Notions of generalized s convex functions on fractal sets

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R E S E A R C H

Open Access

Notions of generalized

s

-convex functions

on fractal sets

Adem Kılıçman

1*

_{and Wedad Saleh}

2

*_{Correspondence:}

akilic@upm.edu.my

1_{Department of Mathematics and}

Institute for Mathematical Research, University Putra Malaysia, Serdang, 43400, Malaysia

Full list of author information is available at the end of the article

Abstract

The purpose of this article is to present some new inequalities for products of generalized convex and generalizeds-convex functions on fractal sets. Furthermore, some applications are given.

MSC: 26A51; 26D07; 26D15; 53C22

Keywords: s-convex functions; fractal space; local fractional derivative

1 Introduction

Letf: I⊂R−→Rα_{. For any}_x

,x∈Iandγ ∈[, ] if the inequality

fγx+ ( –γ)x

≤γαf(x) + ( –γ)αf(x)

holds, thenf is called a generalized convex function onI []. Inα= , we have convex function, convexity is deﬁned only in geometrical terms as being the property of a function whose graph bears tangents only under it [].

The convexity of functions plays a signiﬁcant role in many ﬁelds, for example, in biolog-ical system, economy, optimization, and so on [–].

In recent years, the fractal theory has received significantly remarkable attention from scientists and engineers. In the sense of Mandelbrot, a fractal set is one whose Hausdorff dimension strictly exceeds the topological dimension [, ]. Many researchers studied the properties of functions on fractal space and constructed many kinds of fractional calcu-lus by using different approaches [–]. Particularly, in [], Yang gave the analysis of local fractional functions on fractal space systematically, which includes local fractional calculus and the monotonicity of function.

LetRα_{be the real line numbers on fractal space. Then by using Gao-Yang-Kang’s}

con-cept one can explain the deﬁnitions of the local fractional derivative and local fractional integral as in [–]. Now ifrα

,rαandrα∈Rα( <α≤), then

() rα+rα∈Rα,rαrα∈Rα,

() rα

 +rα=rα+rα = (r+r)α= (r+r)α,

() rα

 + (rα+rα) = (rα+rα) +rα,

() rα

rα=rαrα = (rr)α= (rr)α,

() rα

(rαrα) = (rαrα)rα,

() rα

(rα+rα) = (rαrα) + (rαrα),

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() rα

 + α= α+rα =rα andrα·α= α·rα =rα.

Let us start with some deﬁnitions as regards the local fractional calculus onRα_.

Deﬁnition .[] Letybe a local fractional continuous function on the interval [a,a].

The local fractional integral of the functiony(m) of orderαis deﬁned by

aI

(α)

ay(m) =

( +α)– a

a

y(μ)(dμ)α

=( +α)– lim

μ−→

n

i=

y(μi)(μi)α

withμi=μi+–μiandμ=max{μi:i= , , . . . ,n– }where [μi,μi+],i= , , . . . ,n–

,μ=a <μ<· · ·<μn–<μn=a is a partition of the interval [a,a] and is the

well-known Gamma function(m) =_∞tm–_e–t_dt.

In [], Mo and Sui introduced the deﬁnitions of two kinds of generalizeds-convex func-tions on fractal sets as follows.

Deﬁnition .

(i) A functionf:R+−→Rα, is called a generalizeds-convex ( <s< ) in the ﬁrst sense

if

f(γa+γa)≤γsαf(a) +γsαf(a) ()

for alla,a∈R+and allγ,γ≥withγs+γs= . This class of functions is

denoted byGK

s.

(ii) A functionf:R+−→Rα, is called a generalizeds-convex ( <s< ) in the second

sense if () holds for alla,a∈R+and allγ,γ≥withγ+γ= . This class of

functions is denoted byGK

s.

In the same paper, [], Mo and Sui proved that all functions fromGK_s, fors∈(, ), are nonnegative.

