Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
DOI 10.1186/s13662-015-0639-8
RESEARCH
Open Access
Some generalized Hermite-Hadamard
type integral inequalities for generalized
s-convex functions on fractal sets
Adem Kılıçman1* and Wedad Saleh2
*
Correspondence:
[email protected]
1
Department of Mathematics,
Institute for Mathematical Research,
University Putra Malaysia, Serdang,
Malaysia
Full list of author information is
available at the end of the article
Abstract
In this article, some new integral inequalities of generalized Hermite-Hadamard type
for generalized s-convex functions in the second sense on fractal sets have been
established.
MSC: 26A51; 26D07; 26D15; 53C22
Keywords: s-convex functions; fractal space; local fractional derivative
1 Introduction
The convexity of functions is an important concept in the class mathematical analysis
course, and it plays a significant role in many fields, for example, in biological system,
economy, optimization, and so on [–]. Furthermore, there are a lot of several inequalities
related to the class of convex functions. For example, Hermite-Hadamard’s inequality is
one of the well-known results in the literature, which can be stated as follows.
Theorem . (Hermite-Hadamard’s inequality) Let f be a convex function on [a , a ] with
a < a . If f is integral on [a , a ], then
f
a + a

≤

a – a
a
f (x) dx ≤
a
f (a ) + f (a )
.

()
In [], Dragomir and Fitzpatrick demonstrated a variation of Hadamard’s inequality
which holds for s-convex functions in the second sense.
Theorem . Let f : R+ → R+ be an s-convex function in the second sense,  < s <  and
a , a ∈ R+ , a < a . If f ∈ L ([a , a ]), then

s–
a + a
f


≤
a – a
a
a
f (x) dx ≤
f (a ) + f (a )
.
s+
()
In recent years, fractional calculus played an important part in fractal mathematics and
engineering. In the sense of Mandelbrot, a fractal set is the 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 calculus by
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Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
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using different approaches [–]. Particularly, in [], Yang stated the analysis of local
fractional functions on fractal space systematically, which includes local fractional calculus and the monotonicity of function.
The outline of this article is as follows. In Section , we state the operations with real
line number fractal sets and some definitions are given. Some integral inequalities of generalized Hermite-Hadamard type for generalized s-convex functions in the second sense
are studied in Section . Finally, some applications are also illustrated in Section . The
conclusions are in Section .
2 Preliminaries
Let Rα be the real line numbers on fractal space. Then, by using Gao-Yang-Kang’s concept,
one can explain the definitions of the local fractional derivative and local fractional integral
as in [–]. Now, if rα , rα and rα ∈ Rα ( < α ≤ ), then
() rα + rα ∈ Rα , rα rα ∈ Rα ,
() rα + rα = rα + rα = (r + r )α = (r + r )α ,
() rα + (rα + rα ) = (rα + rα ) + rα ,
() rα rα = rα rα = (r r )α = (r r )α ,
() rα (rα rα ) = (rα rα )rα ,
() rα (rα + rα ) = (rα rα ) + (rα rα ),
() rα + α = α + rα = rα and rα · α = α · rα = rα .
Let us state some definitions about the local fractional calculus on Rα .
Definition . [] A non-differentiable function y : R → Rα is called local fractional continuous at x if, for any ε > , there exists δ >  such that
y(x) – y(x ) < εα
holds for |x – x | < δ, where ε, δ ∈ R. y ∈ Cα (a , a ) if it is local fractional continuous on
the interval (a , a ).
Definition . [] The local fractional derivative of y(m) of order α at m = m is defined
by
yα (m ) =
dα y(m) ( + α)(y(m) – y(m ))
= lim
,
dmα m=m m→m
(m – m )α
∞
where (m) =  mz– e–m dm. If there exists y(n+)α (m) = Dαm · · · Dαm y(m) (n +  times) for
any m ∈ I ⊆ R, then y ∈ D(n+)α (I), n = , , , . . . .
Definition . [] The local fractional integral of function y(m) of order α is defined by,
where y ∈ Cα [a , a ],
(α)
a Ia y(m)
=

