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Exploring the concept of S-Convexity

Authors:
  • IICSE University

Abstract

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]).
Aequationes Math. 74 (2007) 201–209
0001-9054/07/030201-9
DOI 10.1007/s00010-007-2891-9
c
°Birkh¨auser Verlag, Basel, 2007
Aequationes Mathematicae
Research papers
Exploring the concept of s-convexity
Marcia R. Pinheiro
Summary. 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]).
Mathematics Subject Classification (2000). 26A51.
Keywords. Convex, s-convex, function.
1. Introduction
Recently, Hudzik and Maligranda ([1]) studied some classes of functions, the classes
of s-convex functions. Although they claim, in their abstract, to provide several
examples and clarify the idea introduced by Orlicz further, their work leaves plenty
of room to build over the concept. Besides, their reference to Orlicz seems to be
a bit mistaken and they seem to have created a new concept of convexity without
noticing it.
The old conclusions presented here are: definitions of convex/s-convex func-
tions; Orlicz’s symbols are good to represent the classes of s-convex functions; a
theorem which acts as a generator of s-convex functions.
The new conclusions arising from this paper are: a rephrasing of the definitions
of s-convex functions to look more similar to the definition of convex function; an
identity between the class of 1-convex functions and the class of convex functions; a
conjecture about the geometry of a s-convex function; some theorems on functions
that are s-convex in both senses; a few other side results that might lead to future
work or are, at least, useful to clarify similarities and differences between functions
that are s-convex in the first sense and the ones which are s-convex in the second
sense.
The paper further tries to explain why the first s-convexity sense was aban-
doned by the literature in the field.
The paper is organized as follows: Introduction, Elementary results for s-convex
202 M. R. Pinheiro AEM
functions, Inequalities for functionals generated by s-convex functions, some clos-
ing remarks.
The concept of convexity that is mostly cited in the bibliography is (as an
example, [4]):
Definition 1. The function f:XRRis called convex if the inequality
f(λx + (1 λ)y)λf(x) + (1 λ)f(y)
holds for all λ[0,1], for all x, y Xsuch that the right-hand side is well defined.
It is called strictly convex if the above inequality strictly holds for all λ]0,1[ and
for all pairs of distinct points x, y Xwith f(x)<and f(y)<.
In some sources, such as [2], convexity is defined only in geometrical terms as
being the property of a function whose graph bears tangents only under it. In
their words, “fis called convex if its graph lies below the chord between any two
points, that is, for every compact interval JI, with boundary ∂J , and every
linear function L, we have
sup
J
(fL)sup
∂J
(fL).
One calls fconcave if fis convex.”
The concept of s-convexity, on the other hand, is split into two notions which
are described below with the basic condition that 0 < s 1 ([1]).
Definition 2. A function f: [0,)Ris said to be s-convex in the first sense
if f(ax +by)asf(x) + bsf(y), for all x, y [0,) and for all a, b 0 with
as+bs= 1.
Definition 3. A function f: [0,)Ris said to be s-convex in the second
sense if f(ax +by)asf(x) + bsf(y), for all x, y [0,) and for all a, b 0 with
a+b= 1.
Some criticisms to the current way of presenting the definition of s-convex
functions should be written:
It takes us a long time, the way the definition is written now, to work out
the true difference between convex and s-convex functions;
So far, we did not find references, in the literature, to the geometry of an
s-convex function, what, once more, makes it less clear to understand the
difference between an s-convex and a convex function whilst there are clear
references to the geometry of the convex functions.
We now propose that the standard symbology for s-convex functions gains roots in
([3]): In this paper, we denote by K1
sthe class of functions fwhich are s-convex in
the first sense and by K2
sthe class of functions fwhich are s-convex in the second
sense; We call s1the generic class constant for those functions that are s-convex
in the first sense; We call s2the generic class constant for those functions that are
s-convex in the second sense.
Vol. 74 (2007) Exploring the concept of s-convexity 203
2. Elementary results for s-convex functions
2.1. Re-writing the definition of s-convex function
It is trivial to prove that a, b [0,1] is a consequence of the present definition of
s-convexity.
Lemma 2.1. If fK1
sor fK2
sthen
f(au +bv)asf(u) + bsf(v)
with a, b [0,1], exclusively.
