In this note, we discuss the definition of the S 1 -convexity Phenomenon. We first make use of some results we have attained for in the past, such as those contained in [1] , to refine the definition of the phenomenon. We then observe that easy counter-examples to the claim extends K 0 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 does not properly extend K0 in both its original and its revised version.
According to the scientific literature, Hudzik and Maligranda [
We had contact with the phenomenon because of the work of Dragomir and Pearce [
Sofo, who worked in the same university as Dragomir in 2001, when we met both, also asked us to work with the topic.
The university where we all worked (Pinheiro, Dragomir, and Sofo) in that 2001 was called Victoria University of Technology.
We actually tried to communicate with both Hudzik and Maligranda in that 2001 by means of the electronic addresses that we found on the Internet for them. Even though the addresses seemed to work (the electronic letters never bounced), they never replied.
Hudzik and Maligranda published their paper in 1994 and we started working with the topic in 2001.
Dragomir’s book dates from 2002, but we helped revise it in 2001.
Some interesting results regarding this phenomenon have been attained by Dragomir in 1999 [
Our first results had to do with the shape of S-convexity
We submitted the same paper we published in 2007 with Aequationes Mathematicae [
Because of that, our first publication on the topic was [
We use the symbols from [
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Remark 1. The class 1-convex functions is simply a subclass of the class Convex Functions. If we make the domain of the convex functions be inside of the set of the non-negative real numbers, we then have the class 1-convex functions:
The definition, for
Definition 1. A function
holds
Remark 2. If the inequality is obeyed in the reverse1 situation by f, then f is told to be s1-concave.
Trivially, we need to get rid of one of the variables in this definition, just like we did in [
After doing that, our definition will look like this:
Definition 2. A function
holds
As seen in [
Definition 3. A function
holds
Because we know that s1 should be between 0 and 1 and there is no reason to exclude
Definition 4. A function
holds
The original definition, for
Definition 5. A function
To go from the original definition to our modified version, we not only did all that we have already written about, but we also considered the results attained in [
call
・ The left side of the definition inequality becomes
・ The right side of the definition inequality becomes
・ Because
・ The same will happen to the other addend:
・ We conclude that
which is precisely the opposite to what we needed to get to be able to assert that
(**) We will, on the next paragraph, prove the supplementary theorem to the theorem whose proof we have rewritten in [
Theorem 1. Let
Proof. When we apply the definition of s1-convexity to a function that satisfies the conditions of this theorem,
In replacing
Following the reasoning we have presented in [
Our inequality then becomes
Because
In this case, we can only have a non-increasing function: (
Because the Convexity Phenomenon covers all types of functions in what comes to growth, S1-convexity cannot be told to be an extension of convexity: According to (**), it is not covering decreasing real functions inside of the interval
We believe that we have now proven, once and for all, that the S1-convexity Phenomenon cannot possibly be a proper extension of the Convexity Phenomenon: Easy counter-examples are found, and at least two theorems that make us be able to generate an infinity of convex functions that are not contained in the set of s1-convex functions exist and seem to be very sound.
We shall, therefore, and from now onwards, refer to exclusively
We may still try to determine the exact shape of the S1-convexity Phenomenon because it is an interesting creation, and several researchers, some of them with hundreds of publications, have already produced results involving it.