( + α)
a
y(t)(dt)α
a

lim
y(ti )(ti )α
( + α) t→ i=
n
=
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
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with ti = ti+ – ti and t = max{ti : i = , , . . . , n – }, where [ti , ti+ ], i = , , . . . , n –  and
t = a < t < · · · < tn– < tn = a is a partition of the interval [a , a ].
In [], the authors introduced the generalized convex function and established the
generalized Hermite-Hadamard’s inequality on fractal space. Let f : I ⊂ R → Rα for any
x , x ∈ I and γ ∈ [, ] if the following inequality
f γ x + ( – γ )x ≤ γ α f (x ) + ( – γ )α f (x )
holds, then f is called a generalized convex function on I. In α = , we have a convex
function, convexity is defined only in geometrical terms as being the property of a function
whose graph bears tangents only under it [].
Theorem . (Generalized Hermite-Hadamard’s inequality) Let f ∈ a Ia(α)
 be a generalized
convex function on [a , a ] with a < a . Then
a + a
f

≤
( + α)
f (a ) + f (a )
(α)
.
a Ia f (x) ≤
α
(a – a )
α
Note that it will be reduced to the class Hermite-Hadamard’s inequality () if α = .
In [], Mo and Sui introduced the definitions of two kinds of generalized s-convex functions on fractal sets as follows.
Definition .
(i) A function f : R+ → Rα is called generalized s-convex ( < s < ) in the first sense if
f (γ x + γ x ) ≤ γsα f (x ) + γsα f (x )
()
for all x , x ∈ R+ and all γ , γ ≥  with γs + γs = , we denote this class of functions
by GKs .
(ii) A function f : R+ → Rα is called generalized s-convex ( < s < ) in the second sense
if inequality () holds for all x , x ∈ R+ and all γ , γ ≥  with γ + γ = , we denote
this class of functions by GKs .
In the same paper [], Mo and Sui proved that all functions from GKs , s ∈ (, ), are
non-negative.
3 Main results
In [], the authors demonstrated a variation of generalized Hadamard’s inequality which
holds for a generalized s-convex function in the second sense. Now, we will give another
proof for generalized s-Hadamard’s inequality.
Theorem . Let f : R+ → Rα+ be a generalized s-convex function in the second sense,  <
s <  and a , a ∈ R+ with a < a . If f ∈ L ([a , a ]), then
α(s–) f
a + a

≤
( + α)
(α)
a I f (x)
(a – a )α  a
≤
( + sα)( + α) f (a ) + f (a ) .
( + (s + )α)
()
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
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Proof Since f is generalized s-convex in the second sense, then
f γ a + ( – γ )a ≤ γ αs f (a ) + ( – γ )αs f (a ),
∀γ ∈ [, ].
Integrating the above inequality with respect to γ on [, ], we have
( + α) I(α) f γ a + ( – γ )a ≤ f (a )( + α) I(α) γ αs
+ f (a )( + α) I(α) ( – γ )αs
=
( + sα)( + α) f (a) + f (b) .
( + (s + )α)
Let x = γ a + ( – γ )a . Then we have
( + α)
(α)
( + α) I(α) f γ a + ( – γ )a =
a I f (x)
(a – a )α  a
=
( + α)
(α)
a I f (x).
(a – a )α  a
Now, it follows that
( + α)
( + sα)( + α) (α)
f (a ) + f (a ) .
a Ia f (x) ≤
α
(a – a )
( + (s + )α)
Then the second inequality in () is proved.
In order to prove the first inequality in (), we use the following inequality:
x + x
f

≤
f (x ) + f (x )
,
αs
∀x , x ∈ I.
()
Now, assume that x = γ a + ( – γ )a and x = ( – γ )a + γ a with γ ∈ [, ].
Then we get by inequality () that
f
a + a

≤
f (γ a + ( – γ )a ) + f (( – γ )a + γ a )
,
αs
∀γ ∈ [, ].
By integrating both sides of the above inequalities over [, ], we have

( + α)

f

a + a

(α)
(dγ )α ≤ α(s–)
a I f (x).


