R E S E A R C H
Open Access
Fractional Hermite-Hadamard inequalities for
(
α
,
m
)
-logarithmically convex functions
Jianhua Deng
1and JinRong Wang
1,2**Correspondence:
wjr9668@126.com
1Department of Mathematics,
Guizhou University, Guiyang, Guizhou 550025, P.R. China 2School of Mathematics and
Computer Science, Guizhou Normal College, Guiyang, Guizhou 550018, P.R. China
Abstract
By means of two fundamental fractional integral identities, we derive two classes of new Hermite-Hadamard type inequalities involving Riemann-Liouville fractional integrals for once and twice differentiable (
α
,m)-logarithmically convex functions, respectively. The main novelty of this paper is that we use powerful series to describe our estimations.MSC: 26A33; 26A51; 26D15
Keywords: Hermite-Hadamard type inequalities; Riemann-Liouville fractional integrals; (
α
,m)-logarithmically convex functions1 Introduction
Fractional calculus was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of non-integer calculus, as a generalization of the traditional in-teger order calculus, was mentioned already in by Leibnitz and L’Hospital. The sub-ject of fractional calculus has become a rapidly growing area and has found applications in diverse fields ranging from physical sciences and engineering to biological sciences and economics. For more recent development on fractional calculus, one can see the mono-graphs [–].
Due to the wide application of fractional integrals and importance of Hermite-Hadamard type inequalities, some authors extended to study fractional Hermite-Hadamard type in-equalities according to the Hermite-Hadamard type inin-equalities for functions of different classes. For example, see for convex functions [, ] and nondecreasing functions [], form-convex functions [–] and (s,m)-convex functions [], for functions satisfying
s-e-condition [] and the references therein.
Very recently, the authors [] raised the new concept of (α,m)-logarithmically convex functions and established some interesting Hermite-Hadamard type inequalities of such functions. The main results can be improved if we replaceEi,i= , , , by suitable series.
Motivated by [, , ], we study Hermite-Hadamard type inequalities for (α,m )-logarithmically convex functions involving Riemann-Liouville fractional integrals. Thus, the purpose of this paper is to establish fractional Hermite-Hadamard type inequalities for (α,m)-logarithmically convex functions.
2 Preliminaries
In this section, we introduce notations, definitions, and preliminary facts.
Definition .(see []) Letf∈L[a,b]. The symbolsRLJaα+fandRLJbα–f denote the left-sided
and right-sided Riemann-Liouville fractional integrals of the orderα∈R+and are defined by
RLJaα+f
(x) =
(α) x
a
(x–t)α–f(t)dt (≤a<x≤b)
and
RLJbα–f
(x) =
(α) b
x
(t–x)α–f(t)dt (≤a≤x<b),
respectively. Here(·) is the gamma function.
Definition .(see []) The functionf : [,b]→R+is said to be (α,m)-logarithmically convex if for everyx,y∈[,b], (α,m)∈(, ]×(, ], andt∈[, ], we have
ftx+ ( –t)y≤f(x)tαf(y)m(–tα).
The following inequalities results will be used in the sequel.
Lemma .(see []) Forα> and k> ,we have
I(α,k) :=
tα–ktdt=k ∞
i=
(–)i–(lnk)
i–
(α)i
<∞,
where
(α)i=α(α+ )(α+ )· · ·(α+i– ).
Moreover,it holds
I(α,k) –k
m
i=
(–lnk)i– (α)i
≤ |
lnk|
α√π(m– )
|lnk|e m–
m– .
Lemma .(see []) Forα> and k> ,z> ,we have
J(α,k) :=
( –t)α–ktdt=
∞
i=
(lnk)i– (α)i
<∞,
H(α,k,z) := z
tα–ktdt=zαkz
∞
i=
(–zlnk)i– (α)i
<∞.
Lemma . For t∈[, ],we have
Proof Letf(t) =tn+ ( –t)nforn,t∈[, ]. Clearly,f(·) is increasing on the interval [, ] and decreasing on the interval [, ]. So,f(t)≤f() = –nfor allt∈[, ]. Then we have
the first statement. Similarly one can obtain the second one.
3 The first main results
In this section, we apply the fractional integral identity from Sarikayaet al.[] to derive some new Hermite-Hadamard type inequalities for differentiable (α,m)-logarithmically convex functions.
Lemma .(see []) Let f : [a,b]→R be a differentiable mapping on(a,b).If f∈L[a,b],
then the following equality for fractional integrals holds:
f(a) +f(b)
–
(α+ ) (b–a)α RLJ
α
a+f(b) +RLJbα–f(a)
=b–a
( –t)α–tαfta+ ( –t)bdt. ()
By using Lemma ., one can extend to the following result.
