Some generalized Riemann Liouville k fractional integral inequalities

R E S E A R C H

Open Access

Some generalized Riemann-Liouville

k-fractional integral inequalities

Praveen Agarwal

_{, Jessada Tariboon}

_{and Sotiris K Ntouyas}

*_{Correspondence:}

jessada.t@sci.kmutnb.ac.th

2_{Nonlinear Dynamic Analysis}

Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand Full list of author information is available at the end of the article

Abstract

The focus of the present study is to prove some new Pólya-Szegö type integral inequalities involving the generalized Riemann-Liouvillek-fractional integral operator. These inequalities are used then to establish some fractional integral inequalities of Chebyshev type.

MSC: 26D10; 26A33; 26D15

Keywords: integral inequalities; Chebyshev functional; Riemann-Liouville k-fractional integral operator; Pólya and Szegö type inequalities

1 Introduction and motivation

The celebrated functionals were introduced by the Chebyshev in his famous paper [] and were subsequently rediscovered in various inequalities (for the celebrated functionals) by

numerous authors, including Anastassiou [], Belarbi and Dahmani [], Dahmaniet al.

[], Dragomir [], Kalla and Rao [], Lakshmikantham and Vatsala [], Ntouyaset al.[],

Öˇgünmez and Özkan [], Sudsutadet al.[], Sulaiman []; and, for very recent work,

see also Wanget al.[]. This type of functionals is usually defined as

T(f,g) =  b–a

b

a

f(x)g(x)dx–

 b–a

b

a f(x)dx

 b–a

b

a g(x)dx

, (.)

wheref andgare two integrable functions which are synchronous on [a,b],i.e.,

f(x) –f(y)g(x) –g(y)≥, (.)

for anyx,y∈[a,b].

The well-known Grüss inequality [] is defined by

T(f,g)≤(M–m)(N–n)

 , (.)

wheref andgare two integrable functions which are synchronous on [a,b] and satisfy the following inequalities:

m≤f(x)≤M and n≤g(y)≤N, (.)

for allx,y∈[a,b] and for somem,M,n,N∈R.

Pólya and Szegö [] introduced the following inequality:

b af(x)dx

b a g(x)dx (_abf(x)dx_abg(x)dx) ≤

 

MN

mn +

mn MN



. (.)

Dragomir and Diamond [] by using the Pólya and Szegö inequality, proved that

T(f,g)≤ (M–m)(N–n)

(b–a)√_mMnN

b

a f(x)dx

b

a

g(x)dx, (.)

wheref andgare two positive integrable functions which are synchronous on [a,b], and

 <m≤f(x)≤M<∞,  <n≤g(y)≤N<∞, (.)

for allx,y∈[a,b] and for somem,M,n,N∈R.

Recently, k-extensions of some familiar fractional integral operator like

Riemann-Liouville have been investigated by many authors in interesting and useful manners (see [–], and []). Here, we begin with the following.

Definition . Letk> , then the generalizedk-gamma andk-beta functions defined by []

k(x) = lim n→∞

n!kn(nk)xk–

(x)n,k

, (.)

where (x)n,k, is the Pochhammerk-symbol defined by

(x)n,k=x(x+k)(x+ k)· · ·

x+ (n– )k (n≥).

Definition . Thek-gamma function is defined by

k(x) =

∞



tx–e–tkk dt, (x) > .

It is well known that the Mellin transform of the exponential functione–tkk _{is the}

k-gamma function. Clearly

(x) =lim

k→k(x), k(x) =k

x k–

x k

and k(x+k) =xk(x).

Definition . Ifk> , letf ∈L(a,b),a≥, then the Riemann-Liouvillek-fractional integralRα

a,kof orderα>  for a real-valued continuous functionf(t) is defined by ([]; see also [])

Rα_a,kf(t)=  kk(α)

t

a

(t–τ)αk–_{f(τ)dτ t}∈_{[a,b]. (.)}

Definition . Ifk> , letf ∈L,r_{[a,b],a≥,r∈R\ {–}then the generalized} Riemann-Liouvillek-fractional integralRα_a,,_krof orderα>  for a real-valued continuous functionf(t) is defined by ([])

Rα_a,_,krf(t)=( +r)

–α_k

kk(α)

t

a

tr+–τr+αk–_{f(τ)dτ t}∈_{[a,b], (.)}

wherekis the Euler gammak-function.