In this article, recall thatgsandgs are the generic classes consistent for those

func-tions that are generalizeds-convex in the ﬁrst sense, and in the second sense, respectively. It is well known that there are many important established inequalities for the class of generalized convex functions, however, one of the most famous is known as the general-ized Hermit-Hadamard inequality, or the ‘generalgeneral-ized Hadamard inequality’ and stated as follows (see []): letf be a generalized convex function on [a,a]⊆R,a<a, then

f

a+a



≤ ( +α) (a–a)αa

I_a(α_)f(x)≤f(a) +f(a) α .

2 Basic results for generalizeds-convex functions Lemma . If f ∈GK

s or f ∈GKs,then f(γx+γx)≤γαsf(x) +γαsf(x)

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So, we might re-write the deﬁnitions of generalizeds-convexity in the ﬁrst sense and the second sense as follows.

Deﬁnition . A functionf:I⊂R+−→Rα is called a generalizeds-convex in the ﬁrst

sense if

fγx+

 –γs



s_x



≤γαsf(x) +

 –γsαf(x)

for allx,x∈Iand for all ≤γ ≤.

Deﬁnition . A functionf:I⊂R+−→Rαis called a generalizeds-convex in the second

sense if

fγx+ ( –γ)x

≤γαsf(x) + ( –γ)αsf(x)

for allx,x∈Iand for all ≤γ ≤.

Theorem . The classes GK_, GK_ and the class of generalized convex functions are equivalent when the domain is restricted toR+.

Proof Simply an issue of applying the deﬁnitions.

Now, some results as regards generalizeds-convex functions are given.

Remark .

(i) Iff∈GK

s, thenf( x+x

s )≤

f(x)+f(x)

α .

(ii) Iff∈GK

s, thenf(x+x)≤

f(x)+f(x)

αs .

(iii) For a function that is both generalizedgsandgs, there is a one-to-one

correspondence between the set of all pairs of the sort(γ,η)with respect togsand

the set of all pairs of the sort(γ,η)with respect togs. (So we may write eachγ as a

γ_sand eachηas aηs_andvice versa. This happens in the light of the fact that

≤γ,η≤,≤s≤.)

Theorem . If a function f ∈GK

s and f ∈GKs,then f(γx+ηx)≤γαsf(x) +ηαsf(x)≤γαsf(x) +ηαsf(x)

for some{γi,ηi,i= , } ⊂[, ].

Proof It follows from the one-to-one correspondence proved before. To eachγ,ηsuch

that γ+η = , there correspondsγ, η whereγs+ηs=  andγ ≥γ, η≥η since

{γi,ηi,i= , } ⊂[, ].

Theorem . If a function f ∈GK

sand f ∈GKsand its domain concurs with its counter-domain,then fof ∈GK

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Proof f(γx+ ( –γs)



s_x_₎≤_γαs

 f(x) + ( –γs)αf(x), then

fγ_αsf(x) +

 –γ_sαf(x)

≤γ_αsαsff(x)

+ –γ_sααsff(x)

=γ_αsfof(x) +βαsfof(x).

3 Inequalities for generalizeds-convex functions Theorem . Let g: [, ]−→Rα_{be a function such that}

g(γ) = 

(a–a)α(( +α))

a

a

a

a

fγx+ ( –γ)x

(dx)α(dx)α,

where f: [a,a]−→Rαand f ∈GKs.Then:

(i) g∈GK

s in[, ].Iff ∈GKs,theng∈GKs.

(ii) α_{g() = }α_{g() =} α

(a–a)α(+α)aI

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af(x)is an upper bound forg(γ).

(iii) α(–s)_g(γ₎_≥_g( ) =

 (a–a)α((+α))

a

a

a

a f(

x+x

 )(dx)

α_(dx

)α,γ ∈[, ].

Proof (i) Take{γ,γ} ⊂[, ],γ+γ= ,t,t∈Dandf ∈GKs, then we have g(γt+γt)

= 

(a–a)α(( +α))

a

a

a

a

f(γt+γt)x

+ – (γt+γt)

x

(dx)α(dx)α

≤ 

(a–a)α(( +α))

a

a

a

a

γ_αsf(tx+x–tx)

+γ_αsf(tx+x–tx)

(dx)α(dx)α

=γ_αs 

(a–a)α(( +α))

a

a

a

a

ftx+ ( –t)x

(dx)α(dx)α

+γ_αs 

(a–a)α(( +α))

a

a

a

a

ftx+ ( –t)x

(dx)α(dx)α

=γ_αsg(t) +γαsg(t),

which implies thatg∈GK

s in [, ].

