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First Note on the New Shape of S−convexity

Dr. M. R. Pinheiro∗

October 19, 2016

Abstract

In this note we copy the work we presented in Second Note on the

Shape of S-convexity, but apply the reasoning to one of the new limit-

ing curves, limiting curves we presented in Summary and Importance

of the Results Involving the Deﬁnition of S-Convexity. This is about

Possibility 1, second part of the deﬁnition, that is, the part that deals

with negative real functions. The ﬁrst part has already been dealt with

in Second Note on the Shape of S-convexity.

MSC(2010): 26A51

Key-words: Analysis, Convexity, Deﬁnition, S-convexity, geometry, shape.

I. Introduction

In [8], 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 [9].

So far, we have:

S1

Deﬁnition 1. A function f:X−><, where |f(x)|=f(x), is told to

belong to K1

sif, for each x∈Xwe select, and for all of them, the inequality

f((1 −λ)x+λ(x+δ))

∗Postal address: P.O. Box 12396 A’Beckett St, Melbourne, Victoria, Australia, 8006.

Electronic address: illmrpinheiro@gmail.com.

1

≤(1 −λ)sf(x) + λsf(x+δ)

holds ∀λ/λ ∈[0,1]; s=s1/0< s1<1;X/X ⊆ < ∧ X= [a, b];∀δ/0< δ ≤

(b−x).

Deﬁnition 2. A function f:X−><, where |f(x)|=−f(x), is told to

belong to K1

sif, for each x∈Xwe select, and for all of them, the inequality

f((1 −λ)x+λ(x+δ))

≤(1 −λ)

log21

1−2−sf(x) + λ

log21

1−2−sf(x+δ)

holds ∀λ/λ ∈[0,1]; s=s1/0< s1<1;X/X ⊆ < ∧ X= [a, b];∀δ/0< δ ≤

(b−x).

Remark 1.If the inequalities are obeyed in the reverse1situation by f, then

fis said to be s1−concave.

We are now going to study continuity, arc length, and maximum height of

the just-mentioned limiting line. We have discarded the original deﬁnition

of K1

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.

II. Continuity

We now prove that the function

f(λ) = (1 −λ)

log21

1−2−sf(x) + λ

log21

1−2−sf(x+δ)

is continuous through a few theorems from Real Analysis.

We know, for instance, that both the sum and the product of two con-

tinuous functions are continuous functions (see, for instance, [3]). Notice

that λ

log21

1−2−sis continuous, given that 0 ≤λ≤1 and 0 < s ≤1.

y1=f(x) and y2=f(x+δ) are constants, therefore could be seen as

constant functions, which are continuous functions. (1 −λ)

log21

1−2−s

1Reverse here means ‘>’, not ‘≥’.

2

is continuous due to the allowed values for λand s. As a consequence,

(1 −λ)

log21

1−2−sf(x) + λ

log21

1−2−sf(x+δ) is continuous.

Notice that f(λ) = (1 −λ)

log21

1−2−sf(x) + λ

log21

1−2−sf(x+δ) is C∞,

that is, is smooth (see [4], for instance).

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 −(1 −λ)

log21

1−2−s≥ −(1 −λ) and −λ

log21

1−2−s≥

−λ(we are using the negativity of the function here), we know that the

limiting line for S−convexity lies always above or over the limiting line for

convexity, and contains two points that always belong to both the convexity

and the s−convexity 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 S−convexity phenomenon is smooth, contin-

uous, and located above or over the limiting line for the convexity phe-

nomenon. Our S−convexity 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).

III. Arc Length

Arc length is deﬁned as the length along a curve,

s≡Zγ

|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

p≡Zb

ap1 + f02(x)dx

whenever the curve is written in the shape r(x) = xˆx+f(x)ˆy.

Our limiting curve for S−convexity could be expressed as a function of λin

the following way:

f(λ) = (1 −λ)

log21

1−2−sy1+λ

log21

1−2−sy2.

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In deriving the above function in terms of λ, we get:

f0(λ) = −log21

1−2−s(1 −λ)

log21

2(1−2−s)y1

+ log21

1−2−sλ

log21

2(1−2−s)y2.

With this, our arc length formula will return:

p≡Z1

0

v

u

u

u

t1+[−log21

1−2−s(1 −λ)

log21

2(1−2−s)y1+ log21

1−2−sλ

log21

2(1−2−s)y2]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 S−convexity better.

We choose f(x) = −1 to work with (this function is suitable because |f(x)|=

| − 1|=1=−f(x)).

We then have:

p≡Z1

0

v

u

u

u

t1 + [log21

1−2−s(1 −λ)

log21

2(1−2−s)−log21

1−2−sλ

log21

2(1−2−s)]2dλ.

Notice that s→0⇒p→indeterminate and s−>1⇒p−>≈1.

See:

Figure 1: Maple Plot, s=0.9

4

Notice that 0.25 will become 0.09 when raised to log21

1−2−0.5and its

supplement through the formula (1 −λ), 0.75, will become 0.6.

