Master Science (Mathematics)
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Lecturer at IICSE University
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Master Yaser, have you heard of Hudzik,
Maligranda, and their S1-Convexity before? Have you heard of convex
functions?
Yes, I know convex functions but I haven't heard
anything about S1-Convexity.
Usually people like the concept of convex
function because it is a lot graphical, is it not? In the Universe of the Real
Numbers, a convex function is a function with a graph that is built in such a
way that regardless of which couple of points we pick on its line (it will be a
line for our eyes, right?), the curve representing the function will always be
either over or under that line.
Yes
it is graphical and this property helps us understand convex functions better
than other types of function. Also people who do not have a mathematical
knowledge like convex function more than other in fact they understand their
eyes. But if we want to talk about Mathematics, in particular Pure Mathematics,
we know that it is not necessary for a function to have a graphical property,
but yes it is interesting.
As we
can see in the graph above, whatever that is part of the curve (blue) between
one intersection of the straight line (red) with the curve and another is under
the straight line. It does not matter where we draw this straight line that
contains two points of our original curve; it is always going to be the same:
It is all under the line, at most over it, that is, never above it. We can then
say that, in analytical terms, that, for any two elements of the domain of the
function we pick, so say x and y, it is always true that the image of ax + by is always less than or equal to a times the image of x plus b times the image of y if a+b=1.
In our
first internationally known paper on S-convexity, which got published by the
WSEAS group, due to their conference in Cancun, which we did not actually
attend, we wrote:
That was in 2004, Master Yaser. You can already notice some difference between this picture, from the WSEAS paper, and what I wrote before. Can you?
No, I can't see any differences between
them: Everything you say is shown in this definition. I just zoom on the
(f:X--->R)2. 2 is for footnote or it is part of the definition?
I think you are talking about a footnote,
Master Yaser. You are right, it is all very subtle, but I have been working on
this for a while because I think it all matters quite a lot. Please observe the
coefficients: In one definition, you see a and b,
therefore two constants. In another definition, you see lambda only. You are also immediately told, in
the second definition, the one presented at the WSEAS, that it is lambda and 1-lambda, and therefore the sum of the coefficients leads to 1, so that you don't need to write that
down, like not only we have reduced the constants to one (we had two constants,
now we have one), in terms of coefficients, but we also deleted the extra piece
of information: That the coefficients together give us 1. That saves us and makes the definition
look more elegant, which can only be part of the objectives of Science when we
talk about refining mathematical definitions: more objectivity, simplest
presentation as possible, more immediate application, etc.
The first difference between convex and
S-convex functions is their domain. The second difference is the way of
choosing a and
b, and the third is the conditions on a and b. For instance, a+b=1. The difference between S1-convex and S2-convex
is in the condition on a+b, perhaps that a+b=1 for one of them. I can imagine some things, as you
can see. Are they true? In the convex function, we see that if we choose
two elements in the domain, x and y, then the image of
the graph for all points between x and y is under the line (red line) across f(x) and f(y), but I imagine that
this S-convex concept does not lead to a line!! I think it is a curve and its
Curvature depends on S.
You are very right, Master Yaser. It is
precisely that, the main points are precisely those. I saw things in the same
way you see them in that 2001, since it is all very much obvious. One more
detail catches our eyes: Because S is between 0 and 1, and both a and b would
have to be between 0 and 1, and you will notice that I actually
produced a proof for that in my paper with WSEAS, the first one on the topic
that got published by a major vehicle, and the proof follows this paragraph, a to the s would have to be greater than a. That was the key for my understanding of the shape of S-convexity: It
is actually a lift on the limiting line for Convexity, and that is why both
Hudzik and Maligranda thought that they had a proper extension of the concept.
See the proof regarding the coefficients first:
Before we talk about the rest, I think I
would like to know if you agree with what I stated before after seeing the
proof we presented at the WSEAS (a and b would both be between 0 and 1, and, therefore, we can rewrite the definition of S-convexity given by
Hudzik and Maligranda in the way you see in the last picture of the paper
presented at the WSEAS in that 2004). Do you agree with that, please, Master
Yaser?
I understand the new definition of
S2-convex and agree with this, but I don't understand S1-convexity.
In fact I don't understand why you use (1-lambda
^ s)^ (1/s)!!!!
Yes, exactly. Perhaps the first intuition
is that it is important to keep the coefficients unaltered because we want to
keep the percentages we take in the mix unaltered in terms of base. You will
notice however that easy counter-examples exist to prove to us that S1-Convexity
is not a proper extension of Convexity. Perhaps the main question to be asked
was always what does extension mean? When we extend something in Mathematics,
that means that we have included what we had before in what we have after the
extension and we have added a little bit, as a minimum thing, is it not? If we
lose something that was part of the something we claim to now be extending,
then we must not be extending: We must be creating another class instead.
See the counter-example to the claim that
S1 extended convexity, which is right below this line. I want to
know if you agree with all that is said here. Perhaps you could give your take
on extension as well.
I agree with your take on extension in
Mathematics. We can also notice that if we find a new class of something, then
we also extend the Mathematics involved. However maybe we don't extend an
old definition. Now something to make my mind busy: Is S-convex function a
subclass of convex function? In other words, is every S-convex function a
convex function?
Master Yaser, the reason for your
confusion is probably the fact that you agreed that we have a genuine
counter-example to the claim that S1-convexity extends convexity. I
wonder if you have validated my every step in the proof above. Please
confirm.
Yes.
Every part of the above extract proves that it is all true and I can't add any
comments to that.
Great,
Master Yaser! I think I was eager to get more people saying yes to my results
in a meaningful manner, people who are not journal editors. Thanks for that.
That means we both agreed that S1-convexity cannot extend Convexity
because, for instance, the group of functions we have just mentioned is part of
the class convex real functions but is not part of the class S1-convex
functions. As said before, if a class extends another, we should have at least
the group we claim to be extending inside of it. As for convexity, please
observe that whenever s=1 you
would be recovering this notion both in the definition of S1- and in
the definition of S2-convexity.
References
Pinheiro,
M. R. (2004). Exploring the Concept of S-convexity – Proceedings of the 6th WSEAS Int. Conf. on Mathematics
and Computers in Physics (MCP '04).
Pinheiro, M. R. (2015).
Second Note on the Definition of S1-Convexity. Advances in Pure Mathematics,
5, 127–130.