The generalized Riemann-Liouvillek-fractional integral (.) has the properties

R_aα,_,krR_aβ_,,_krf(t)=R_aα_,+_kβ,rf(t)=R_aβ_,,_krR_aα_,,_krf(t) (.)

and

Rα_a,_,kr{}= (t

r+_–ar+₎αk

(r+ )αk_k(α+k)

, α> . (.)

In this paper, we derive some new Pólya-Szegö type inequalities by making use of the

generalized Riemann-Liouvillek-fractional integral operators and then use them to

estab-lish some Chebyshev type integral inequalities.

We organize the paper as follows: in Section , we prove some generalized Pólya-Szegö

type integral inequalities involving the generalized Riemann-Liouvillek-fractional

inte-gral operators that we need to establish main theorems in the sequel and Section  contains

some Chebyshev type integral inequalities via generalized Riemann-Liouvillek-fractional

integral operators.

2 Some Pólya-Szegö types inequalities

In this section, we prove some Pólya-Szegö type integral inequalities for positive

inte-grable functions involving the generalized Riemann-Liouvillek-fractional integral

opera-tor (.).

Lemma . Let f and g be two positive integrable functions on[a,∞).Assume that there exist four positive integrable functionsϕ,ϕ,ψ,andψon[a,∞)such that:

(H)  <ϕ(τ)≤f(τ)≤ϕ(τ), <ψ(τ)≤g(τ)≤ψ(τ)(τ∈[a,t],t>a).

Then,for t>a,k> ,a≥,α> ,and r∈R\ {–},the following inequality holds: Rα_a,,r_k{ψψf}(t)Rαa,,kr{ϕϕg}(t)

(Rα_a,,_kr{(ϕψ+ϕψ)fg}(t))

≤ 

. (.)

Proof From (H), forτ∈[a,t],t>a, we have

f(τ) g(τ)≤

ϕ(τ)

ψ(τ)

, (.)

which yields

ϕ(τ)

ψ(τ)

–f(τ) g(τ)

Analogously, we have

ϕ(τ)

ψ(τ)

≤f(τ)

g(τ), (.)

from which one has

f(τ) g(τ)–

ϕ(τ)

ψ(τ)

≥. (.)

Multiplying (.) and (.), we obtain

ϕ(τ)

ψ(τ)

–f(τ) g(τ)

f(τ) g(τ)–

ϕ(τ)

ψ(τ)

≥,

or

ϕ(τ)

ψ(τ)

+ϕ(τ)

ψ(τ)

f(τ) g(τ)≥

f_(τ)

g_(τ)+

ϕ(τ)ϕ(τ)

ψ(τ)ψ(τ)

. (.)

The inequality (.) can be written as

ϕ(τ)ψ(τ) +ϕ(τ)ψ(τ)

f(τ)g(τ)≥ψ(τ)ψ(τ)f(τ) +ϕ(τ)ϕ(τ)g(τ). (.)

Now, multiplying both sides of (.) by (+r)– α

k(tr+_–τr+₎αk–

kk(α) and integrating with respect to

τ fromatot, we get

Rα_a,_,kr(ϕψ+ϕψ)fg

(t)≥Rα_a,,r_kψψf

(t) +Rα_a,,_krϕϕg

(t).

Applying the AM-GM inequality,i.e.,a+b≥√ab,a,b∈R+_{, we have}

Rα_a,_,kr(ϕψ+ϕψ)fg

(t)≥

Rα_a,,_krψψf

(t)Rα_a,_,krϕϕg

(t),

which leads to

Rα_a,_,krψψf

(t)Rα_a,,r_kϕϕg

(t)≤  

Rα_a,,_kr(ϕψ+ϕψ)fg

(t).

Therefore, we obtain the inequality (.) as required.

Lemma . Let f and g be two positive integrable functions on[a,∞).Assume that there exist four positive integrable functionsϕ,ϕ,ψ,andψsatisfying(H)on[a,∞).Then,

for t>a,k> ,a≥,α> ,β> ,and r∈R\ {–},the following inequality holds: Rα_a,,_kr{ϕϕ}(t)Rβ_a,,_kr{ψψ}(t)Rα_a,,_kr{f}(t)Rβ_a,,_kr{g}(t)

(Rα_a,,_kr{ϕf}(t)Rβa,,kr{ψg}(t) +Rαa,,kr{ϕf}(t)Raβ,,kr{ψg}(t))

≤

. (.)

