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The purpose of this paper is to distinguish, as much as possible, the
concept of s-convexity from the concept of convexity and the concept of s-convexity
in the first sense from the concept of s-convexity in the second sense. In this respect,
the present work further develops a previous study by Orlicz(1961, [3]), Hudzik and
Maligranda (1994, [1]).

In this note, we present a few more important scientific remarks regarding the Sâconvexity phenomenon. We talk about examples and fixings. We sustain the example we gave in Really Short Note, but fix the way we presented one of the counterexamples to the existing theorem that we there mentioned. We also present another two generators of elements for K 2 s. MSC(2010): 26A51

The purpose of this paper is to distinguish, as much as possible, the concept of s-convexity from the concept of convexity and the concept of s-convexity in the first sense from the concept of s-convexity in the second sense. In this respect, the present work further develops a previous study by Hudzik and Maligranda
(1994, [1]).

In this piece of work, we start building some foundational theory about S convexity (sets and points). We also define S convex generaliza- tions (for more dimensions, functions).

We simply extend Lazharâs work on inequalities for convex functions to those a little bit beyond: S-convex functions.

It was supposed to be a finger-friendly catalog of convexity types for functions where we could also find a history of the progresses with full reference list and etc. Trafford did not have the resource to make the finger-friendly index. The introduction is in very poor English and we regret having written it. With the fight support. we can certainly prepare a much better version of this material. Researchers were supposed to find plenty of material for innovative research and use the book as a tool to save time and effort. They should be able to simply copy the entire history of the convexity type from the book when producing their own material.

We here study the inequality by Jensen for the case of S- convexity.

Convexity has captured the hearts of pure and applied mathematicians because it is a very geometric concept. Professor Stanley F. Gudder, already in 1977, discussed using convexity in studies that connect to social, behavioural, and physical sciences. Professor Robert E. Jamison-Waldner talked about mathematical methods that are based on the convexity phenomenon in 1983. S-convexity was not a proper extension of convexity until 2001, when we started working with the concept. We have found many fallacies and inaccuracies that we have been addressing since then. The paper Minima Domain Intervals ended up fixing things in the realm of convexity as well, basic items, such as the analytical definition, and the paper First Note ended up fixing the geometric definition for convexity, so that S-convexity became a really useful and beautiful concept for Pure and Applied Mathematics. In this talk, we would like to discuss our final results on the analysis of the shape and the re-wording of the definitions.

This is the last presentation of mine involving the shape of S-convexity, I believe. We now have a permanent choice of shape and definition. It is in the poster and it is also in the slides.

Abstract In this note, we copy the work we presented in Second Note on the Shape of S-convexity [1], but apply the reasoning to one of the new limiting lines, limiting lines we presented in Summary and Importance of the Results Involving the Definition of S-Convexity [2]. That line was called New Positive System in Third Note on the Shape of S-convexity [3] because, on that instance, the images of the points of the domain had been replaced with a positive constant, which we
called A . This is about Possibility 2 of Summary and Importance of the Results Involving the Definition of S-Convexity. We have called it S2 in Summary and Importance of the Results Involving the Definition of S-Convexity, and New Positive System in Third Note on the Shape of S-convexity. The second part has already been dealt with in Second Note on the Shape of S-convexity. In Second Note on the Shape of S-convexity, we have already performed the work we performed in First Note on the Shape of S-convexity [4] over the case in which the modulus does not equate the function in the system S2 from Summary and Importance of the Results Involving the Definition of S-Convexity. This paper is about progressing toward the main target: Choosing the best limiting lines amongst our candidates.

In this short, but fundamental, note, we start progressing towards a mathematically sound definition of the real functional classes K s 2 .

In this revisional article, we criticize (strongly) the use made by Medar et al., and those whose work they base themselves
on, of the name âconvexityâ in definitions which intend to relate to convex functions, or cones, or sets, but actually seem
to be incompatible with the most basic consequences of having the name âconvexityâ associated to them. We then believe to
have fixed the âdenominationsâ associated with Medarâs (et al.) work, up to a point of having it all matching the existing
literature in the field [which precedes their work (by long)]. We also expand his work scope by introducing s
1-convexity concepts to his group of definitions, which encompasses only convex and its proper extension, s
2-convex, so far. This article is a long version of our previous review of Medarâs work, published by FJMS (Pinheiro, M.R.:
S-convexity revisited. FJMS, 26/3, 2007).