(ii) Sincef(γx+ ( –γ)x)≤γαsf(x) + ( –γ)αsf(x) we have

A=  (( +α))

a

a

a

a

fγx+ ( –γ)x

(dx)α(dx)α

≤ 

(( +α))

a

a

a

a

γαsf(x)(dx)α(dx)α

+ 

(( +α))

a

a

a

a

( –γ)αsf(x)(dx)α(dx)α

=γαs 

( +α)(a–a)

α

aI

(α)

af(x) + ( –γ)

αs 

( +α)(a–a)

α

aI

(α)

af(x)

= (γ

αs_{+ ( –}_γ₎αs₎

( +α) (a–a)

α

aI

(α)

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sinceγαs_≤_α_{. Because ( –}_γ₎αs_≤_α_{as well, we have}

A≤ 

α

( +α)(a–a)

α

aI

(α)

af(x).

(iii) Sincef is generalizeds-convex in the second sense,

f(γx+ ( –γ)x) +f(( –γ)x+γx)

αs ≥f

x+x



for allγ ∈[, ] andx,x∈[a,b].

If we integrate on [a,a]×[a,a], we obtain

 αs_{(( +}_α))

a

a

a

a

fγx+ ( –γ)x

+f( –γ)x+γx

(dx)α(dx)α

≥ 

(( +α))

a

a

a

a

f

x+x



(dx)α(dx)α

⇒

 α(s–)_(a

–a)α(( +α))

a

a

a

a

fγx+ ( –γ)x

(dx)α(dx)α

≥ 

(a–a)α(( +α))

a

a

a

a

f

x+x



(dx)α(dx)α,

which means that

α(–s)_g(γ₎_≥_g

 

.

The proof is complete.

Assume the following functions:

(i) gF: [, ]−→R

α_{, deﬁned by}

gF(γ) =

 (( +α))

a

a

a

a

fγx+ ( –γ)x

F(x)F(x)(dx)α(dx)α,

wheref: [a,a]⊂R+−→Rαandf ∈GKs.

(ii) gF: [, ]−→R

α_{, deﬁned by}

gF(γ) =

 (( +α))

a

a

a

a

fγx+ ( –γ)x

F(x)F(x)(dx)α(dx)α,

wheref: [a,a]⊂R+−→Rαandf ∈GKs. Theorem . The following holds:

(i) gFandgF are both symmetric aboutγ =

 .

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(iii) We have the upper bound

α_g

F() =

α

(( +α))

a

a

a

a

f(x)F(x)F(x)(dx)α(dx)α

for the functiongF(γ).

Proof (i) gF and gF are both symmetric aboutγ =



 because gF(γ) =gF( –γ) and

gF(γ) =gF( –γ).

(ii) We get

gF

γt+ ( –γ)t

= 

(( +α))

a

a

a

a

fγt+ ( –γ)t

x

+ –γt+ ( –γ)t

x

F(x)F(x)(dx)α(dx)α.

But

γt+ ( –γ)t

x+ ( –

γt+ ( –γ)t

x

=γtx+ ( –γ)tx+γx+ ( –γ)x–γtx– ( –γ)tx

=γ(tx+x–tx) + ( –γ)(tx+x–tx).

Sincef∈GK

s,

gF

γt+ ( –γ)t

= 

(( +α))

a

a

a

a

fγt+ ( –γ)t

x

+ –γt+ ( –γ)t

x

F(x)F(x)(dx)α(dx)α

≤γαs  (( +α))

a

a

a

a

ftx+ ( –t)x

+ ( –γ)αs  (( +α))

a

a

a

a

ftx+ ( –t)x

=γαsgF(t) + ( –γ)

αs_g F(t),

which proves thatgF∈GK



s.