Proof To prove (.), using the condition (H), we obtain

ϕ(τ)

ψ(ρ)

–f(τ) g(ρ)

and

f(τ) g(ρ)–

ϕ(τ)

ψ(ρ)

≥, (.)

which imply that

ϕ(τ)

ψ(ρ)

+ ϕ(τ)

ψ(ρ)

f(τ) g(ρ)≥

f_(τ)

g_(ρ)+

ϕ(τ)ϕ(τ)

ψ(ρ)ψ(ρ)

. (.)

Multiplying both sides of (.) byψ(ρ)ψ(ρ)g(ρ), we have

ϕ(τ)f(τ)ψ(ρ)g(ρ) +ϕ(τ)f(τ)ψ(ρ)g(ρ)

≥ψ(ρ)ψ(ρ)f(τ) +ϕ(τ)ϕ(τ)g(ρ). (.)

Multiplying both sides of (.) by

( +r)–αk_{( +r)}–

β

k_(tr+_–τr+₎kα–_(tr+_–ρr+₎

β

k–

kk(α)kk(β) ,

and double integrating with respect toτandρfromatot, we have

Rα_a,_,kr{ϕf}(t)Rβ_a,,_kr{ψg}(t) +Rα_a,,_kr{ϕf}(t)R_aβ_,,_kr{ψg}(t)

≥Rα_a,,r_kf(t)Rβ_a,,_kr{ψψ}(t) +Rα_a,,_kr{ϕϕ}(t)Rβ_a,,_kr

g(t).

Applying the AM-GM inequality, we get

Rα_a,_,kr{ϕf}(t)R

β,r

a,k{ψg}(t) +Raα,,kr{ϕf}(t)R

β,r

a,k{ψg}(t)

≥

Rα_a,,_krf_(t)Rβ,r

a,k{ψψ}(t)Raα,,kr{ϕϕ}(t)Rβa,,kr

g_(t),

which leads to the desired inequality in (.). The proof is completed.

Lemma . Let f and g be two positive integrable functions on[a,∞).Assume that there exist four positive integrable functionsϕ,ϕ,ψ,andψsatisfying(H)on[a,∞).Then,

for t>a,k> ,a≥,α> ,β> ,and r∈R\ {–},the following inequality holds: Rα_a,_,krf(t)Rβ_a,,_krg(t)≤R_aα_,,_kr(ϕfg)/ψ

(t)Rβ_a,,_kr(ψfg)/ϕ

(t). (.)

Proof From (.), we have

( +r)–αk

kk(α)

t

a

tr+–τr+αk–_f_(τ)dτ

≤( +r)–

α

k

kk(α)

t

a

tr+–τr+αk–ϕ(τ) ψ(τ)

f(τ)g(τ)dτ,

which implies

Rα_a,_,krf(t)≤Rα_a,,_kr(ϕfg)/ψ

By (.), we get

( +r)–βk

kk(β)

t

a

tr+_–ρr+βk–_g_(ρ)dρ

≤( +r)–

β

k

kk(β)

t

a

tr+–ρr+

β

k–ψ(ρ) ϕ(ρ)

f(ρ)g(ρ)dρ,

from which one has

R_,β_tg(t)≤R_aβ_,,_kr(ψfg)/ϕ

(t). (.)

Multiplying (.) and (.), we get the desired inequality in (.).

Corollary . Let f and g be two positive integrable functions on[,∞)satisfying (H)  <m≤f(τ)≤M<∞, <n≤g(τ)≤N<∞(τ∈[a,t],t>a).

Then,for t>a,k> ,a≥,α> ,β> ,and r∈R\ {–},we have (Rα_a,,_kr{f_}(t))(Rβ,r

a,k{g}(t)) R_aα,_,kr{fg}(t)R_aβ_,,_kr{fg}(t) ≤

MN

mn. (.)

3 Chebyshev type integral inequalities

In the sequel, we establish our main Chebyshev type integral inequalities involving the

generalized Riemann-Liouvillek-fractional integral operator (.), with the help of the

Pólya-Szegö fractional integral inequality in Lemma . as follows.

