Third Note on the Shape of S−convexity

I.M.R. Pinheiro∗

April 22, 2013

Abstract

As promised in Second Note on the Shape of s−convexity, we now dis-cuss the exponent of the definition of the s−convex class of functionsthat deals with negative images of real functions. We also present aseverely improved version of the definition of the phenomenon s−convexity.

MSC(2000): 26A51Key-words: Analysis, Convexity, Definition, S-convexity, s−convexity, ge-ometry, shape.

I. Introduction

In First Note on the Shape of S−convexity, we have presented more evidenceon our re-wording of the piece of definition of the phenomenon s−convexitythat deals with non-negative real functions being of fundamental impor-tance for Mathematics and have proposed a geometric definition for thephenomenon.In Second Note on the Shape of S-convexity, we have once more proved thatour proposed modifications to the definition of the phenomenon s−convexity,this time for the negative share of the real functions, constitute a major steptowards making the phenomenon be a proper extension of Convexity. Wehave also proposed a geometric definition for the negative case.We imagine that our limiting curve for s−convexity, when the real functionis negative, may be a bit bigger than the limiting curve for s−convexitywhen the real function is not negative in terms of length, what means that

∗Postal address: P.O. Box 12396 A’Beckett St, Melbourne, Victoria, Australia, 8006.Electronic address: illmrpinheiro@gmail.com.

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our lift may not be the same for both cases.Our perimeters seem to be close enough in dimension, however, the differ-ence being noticed by the first decimal digit only and being less than 0.5(considering our approximation for pi, our manual calculations, and the ap-proximation for the perimeter via elliptical curve).In Third Note on the Shape of s−convexity, the present note, we studythe perimeter of our limiting curve for the phenomenon s−convexity aimingequal perimeters for geometrically equivalent situations.As a consequence, we propose new refinements to the definition of the phe-nomenon at the end of this note.The limiting curve originates in the application of the definition of the phe-nomenon.Piece of the analytical definition of the phenomenon s−convexity that wehere deal with (see [Pin11] and [Pin13], for instance)

Definition 1. A function f : X− > ℜ, where |f(x)| = −f(x), is told tobelong to K2

s if the inequality

f((1− λ)x+ λ(x+ δ))

≤ (1− λ)1s f(x) + λ

1s f(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).

Piece of the geometrical definition that we here deal with

Definition 2. A real function f : X− > Y , for which |f(x)| = −f(x), iscalled s−convex if and only if, for all choices (x1; y1) and (x2; y2), where{x1, x2} ⊂ X, {y1, y2} ⊂ Y , Y = Imf , and x1 ̸= x2, it happens that theline drawn between (x1; y1) and (x2; y2) by means of the expression (1 −λ)

1s y1 + λ

1s y2, where λ ∈ [0, 1], does not contain any point with height,

measured against the vertical Cartesian axis, that is inferior to the height ofits horizontal equivalent in the curve representing the ordered pairs of f inthe interval considered for the line in terms of distance from the origin ofthe Cartesian axis.

II. Chasing equal lengths for both cases

In First Note on the Shape of s−convexity, we have reached the followingexpression for the Arc Length of our limiting curve (non-negative real func-

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tions):

p ≡∫ 1

0

√1 + [−s(1− λ)s−1y1 + sλs−1y2]2dλ.

In Second Note on the Shape of s−convexity, the expression for the ArcLength was (negative real functions):

p ≡∫ 1

0

√1 + [−1

s(1− λ)

1−ss y1 +

1

sλ

1−ss y2]2dλ.