(a – a )α  a
Then it follows that

α(s–)
a + a
f

≤
( + α)
(α)
a I f (x).
(a – a )α  a
This completes the proof.
Remark . If we set c =
inequality of ().
(+sα)(+α)
(+(s+)α)
for s ∈ (, ], then it is best possible in the second
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
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As the function f : [, ] → [α , α ] given by f (x) = xsα is generalized s-convex in the
second sense,
( + α)

(α)
a I f (x) = ( + α)
(a – a )α  a
( + α)
=

xαs (dx)α

( + sα)( + α)
( + (s + )α)
and
( + sα)( + α)
( + sα)( + α) .
f () + f () =
( + (s + )α)
( + (s + )α)
Similarly, if α = , then inequalities () reduce to inequalities ().
Theorem . Let A : [, ] → Rα be a function such as
( + α)
a + a
(α)
A(γ ) =
,
a I f γ x + ( – γ )
(a – a )α  a

γ ∈ [, ],
where f : [a , a ] → Rα is a generalized s-convex function in the second sense, s ∈ (, ],
a , a ∈ R+ , a < a and f ∈ L ([a , a ]). Then
(i) A ∈ GKs on [, ],
(ii) we have the inequality
A(γ ) ≥ α(s–) f
a + a
,

∀γ ∈ [, ],
()
(iii) and the following inequality also holds:
A ≤ min A (γ ), A (γ ) ,
γ ∈ [, ],
where
A (γ ) = γ
αs
a + a
( + α)
(α)
αs
a I f (x) + ( – γ ) f
(a – a )α  a

and
A (γ ) =
( + αs)( + α)
a + a
f γ a + ( – γ )
( + (s + )α)

a + a
+ f γ a + ( – γ )

for γ ∈ (, ].
(iv) If à = max{A (γ ), A (γ )}, γ ∈ [, ], then
α
( + αs)( + α) αs a + a
αs
à ≤
γ f (a ) + f (a ) +  ( – γ ) f
.
( + (s + )α)

()
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
Proof (i) Let γ , γ ∈ [, ] and μ , μ ≥  with μ + μ = , then
a + a
( + α)
(α)
I
f
(μ
γ
+
μ
γ
)x
+

–
(μ
γ
+
μ
γ
)
a








(a – a )α  a

a + a
( + α)
(α)
μαs
≤
a I
 f γ x + ( – γ )
(a – a )α  a

a + a
+ μαs
 f γ x + ( – γ )

A(μ γ + μ γ ) =
αs
= μαs
 A(γ ) + μ A(γ ),
which implies that A ∈ GKs on [, ].

, we have
(ii) Let γ ∈ (, ] and by the change of variable m = γ x + ( – γ ) a +a

A(γ ) =
( + α)
( + α)
(α)
(α)
a +a I
a +a f (m) =
b I f (m).
γ α (a – a )α γ a +(–γ )  γ a +(–γ )   
(b – b )α  b
By using the first generalized Hermite-Hadamard inequality, we have
b + b
a + a
( + α)
(α)
α(s–)
α(s–)
=

,
I
f
(m)
≥

f
f
b
(b – b )α  b


and inequality () is obtained.
If γ = , the inequality
f
a + a

≥ α(s–) f
a + a

also holds.
(iii) By using the second part of generalized Hadamard’s inequality, we get
( + sα)( + α) ( + α)
(α)
f (b ) + f (b )
b Ib f (m) ≤
α
(b – b )
( + (s + )α)
( + sα)( + α)
a + a
=
f γ a + ( – γ )
( + (s + )α)

a + a
+ f γ a + ( – γ )