Theorem . Let f and g be two positive integrable functions on[a,∞),a≥.Assume that there exist four positive integrable functionsϕ,ϕ,ψ,andψsatisfying(H).Then,

for t>a,k> ,a≥,α> ,and r∈R\ {–},the following inequality is fulfilled:

(tr+–ar+)

α

k

(r+ )αk_k(α+k)

R_aα_,,_kr{fg}(t) –Rα_a,,_kr{f}(t)R_aα_,,_kr{g}(t)

≤G(f,ϕ,ϕ)(t)G(g,ψ,ψ)(t)



_{, (.)}

where

G(u,v,w)(t) = (t

r+_–ar+₎αk

(r+ )αk_k(α+k)

(Rα_a,_,kr{(v+w)u}(t))

Rα_a,_,kr{vw}(t) –

Rα_a,_,kr{u}(t). (.)

Proof Letfandgbe two positive integrable functions on [a,∞). Forτ,ρ∈(a,t) witht>a, we defineA(τ,ρ) as

A(τ,ρ) =f(τ) –f(ρ)g(τ) –g(ρ), (.)

or, equivalently,

Multiplying both sides of (.) by (+r)

(–α

k)(tr+_–τr+₎αk–(tr+_–ρr+₎αk–

(kk(α)) and double integrating

with respect toτ andρfromatot, we get

( +r)(–αk)

(kk(α))

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+αk–_{A(τ,ρ)dτdρ}

=  (t

r+_–ar+₎α_k

(r+ )αk_k(_α+k)

Rα_a,_,kr{fg}(t) – Rα_a,,_kr{g}(t)R_aα_,,_kr{f}(t). (.)

By using the Cauchy-Schwartz inequality for double integrals, we have

( +r)(–

α

k)

(kk(α))

t

a

t

a

tr+_–τr+αk–_tr+_–ρr+αk–_{A(τ,ρ)dτdρ}

≤

( +r)(–αk)

(kk(α))

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+αk–_f_(τ)dτdρ

+( +r)

(–α_k)

(kk(α))

t

a

t

a

tr+–τr+αk–_tr+_–ρr+αk–_f_(ρ)dτdρ

– ( +r)

(–α

k)

(kk(α))

t

a

t

a

tr+–τr+αk–_tr+_–ρr+αk–_{f(τ)f(ρ)dτdρ} 



×

( +r)(–αk)

(kk(α))

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+αk–_g_(τ)dτdρ

+( +r)

(–α_k)

(kk(α))

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+αk–_g_(ρ)dτdρ

– ( +r)

(–α_k)

(kk(α))

t

a

t

a

tr+–τr+αk–_tr+_–ρr+αk–_{g(τ)g(ρ)dτdρ} 



. (.)

Therefore, we obtain

( +r)(–

α

k)

(kk(α))

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+

α

k–_{A(τ,ρ)dτdρ}

≤

(tr+_–ar+₎αk

(r+ )αk_k(α+k)

Rα_a,,_krf(t) –Rα_a,,_kr{f}(t)

 

×

(tr+_–ar+₎αk

(r+ )αk_k(α+k)

Rα_a,,r_kg(t) –Rα_a,,_kr{g}(t)

 

. (.)

By applying Lemma ., forψ(t) =ψ(t) =g(t) = , we get

Rα_a,_,krf(t)≤  

(Rα_a,,_kr{(ϕ+ϕ)f}(t))

Rα_a,,_kr{ϕϕ}(t)

which leads to

(tr+_–ar+₎α_k

(r+ )αk_k(_α+k)

Rα_a,,_krf(t) –Rα_a,,r_k{f}(t)

≤ (tr+–ar+)

α

k

(r+ )αk_k(α+k)

(Rα_a,,_kr{(ϕ+ϕ)f}(t))

Rα_a,,_kr{ϕϕ}(t)

–Rα_a,_,kr{f}(t)

=G(f,ϕ,ϕ)(t). (.)

Similarly, we get

(tr+_–ar+₎αk

(r+ )αk

k(α+k)

Rα_a,_,krg(t) –Rα_a,,_kr{g}(t)

≤ (tr+–ar+)

α

k

(r+ )αk_k(α+k)

(Rα_a,,_kr{(ψ+ψ)g}(t))

Rα_a,,_kr{ψψ}(t)

–Rα_a,,_kr{g}(t)

=G(g,ψ,ψ)(t). (.)