Lemma . Let f : [a,b]→R be a differentiable mapping on (a,b)with a<mb≤b.If f∈L[a,b],then the following equality for fractional integrals holds:
f(a) +f(mb)
–
(α+ ) (mb–a)α RLJ
α
a+f(mb) +RLJmbα –f(a)
=mb–a
( –t)α–tαfta+m( –t)bdt. ()
Proof This is just Lemma . on the interval [a,mb]⊂[a,b]. By using Lemma ., we can obtain the main results in this section.
Theorem . Let f : [,b]→R be a differentiable mapping. If|f| is measurable and
(α,m)-logarithmically convex on[a,b]for some fixedα∈(, ], ≤a<mb≤b,then the following inequality for fractional integrals holds:
f(a) +f(mb)– (α+ ) (mb–a)α RLJ
α
a+f(mb) +RLJmbα –f(a)≤Ik,
where
Ik=
(mb–a)|f(b)|m
α
∞
i=
(–lnk)i–αi
[α;i]
k–k –α
iα
+α
i(–lnk)i– [α;i– ]
k–α iα+ +
k–α iα–α+–
k
for k= ,
Ik=|
f(b)|m(mb–a)(α– )
(α+ )α for k= ,
k= |f
(a)|
and
[α; ] := , [α;i] := (α+ )(α+ )· · ·(iα+ ), i∈N.
Proof (i) Case :k= . By Definition ., Lemma . and Lemma ., we have
f(a) +f(mb)– (α+ ) (mb–a)α RLJ
α
a+f(mb) +RLJmbα –f(a)
≤mb–a
( –t)α–tαfta+m( –t)bdt
≤mb–a
( –t)α–tαf(a)tαf(b)m–mtαdt
≤(mb–a)|f(b)|m
( –t)α–tαktαdt
≤(mb–a)|f(b)|m
( –t)α–tαktαdt
+(mb–a)|f
(b)|m
( –t)α–tαktαdt
≤(mb–a)|f(b)|m
( –t)α–tαktα+k(–t)αdt
≤(mb–a)|f(b)|m
( –t)αktα+ ( –t)αk(–t)α
dt
+(mb–a)|f
(b)|m
–tαktα–tαk(–t)αdt
≤(mb–a)|f(b)|m
( –t)αktα
dt
+(mb–a)|f
(b)|m
( –t)αk(–t)αdt
–(mb–a)|f
(b)|m
tαktα
dt–(mb–a)|f
(b)|m
tαk(–t)α
dt
≤(mb–a)|f(b)|m
–α–tαktαdt+(mb–a)|f
(b)|m
tαktαdt
–(mb–a)|f
(b)|m
tαktαdt–(mb–a)|f
(b)|m
( –t)αktαdt
≤(mb–a)|f(b)|m
–α–tαktαdt+(mb–a)|f
(b)|m
tαktαdt
–(mb–a)|f
(b)|m
tαktαdt–(mb–a)|f
(b)|m
–tαktαdt
≤(mb–a)|f(b)|m
tαktα
dt–(mb–a)|f
(b)|m
tαktα
+ –α(mb–a)|f
(b)|m
ktαdt–(mb–a)|f
(b)|m
ktαdt
≤(mb–a)f(b)m
tαktαdt– (mb–a)f(b)m
tαktαdt
+ –α(mb–a)f(b)m
ktαdt–(mb–a)|f
(b)|m
ktαdt
≤(mb–a)f(b)m
α
α
t(α+)–ktdt– (mb–a)|f(b)|m· α
α
t(α+)–ktdt
+ –α(mb–a)f(b)m·
α
α
tα–ktdt–(mb–a)|f
(b)|m
· α α
tα–ktdt
≤(mb–a)|f(b)|m
α
α
t(α+)–ktdt–
α
t(α+)–ktdt
+
–α(mb–a)|f(b)|m
α
α
tα–ktdt–(mb–a)|f
(b)|m
α
α
tα–ktdt
≤(mb–a)|f(b)|m
α
t(α+)–ktdt–
α
t(α+)–ktdt
+
–α(mb–a)|f(b)|m
α
α
tα–ktdt
–(mb–a)|f
(b)|m
α
tα–ktdt–
α
tα–ktdt
≤(mb–a)|f(b)|m
α k ∞ i=
(–lnk)i– (α+ )i
– α α+
kα
∞
i= (–
αlnk)i–
(α + )i
+
–α(mb–a)|f(b)|m
α α α
kα
∞
i=
(–αlnk)i–
(α)i
–(mb–a)|f
(b)|m
α k ∞ i=
(–lnk)i– (
α)i
– α α
kα
∞
i=
(–αlnk)i–
(
α)i
≤(mb–a)|f(b)|m
α k ∞ i=
(–lnk)i–αi
[α;i] –k –α
∞
i=
(–lnk)i–αi
[α;i]iα
+k –α
(mb–a)|f(b)|m
α
∞
i=
αi(–lnk)i– [α;i– ]iα+
–(mb–a)|f
(b)|m
α k ∞ i=
αi(–lnk)i– [α;i– ] –k
–α∞
i=
αi(–lnk)i– [α;i– ]iα–α+
≤(mb–a)|f(b)|m
α
∞
i=
(–lnk)i–αi
[α;i]
k–k –α
iα
+α
i(–lnk)i– [α;i– ]
k–α
iα+ +
k–α
iα–α+ –
k
.