(iii) From the deﬁnition and the assumptions, we get

gF(γ) =

 (( +α))

a

a

a

a

fγx+ ( –γ)x

≤γαs  (( +α))

a

a

a

a

+ ( –γ)αs  (( +α))

a

a

a

a

f(x)F(x)F(x)(dx)α(dx)α

=γαs+ ( –γ)αs  (( +α))

a

a

a

a

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≤ α (( +α))

a

a

a

a

= αgF().

Theorem . Let f,f: [a,a]⊂R+−→Rα,a<a,be nonnegative,and generalized

s-convex functions in the second sense.If f∈GKsand f∈GK



son[a,a]for someγ∈[, ]

and s,s∈(, ],then

 (( +α))

a

a

a

a

 

f

γx+ ( –γ)x

f

γx+ ( –γ)x

(dγ)α(dx)α(dx)α

≤ α( + (γ+γ)α)

( + (γ+γ+ )α)



(( +α))(a–a)

α

a

a

f(x)f(x)(dx)α

+ 

α_{(( +}_α))βα(a–a) α _T

(a,b) +T(a,b)

,

where

βα=

 

γαs_{( –}_γ₎αs_(dγ₎α_,

T(a,a) =f(a)f(a) +f(a)f(a),

and

T(a,a) =f(a)f(a) +f(a)f(a).

Proof Sincef∈GKsandf∈GK



son [a,a],

f

γx+ ( –γ)x

≤γαs_f

(x) + ( –γ)αsf(x),

f

γx+ ( –γ)x

≤γαs_f

(x) + ( –γ)αsf(x),

for allγ ∈[, ]. Sincefandfare nonnegative,

f

γx+ ( –γ)x

f

γx+ ( –γ)x

≤γα(s+s)_f

(x)f(x) + ( –γ)α(s+s)f(x)f(x)

+γαs_{( –}_γ₎αs_f

(x)f(x) +γαs( –γ)αsf(x)f(x).

Integrating both sides of the above inequality over [, ], we obtain

 ( +α)

 

f

γx+ ( –γ)x

f

γx+ ( –γ)x

(dγ)α

≤ 

( +α)f(x)f(x) 



γα(s+s)_(dγ₎α

+ 

( +α)f(x)f(x) 



( –γ)α(s+s)_(dγ₎α

+ 

( –α)f(x)f(x) 



(8)

+ 

( +α)f(x)f(x) 



γαs_{( –}_γ₎αs_(dγ₎α

= ( + (s+s)α) ( + (s+s+ )α)

f(x)f(x) +

( + (s+s)α)

( + (s+s+ )α)

f(x)f(x)

+ 

( +α)βα f(x)f(x) +f(x)f(x)

.

If we integrate both sides of the above inequalities on [a,a]×[a,a], we obtain

 (( +α))

a

a

a

a

 

f

γx+ ( –γ)x

f

γx+ ( –γ)x

≤ ( + (s+s)α)

( + (s+s+ )α)

 (( +α))

a

a

a

a

f(x)f(x)(dx)α(dx)α

+ a

a

a

a

f(x)f(x)(dx)α(dx)α

+ 

(( +α))βα

a

a

a

a

f(x)f(x)(dx)α(dx)α

+ a

a

a

a

f(x)f(x)(dx)α(dx)α

= ( + (s+s)α) ( + (s+s+ )α)



(( +α))(a–a)

α

× a

a

f(x)f(x)(dx)α+

a

a

f(x)f(x)(dx)α

+ 

(( +α))βα

a

a

f(x)(dx)α

a

a

f(x)(dx)α

+ a

a

f(x)(dx)α

a

a

f(x)(dx)α

.

By using the right half of the generalized Hadamard inequality on the right side of the above inequality, we have

 (( +α))

a

a

a

a

 

f

γx+ ( –γ)x

f

γx+ ( –γ)x

≤ α( + (s+s)α)

( + (s+s+ )α)



(( +α))(a–a)

α

a

a

f(x)f(x)(dx)α

+ 

α_{(( +}_α))βα(a–a) α

T(a,a) +T(a,a)

.

Remark .