Finally, combining (.), (.), (.), and (.), we arrive at the desired result in (.). This

completes the proof.

Remark . Ifϕ=m,ϕ=M,ψ=n, andψ=N, then we have

G(f,m,M)(t) =(M–m)



mM

Rα_a,,r_k{f}(t), (.)

G(g,n,N)(t) = (N–n)



nN

Rα_a,,_kr{g}(t). (.)

Theorem . Let f and g be two positive integrable functions on[a,∞),a≥.Assume that there exist four positive integrable functionsϕ,ϕ,ψ,andψsatisfying(H).Then,

for t>a,k> ,a≥,α> ,β> ,and r∈R\ {–},the following inequality is true:

(tr+–ar+)

α

k

(r+ )αk_k(α+k)

Rβ_a,,_kr{fg}(t) + (t

r+_–ar+₎β_k

(r+ )βk_k(β+k)

Rα_a,,_kr{fg}(t)

–Rα_a,,_kr{f}(t)Rβ_a,,_kr{g}(t) –Rα_a,,r_k{g}(t)R_aβ_,,_kr{f}(t)

≤G(f,ϕ,ϕ)(t) +G(f,ϕ,ϕ)(t)

 

×G(g,ψ,ψ)(t) +G(g,ψ,ψ)(t)



_{, (.)}

where

G(u,v,w)(t) =

(tr+_–ar+₎βk

(r+ )βk_k(β+k)

(Rα_a,,r_k{(v+w)u}(t)) Rα_a,,_kr{vw}(t) –R

α,r a,k{u}(t)R

β,r a,k{u}(t),

G(u,v,w)(t) =

(tr+_–ar+₎αk

(r+ )αk_k(α+k)

(Rβ_a,,_kr{(v+w)u}(t)) Rβ_a,,_kr{vw}(t) –R

α,r a,k{u}(t)R

Proof Multiplying both sides of (.) by(+r)

–α+β

k (tr+_–τr+₎αk–(tr+_–ρr+₎βk–

k

k(α)k(β) and double

inte-grating with respect toτandρfromatot, we obtain

( +r)–α+kβ

k_k(α)k(β)

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+

β

k–_{A(τ,ρ)dτdρ}

= (t

r+_–ar+₎αk

(r+ )αk_k(α+k)

Rβ_a,,_kr{fg}(t) + (t

r+_–ar+₎βk

(r+ )βk_k(β+k)

Rα_a,_,kr{fg}(t)

–Rα_a,,_kr{f}(t)Rβ_a,,_kr{g}(t) –Rβ_a,,_kr{f}(t)R_aα_,,r_k{g}(t). (.)

By using the Cauchy-Schwartz inequality for double integrals, we have

( +r)–

α+β

k

k_k(α)k(β)

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+

β

k–_{A(τ,ρ)dτdρ}

≤

( +r)–α+kβ

k_k(α)k(β)

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+

β

k–_f_(τ)dτdρ

+ ( +r)

–α+_kβ

k_k(α)k(β)

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+

β

k–_f_(ρ)dτdρ

–  ( +r)

–α+_kβ

k_k(α)k(β)

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+

β

k–_f(ρ)dτdρ 



×

( +r)–α+kβ

k

k(α)k(β)

t

a

t

a

tr+–τr+αk–_tr+_–ρr+

β

k–_g_(τ)dτdρ

+ ( +r)

–α+_kβ

k_k(α)k(β)

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+

β

k–_g_(ρ)dτdρ

–  ( +r)

–α+_kβ

k_k(α)k(β)

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+

β

k–_{g(τ)g(ρ)dτdρ} 



.