(ii) Case :k= . By Definition ., we have
f(a) +f(mb)– (α+ ) (mb–a)α RLJ
α
a+f(mb) +RLJmbα –f(a)
≤mb–a
( –t)α
–tαfta+m( –t)bdt
≤mb–a
( –t)α–tαf(a)tαf(b)m–mtαdt
≤(mb–a)|f(b)|m
( –t)α–tαdt
≤(mb–a)f(b)m
( –t)α
–tαdt
≤|f(b)|m(mb–a)(α– ) (α+ )α .
The proof is completed.
Theorem . Let f : [,b]→R be a differentiable mapping and <q<∞.If|f|qis
mea-surable and(α,m)-logarithmically convex on[a,b]for some fixedα∈(, ], ≤a<mb≤ b,then the following inequality for fractional integrals holds:
f(a) +f(b)– (α+ ) (b–a)α RLJ
α
a+f(b) +RLJbα–f(a)≤Ik,
where
Ik=
(mb–a)|f(a)| qαq
– –pα
pα+
p∞
i=
(mqln|f(b)|–qln|f(a)|)i–αi
[α;i– ]
q
for k= ,
Ik=
(b–a)|f(b)|m q
– –pα
pα+
p
for k=
and k=||ff(b(a)|)mq|q ,p+q= .
Proof (i) Case :k= . By Definition ., Lemma ., and using the Hölder inequality, we have
f(a) +f(mb)– (α+ ) (mb–a)α RLJ
α
a+f(b) +RLJmbα –f(a)
≤mb–a
( –t)α–tαfta+m( –t)bdt
≤mb–a
( –t)α–tαpdt
p
fta+m( –t)bqdt
q
=mb–a
( –t)α–tαpdt
p
fta+m( –t)bqdt
q
≤mb–a –p
( –t)pα–tpαdt
p
fta+m( –t)bqdt
=mb–a –p
– –pα
pα+
p
fta+m( –t)bqdt
q
≤mb–a q
– –pα
pα+
p
f(a)qtαf(b)mq–mqtαdt
q
≤(mb–a)|f(b)|m q
– –pα
pα+
p
ktαdt
q
≤(mb–a)|f(b)|m qαq
– –pα
pα+
p
k ∞
i=
(–lnk)i– (
α)i
q
≤(mb–a)|f(a)| qαq
– –pα
pα+
p∞
i=
(–lnk)i– (α)i
q
≤(mb–a)|f(a)| qαq
– –pα
pα+
p∞
i=
(mqln|f(b)|–qln|f(a)|)i–αi
[α;i– ]
q
.
The proof is done.
(ii) Case :k= . By Definition ., Lemma ., and using the Hölder inequality, we have
f(a) +f(b)– (α+ ) (b–a)α RLJ
α
a+f(b) +RLJbα–f(a)
≤b–a q
– –pα
pα+
p
f(a)qtαf(b)mq–mqtαdt
q
≤(b–a)|f(b)|m q
– –pα
pα+
p
.
The proof is done.
4 The second main results
In this section, we apply a fractional integral identity to derive some new Hermite-Hadamard type inequalities for twice differentiable (α,m)-logarithmically convex func-tions.
We need the following result.
Lemma .(see []) Let f : [a,b]→R be a twice differentiable mapping on(a,b)with a<mb≤b.If f∈L[a,b],then the following equality for fractional integrals holds:
f(a) +f(mb)
–
(α+ ) (mb–a)α RLJ
α
a+f(mb) +RLJmbα –f(a)
=(mb–a)
– ( –t)α+–tα+
α+ f
ta+m( –t)bdt.
Theorem . Let f : [,b]→R be a differentiable mapping. If|f| is measurable and
(α,m)-logarithmically convex on[a,b]for some fixedα∈(, ], ≤a<mb≤b,then the following inequality for fractional integrals holds:
f(a) +f(mb)– (α+ ) (mb–a)α RLJ
α
a+f(mb) +RLJmbα –f(a)≤Ik,
where
Ik=|
f(a)|(mb–a)(α– )
α+(α+ )
∞
i=
(mαln|f(b)|–αln|f(a)|)i–
[α;i– ] for k= ,
Ik=
(b–a)|f(b)|m
(α+ )
–
pα+p+
p
for k=
and k=|f|f(b(a))|m| .