= A (γ ),
∀γ ∈ [, ].
If γ = , then the inequality
a + a
f

α ( + sα)( + α) a + a
= A() ≤ A () =
f
( + (s + )α)

holds as it is equivalent to
( + (s + )α)
a + a
– α f
≤ α
( + sα)( + α)

Page 6 of 15
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Page 7 of 15
and we know that for s ∈ (, ),
f
a + a

≥ α .
Since
a + a
a + a
αs
αs
f γ x + ( – γ )
≤ γ f (x) + ( – γ ) f


for ∀γ ∈ [, ] and x ∈ [a , a ], then we obtain
a + a
( + α)
(α)
I
f
γ
x
+
(
–
γ
)
a
(a – a )α  a

a + a
( + α)
(α)
αs
I
f
(x)
+
(
–
γ
)
f
≤ γ αs
a
(a – a )α  a

A(γ ) =
= A (γ ).
Then, the proof of inequality () is complete.
(iv) We have
( + sα)( + α)
a + a
a + a
f γ a + ( – γ )
+ f γ a + ( – γ )
( + (s + )α)


a + a
( + sα)( + α) αs
γ f (a ) + ( – γ )αs f
≤
( + (s + )α)

a + a
+ γ αs f (a ) + ( – γ )αs f

( + sα)( + α) αs a + a
γ f (a ) + f (a ) + α ( – γ )αs f
,
=
( + (s + )α)

A (γ ) =
∀γ ∈ [, ].
Since
( + α)
( + sα)( + α) (α)
f (a ) + f (a )
a Ia f (x) ≤
α
(a – a )
( + (s + )α)
and
( – γ )αs f
a + a

≤ α ( – γ )αs
( + sα)( + α) a + a
f
,
( + (s + )α)

then
( + sα)( + α) f (a ) + f (a )
( + (s + )α)
( + sα)( + α) a + a
+ α ( – γ )αs
f
( + (s + )α)

A (γ ) ≤ γ αs
and the proof of Theorem . is complete.
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
Remark . In particular:
. If we choose s =  in Theorem ., then we get:
(a)
a + a
( + α)
(α)
a I f γ x + ( – γ )
(a – a )α  a

a + a
( + α)
(α)
α
,
I
f
(x)
+
(
–
γ
)
f
≤ min γ α
a
(a – a )α  a

a + a
(( + α))
f γ a + ( – γ )
( + α)

a + a
.
+ f γ a + ( – γ )

(b) Since
a + a
( + α)
(α)
α
à = max γ α
I
f
(x)
+
(
–
γ
)
f
,
a
(a – a )α  a

a + a
a + a
(( + α))
f γ a + ( – γ )
+ f γ a + ( – γ )
,
( + α)


we have
α
(( + α)) α a + a
α
Ã(γ ) ≤
γ f (a ) + f (a ) +  ( – γ ) f
.
( + α)

. Now if one chooses α =  in Theorem ., then we can easily obtain:
(a)
a a + a

dx
f γ x + ( – γ )
(a – a ) a

a

a + a
,
f (x) dx + ( – γ )s f
≤ min γ s
(a – a ) a


a + a
a + a
f γ a + ( – γ )
+ f γ a + ( – γ )
.
s+


(b) Similarly we have
à = max γ s
a
a + a

s
,
f (x) dx + ( – γ ) f
(a – a ) a

a + a
a + a

f γ a + ( – γ )
+ f γ a + ( – γ )
s+


and

a + a
s
s
Ã(γ ) ≤
γ f (a ) + f (a ) + ( – γ ) f
for ∀γ ∈ [, ].
s+

. If one considers α =  and s =  in Theorem ., then we get:
Page 8 of 15
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
Page 9 of 15
(a)
a 
a + a
dx
f γ x + ( – γ )
(a – a ) a

a
a + a

,
f (x) dx + ( – γ )f
≤ min γ
(a – a ) a

a + a
a + a

f γ a + ( – γ )
+ f γ a + ( – γ )
.



