Advances in Pure Mathematics, 2011, 1, 1-2
doi:10.4236/apm.2011.11001 Published Online January 2011 (http://www.scirp.org/journal/apm)
First Note on the Definition of s2-Convexity
I. M. R. Pinheiro
A'beckett st, Melbourne, Victoria, Australia
Received January 7, 2011; revised January 18, 20 1 1; accepted January 25, 2011
Abstract
In this short, but fundamental, note, we start progressing towards a mathematically sound definition of the
real functional classes 2
s
K .
Keywords: Convex, S-convex, s-convex, s2-convex, S2-Convex, Real Function
1. Introduction
As seen in [1], the boundaries of each one of the func-
tional classes 2
s
K are currently determined through the
following definition:
Definition 1. A function
f
: is said to be
s2-convex if the inequality X
   
11
s
s
f
xyfx f  y
holds


0,1;,; : 0 < 1 ; .xyX ssX

 
Remarks:
 If the complementary inequality to the above in-
equality is verified then f is told to be s2-concave;
 To recover the definition of real convex functions,
it suffices making s = 1 in the above inequality, de-
leting the symbol ‘s2-’ in the definition, and replac-
ing with  in the definition.

Unfortunately, a few real convex functions have been
left out of the just mentioned functional classes because
of the wording of the cu rrent definition of such classes.

It is just that each one of the functional classes 2
s
K is
supposed to contain, in itself, the entire class formed by
the real convex functions!
We are then facing an impossible-to-deny-urgency of
fixing the above mentioned definition.
The titles of the sections of this paper are:
1. Introduction;
2. Counter-examples;
3. Propose d f i xing;
4. Conclusions;
5. References.
2. Counter-Examples
The easiest way to produce evidence to the claim that the
current definition of the classes 2
s
K is equivocated is to
present a convex function that does not belong to 2
s
K
for some allowed value of s. We here will actually ex-
hibit a model from which to generate such convex func-
tions, so that we can choose one of them to present.
Consider a real convex function (any) that is negative
in its entire domain.
Take its domain to be the real interval [a, b] and the
function to be
f
.
Because f is convex, it is definitely true that, for each


,
f
x
yD, we have:



  
11,
f
xyfx f  y


0, 1 .
Assume that f is also s2-convex, for all allowed values
of s2, as it would have to be the case in order for us to be
entitled to state that 2
s
K extends the class of convex
functions.
After considering the just mentioned assumption, we
will then have that



 
s
s
11
f
xyfxf  y
,


0, 1 .
Because




fx fx, it is true that



sfx fx

11
and it is also true that

s
f
yf y
.
 .
Consequently, it is true that
     
s
s
11 1fxyfxfyfxfy   

I. M. R. PINHEIRO 2
However, the last inequality frontally contradicts the
assumption that 2
s
K extends 2
1
K
, that is, that 2
s
K
extends the class of co nvex functions!
Therefore, it can only be the case that all the fully
negative real convex functions have been left out of the
current definition of s2-convexity, and at least all of those
have been left out of the current definition, which, there-
fore, cannot be a mathematically acceptable definition at
least in terms of its analytical description.
To exemplify the just mentioned missing convex func-
tions, we consider the function when

25fx x


0,2x
f
. is clearly a convex function, due to its
graph and the geometric definition of convex functions,
which does match their analytical definition. In this case,
should also be s2-convex, as explained before. How-
ever, see:
f
   
s
s
11fxy fxfy   
 




2s
s2 2
1551xy xy  5.
The last step is clearly untrue for at least
and .
0, 1,xy
0.5, 0.5s
3. Proposed Fixing
We propose two definitions to replace the current defini-
tion of the classes 2
s
K .
Our definitions should keep the intended geometry of
the functions that are supposed to extend the convex
functions and, at the same time, provide us with a more
mathematically acceptable analytical interpretation of the
phenomenon s2-convexity.
The proposed definitions are:
Definition 2. A function f : , for which X
 
fx fx, is told to belong to 2
s
K , for some al-
lowed and fixed value of s, if the inequality
   
11
s
s
f
xyfx f  
holds


0,1;,;: 01;xy XssX


 .
Definition 3. A function : , for which fX 




fx fx , is told to belong to 2
s
K , for some
allowed and fixed value of s, if the inequality
   
s
11
s
11
f
xyfxf  y
holds


0,1;,;: 01;.xy XssX



Remark:
If the complementary inequality to one of the inequalities
above is verified for some function f then such a function
is told to be s2-concave.
4. Conclusions
In this short note, we have started to fix the analytical
definition of the phenomenon s2-convexity, which has
been created with the intention of extending the phe-
nomenon convexity, once if the geometric idea of the
phenomenon has ever been correct then something went
really wrong when transferring that idea to the ‘Real
Analysis world’.
We have exhibited an entire set of convex functions
that had been left out of the classes 2
s
K because of the
inappropriate wording of its current analytical definition.
We then progressed to present a member of that set, the
set of the ‘forgotten convex functions’, and, finally, to
re-word the analytical definition of s2-convexity so that
the just mentioned set, formed by certain convex func-
tions, were finally included in the classes that extend it.
5. References
[1] M. R. Pinheiro, “Convexity Secrets,” Trafford Publishing,
Canada, 2008. ISBN: 1425138217.
y