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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 S−CONVEXITY

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 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.

AMS Subject Classiﬁcation: 26A51

Key Words: analysis, convexity, deﬁnition, S-convexity, geometry, shape

1. Introduction

In First Note on the Shape of S−convexity, we conﬁrmed the value of our re-

wording of the piece of deﬁnition for the phenomenon S−convexity 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 S−convexity, 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 S−convexity,

when the real function is negative, is a bit bigger than the limiting curve for

S−convexity 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 S−convexity, we study the perimeter of our

limiting curve for the phenomenon S−convexity 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 S−convexity 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< s2≤1;X/X ⊆ ℜ+∧X= [a, b];

∀δ/0< δ ≤(b−x).

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< s2≤1;X/X ⊆ ℜ+∧X= [a, b];

∀δ/0< δ ≤(b−x).

Remark 1. If the inequality is obeyed in the reverse1situation by f, then

fis told to be s2−concave.

Pieces of the geometrical deﬁnition that we here deal with ([3] and [4])

Deﬁnition 4. A real function f:X→Y, for which |f(x)|=f(x),

is called S−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 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 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 S−CONVEXITY 731

Deﬁnition 5. A real function f:X→Y, for which |f(x)|=−f(x),

is called S−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 line

drawn between (x1;y1) and (x2;y2) by means of the expression (1−λ)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):

p≡Z1

0p1 + [−s(1 −λ)s−1y1+sλs−1y2]2dλ.

In Second Note on the Shape of S−convexity, the expression for the Arc Length

was (negative real functions):

p≡Z1

0r1 + [−1

s(1 −λ)1−s

sy1+1

sλ1−s

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 S−convexity 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 S−convexity),

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 ) = (1−λ)δ1A+λδ1A,f(λcn) =

(λ−1)A−λA, and f(λscn) = −(1 −λ)1

sA−λ1

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

sA−λ1

sA−[(λ−1)A−λA]|.

What follows is:

|(1 −λ)δ1+λδ1−1|=| − (1 −λ)1

s−λ1

s+ 1|.

Making λ=1

2for practical purposes, we have:

|2−δ1+ 2−δ1−1|=| − 2−1

s−2−1

s+ 1|.

What follows is:

|2∗2−δ1−1|=| − 2∗2−1

s+ 1|.

Then:

(A) 2 ∗2−δ1−1 = −2∗2−1

s+ 1 or

(B) 2 ∗2−δ1−1 = 2 ∗2−1

s−1.

From A, we get:

21−δ1−1 = −21−1

s+ 1,

THIRD NOTE ON THE SHAPE OF S−CONVEXITY 733

∴21−δ1= 2 −21−1

s,

∴2δ1=2

2−21−1

s

,

∴2δ1=1

1−2−1

s

.

A then leads to our sbeing replaced with log21

1−2−1

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

1−2−1

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

1−2−1

s+λ

log21

1−2−1

s,

f′(λscp) = −log21

1−2−1

s(1 −λ)

log21

1−2−1

s−1

+ log21

1−2−1

sλ

log21

1−2

−1

s−1

,

(f′(λscp))2=

−log21

1−2−1

s(1 −λ)

log21

1−2

−1

s−1

+ log21

1−2−1

sλ

log21

1−2

−1

s−1

2

,

p≡Z1

0

1 + −log21

1−2−1

s(1 −λ)

log21

1−2

−1

s−1

734 I.M.R. Pinheiro

+ log21

1−2−1

sλ

log21

1−2

−1

s−12

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) =

(λ−1)A−λA, and f(λscn) = −(1 −λ)δ2A−λδ2A.

We then have |f(λscp)−f(λcp)|=|f(λscn)−f(λcn )|and, with this, |(1 −

λ)sA+λsA−[(1 −λ)A+λA]|=| − (1 −λ)δ2A−λδ2A−[(λ−1)A−λA]|.

What follows is:

|(1 −λ)s+λs−1|=| − (1 −λ)δ2−λδ2+ 1|.

Making λ=1

2for practical purposes, we have:

|2−s+ 2−s−1|=| − 2−δ2−2−δ2+ 1|.

What follows is:

|2∗2−s−1|=| − 2∗2−δ2+ 1|.

Then:

(A) 2 ∗2−s−1 = −2∗2−δ2+ 1 or

(B) 2 ∗2−s−1 = 2 ∗2−δ2−1.

From A, we get:

21−s−1 = −21−δ2+ 1,

∴21−δ2=−21−s+ 2,

∴2δ2=2

2−21−s,

∴2δ2=1

1−2−s.

A then leads to our sbeing replaced with log21

1−2−sand B leads to our

replacement being s.

THIRD NOTE ON THE SHAPE OF S−CONVEXITY 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

1−2−s.

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

1−2−s−λ

log21

1−2−s,

f′(λscn) = log21

1−2−s(1 −λ)

log21

1−2−s−1

−log21

1−2−sλ

log21

1−2−s−1

,

(f′(λscn))2=log21

1−2−s(1 −λ)

log21

1−2−s−1

−log21

1−2−sλ

log21

1−2−s−12

,

p≡Z1

0

1 + log21

1−2−s(1 −λ)

log21

1−2−s−1

−log21

1−2−sλ

log21

1−2−s−12

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

1−2

−1

sA+λ

log21

1−2

−1

sA

for when |f|=fand

f(λscn) = −(1 −λ)1

sA−λ1

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

1−2−sA−λ

log21

1−2−sA

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 S−CONVEXITY 737

7. Interval of Interest

For the sin S−convexity 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 S−convexity.

8. Conclusion

We have reﬁned the deﬁnition of the phenomenon S−convexity 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 S−convexity 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 S−convex functions to ℜbecause 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 S−convexity, 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 ≤(b−x).

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

1−2−sf(x) + λ

log21

1−2−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 ≤(b−x).

Remark 2. If the inequalities are obeyed in the reverse situation by f,

then fis said to be s2−concave.

Geometric Deﬁnition

Deﬁnition 11. A real function f:X→Y, for which |f(x)|=f(x),

is called S−convex3if 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:X→Y, for which |f(x)|=−f(x),

is called S−convex4if 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

1−2−sy1+λ

log21

1−2−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 3. If all the points deﬁning the function are located above the

limiting line instead, then fis called S−concave.

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 S−CONVEXITY 739

References

[1] M.R. Pinheiro, First note on the deﬁnition of S2−convexity, 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.

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