(b)
à = max γ

(a – a )
a
f (x) dx + ( – γ )f
a
a + a
,

a + a
a + a

f γ a + ( – γ )
+ f γ a + ( – γ )



and
Ã(γ ) ≤
 a + a
γ f (a ) + f (a ) + ( – γ )f
for ∀γ ∈ [, ].


Theorem . Let g : [, ] → Rα be a function such as


g(γ ) =

(( + α)) (a – a )α
a
a
a
f γ x + ( – γ )x (dx )α (dx )α ,
γ ∈ [, ],
a
where f : [a , a ] → Rα+ is a generalized s-convex function in the second sense, s ∈ (, ],
a , a ∈ R+ which a < a and f ∈ L ([a , a ]). Then:
(i) g ∈ GKs in [, ]. If f ∈ GKs , then g ∈ GKs .
(ii) g(γ +  ) = g(  – γ ) for all γ = [,  ] and g(γ ) is symmetric about γ =  .
(iii) We have the inequality
g(γ ) ≥
α(s–)
α(s–)
a + a
A(γ
)
≥
f
(( + α))
(( + α))

for ∀γ ∈ [, ].
()
(iv) We have the inequality
g(γ ) ≤ min g (γ ), g (γ ) ,
()
where
g (γ ) = γ αs + ( – γ )αs

(α)
a I f (x )
( + α)(a – a )α  a
and
g (γ ) =
for ∀γ ∈ [, ].
( + sα)
( + (s + )α)

f (a )+f γ a +(–γ )a +f (–γ )a +γ a +f (a )
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
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Proof (i) Take {γ , γ } ⊂ [, ], γ + γ = , t , t ∈ D and f ∈ GKs , then we have
g(γ t + γ t ) =

)α (( + α))
(a – a
a ×
a
≤
a
f (γ t + γ t )x +  – (γ t + γ t ) x (dx )α (dx )α
a

)α (( + α))
(a – a
a ×
a
a a
γαs f (t x + x – t x )
+ x – t x ) (dx )α (dx )α
a a

αs
= γ
f (t x + x – t x )(dx )α (dx )α
(a – a )α (( + α)) a a
a a

+ γαs
f (t x + x – t x )(dx )α (dx )α
(a – a )α (( + α)) a a
+ γαs f (t x
= γαs g(t ) + γαs g(t ),
which implies that g ∈ GKs in [, ].
(ii) Let γ ∈ [,  ], then


g γ+
=

(a – a )α (( + α))
a a 

x +  – γ –
x (dx )α (dx )α
×
f
γ+


a
a

(a – a )α (( + α))
a a 

– γ x +
+ γ x (dx )α (dx )α
×
f


a
a

–γ .
=g

=
g(γ ) is symmetric about γ =
(iii) Let us observe that
g(γ ) =


because g(γ ) = g( – γ ).


(( + α)) (a – a )α
a a

(a – a )α
a
a
Now, since x is fixed in [a , a ], then the function
Ax : [, ] → Rα
can be given by

Ax (γ ) =
(a – a )α
=
a
f γ x + ( – γ )x (dx )α
a
( + α)
(α)
a Ia f γ x + ( – γ )x .
α
(a – a )
f γ x + ( – γ )x (dx )α (dx )α .
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
Page 11 of 15
As it was shown in the proof of Theorem ., for γ ∈ [, ], we have equality
Ax (γ ) =
( + α)
(α)
b I f (m),
(b – b )α  b
where b = γ a + ( – γ )x and b = γ a + ( – γ )x . By using the generalized HermiteHadamard inequality, we have
b + b
( + α)
(α)
α(s–)
I
f
(m)
≥