(i) Ifα= , in Theorem ., then

β= 



γs_{( –}_γ₎s_(dγ_{) =}(s+ )(s+ )

(9)

and

a

a

a

a

 

f

γx+ ( –γ)x

f

γx+ ( –γ)x

(dγ)(dx)(dx)

≤ 

s+s+ 

(a–a)

a

a

f(x)f(x)dx

+(s+ )(s+ ) (s+s+ )

(a–a) T(a,a) +T(a,a)

.

(ii) Ifα= , ands=s= , then(s₍+)_s_₊_s_(_+)s+)=_. Also,

β= 



γs_{( –}_γ₎s_(dγ_{) =}(s+ )(s+ )

(s+s+ )

,

which implies that

a

a

a

a

 

f

γx+ ( –γ)x

f

γx+ ( –γ)x

(dγ)(dx)(dx)

≤

(a–a) a

a

f(x)f(x)dx

+ 

(a–a)

 _T

(a,a) +T(a,a)

.

Theorem . Let f,f: [a,a]⊂R+−→Rα,a<a,be nonnegative and generalized



and s,s∈(, ],then

 (( +α))

a

a

 

f

γx+ ( –γ)

a+b 

f

γx+ ( –γ)

a+b 

(dγ)α(dx)α

≤ ( + (s+s)α)

( + (s+s+ )α)

 ( +α)

a

a

f(x)f(x)(dx)α

+ 

α_{( +}_α)

( + (s+s)α)

α_{( + (s}

+s+ )α)

+ 

( +α)βα

×(a–a)α T(a,a) +T(a,a)

,

whereβα,T(a,a),and T(a,a)are deﬁned in Theorem..

Proof Sincef∈GKsandf∈GK



son [a,a],

f

γx+ ( –γ)

a+a



≤γαs_f

(x) + ( –γ)αsf

a+a



,

f

γx+ ( –γ)

a+a



≤γαs_f

(x) + ( –γ)αsf

a+a



,

(10)

f

γx+ ( –γ)

a+a



f

γx+ ( –γ)

a+a



≤γα(s+s)_f

(x)f(x) + ( –γ)α(s+s)f

a+a



f

a+a



+γαs_{( –}_γ₎αs_f

(x)f

a+a



+γαs_{( –}_γ₎αs_f



a+a



f(x).

 ( +α)

 

f

γx+ ( –γ)

a+a



f

γx+ ( –γ)

a+a



(dγ)α

≤ 

( +α)f(x)f(x) 



γα(s+s)_(dγ₎α

+ 

( +α)f

a+a



f

a+a



 

( –γ)α(s+s)_(dγ₎α

+ 

( +α)f(x)f

a+a



 

γαs_{( –}_γ₎αs_(dγ₎α

+ 

( +α)f

a+a



f(x)

 

γαs_{( –}_γ₎αs_(dγ₎α

= ( + (s+s)α) ( + (s+s+ )α)

f(x)f(x)

+ ( + (s+s)α) ( + (s+s+ )α)

f

a+a



f

a+a



+ 

( +α)βα

f(x)f

a+a



+f

a+a



f(x)

.

If we integrate both sides of the above inequalities on [a,a], we obtain

 (( +α))

a a   f

γx+ ( –γ)

a+a



f

γx+ ( –γ)

a+a



(dγ)α_(dx

)α

≤ ( + (s+s)α)

( + (s+s+ )α)

 ( +α)

a

a

f(x)f(x)(dx)α

+ ( + (s+s)α) ( + (s+s+ )α)



( +α)(a–a)

α_f



a+a



f

a+a



+ 

(( +α))βα

f

a+a



a

a

f(x)(dx)α+f

a+a



a

a

f(x)(dx)α

≤ ( + (s+s)α)

( + (s+s+ )α)

 ( +α)

a

a

f(x)f(x)(dx)α

+ ( + (s+s)α) ( + (s+s+ )α)



( +α)(a–a)

αf(a) +f(a)

α

f(a) +f(a)

α

+ 

(( +α))(a–a)

α

βα

αf(a) +f(a) α

f(a) +f(a)

α

= ( + (s+s)α) ( + (s+s+ )α)

 ( +α)

a

a

(11)

+ ( + (s+s)α) α_{( + (s}

+s+ )α)



( +α)(a–a)

α

T(a,a) +T(a,a)

+ 

α_{(( +}_α))(a–a)

α_β

α T(a,a) +T(a,a)

.