(Simply consider the sum and the limit of both aand aswhen s=1
n,nN.)
With this, we may re-write the definitions of s-convexity in each of the senses
as being:
Definition 4. A function f:XRis said to be s-convex in the first sense if
f(λx+ (1 λs)1
sy)λsf(x) + (1 λs)f(y), for all x, y Xand for all λ[0,1],
where XR+.
Definition 5. A function f:XRis said to be s-convex in the second sense
if f(λx+ (1 λ)y)λsf(x) + (1 λ)sf(y), for all x, y Xand for all λ[0,1]
where XR+.
Remark 1. The classes K1
1,K2
1, and convex coincide when the domains are
restricted to R+.
Theorem 2.2. The classes K1
1,K2
1, and the class of convex functions are equiv-
alent when the domain is restricted to R+.
Proof. Just a matter of applying the definitions. ¤
Natural implication: All 1-convex functions are convex.
A few consequences of the definition of s-convex functions are verified:
Remark 2. i) fK1
s=fµu+v
21
sf(u) + f(v)
2.
(Simply consider the case where as=bs=1
2.)
ii) fK2
s=fµu+v
2f(u) + f(v)
2s.
(Simply consider the case where a=b=1
2.)
204 M. R. Pinheiro AEM
iii) For a function that is both s1and s2-convex, there is a bijection between
the set of all pairs of the sort (a, b) with respect to s1and the set of all pairs of
the sort (a, b) with respect to s2. (Each amay be written as an as
1and each bas
abs
1and vice-versa. This happens because a, b [0,1], s[0,1] (each 1
s-root in
(0,1) will give us a number in (0,1)).)
Theorem 2.3. If a function belongs to both K1
sand K2
s, then
f(a1u+b1v)as
1f(u) + bs
1f(v)as
2f(u) + bs
2f(v)
for some {a1, b1, a2, b2} ⊂ [0,1] and such that it occurs to each and all of them.
Proof. It follows from the bijection proved before. For each a2, b2such that a2+
b2= 1, it corresponds a1, b1such that as
1+bs
1= 1 and a2a1,b2b1since
{a, b} ⊂ [0,1]. ¤
Theorem 2.4. If a function belongs to both K1
sand K2
sand its domain coincides
with its counter-domain then the composition ffis s2
1-convex.
Proof. f(a1u+(1as
1)1
sv)as
1f(u)+(1as
1)f(v) =f(as
1f(u)+(1as
1)f(v))
(as
1)sf(f(u)) + (1 as
1)sf(f(v)) = as
2f(f(u)) + bs
2f(f(v)). ¤
3. Inequalities for functionals generated by s-convex functions
Consider the following functions:
i) F: [0,1] R,defined by
F(t) = 1
(ba)2Zb
aZb
a
f(tx + (1 t)y)dxdy
([5]);
ii) p:R[0,+] defined freely such that
p(λx + (1 λ)y)p(x) + p(y) for all λ(0,1),{x, y} ⊂ R
(p-function).
Theorem 3.1. Consider the function F, as described above, where f: [a, b]R
is s2-convex. Then:
1. Fis s2-convex in [0,1]. If fis s1-convex instead, then Fis s1-convex as
well.
2. 2F(0) = 2F(1) = 2
baRb
af(x)dx is an upper bound for F(t).
3. F(t)is symmetric about t=1
2.
Proof. 1. Take {λ, β} ⊂ [0,1], λ+β= 1, x1, y1D,fbeing s2-convex.
Vol. 74 (2007) Exploring the concept of s-convexity 205
We then have
F(λx1+βy1) = 1
(ba)2Zb
aZb
a
f(λx1x+βy1x+ (1 λx1βy1)y)dxdy
1
(ba)2Zb
aZb
a
(λsf(x1xx1y+y) + βsf(y1x+yy1y)dxdy
=λsF(x1) + βsF(y1)
which implies that Fis s2-convex in [0,1].