Our tables, containing some of the allowed values for s and their respectiveArc Lengths, had been put together through approximating values to nodecimals in the third listed value for s, 0.25, and through approximatingthe value of pi to 3.141516 plus the result of the calculation to two decimaldigits in the second listed value for s, 0.5.There is some chance that the negative case ‘give more rope’ than the non-negative case, for instance. That would imply that we would get more func-tions in each ‘s-group’ if we chose to make the non-negative functions limitingline be ‘the same’ as the negative functions limiting line (in a relative man-ner).To chase equal lengths for both cases, we first observe that if the percent-age we select inside of the brackets, when the definition of the s−convexitylimiting line is used, is the same, what commands the result is the value ofthe function, first of all, and then the value of the exponent we raise thepercentage to.This way, to compare both cases and chase the same result, we should startby finding functions that hold the same value in modulus in the interval wechoose to consider.Whilst in the non-negative case we increase the value of the percentage whenraising it to a fractionary exponent, given that it is always at most 100% andat least 0%; in the negative case, we decrease it if doing the same becauseincreasing the portion we take from a negative value is making the resultsmaller, not bigger.That tells the reader why we have guessed 1

s for the second part of our def-inition.To chase equal lengths, we use the convexity limiting line as a reference linefor both cases and equate the moduli of the differences between the limitinglines for the extension of convexity and the limiting lines for convexity.We choose the constant functions to work with because they are the nicestfunctions 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

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limiting line for convexity is equal to the limiting line for s−convexity), andh(x) = −A.Also suppose that f(λcp) = (1 − λ)A + λA, f(λscp) = (1 − λ)δ1A + λδ1A,

f(λcn) = (λ − 1)A − λA, f(λscn) = −(1 − λ)1sA − λ

1sA, f1(λscp) = (1 −

λ)sA+ λsA, and f1(λscn) = −(1− λ)δ2A− λδ2A.We then have |f(λscp)−f(λcp)| = |f(λscn)−f(λcn)| and |f1(λscp)−f(λcp)| =|f1(λscn)− f(λcn)|.With this:A) |(1−λ)δ1A+λδ1A−[(1−λ)A+λA]| = |−(1−λ)

1sA−λ

1sA−[(λ−1)A−λA]|;

and B) |(1− λ)sA+ λsA− [(1− λ)A+ λA]| = | − (1− λ)δ2A− λδ2A− [(λ−1)A− λA]|What follows is:|(1 − λ)δ1 + λδ1 − 1| = | − (1 − λ)

1s − λ

1s + 1| and |(1 − λ)s + λs − 1| =

| − (1− λ)δ2 − λδ2 + 1|.Making λ = 1

2 for practical purposes, we have:

|2−δ1+2−δ1−1| = |−2−1s −2−

1s +1| and |2−s+2−s−1| = |−2−δ2−2−δ2+1|.

What follows is two possibilities for each case, that is, four possibilities:A) |2 ∗ 2−δ1 − 1| = | − 2 ∗ 2−

1s + 1|.

Then:(A1) 2 ∗ 2−δ1 − 1 = −2 ∗ 2−

1s + 1 or (A2) 2 ∗ 2−δ1 − 1 = 2 ∗ 2−

1s − 1.

B) |2 ∗ 2−s − 1| = | − 2 ∗ 2−δ2 + 1|.Then:(B1) 2 ∗ 2−s − 1 = −2 ∗ 2−δ2 + 1 or (B2) 2 ∗ 2−s − 1 = 2 ∗ 2−δ2 − 1.(A1) returns δ1 = log2

1

1−2−1sand (A2) returns δ1 =

1s .

(B1) returns δ2 = log21

1−2−s and (B2) returns δ2 = s.

When we use Mathematica and the Arc Length function, we notice that 1s

wins over s up to a value (more than 0.5) and loses after that, and, if weconsider the variation of the differences, we will prefer 1

s over s, like it will‘give us more rope’.The main problem that appears here is that we cannot extend a functionthat is basically founded on a geometric idea (under a line or over it, linethat is symmetric if we consider its middle part), in a way to get a nongeometrically-alike limiting line, and we have just become suspicious of thisline, like we have just started to think that that (absence of symmetry) mightbe the case with both s and 1

s .Testing such a hypothesis is not difficult. We just have to get one examplefor each case, exhibit their limiting lines, and prove that the first derivativeapplied to points that are equally distant from the center of the limiting line(λ = 0.5) will generate different results (in terms of modulus).

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One of the most trivial examples of s−convex functions of the sort |f(x)| =f(x) is f(x) =

√x.

