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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 S−CONVEXITY

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 S−convexity.

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 S−convex 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 S−convexity phenomenon). One of the resulting analytical

deﬁnitions in [1] is:

Deﬁnition 1. 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)) ≤λ1

sf(x) + (1 −λ)1

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).

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 s2−concave.

We now need to have a geometric deﬁnition for this group of S−convex

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 S−convex functions

in a similar way:

Deﬁnition 3. 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.

To prove that the geometric rule for pertinence to the S−convex class of

functions is the one that we here present, we remind the reader that (1−λ)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 S−convexity 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

S−convexity is a proper extension of convexity, also geometrically speaking.

1Supplementary here means ’>’, not ’≥’.

SECOND NOTE ON THE SHAPE OF S−CONVEXITY 299

2. Continuity

We now prove that the function f(λ) = (1−λ)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 ≤λ≤1 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) = y2−y1.

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)1

s≥ −(1 −λ) and −λ1

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, continuous,

and located above or over the limiting line for the convexity phenomenon. 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), 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,

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 + f′2(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 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)1−s

sy1+1

sλ1−s

sy2.

With this, our arc length formula will return:

p≡Z1

0r1 + [−1

s(1 −λ)1−s

sy1+1

sλ1−s

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 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

0r1 + [1

s(1 −λ)1−s

s−1

sλ1−s

s]2dλ.

Notice that s→0⇒p→indeterminate 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 S−convexity, respectively.

SECOND NOTE ON THE SHAPE OF S−CONVEXITY 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 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.

5. Conclusion

This paper and [8], together, provide us with a geometric deﬁnition for the

phenomenon S−convexity 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 S−convexity

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 S−convexity 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 S−convexity,

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 S−convexity, as well as

the convexity, phenomenon, Advances in Pure Mathematics,3(2013).

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