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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:X⊂R→Ris 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 J⊂I, with boundary ∂J , and every

linear function L, we have

sup

J

(f−L)≤sup

∂J

(f−L).

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 f∈K1

sor f∈K2

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,n∈N.)

With this, we may re-write the deﬁnitions of s-convexity in each of the senses

as being:

Deﬁnition 4. A function f:X→Ris 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 X⊂R+.

Deﬁnition 5. A function f:X→Ris 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 X⊂R+.

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) f∈K1

s=⇒fµu+v

21

s¶≤f(u) + f(v)

2.

(Simply consider the case where as=bs=1

2.)

ii) f∈K2

s=⇒fµu+v

2¶≤f(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 a2≥a1,b2≥b1since

{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 f◦fis s2

1-convex.

Proof. f(a1u+(1−as

1)1

sv)≤as

1f(u)+(1−as

1)f(v) =⇒f(as

1f(u)+(1−as

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

(b−a)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

b−aRb

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, y1⊂D,fbeing s2-convex.

Vol. 74 (2007) Exploring the concept of s-convexity 205

We then have

F(λx1+βy1) = 1

(b−a)2Zb

aZb

a

f(λx1x+βy1x+ (1 −λx1−βy1)y)dxdy

≤1

(b−a)2Zb

aZb

a

(λsf(x1x−x1y+y) + βsf(y1x+y−y1y)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)(b−a)dx + (1 −t)sZb

a

f(y)(b−a)dy

=ts(b−a)Zb

a

f(x)dx + (1 −t)s(b−a)Zb

a

f(y)dy

= (ts+ (1 −t)s)(b−a)Zb

a

f(x)dx

≤(1 + (1 −t)s)(b−a)Zb

a

f(x)dx

since ts≤1. Because (1 −t)s≤1 as well, this implies A≤2(b−a)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+(1−λ)t2) = Zb

aZb

a

f((λt1+(1−λ)t2)x+(1−(λt1+(1−λ)t2))y)g(x)g(y)dxdy.

But

(λt1+ (1 −λ)t2)x+ (1 −(λt1+ (1 −λ)t2))y

=λt1x+xt2−λt2x+y−λt1y−t2y+λt2y

=λ(t1x+y−t1y) + (1 −λ)(t2x−t2y+y).

Since fis s2-convex,

Zb

aZb

a

f(λ(t1x+y−t1y) + (1 −λ)(t2x−t2y+y))g(x)g(y)dxdy

≤Zb

aZb

a

(λsf(t1x+y−t1y) + (1 −λ)sf(t2x−t2y+y))g(x)g(y)dxdy

=λsZb

aZb

a

f(t1x+y−t1y)g(x)g(y)dxdy

+ (1 −λ)sZb

aZb

a

f(t2x−t2y+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 g≡1).

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+ (1−t)s]Zb

aZb

a

fm(x)dxdy ≤2Zb

aZb

a

fm(x)dxdy = 2Am(0) = 2Am(1).

2. For each m,Am(t) = Am(1−t). Thus, Amis symmetric about t=1

2. Thus,

S(1 −t) = S(t) and Salso is.

3. Take λ∈[0,1], then

Am(λt1+(1−λ)t2) = Zb

aZb

a

fm((λt1+(1−λ)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 J⊂I, 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

(f−L)≥sup

∂J

(f−L),

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.