√x is 0.5−convex, for instance. Accompany the proof:

Proof. A = f((1− λ)x+ λ(x+ δ)) =√(1− λ)x+ λ(x+ δ).

B = (1− λ)0.5f(x) = (1− λ)0.5√x.

C = λ0.5f(x+ δ) = λ0.5√x+ δ.

A2 = (1− λ)x+ λ(x+ δ).(B + C)2 = (1− λ)x+ λ(x+ δ) + 2[(1− λ)0.5

√x](λ0.5

√x+ δ).

A2 ≤ (B + C)2 because A, B, and C are all non-negative. For the samereason, A2 ≤ (B + C)2 =⇒ A ≤ B + C.

Now, we will worry about our limiting line only, the 0.5-line. We call it L(λ).This way, L(λ) = (1− λ)0.5

√x+ λ0.5

√x+ δ.

∴ L′(λ) = 0.5(1− λ)−0.5(−1)y1 + 0.5λ−0.5y2.

With this,

L′(0.3) = −0.5(0.7)−0.5y1 + 0.5(0.3)−0.5y2

and

L′(0.7) = −0.5(0.3)−0.5y1 + 0.5(0.7)−0.5y2.

Notice that everything looks promising at this stage. It suffices that we con-sider a function that is constant, as we have done, and our limiting curve willbe symmetric. However, if y1 differs from y2, then we do not have symmetry.In the case of our chosen function here, let’s choose δ = 1 and x1 = 4 to makeit easier. This way, x1+ δ = 5. y1 = 2 and y2 =

√5. L′(0.3) = 0.846013 and

L′(0.7) = −0.489436. The moduli are so different that one is almost twicethe other. Symmetry is obviously not a possibility here.It is interesting how we never have questioned, this far, the capability ofHudzik and Maligranda in terms of using the mathematical terms correctly.An extension of a set in Mathematics is, for instance, the real numbers.The Real Numbers Set extends the Integer Numbers Set. How do we provethat? Do we simply prove that the Real Numbers Set contains the Inte-ger Numbers Set and that the difference between the Real Numbers Setand the Integer Numbers Set is non-null? That is all we need to havean extension of a set, or of a class, according to the linguists (see http ://dictionary.reference.com/browse/ex tension, for instance).

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One should notice that there is a difference between a bigger set and an exten-sion of a set. In Mathematics, when we say that we have extended something,we usually mean that we have uniformly extended something, contrary towhat we read from http : //www.math.utah.edu/on line/1010/numbers/,for instance. Hudzik and Maligranda, who blame Breckner for that, butwere the ones to first come up with the definitions we work with since 2001,thought that, to extend the set of convex functions, they had to preservethe analytical shape of the limiting line for convexity, what then made themthink of the current shape of the definition (we have preserved one hundredpercent of their chosen shape for the non-negative case). It did not occur tothem that ‘percentage’ is a linear operation. In Vector Analysis with Analyt-ical Geometry, at the university, we learn that some operators are ‘linear op-erators’ (see http : //mathworld.wolfram.com/LinearOperator.html, forinstance), and that is the case with the ‘percentage’ operation. However,exponential operators are not linear. When Hudzik and Maligranda usedthe exponent ‘s’ to ‘extend’ convexity, they changed a linear operator intoa non-linear one. That gave us something that is not symmetric and growsin a very different manner. They made the line depend on the function val-ues, but the linear operator we had that far returned always a straight line.They did get a set that included the set of convex functions as a result, alsoa set that gave us a difference in its favor when compared with the set of theconvex functions, so that everything was looking OK in those regards. If wedo not care about symmetry or uniformity, however, zillions of sets wouldextend the set of convex functions, for it suffices having one point more thatwe have an extension. Uniformity of our extensional elements, however, willallow us to derive nice theorems in Real Analysis, theorems of the sort ofthose that we have in convexity. Notice that every unit interval in the RealNumbers Set is equal to any other unit interval in the Real Numbers Setshape-wise. This repetition, these patterns, is what allow us to have booksof more than 200 pages with theorems involving real numbers (Real Anal-ysis books, for instance). Even though the name of the numbers changefrom unit interval to unit interval, the moduli of the differences between theelements found in similar geometric situations and the first element of theunit intervals are equal. We should be after the same sort of situation hereso that we have the largest number of analytical theorems as possible.Also notice that convexity is essentially a geometric concept, so that Hudzikand Maligranda should have privileged the geometrical description of theclass of functions over the analytical description when creating their exten-sion.If we care about the geometrical description more than we care about the