f
b
(b – b )α  b

a + a
α(s–)
f γ
=
+ ( – γ )x

for all γ ∈ (, ) and x ∈ [a , a ]. Integrating on [a , a ] over x , we have
g(γ ) ≥
α(s–)
A( – γ ) for ∀γ ∈ (, ).
(( + α))
Further, since g(γ ) = g( – γ ), then the proof of inequality () is done for γ ∈ (, ). If γ = 
or γ = , then inequality () also holds.
(iv) Since f (γ x + ( – γ )x ) ≤ γ αs f (x ) + ( – γ )αs f (x ) for all x , x ∈ [a , a ] and γ ∈ [, ],
integrating the above inequality on [a , a ] , we have

(a – a )α
≤ γ αs
a
a
a
f γ x + ( – γ )x (dx )α (dx )α
a
(( + α))

α
(a – a ) (( + α))
+ ( – γ )αs
= γ αs
a
a
a
a
(( + α))

α
(a – a ) (( + α))
f (x )(dx )α (dx )α
a
a
a
f (x )(dx )α (dx )α
a
( + α)
(α)
αs ( + α)
(α)
a I f (x ) + ( – γ )
a I f (x )
(a – a )α  a
(a – a )α  a
( + α)
(α)
= γ αs + ( – γ )αs
a I f (x ).
(a – a )α  a
The proof of the first part in () is done.
By the second part of the generalized Hermite-Hadamard inequality, we obtain
Ax (γ ) =
≤
( + α)
(α)
b I f (m)
(b – b )α  b
( + sα)( + α) f γ a + ( – γ )x + f γ a + ( – γ )x ,
( + (s + )α)
where b = γ a + ( – γ )x and b = γ a + ( – γ )x , γ ∈ [, ]. Integrating this inequality
on [a , a ] over x , then
( + sα)
( + α)
(α)
g(γ ) ≤
a I f γ a + ( – γ )x
( + α)( + (s + )α) (a – a )α  a
( + α)
(α)
I
f
γ
a
+
(
–
γ
)x
+
a

 .
(a – a )α  a
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
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A simple calculation shows that
( + α)
(α)
a Ia f γ a + ( – γ )x
α
(a – a )
=
( + α) (α)
c I f (m)
(c – c )α  c
≤
( + sα)( + α) f (c ) + f (c )
( + (s + )α)
=
( + sα)( + α) f (a ) + f γ a + ( – γ )a ,
( + (s + )α)
where c = a , c = γ a + ( – γ )a and γ ∈ (, ). Similarly, for γ ∈ (, ),
( + α)
(α)
a Ia f γ a + ( – γ )x
α
(a – a )
≤
( + sα)( + α) f (a ) + f γ a + ( – γ )a .
( + (s + )α)
Then
g(γ ) ≤

( + sα)
( + (s + )α)
f (a ) + f γ a + ( – γ )a + f ( – γ )a + γ a + f (a ) .
If γ =  or γ = , then this inequality also holds.
Remark . If α =  in the above theorem, then
g(γ ) =

(a – a )
a
a
a
f γ x + ( – γ )x (dx )(dx ),
γ ∈ [, ]
a
and
g(γ ) ≤ min γ s + ( – γ )s

(a – a )
a
f (x )(dx ),
a
 f (a ) + f γ a + ( – γ )a + f ( – γ )a + γ a + f (a ) .
( + s)
Theorem . Let us consider that a sum of A belongs to GKs ,
A=
n
ai (γ ),
i=
where

ai (γ ) =
(( + α))
a
a
a
fi γ x + ( – γ )x (dx )α (dx )α ,
a
then
(i) sup(A) = α ni= ai () = α ni= ai (),
(ii) A is symmetric about γ =  ,
(iii) A ∈ GKs .
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
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Proof (i)
ai (γ ) ≤ γ αs