2. f(tx + (1 t)y)tsf(x) + (1 t)sf(y) implies that
A=Zb
aZb
a
f(tx + (1 t)y)dxdy
Zb
aZb
a
tsf(x)dxdy +Zb
aZb
a
(1 t)sf(y)dxdy
=tsZb
a
f(x)(ba)dx + (1 t)sZb
a
f(y)(ba)dy
=ts(ba)Zb
a
f(x)dx + (1 t)s(ba)Zb
a
f(y)dy
= (ts+ (1 t)s)(ba)Zb
a
f(x)dx
(1 + (1 t)s)(ba)Zb
a
f(x)dx
since ts1. Because (1 t)s1 as well, this implies A2(ba)Rb
af(x)dx.
3. F(t) is symmetric about t=1
2because F(t) = F(1 t). ¤
Consider now the following functions:
i) Fg1: [0,1] R, defined by
Fg1(t) = Zb
aZb
a
f(tx + (1 t)y)g(x)g(y)dxdy,
where f: [a, b]R+Ris s1-convex.
ii) Fg2(t) : [0,1] R, defined by
Fg2(t) = Zb
aZb
a
f(tx + (1 t)y)g(x)g(y)dxdy,
where f: [a, b]R+Ris s2-convex.
206 M. R. Pinheiro AEM
Theorem 3.2. The following hold true:
1. Fg1and Fg2are both symmetric about t=1
2;
2. Fg2is s2-convex in [0,1];
3. We have the upper bound
2Fg2(1) = 2 Zb
aZb
a
f(x)g(x)g(y)dxdy
for the function Fg2(t).
Proof. 1. If we replace twith (1 t) in the definition of Fg1and Fg2, we get the
same result, thus both Fg1and Fg2are symmetric about t=1
2.
2. We have
Fg2(λt1+(1λ)t2) = Zb
aZb
a
f((λt1+(1λ)t2)x+(1(λt1+(1λ)t2))y)g(x)g(y)dxdy.
But
(λt1+ (1 λ)t2)x+ (1 (λt1+ (1 λ)t2))y
=λt1x+xt2λt2x+yλt1yt2y+λt2y
=λ(t1x+yt1y) + (1 λ)(t2xt2y+y).
Since fis s2-convex,
Zb
aZb
a
f(λ(t1x+yt1y) + (1 λ)(t2xt2y+y))g(x)g(y)dxdy
Zb
aZb
a
(λsf(t1x+yt1y) + (1 λ)sf(t2xt2y+y))g(x)g(y)dxdy
=λsZb
aZb
a
f(t1x+yt1y)g(x)g(y)dxdy
+ (1 λ)sZb
aZb
a
f(t2xt2y+y)g(x)g(y)dxdy
=λsZb
aZb
a
f(t1x+ (1 t1)y)g(x)g(y)dxdy
+ (1 λ)sZb
aZb
a
f(t2x+ (1 t2)y)g(x)g(y)dxdy
=λsFg2(t1) + (1 λ)sFg2(t2)
which proves that Fg2is s2-convex.
3. From the definition and the assumptions we get
Fg2(t) = Zb
aZb
a
f(tx + (1 t)y)g(x)g(y)dxdy
tsZb
aZb
a
f(x)g(x)g(y)dxdy + (1 t)sZb
aZb
a
f(y)g(x)g(y)dxdy
Vol. 74 (2007) Exploring the concept of s-convexity 207
= (ts+ (1 t)s)Zb
aZb
a
f(x)g(x)g(y)dxdy
2Zb
aZb
a
f(x)g(x)g(y)dxdy = 2Fg2(1).¤
Theorem 3.3. Consider a sum Sof s2-convex functions,
S=
n
X
m=1
Am(t),
where
Am(t) = Zb
aZb
a
fm(tx + (1 t)y)dxdy.
In this context, we have:
1. Sup(S) = 2 Pn
m=1 Am(0) = 2 Pn
m=1 Am(1);
2. Sis symmetric about t=1
2;
3. Sis a p-function;
4. Sis s2-convex (special case of the previous theorem, with g1).
Proof. 1. For each m,
Am(t)tsZb
aZb
a
fm(x)dxdy + (1 t)sZb
aZb
a
fm(y)dxdy
since fmis s-convex. Thus
Am[ts+ (1t)s]Zb
aZb
a
fm(x)dxdy 2Zb
aZb
a
fm(x)dxdy = 2Am(0) = 2Am(1).
2. For each m,Am(t) = Am(1t). Thus, Amis symmetric about t=1
2. Thus,
S(1 t) = S(t) and Salso is.