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analytical description, we will have a linear operator, or at least a symmetriclimiting curve instead of what we have this far.If we fix the limiting curve suggested by Hudzik and Maligranda in a wayto make it be symmetric, we will still not have the desired uniformity inthe extension, since we will not interpolate values uniformly for each unitinterval of each limiting line, like we will not do it in the same way for eachlimiting line we draw, for instance.Notice that we can easily get examples in which the inclination of the limit-ing curve differs if we consider the middle point (even for the same functionin the same sort of situation). Notice that the convex limiting line is alwaysa straight line connecting starting to finishing point in the interval underconsideration instead. That is what we would like to get as a limiting linefor s−convexity: The same geometric shape always for all complete limitingcurves (drawn from starting to finishing point). That would be mimickingthe ‘work’ of the convex limiting line, ‘the geometry’ involved.We can however finish our work on the definitions proposed by Hudzik andMaligranda by presenting the most acceptable result when the analyticaldescription of the phenomenon convexity is privileged over its geometric de-scription. And that is what we shall do.

III. Conclusion

There might be exponents that are nicer than our chosen δ1 and δ2 and arestill acceptable.We have decided to deal with the phenomenon s−convexity as if it were anexclusively extensional concept for issues that have to do with practicalityand accuracy (we now forbid s to assume the value 1 in our definition).We have extended the domain of the s−convex functions to ℜ because thedefinition should only bring necessary limitations, and we have found prob-lems only with the image of the functions so far in what regards the currentshape of the definition of the phenomenon we study, not the domain.We have decided to swap the coefficients in our definition because λ = 0should bring f(x) ‘to life’, trivially, not f(x+ δ).We have added the interval of definition of s to our geometric definition tomake it be independent from the analytical definition.Because of our new findings and decisions, we have produced a new updatefor our definition of the phenomenon s−convexity, and we present it here,after the third paragraph from this one.

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This paper should leave the readers with certainty that we cannot have any-thing suitable for inclusion in Real Analysis coming from the mathematicalwork of Hudzik and Maligranda in what regards s−convexity.Their idea, however, of extending convexity through drawing a bigger limit-ing line, is highly interesting.We promise presenting a final suggestion, this time one that is fully accept-able for inclusion in the body of Real Analysis, in future work.

Update on our refinements of the definitions proposed by Hudzik and Ma-ligranda

1) Analytical Definition

We have two possibilities so far for each piece of the analytical definition ofthe phenomenon if we think of including the definition of Hudzik and Ma-ligranda, and those stated that the definition had actually been created byBreckner, properly in Mathematics.

1.A) Possibility 1

Definition 3. A function f : X− > ℜ, where |f(x)| = f(x), is told tobelong to K2

s if 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).

Definition 4. A function f : X− > ℜ, where |f(x)| = −f(x), is told tobelong to K2

s if the inequality

f((1− λ)x+ λ(x+ δ))

≤ (1− λ)log2

(1

1−2−s

)f(x) + λ

log2

(1

1−2−s

)f(x+ δ)

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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 inequalities are obeyed in the reverse1 situation by f , thenf is said to be s2−concave.

1.B) Possibility 2

Definition 5. A function f : X− > ℜ, where |f(x)| = f(x), is told tobelong to K2

s if the inequality

f((1− λ)x+ λ(x+ δ))

≤ (1− λ)log2

1

1−2− 1

s f(x) + λlog2

1

1−2− 1

s f(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 6. A function f : X− > ℜ, where |f(x)| = −f(x), is told tobelong to K2

s if the inequality

f((1− λ)x+ λ(x+ δ))

≤ (1− λ)1s f(x) + λ

1s f(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 reverse2 situation by f , thenf is said to be s2−concave.

