl|,
where dl is a differential displacement vector along a curve γ(see [6]).
In Cartesian coordinates, that means that the Arc Length of a curve is given
by
pZb
ap1 + f2(x)dx
whenever the curve is written in the shape r(x) = xˆx+f(x)ˆy.
Our limiting curve for Sconvexity could be expressed as a function of λ
in the following way:
f(λ) = (1 λ)1
sy1+λ1
sy2.
In deriving the above function in terms of λ, we get:
f(λ) = 1
s(1 λ)1s
sy1+1
sλ1s
sy2.
With this, our arc length formula will return:
pZ1
0r1 + [1
s(1 λ)1s
sy1+1
sλ1s
sy2]2dλ.
We will make use of a constant function, and we know that every constant
function is convex, therefore also S-convex (for every allowed value of s), to
study the limiting line for Sconvexity better.
We choose f(x) = 1 to work with (this function is suitable because
|f(x)|=| − 1|= 1 = f(x)).
We then have:
pZ1
0r1 + [1
s(1 λ)1s
s1
sλ1s
s]2dλ.
Notice that s0pindeterminate and s= 1 p= 1.
Notice that 0.25 will become 0.0625 when raised to 1
0.5and its supplement
through the formula (1 λ), 0.75, will become 0.5625.
In convexity, our results would have been 0.25 and 0.75 instead, that is,
75% and 25% less in negativity is gotten with Sconvexity, respectively.
SECOND NOTE ON THE SHAPE OF SCONVEXITY 301
We finish this section with a table2containing three of the possible values
for sand their respective arc lengths (good approximations. Only the first value
is precise) for the situation in which f(x) is replaced with 1 in the arc length
formula:
sArc Length
1 1
0.5 1.57
0.25 2
We notice that the naive formula from [7] gives us an excellent approxima-
tion to our arc length for the s-convexity limiting line, so that there is a good
chance that we can use the naive formula to calculate the arc length in this
situation always.
4. Maximum Height
The maximum height of the sconvexity limiting curve is reached when λ= 0.5
if fis constant and |f(x)|=f(x) because the first derivative of the function
describing the limiting line gives us zero for λ= 0.5 and changes sign from
positive to negative there.
5. Conclusion
This paper and [8], together, provide us with a geometric definition for the
phenomenon Sconvexity when it happens inside of the Universe of the Real
Numbers.
We here supplement [8] by dealing with the cases in which the function is
negative, present alternative argumentation as to why the shape of Sconvexity
is what we have declared it to be in our 2001 talk at the Victoria University of
Technology, talk given to the members of the Research Group on Inequalities
and Applications that there were working on a regular basis, and in our 2005
talk at the University of Adelaide, talk given to an academic audience working,
2The first value for Arc Length in the table has been attained through simple substitution
in the formula. The second value has been attained through using the formula for circum-
ference length. The third value has been attained through the naive formula from [7]. Hand
measurement has returned 2 as a result for the third value as well.
302 I.M.R. Pinheiro
or interested in working, with Geometry, and present a conjecture of major
importance that connects the sin Sconvexity with the arc length function
and gives us a relationship that determines how bent the limiting curve is in
function of the value of s(the conjecture is formed from the analysis of a few
experimental results).
Because of our calculations, and also because of a few graphical simulations,
we get to the conclusion that the definition of the phenomenon Sconvexity,
in what regards negative real functions, is not yet perfect, since the exponent
referring to the percentages taken from the main heights, let’s say, is not the
best (for instance, we are getting a bit more rope here than what we got in
the first case). Our definition for the negative case seems to be a very good
approximation, however, like it seems to be the closest thing that we can get
with nice exponents.
Future work may present alternative exponents.
References
[1] M.R. Pinheiro, Minima domain intervals and the Sconvexity, as well as
the convexity, phenomenon, Advances in Pure Mathematics,3(2013).
[2] M.R. Pinheiro, First note on the definition of S2convexity, Advances in
Pure Mathematics,1(2011), 1-2.
[3] L.S. Hush, Continuous Functions, Retrieved December 8, 2011 from
http://archives.math.utk.edu/visual.calculus/1/continuous.5/index.html
(1995).
[4] Planetmath authors, Continuously Differentiable, Retrieved December
9, 2011 from http://planetmath.org/encyclopedia/ ContinuouslyDifferen-
tiable.html (2011).
[5] E.W. Weisstein, Indeterminate, Retrieved December 12, 2010 from
http://mathworld.wolfram.com/Indeterminate.html (2013).
[6] E.W. Weisstein, Arc Length, Retrieved December 5, 2010 from
http://mathworld.wolfram.com/ArcLength.html (2002).
[7] G.P. Michon, Perimeter of an Ellipse, Retrieved December 12, 2011 from
http://www.numericana.com/answer/ellipse.htm (2000).
SECOND NOTE ON THE SHAPE OF SCONVEXITY 303
[8] M.R. Pinheiro, First note on the shape of S-convexity, International Jour-
nal of Pure and Applied Mathematics,90 (2014), 101-107, doi: 10.12732/ij-
pam.v90i1.12.
304
... Pieces of the geometrical definition that we here deal with ( [3] and [4]) ...
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Continuous Functions
  • L S Hush
L.S. Hush, Continuous Functions, Retrieved December 8, 2011 from http://archives.math.utk.edu/visual.calculus/1/continuous.5/index.html (1995).
  • Continuously Planetmath Authors
  • Differentiable
Planetmath authors, Continuously Differentiable, Retrieved December 9, 2011 from http://planetmath.org/encyclopedia/ ContinuouslyDifferentiable.html (2011).
Perimeter of an Ellipse
  • G P Michon
G.P. Michon, Perimeter of an Ellipse, Retrieved December 12, 2011 from http://www.numericana.com/answer/ellipse.htm (2000).
  • Continuously Planetmath Authors
  • Differentiable
Planetmath authors, Continuously Differentiable, Retrieved December 9, 2011 from http://planetmath.org/encyclopedia/ ContinuouslyDifferentiable.html (2011).