Proof (i) Case :k= . By Definition ., Lemma . and Lemma ., we have
f(a) +f(mb)
–
(α+ ) (mb–a)α RLJ
α
a+f(mb) +RLJmbα –f(a)
≤(mb–a)
– ( –t)α+–tα+
α+ f
ta+m( –t)bdt
≤(mb–a)
– ( –t)α+–tα+
α+ f
(a)tα
f(b)m–mtαdt
≤(mb–a)|f(b)|m (α+ )
––α–tα+–tα+ktα
dt
≤(mb–a)|f(b)|m(α– ) α+(α+ )
ktαdt
≤(mb–a)|f(b)|m(α– ) α+α(α+ )
tα–ktdt
≤(mb–a)|f(b)|m(α– ) α+α(α+ ) k
∞
i=
(–lnk)i– (
α)i
≤|f(a)|(mb–a)(α– ) α+α(α+ )
∞
i=
(mln|f(b)|–ln|f(a)|)i–αi
[α;i– ] ≤|f(a)|(mb–a)(α– )
α+(α+ )
∞
i=
(mαln|f(b)|–αln|f(a)|)i– [α;i– ] .
The proof is done.
(ii) Case :k= . By Definition ., Lemma ., we have
f(a) +f(mb)– (α+ ) (mb–a)α RLJ
α
a+f(mb) +RLJmbα –f(a)
≤(mb–a)
– ( –t)α+–tα+
α+ f
(a)tα
≤(mb–a)|f(b)|m (α+ )
––α–
tα+–tα+dt
≤(mb–a)|f(b)|m(α– ) α+(α+ ) .
The proof is done.
Theorem . Let f : [,b]→R be a differentiable mapping and <q<∞.If|f|qis
mea-surable and(α,m)-logarithmically convex on[a,b]for some fixedα∈(, ], ≤a<mb≤ b,then the following inequality for fractional integrals holds:
f(a) +f(mb)– (α+ ) (mb–a)α RLJ
α
a+f(mb) +RLJmbα –f(a)≤Ik,
where
Ik=
(mb–a)|f(a)| (α+ )
–
pα
p
∞
i=
(mqαln|f(b)|–qαln|f(a)|)i– [α;i– ]
q
for k= ,
Ik=
(mb–a)|f(b)| (α+ )
–
pα
p
for k=
and k=||ff(b(a))||mqq , p+q= .
Proof (i) Case :k= . By Definition ., Lemma ., Lemma ., and using the Hölder inequality, we have
f(a) +f(mb)– (α+ ) (mb–a)α RLJ
α
a+f(mb) +RLJmbα –f(a)
≤(mb–a)
– ( –t)α+–tα+
α+ f
ta+m( –t)bdt
≤(mb–a) (α+ )
– ( –t)α+–tα+pdt
p
fta+m( –t)bqdt
q
≤(mb–a) (α+ )
–
pα
p
f(a)qtαf(b)mq–mqtαdt
q
≤(mb–a)|f(b)|m (α+ )
–
pα
p
ktαdt
q
≤(mb–a)|f(a)| (α+ )αq
–
pα
p∞
i=
(–lnk)i– (
α)i
q
≤(mb–a)|f(a)| (α+ )αq
–
pα
p∞
i=
(mqln|f(b)|–qln|f(a)|)i–αi
[α;i– ]
q
≤(mb–a)|f(a)| (α+ )
–
pα
p∞
i=
(mqαln|f(b)|–qαln|f(a)|)i– [α;i– ]
q
where we use the following inequality:
– ( –t)α+–tα+p≤ – ( –t)α++tα+p
≤ ––αp
= – pα.
The proof is done.
(ii) Case :k= . By Definition ., Lemma ., and using the Hölder inequality, we have
f(a) +f(mb)– (α+ ) (mb–a)α RLJ
α
a+f(mb) +RLJmbα –f(a)
≤(mb–a)
– ( –t)α+–tα+
α+ f
ta+m( –t)bdt
≤(mb–a) (α+ )
– ( –t)α+–tα+pdt
p
fta+m( –t)bqdt
q
≤(mb–a) (α+ )
–
pα
p
f(a)qtαf(b)mq–mqtαdt
q
≤(mb–a)|f(b)|m (α+ )
–
pα
p
.
The proof is done.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
This work was carried out in collaboration between all authors. JRW raised these interesting problems in this research. JD and JRW proved the theorems, interpreted the results and wrote the article. All authors defined the research theme, read and approved the manuscript.
Acknowledgements
The authors thank the referee for valuable comments and suggestions which improved their paper. This work is supported by the National Natural Science Foundation of China (11201091) and Key Support Subject (Applied Mathematics) of Guizhou Normal College.
Received: 10 June 2013 Accepted: 19 July 2013 Published: 5 August 2013 References
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