Therefore, we get

( +r)–

α+β

k

k_k(α)k(β)

t

a

t

a

tr+–τr+

α

k–_tr+_–ρr+

β

k–_{A(τ,ρ)dτdρ}

≤

(tr+_–ar+₎αk

(r+ )αk_k(α+k)

Rβ_a,,_krf(t)+ (t

r+_–ar+₎βk

(r+ )βk_k(β+k)

Rα_a,,_krf(t)

– Rβ_a,,_krf(t)Rα_a,,_krf(t)

 

×

(tr+–ar+)αk

(r+ )αk_k(α+k)

Rβ_a,,_krg(t)+ (t

r+_–ar+₎β_k

(r+ )βk_k(_β+k)

Rα_a,,_krg(t)

– Rβ_a,,_krg(t)Rα_a,,r_kg(t)

 

Applying Lemma . withψ(t) =ψ(t) =g(t) = , we have

(tr+_–ar+₎βk

(r+ )βk_k(β+k)

Rα_a,,r_kf(t)≤ (t

r+_–ar+₎βk

(r+ )βk_k(β+k)

(Rα_a,_,kr{(ϕ+ϕ)f}(t))

Rα_a,,_kr{ϕϕ}(t)

.

This implies that

(tr+_–ar+₎βk

(r+ )βk_k(β+k)

Rα_a,,r_kf(t) –R_aα_,,_kr{f}(t)R_aβ_,,_kr{f}(t)

≤ (tr+–ar+)

β

k

(r+ )βk_k(_β+k)

(Rα_a,_,kr{(ϕ+ϕ)f}(t))

Rα_a,,_kr{ϕϕ}(t)

–Rα_a,,_kr{f}(t)Rβ_a,,_kr{f}(t)

=G(f,ϕ,ϕ)(t) (.)

and

(tr+_–ar+₎α_k

(r+ )αk_k(α+k)

Rβ_a,,_krf(t) –R_aα_,,_kr{f}(t)R_aβ_,,_kr{f}(t)

≤ (tr+–ar+)

α

k

(r+ )αk_k(α+k)

(Rβ_a,,_kr{(ϕ+ϕ)f}(t))

Rβ_,t{ϕϕ}(t)

–Rα_a,_,kr{f}(t)Rβ_a,,_kr{f}(t)

=G(f,ϕ,ϕ)(t). (.)

Also, applying the same procedure withφ(t) =φ(t) =f(t) = , we get

(tr+–ar+)βk

(r+ )βk_k(_β+k)

Rα_a,,r_kg(t) –R_aα_,,_kr{g}(t)R_aβ_,,_kr{g}(t)

≤G(g,ψ,ψ)(t) (.)

and

(tr+_–ar+₎αk

(r+ )αk_k(α+k)

Rβ_a,,_krg(t) –R_aα,_,kr{g}(t)R_aβ_,,_kr{g}(t)

≤G(g,ψ,ψ)(t). (.)

Finally, considering (.) to (.), we arrive at the desired result in (.). This completes

the proof of Theorem ..

Remark . We conclude the present investigation by remarking that if we follow Sarikaya and Karaca [] then our main results become the results recently given by Ntouyaset al.[]. Similarly, after some parametric changes our results reduce to numerous well-known results presented in the literature.

4 Examples

In this section, we show some approximations of unknown functions by using four linear functions. Let us define the constantsm,m,M,M,n,n,N,N∈Rsuch that

Proposition . Suppose that f and g are two positive integrable functions on[a,∞),a≥ satisfying(H).Then,for t>a,k> ,a≥,α> ,β> ,and r∈R\ {–},we have

nNRαa,,rk

τf(t) + (nN+nN)Rαa,,rk

τf(t) +nNRαa,,kr

f(t)

×mMRαa,,kr

τg(t) + (mM+mM)Rαa,,kr

τg(t) +mMRαa,,kr

g(t)

≤ 



(mn+MN)Rαa,,kr

τfg(t) + (mn+mn+MN+MN)Rαa,,kr{τfg}(t) + (mn+MN)Raα,,rk{fg}(t)



. (.)

Proof Settingϕ(τ) =mτ+m,ϕ(τ) =Mτ+M,ψ(τ) =nτ+n, andψ(τ) =Nτ+N,

and applying Lemma ., we obtain (.) as desired.

Corollary . Let all assumptions of Proposition.be fulfilled with m =M =n =

N= .Then,for t>a,k> ,a≥,α> ,β> ,and r∈R\ {–},the following inequality

holds:

Rα_a,,r_k{f_}(t)Rα,r a,k{g}(t) (Rα_a,,_kr{fg}(t)) ≤

 

mn

MN

+

MN

mn



. (.)