2) Geometric Definition

We then have two possibilities for each piece of the geometric definition aswell.

1Reverse here means ‘>’, not ‘≥’.2See the previous footnote.

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2.A) Possibility 1

Definition 7. A real function f : X− > Y , for which |f(x)| = f(x), iscalled s−convex3 if and only if, for all choices (x1; y1) and (x2; y2), where{x1, x2} ⊂ X, {y1, y2} ⊂ Y , Y = Imf , and x1 ̸= x2, it happens that theline 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, mea-sured against the vertical Cartesian axis, that is inferior to the height of itshorizontal equivalent in the curve representing the ordered pairs of f in theinterval considered for the line in terms of distance from the origin of theCartesian axis.

Definition 8. A real function f : X− > Y , for which |f(x)| = −f(x), iscalled s−convex4 if and only if, for all choices (x1; y1) and (x2; y2), where{x1, x2} ⊂ X, {y1, y2} ⊂ Y , Y = Imf , and x1 ̸= x2, it happens that theline drawn between (x1; y1) and (x2; y2) by means of the expression (1 −

λ)log2

(1

1−2−s

)y1 + λ

log2

(1

1−2−s

)y2, where λ ∈ [0, 1], does not contain any

point with height, measured against the vertical Cartesian axis, that is infe-rior to the height of its horizontal equivalent in the curve representing theordered pairs of f in the interval considered for the line in terms of distancefrom the origin of the Cartesian axis.

Remark 3. If all the points defining the function are located above the lim-iting line instead, then f is called s−concave.

2.B) Possibility 2

Definition 9. A real function f : X− > Y , for which |f(x)| = f(x), iscalled s−convex5 if and only if, for all choices (x1; y1) and (x2; y2), where{x1, x2} ⊂ X, {y1, y2} ⊂ Y , Y = Imf , and x1 ̸= x2, it happens that theline drawn between (x1; y1) and (x2; y2) by means of the expression (1 −

3s must be replaced, as needed, with a fixed constant located between 0 and 1 butdifferent from 0 and 1. For instance, if the chosen constant is 0.5, then the function willbe 0.5-convex or 1

2-convex and s will be 0.5 in the expression that defines the limiting line.

4See the previous footnote.5See the previous footnote.

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λ)log2

(1

1−2− 1

s

)y1 + λ

log2

(1

1−2− 1

s

)y2, where λ ∈ [0, 1], does not contain any

point with height, measured against the vertical Cartesian axis, that is infe-rior to the height of its horizontal equivalent in the curve representing theordered pairs of f in the interval considered for the line in terms of distancefrom the origin of the Cartesian axis.

Definition 10. A real function f : X− > Y , for which |f(x)| = −f(x), iscalled s−convex6 if and only if, for all choices (x1; y1) and (x2; y2), where{x1, x2} ⊂ X, {y1, y2} ⊂ Y , Y = Imf , and x1 ̸= x2, it happens that theline drawn between (x1; y1) and (x2; y2) by means of the expression (1 −λ)

1s y1 + λ

1s y2, where λ ∈ [0, 1], does not contain any point with height,

measured against the vertical Cartesian axis, that is inferior to the height ofits horizontal equivalent in the curve representing the ordered pairs of f inthe interval considered for the line in terms of distance from the origin ofthe Cartesian axis.

Remark 4. If all the points defining the function are located above the lim-iting line instead, then f is called s−concave.

6See the previous footnote.

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References

[Pin11] M. R. Pinheiro. First Note on the Definition of S2−convexity. Ad-vances in Pure Mathematics, 2011. Vol. 1, pp. 1− 2.

[Pin13] M. R. Pinheiro. Minima Domain Intervals and the S−convexity, aswell as the Convexity, Phenomenon. Advances in Pure Mathematics,2013. Vol. 3, no. 1.

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