Then

 (( +α))

a

a

 

f

γx+ ( –γ)

a+a



f

γx+ ( –γ)

a+a



(dγ)α_(dx

)α

≤ ( + (s+s)α)

( + (s+s+ ))α

 ( +α)

a

a

f(x)f(x)(dx)α

+ 

α_{( +}_α)

( + (s+s)α)

α_{( + (s}

+s+ ))α

+ 

( +α)βα

×(a–a)α T(a,a) +T(a,a)

.

Remark . Ifα= , then

a

a

 

f

γx+ ( –γ)

a+a



f

γx+ ( –γ)

a+a



(dγ)(dx)

≤ 

s+s+ 

a

a

f(x)f(x)(dx)

+ 

 (s+s+ )

+(s+ )(s+ ) (s+s+ )

(a–a)T(a,a) +T(a,a)

.

Theorem . Let f,f: [a,a]⊂R+−→Rα,a<a,be nonnegative and generalized



and s,s∈(, ],then

 (( +α))

a

a

 

f

γa+a

 + ( –γ)x

f

γa+a

 + ( –γ)x

(dγ)α(dx)α

≤ ( + (s+s)α)

( + (s+s+ )α)

 ( +α)

a

a

f(x)f(x)(dx)α

+ 

α_{( +}_α)

( + (s+s)α)

α_{( + (s}

+s+ )α)

+ 

( +α)βα

×(a–a)α T(a,a) +T(a,a)

,

whereβα,T(a,a)and T(a,a)are deﬁned in Theorem..

Proof The proof of this theorem can easily be given like in Theorem ..

Theorem . Let f,f: [a,a]⊂R+−→Rα,a<a,be nonnegative,and generalized

(12)

 (( +α))

a a   f

γa+a

 + ( –γ)x

f

γa+a

 + ( –γ)x

(dγ)α_(dx

)α

≤( + α) ( + α)

 ( +α)

a

a

f(x)f(x)(dx)α

+ 

α_{( +}_α)

( + α) α_{( + α)}+

 ( +α)βα

(a–a)α T(a,a) +T(a,a)

,

where T(a,a)and T(a,a)are deﬁned in Theorem.,but

βα=

 

γα( –γ)α(dγ)α= ( +α) ( + α)–

( + α) ( + α).

Proof Sincefandfare generalized convex on [a,a],

f

γa+a

 + ( –γ)x

≤γαf

a+a



+ ( –γ)αf(x),

f

γa+a

 + ( –γ)x

≤γαf

a+a



+ ( –γ)α_f

(x),

for allx∈[a,a],γ ∈[, ]. Sincefandfare nonnegative,

f

γa+a

 + ( –γ)x

f

γa+a

 + ( –γ)x

≤γαf

a+a



f

a+a



+ ( –γ)α_f

(x)f(x)

+γα( –γ)αf

a+a



f(x) +γα( –γ)αf(x)f

a+a



.

 ( +α)

 

f

γa+a

 + ( –γ)x

f

γa+a

 + ( –γ)x

(dγ)α

≤ 

( +α)f

a+a



f

a+a



 

γα(dγ)α

+ 

( +α)f(x)f(x) 



( –γ)α_(dγ₎α

+ 

( +α)f

a+a



f(x)

 

γα( –γ)α(dγ)α

+ 

( +α)f(x)f

a+a



 

γα( –γ)α(dγ)α

=( + α) ( + α)f

a+a



f

a+a



+( + α)

( + α)f(x)f(x)

+ 

( +α)βα

f

a+a



f(x) +f(x)f

a+a



(13)

If we integrate both sides of the above inequalities on [a,a], we obtain

 (( +α))

a a   f

γa+a

 + ( –γ)x

f

γa+a

 + ( –γ)x

(dγ)α(dx)α

≤( + α) ( + α)



( +α)(a–a)

α

f

a+a



f

a+a



+( + α) ( + α)