Proposition . Suppose that f and g are two positive integrable functions on [a,∞), a≥satisfying(H).Then,for t>a,k> ,a≥,α> ,and r∈R\ {–},we get the

follow-ing inequality:

(tr+–ar+)

α

k

(r+ )αk_k(_α+k)

R_aα_,,_kr{fg}(t) –Rα_a,,_kr{f}(t)R_aα_,,_kr{g}(t)

≤G∗(f,m,m,M,M)(t)G∗(g,n,n,N,N)(t)



_{, (.)}

where

G∗(u,v,w,x,y)(t)

= (t

r+_–ar+₎αk

(r+ )αk_k(α+k)·

((v+x)Rα_a,,_kr{τu}(t) + (w+y)Rα_a,,r_k{u}(t)) vxRα_a,_,kr{τ}_{(t) + (vy+wx)R}α,r

a,k{τ}(t) +wyR α,r a,k{}(t)

–Rα_a,,_kr{u}(t). (.)

Proof By settingϕ(τ),ϕ(τ),ψ(τ), andψ(τ) as in Proposition . and using Theorem .,

we get the inequality (.).

Remark . Ifm=M=n=N= , then we have

G∗(f, ,m, ,M)(t) =G(f,m,M)(t),

G∗(g, ,n, ,N)(t) =G(g,n,N)(t),

Proposition . Assume that f and g are two positive integrable functions on[a,∞),a≥ satisfying(H).Then,for t>a,k> ,a≥,α> ,β> ,and r∈R\ {–},we obtain the

following estimate:

(tr+–ar+)

α

k

(r+ )αk_k(α+k)

Rβ_a,,_kr{fg}(t) + (t

r+_–ar+₎βk

(r+ )βk_k(β+k)

Rα_a,,_kr{fg}(t)

–Rα_a,,_kr{f}(t)Rβ_a,,_kr{g}(t) –Rα_a,,r_k{g}(t)R_aβ_,,_kr{f}(t)

≤G_∗(f,m,m,M,M)(t) +G∗(f,m,m,M,M)(t)

 

×G∗_(g,n,n,N,N)(t) +G∗(g,n,n,N,N)(t)



_{, (.)}

where

G∗_(u,v,w,x,y)(t)

= (t

r+_–ar+₎β_k

(r+ )βk_k(_β+k)

· ((v+x)R

α,r

a,k{τu}(t) + (w+y)R α,r a,k{u}(t)) vxRα_a,,_kr{τ}_{(t) + (vy+wx)R}α,r

a,k{τ}(t) +wyR α,r a,k{}(t) –Rα_a,,_kr{u}(t)Rβ_a,,_kr{u}(t),

G∗_(u,v,w,x,y)(t)

= (t

r+_–ar+₎αk

(r+ )αk_k(α+k)·

((v+x)Rβ_a,,_kr{τu}(t) + (w+y)Rβ_a,,_kr{u}(t))

vxRβ_a,,_kr{τ}_{(t) + (vy+wx)R}β,r

a,k{τ}(t) +wyR β,r a,k{}(t) –Rα_a,,_kr{u}(t)Rβ_a,,_kr{u}(t).

Proof By setting the four linear functions as in Proposition . and using Theorem ., we

get the estimate (.).

Corollary . If m=M=n=N=v=x= ,then we obtain

G∗_(u, ,w, ,y)(t)

=

 w

y + y w



(tr+_–ar+₎β–kα_k(α+k)

(r+ )β–kα_k(β+k)

·Rα_a,,_kr{u}(t)

–Rα_a,,_kr{u}(t)Rβ_a,,_kr{u}(t), G∗_(u, ,w, ,y)(t)

=

 w

y + y w



(tr+_–ar+₎α–kβ_k(β+k)

(r+ )α–kβ_k(α+k)

·Rβ_a,,_kr{u}(t)

–Rα_a,,_kr{u}(t)Rβ_a,,_kr{u}(t).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Author details

1_{Department of Mathematics, Anand International College of Engineering, Jaipur, 303012, India.}2_{Nonlinear Dynamic}

Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand.3_{Department of Mathematics, University of Ioannina, Ioannina,}

451 10, Greece. 4_{Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics,}

Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.

Received: 3 February 2016 Accepted: 11 April 2016

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