(( + α))
a
a
a
fi (x )(dx )α (dx )α
a

+ ( – γ )αs
(( + α))
a
a
a
fi (x )(dx )α (dx )α ,
∀i.
a
Since fi are generalized s-convex functions, we get
a a

fi (x )(dx )α (dx )α
(( + α)) a a
a a
fi (x )(dx )α (dx )α
ai (γ ) ≤ γ αs + ( – γ )αs
≤
α
(( + α))
α
a
a
α
=  ai () =  ai ().
(ii) ai (γ ) is symmetric about γ =  since ai (γ ) = ai ( – γ ), ∀i.
Then A( – γ ) = A(γ ) and A also is.
(iii) Since ai (γ x + ( – γ )x ) ≤ γ αs ai (x ) + ( – γ )αs ai (x ), then
n
A γ x + ( – γ )x =
ai γ x + ( – γ )x
i=
≤ γ αs
n
ai (x ) + ( – γ )αs
i=
n
ai (x )
i=
= γ αs A(x ) + ( – γ )αs A(x ),
that is, A ∈ GKs .
4 Applications to special means
We now consider the applications of our theorems to the following generalized means:
aα + aα
, a , a ≥ ,
α

α
a + aα

K(a , a ) =
, a , a ≥ 
α
A(a , a ) =
and

G(a , a ) = aα aα  ,
a , a ≥ .
In [], the following example is given.
Let  < s <  and aα , aα , aα ∈ Rα . Define, for x ∈ R+ ,
⎧
⎨aα ,
n = ,

f (n) =
⎩aα nsα + aα , n > .


If aα ≥ α and α ≤ aα ≤ aα , then f ∈ GKs .
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
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Proposition . Let a , a ∈ R+ , a < a and a – a ≤ , then the following inequalities
hold:

 
( + α)

α ( + α)
K (a , a ) + α G (a , a ) ≥ 
A (a , a ),
( + α)( + α)

( + α)
α
 – γ + γ  K  (a , a ) + γ α ( – γ )α G (a , a )

γ α + ( – γ )α ( + α)  
K (a , a ) + α G (a , a )
– α
 ( + α) ( + α)

≥ α γ α ( – γ )α A (a , a ).
()
()
Proof If f ∈ GKs on [a , a ] for some γ ∈ [, ] and s ∈ (, ], then, in Theorem ., if
f : [, ] → [α , α ], f (x) = xα , where x ∈ [a , a ] and s = , so



(( + α)) (a – a )α
a
a
a γ x + ( – γ )x
α
(dx )α (dx )α
a
α
( + α)
a + aα aα + aα

( + α)( + α)

( + α) α α
a + aα .
+ α γ α ( – γ )α
( + α)
= γ α + ( – γ )α
Then, by Theorem ., we get

α
( + α)
( + α)  α
α
α α
a + aα .
a + a + a a ≥
( + α)( + α)
( + α)
Then we obtain inequality ().
By applying Theorem ., we obtain inequality () as follows:
α
α  α a + a
α
α α α
+ γ ( – γ ) a a
–γ +γ
α
α  α α
γ α + ( – γ )α ( + α)  aα
 + a
+ α a a
– α
 ( + α)
( + α)
α

α
a + aα 
≥ α γ α ( – γ )α  α  .

5 Conclusion
In this article, we have established some new integral inequalities of generalized HermiteHadamard type for generalized s-convex functions in the second sense on fractal sets Rα ,
 < α < . In particular, our results extend some important inequalities in a classical situation; when α = , some relationships between these inequalities and the classical inequalities have been established. Finally, we have also given some applications for these
inequalities on fractal sets.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors jointly worked on deriving the results and approved the final manuscript.
Kılıçman and Saleh Advances in Difference Equations (2015) 2015:301
Page 15 of 15
Author details
1
Department of Mathematics, Institute for Mathematical Research, University Putra Malaysia, Serdang, Malaysia.
2
Department of Mathematics, University Putra Malaysia, Serdang, 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: 19 June 2015 Accepted: 15 September 2015
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