In convexity, our results would have been 0.25 and 0.75 instead, that is, 64%

and 20% less in negativity is gotten with S−convexity, respectively.

We now calculate the area under the curve by hand because Maple could not

compute it inside of an acceptable time interval: We notice that the vertex

of the graph that represents the function we are interested in is located on

(0.5; 1) in both cases. We can then draw a triangle on both sides of the

space we are interested in, and ﬁnd an approximation to our target area.

After that, we can subtract an approximation to the piece of the triangle we

cannot consider. From eye observation, we can tell that the second leaf is

about half of the ﬁrst, so that whatever we put for the ﬁrst, we just halve it

for the second.

f(λ) = v

u

u

u

t1 + [log21

1−2−s(1 −λ)

log21

2(1−2−s)−log21

1−2−sλ

log21

2(1−2−s)]2

Figure 2: Maple Plot, s=0.5 Figure 3: Maple Plot, s=0.25

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The top triangle of the ﬁrst graph has base of size 0.5 and height of size 1.

The top triangle of the second graph has base of size 0.5 and height of size

of about 1.8. Base times height will then be equal to 0.5 and 0.9.

The area of the just-mentioned triangles is then 0.25 and 0.45, but we have

two of each. We then have 0.5 and 0.9, respectively, as areas for the top part

of each one of the graphs, but we still have to take away the leaves.

Upon using the grid option from Maple, we concluded that it would be fair

saying that approximately 5% of each superior triangle from the ﬁrst graph

is above the curve, so that we have 95% of the original area left. That would

mean 95% of 0.25 twice or 0.475.

Similar reasoning leads to 0.6 for the second set, that is, 6x(0.5x0.1)x2 (ap-

proximation).

Now we have to add the rectangle we skipped when doing that, which is

what comes before the base line we drew: A rectangle of one unit of length

and one unit of height, and therefore a rectangle of area that is equal to 1.

To get the right approximation, we need to now put it all together: For

the ﬁgure 1, it will then be 0.475 + 1 = 1.475, and, for Figure 2, it will be

0.6 + 1 = 1.6.

We ﬁnish this section with a table containing 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 1.3

0.5 1.475

0.25 1.6

IV. Maximum height

The maximum height of the s−convexity 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.

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V. Conclusion

We here seem to have a more uniform limiting curve than the one we pre-

sented in Second Note because of the ﬁgures we get for the length. Our

studies on length are based on rough approximations, however.

We got 1,1.57, and 2 for the exponent swhen we used f(x) = 1.2. When

our exponent was 1

s, and our function was f(x) = −1, we got 1,1.57, and 2.

We now used the exponent log21

1−2−s, the function f(x) = −1, and got

1.3,1.475, and 1.6.

All the conditions for us to have a proper extension seem to have been sat-

isﬁed, 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 non-

negative case, but such fact was not being veriﬁed with the previous deﬁni-

tion, as seen in [8].

More studies are necessary to reach a ﬁnal conclusion. Perhaps two more

papers.

We need to worry more about the details involved in all.

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VI. References

[1] M. R. Pinheiro. Minima Domain Intervals and the S−convexity, as well

as the Convexity, Phenomenon. Advances in Pure Mathematics, vol. 3,

2013.

[2] M. R. Pinheiro. First Note on the Deﬁnition of S2−convexity. Advances

in Pure Mathematics, vol. 1, pp. 1-2, 2011.

[3] 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.

[4] Planetmath authors. From

http://planetmath.org/encyclopedia/ContinuouslyDiﬀerentiable.html. Ac-

cessed on the 9th of December of 2011.

[5] E. W. Weisstein. Indeterminate. From MathWorld - A Wolfram Web Re-

source. http://mathworld.wolfram.com/Indeterminate.html. Accessed on

the 12th of December of 2010.

[6] E. W. Weisstein. Arc Length. From MathWorld - A Wolfram Web Re-

source. http://mathworld.worlfram.com/ArcLength.html. Accessed on the

5th of December of 2010.

[7] G. P. Michon. Perimeter of an ellipse. From Numericana.com.

http://www.numericana.com/answer/ellipse.htm. Accessed on the 12th of

December of 2011.

[8] Pinheiro, M. R. Summary and importance of the results involving the

deﬁnition of S-Convexity, International Journal of Pure and Applied Math-

ematics, vol. 106, no. 3, pp. 699 − −713, 2016.

[9] Pinheiro, M. R. Third Note on the Shape of S-Convexity, International

Journal of Pure and Applied Mathematics, vol. 93, no. 5, pp. 729 − −739,

2014.

[10] Pinheiro, M. R. Second Note on the Shape of S-convexity, International

Journal of Pure and Applied Mathematics, vol. 92, no. 2, pp. 297–303, 2014.

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[11] Pinheiro, M. R. Second Note on the Deﬁnition of S1-Convexity. Ad-

vances in Pure Mathematics, vol. 5, pp. 127ˆa130, 2015.

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