3. Take λ[0,1], then
Am(λt1+(1λ)t2) = Zb
aZb
a
fm((λt1+(1λ)t2)t1+ (1 (λt1+ (1 λ)t2))t2)dt1dt2
from which we deduce
S(λt1+ (1 λ)t2) =
n
X
m=1
Am(λt1+ (1 λ)t2)
n
X
m=1
Am(t1) +
n
X
m=1
Am(t2)
=S(t1) + S(t2).
Therefore, Sis a p-function.
4. As mentioned, just a special case of the previous theorem. ¤
208 M. R. Pinheiro AEM
4. Some closing remarks
We believe that the reason to why s-convexity sense introduced by Hudzik and
Maligranda got abandoned in the literature is that it is not difficult to notice
that if one considers a=1
4,b=1
4,u=1
2,v= 1, for example, one gets that
au +bv = 0.125 + 0.25 = 0.375. Therefore, if s=1
2,au +bv (a, b are those
from the original definition) would lie outside of the interval [u, v]. With this,
the first sense of s-convexity, as mentioned by Hudzik and Maligranda, becomes a
concept that can only be compared with the concept of convexity if some further
restrictions are imposed to it. (Here, what we are taking into consideration is
that for a convex function, the image considered in the definition is always inside
of the interval, allowing us to have a clear geometrical comparison between that
convexity definition and the s-convexity.)
Taking into account the relationship between asand a, we may wonder whether
the following is true or not:
Conjecture 1. fis called s2-convex, s26= 1, if the graph lies below the ‘bent
chord’ between any two points, that is, for every compact interval JI, with
boundary ∂J, and a special function Lwith a convexly (remember that a convex
function has a concave curve) designed (here, we introduce a new concept, we call
convexly designed curve any complete or incomplete curve, broken or continuous,
which, in having its points connected in a smooth manner, gives a convex curve)
curve of curvature ψ, corresponding to the figure s2[0,1), we have
G(s)sup
J
(fL)sup
∂J
(fL),
where sis the constant that determines the sort of convexity in s-convexity and
G(s)is a function yet to be determined of that constant between 0and 1.
Acknowledgements. We thank Dr. Carlos Gustavo Moreira, from IMPA (RJ,
Brasil), for an early investigation of a not mentioned conjecture, and A/Prof. Pan-
lop Zeephongsekul (RMIT, Australia) for strength and encouragement (Elsevier
preprint). We also would like to thank the welcome support from the editorial
office of Aequationes Mathematicae.
References
[1] H. Hudzik and L. Maligranda, Some remarks on si-convex functions, Aequationes Math.
48 (1994), 100–111.
[2] L. H¨
ormander, Notions of convexity, Birkh¨auser Verlag, Basel, 1994.
[3] W. Orlicz, A note on modular spaces I,Bull. Acad. Polon. Sci. Ser. Math. Astronom.
Phys. 9 (1961), 157–162.
[4] Th. Precupanu and V. Barbu, Convexity and Optimization in Banach Spaces, Editura
academiei – D. Reidel Publishing Company, 1986.
[5] C. E. M. Pearce and S. S. Dragomir, Selected topics on Hermite–Hadamard inequalities
and applications [http://www.rgmia.vu.edu.au/monographs/hermite-hadamard.html].
Vol. 74 (2007) Exploring the concept of s-convexity 209
[6] M. A. Rojas-Medar, M. D. Jimenez-Gamero, R. Osuna Gomez, and Y. Chalco-Cano,
Hadamard and Jensen inequalities for s-convex fuzzy processes, Elsevier preprint, 2003.
M. R. Pinheiro
A’Beckett St.
P.O. Box 12396
Melbourne, Au, 8006
Australia
e-mail: mrpprofessional@yahoo.com
Manuscript received: June 14, 2005 and, in final form, August 18, 2006.
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Notions of convexity
  • L Hö
L. Hö, Notions of convexity, Birkhä Verlag, Basel, 1994.
8006 Australia e-mail: mrpprofessional@yahoo.com Manuscript received
  • Melbourne
  • Au
Melbourne, Au, 8006 Australia e-mail: mrpprofessional@yahoo.com Manuscript received: June 14, 2005 and, in final form, August 18, 2006.