Master Yaser and Dr. Pinheiro: S1-Convexity

Master Yaser Maleki                               Master Science (Mathematics) Tarbiat Modares University Tehrān  E-mail yasermaleki71@gmail.com Web: Research Gate

Dr. Marcia Pinheiro Lecturer at IICSE University Certified Translator and Interpreter Portuguese & English NAATI  40296          Member: PROz, RGMIA, Ancient Philosophy PhD in Philosophy and Mathematics Master in Philosophy Certified TESOL/TEFL professional Licentiate in Mathematics PO Box 12396 A’Beckett St Melbourne, VIC, AU, 8006 Tel 0416915138 E-mail drmarciapinheiro@gmail.com Web Research Gate

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.

       DO YOU WANT TO PLAY LOVE GAMES?