 ( +α)

a

a

f(x)f(x)(dx)α

+ 

(( +α))βα

f

a+a



a

a

f(x)(dx)α+f

a+a



a

a

f(x)(dx)α

≤( + α) ( + α)



( +α)(a–a)

αf(a) +f(a)

α

f(a) +f(a)

α

+( + α) ( + α)

 ( +α)

a

a

f(x)f(x)(dx)α

+ 

(( +α))βα

αf(a) +f(a)

α

f(a) +f(a)

α

(a–a)α

=( + α) ( + α)

 ( +α)

a

a

f(x)f(x)(dx)α+

 α_{( +}_α)

( + α) α_{( + α)}

+ 

( +α)βα

(a–a)α T(a,a) +T(a,a)

.

Remark . Ifα= , then

a a   f

γa+a

 + ( –γ)x

f

γa+a

 + ( –γ)x

(dγ)(dx)

≤ 

a

a

f(x)f(x)(dx) +



(a–a) T(a,a) +T(a,a)

.

4 Applications

Example . Leta,a∈R+,a<a, anda–a≤, then

α

( +α)

( + α) ( + α)



+ α

( +α) ( + α)–

( + α) ( + α)

( +α) ( + α)



K(a,a)

+

α

( +α)

( + α) ( + α)



+ α

( +α) ( + α)–

( + α) ( + α)

( +α) ( + α)



G(a,a)

≤α

( + α) ( +α)( + α)



(a_α–a_α) (a–a)α

+ 

α

(( +α))βαA _(a

(14)

where

βα=

 

γα( –γ)α(dγ)α,

A(a,a) =

aα

 +aα

α , a,a≥,

G(a,a) =

aα

aα

 _, _a

,a≥,

and

K(a,a) =

aα

 +aα

α

 

, a,a≥.

Proof Iff∈GKsandf∈GK



s on [a,a] for someγ ∈[, ] ands,s∈(, ], then, by

Theorem ., iff,f: [, ]−→[α, α],f(x) =xαs,f(x) =xαs, wherex∈[a,a],s=s=

 anda–a≤, then

 (( +α))

a

a

a

a

 

γx+ ( –γ)x

α

≤α( + α) ( + α)



(( +α))(a–a)

α

a

a

x_α(dx)α

+ 

α_{(( +}_α))βα a α

 +a

α

 + 

α

aα_aα_(a–a)α

⇒

(a–a)α

α

( +α)

( + α) ( + α)



+ α

( +α) ( + α)–

( + α) ( + α)

( +α) ( + α)



a_α+a_α

+

α

( +α)

( + α) ( + α)



+ α

( +α) ( + α)–

( + α) ( + α)

( +α) ( + α)



aα

aα

≤α

( + α) ( +α)( + α)



(a–a)α

a_α–a_α

+ 

α_{(( +}_α))βα(a–a) α

aα_+aα_ ⇒

(a–a)α

α

( +α)

( + α) ( + α)



+ α

( +α) ( + α)–

( + α) ( + α)

( +α) ( + α)



K(a,a)

+

α

( +α)

( + α) ( + α)



+ α

( +α) ( + α)–

( + α) ( + α)

( +α) ( + α)



G(a,a)

(15)

≤α

( + α) ( +α)( + α)



(a–a)α

a_α–a_α

+ 

α

(( +α))βα(a–a) α_A_(a

,a).

Remark . Ifα= , then

 K

_(a ,a) +

 G

_(a ,a)≤

 

a

–a

a–a

+

A

_(a ,a)

⇒

K(a,a) + G(a,a)≤

a_+aa+a

+ A(a,a),

where

A(a,a) =

a+a

 , a,a≥, is the arithmetic mean, G(a,a) = (aa)



_, _a__,_a_≥_{, is the geometric mean, and}

K(a,a) =

a_+a_

 



, a,a≥, is the quadratic mean.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally in the design and progress of the study. Both authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, Serdang, 43400,}

Malaysia.2_{Department of Mathematics, University Putra Malaysia, Serdang, 43400, Malaysia.}

Acknowledgements

The authors would like to thank the referees for valuable suggestions and comments, which helped the authors to improve this article substantially.

Received: 24 June 2015 Accepted: 15 September 2015

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