In this note, we present a few important scientific remarks regarding the shape of Sâconvexity.

In this paper, we propose a refinement in the analytical definition of the s 2-convex classes of functions aiming to progress further in the direction of including s 2-convexity properly in the body of Real Analysis.

Summary of results we attained in what regards S-convexity

As promised in Second Note on the Shape of Sâconvexity, we now discuss the exponent of the piece of definition for Sâconvexity that deals with negative images of real functions. We also present a severely improved
definition for the phenomenon.

This note supplements First Note on the Shape of S-convexity. We here deal with the negative pieces of the real functions.

In this note we copy the work we presented on Second Note on the Shape of S-convexity [1], but apply the
reasoning to one of the new limiting lines, limiting lines we presented on Summary and Importance of the Results Involving the Definition of S-Convexity [2]. This is about Possibility 1, second part of the definition, that is, the part that deals with negative real functions. We have called it S1 in Summary [2]. The first part has already been dealt with in First Note on the Shape of S-convexity [3]. This paper is about progressing toward the main target: Choosing the best limiting lines amongst our candidates.

In this note, we discuss the definition of the S1-convexity Phenomenon. We first make use of some results we have attained for s K 2 in the past, such as those contained in [1], to refine the definition of the phenomenon. We then observe that easy counterexamples to the claim s K 1 extends K0 are found. Finally, we make use of one theorem from [2] and a new theorem that appears to be a supplement to that one to infer that s K 1 does not properly extend K0 in both its original and its revised version.

In this note, we try to summarize the results we have so far in terms of the definition of the S-convexity
phenomenon, but we also try to explain in detail the relevance of those. For some of those results, we dare
presenting graphical illustrations to make our point clearer. S-convexity came to us through the work of Prof.
Dr. Dragomir (2001) and Prof. Dr. Dragomir claimed to have had contact with the concept through the hands of
Hudzik and Maligranda, who, in their turn, mention Breckner and Orlicz as an inspiration. We are working in a
professional way with the phenomenon since the year of 2001, and that was when we presented our first talk on the topic. In that talk, we introduced a conjecture about the shape of S-convexity. We have examined possible examples, we have worked with the definition and examples, and we then concluded that we needed to refine the definition by much if we wanted to still call the phenomenon an extensional phenomenon in what regards Convexity. We are now working on the fourth paper about the shape of S-convexity and trying to get both limiting lines (negative and non-negative functions) to be as similar as possible. It is a delicate labour to the side of Real Analysis, Vector Algebra, and even Calculus.

In this note, we try to summarize the results we have so far in terms of the
definition of the S-convexity phenomenon, but we also try to explain in detail the relevance
of those. For some of those results, we dare presenting graphical illustrations to make our
point clearer. S-convexity came to us through the work of Prof. Dr. Dragomir (2001) and
Prof. Dr. Dragomir claimed to have had contact with the concept through the hands of
Hudzik and Maligranda, who, in their turn, mention Breckner and Orlicz as an inspiration.
We are working in a professional way with the phenomenon since the year of 2001, and that
was when we presented our first talk on the topic. In that talk, we introduced a conjecture
about the shape of S-convexity. We have examined possible examples, we have worked with
the definition and examples, and we then concluded that we needed to refine the definition by
much if we wanted to still call the phenomenon an extensional phenomenon in what regards
Convexity. We are now working on the fourth paper about the shape of S-convexity and trying
to get both limiting lines (negative and non-negative functions) to be as similar as possible.
It is a delicate labour to the side of Real Analysis, Vector Algebra, and even Calculus.

We are now in the final stages of publication. You will find the last version close to this paper online. It is a supplementary resource. The Research Gate system now does not allow for us to delete files or update. Shame.