CHEBYSHEV TYPE INEQUALITIES FOR THE SAIGO FRACTIONAL INTEGRALS AND THEIR q–ANALOGUES
ABSTRACT The aim of the present paper is to obtain certain new integral inequalities involving the Saigo fractional integral operator. It is also shown how the various inequalities considered in this paper admit themselves of qextensions which are capable of yielding various results in the theory of qintegral inequalities.

Dataset: CJM (2014)
 SourceAvailable from: Praveen AgarwalChinese Journal of Mathematics. 01/2014; 2014:15.
Page 1
Journal of
Mathematical
Inequalities
Volume 7, Number 2 (2013), 239–249doi:10.7153/jmi0722
CHEBYSHEV TYPE INEQUALITIES FOR THE SAIGO
FRACTIONAL INTEGRALS AND THEIR q–ANALOGUES
S. D. PUROHIT AND R. K. RAINA
(Communicated by S. Samko)
Abstract. The aim of the present paper is to obtain certain new integral inequalities involving
the Saigo fractional integral operator. It is also shown how the various inequalities considered in
this paper admit themselves of qextensions which are capable of yielding various results in the
theory of qintegral inequalities.
1. Introduction
Our work in the present paper is based on a celebrated functional introduced by
Chebyshev [4], which is defined by
T(f,g) =
1
b−a
?b
a
f(x)g(x)dx−
?
1
b−a
?b
a
f(x)dx
??
1
b−a
?b
a
g(x)dx
?
,
(1.1)
where f and g are two integrable functions which are synchronous on [a,b], i.e.
{(f(x)− f(y))(g(x)−g(y))} ? 0,
(1.2)
for any x,y ∈ [a,b].
The functional (1.1) has applications in numerical quadrature, transform theory,
probability and in statistical problems. Motivated by these applications, researchers
have used the functional (1.1) in the theory of fractional integral inequalities (see [3],
[5] and [7]). Recently, Belarbi and Dahmani [3], Dahmani et al. [5], and Kalla and
Rao [7] established certain integral inequalities by using known fractional integral op
erators. Also,¨Oˇ g¨ unmez and¨Ozkan [9] derived certain integral inequalities involving
the fractional qintegral operators.
The object of the present investigationis to obtain certain Chebyshev type integral
inequalities involving the Saigo fractional integral operators ([12]). Further, we con
sider the qextensionsof the main results, and point out also their relevanceswith other
related results.
Beforestatingthe fractionalintegralinequalities, wementionbelowthedefinitions
and notations of some wellknown operators of fractional calculus.
Mathematics subject classification (2010): 26D10, 26A33, 05A30.
Keywords and phrases: Integral inequalities, fractional integral operators and fractional qintegral
operators.
c ?
Paper JMI0722
??
, Zagreb
239
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240
S. D. PUROHIT AND R. K. RAINA
DEFINITION 1. A realvalued function f(t) (t > 0) is said to be in the space
Cµ(µ ∈ R), if there exists a real number p > µ such that f(t) =tpφ(t); where φ(t) ∈
C(0,∞).
DEFINITION 2. Let α > 0, β,η ∈ R, then the Saigo fractional integral Iα,β,η
order α for a realvalued continuous function f(t) is defined by ([12], see also [8, p.
19], [11]):
0,t
of
Iα,β,η
0,t
{f(t)} =t−α−β
Γ(α)
?t
0(t −τ)α−12F1
?
α +β,−η;α;1−τ
t
?
f(τ)dτ,
(1.3)
where, the function2F1(−) in the righthand side of (1.3) is the Gaussian hypergeo
metric function defined by
2F1(a,b;c;t) =
∞
∑
n=0
(a)n(b)n
(c)n
tn
n!,
(1.4)
and (a)nis the Pochhammer symbol
(a)n= a(a+1)···(a+n−1), (a)0= 1.
The integral operator Iα,β,η
Kober fractional integral operators given by the following relationships:
0,t
includes both the RiemannLiouvilleand the Erd´ elyi
Rα{f(t)} = Iα,−α,η
0,t
{f(t)} =
1
Γ(α)
?t
0(t −τ)α−1f(τ)dτ
(α > 0)
(1.5)
and
Iα,η{f(t)} = Iα,0,η
0,t
{f(t)} =t−α−η
Γ(α)
?t
0(t −τ)α−1τηf(τ)dτ
(1.6)
(α > 0,
η ∈ R).
For f(t) =tµin (1.3), we get the known formula [12]:
Iα,β,η
0,t
{tµ} =
Γ(µ +1)Γ(µ +1−β +η)
Γ(µ +1−β)Γ(µ+1+α+η)tµ−β,
(1.7)
(α > 0,
min(µ,µ −β +η) > −1,
t > 0)
which shall be used in the sequel.
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CHEBYSHEV TYPE INEQUALITIES FOR THE SAIGO FRACTIONAL INTEGRALS
241
2. Fractional integral inequalities
In this section, we establish Chebyshev type integral inequalities for the syn
chronous functions involving the Saigo fractional integral operator (1.3).
THEOREM 1. Let f and g be two synchronous functions on [0,∞), then
Iα,β,η
0,t
{f(t)g(t)} ?Γ(1−β)Γ(1+α+η) tβ
Γ(1−β +η)
Iα,β,η
0,t
{f(t)} Iα,β,η
0,t
{g(t)},
(2.1)
for all t > 0, α > max{0,−β}, β < 1, β −1 < η < 0.
Proof. By hypothesis,the functions f and g are synchronousfunctionson [0,∞),
therefore, for all τ, ρ ? 0, we have
{(f(τ)− f(ρ))(g(τ)−g(ρ))} ? 0,
(2.2)
which implies that
f(τ)g(τ)+ f(ρ)g(ρ) ? f(τ)g(ρ)+ f(ρ)g(τ).
(2.3)
Consider
F(t,τ) =t−α−β(t −τ)α−1
Γ(α)
(t −τ)α−1
tα+β
2F1
?
α +β,−η;α;1−τ
t
?
(τ ∈ (0,t); t > 0)
(2.4)
=
1
Γ(α)
+(α +β)(−η)
Γ(α +1)
(t −τ)α
tα+β+1
(t −τ)α+1
tα+β+2
+(α +β)(α +β +1)(−η)(−η +1)
Γ(α +2)
+···.
We observe that the function F(t,τ) remains positive, for all τ ∈ (0,t) (t > 0) since
eachtermoftheaboveseriesis positiveinviewoftheconditionsstatedwithTheorem1.
Multiplying both sides of (2.3) by F(t,τ) (defined above by (2.4)) and integrating
with respect to τ from 0 to t, and using (1.3), we get
Iα,β,η
0,t
{f(t)g(t)}+ f(ρ)g(ρ) Iα,β,η
0,t
{1} ? g(ρ) Iα,β,η
0,t
{f(t)}+ f(ρ) Iα,β,η
0,t
{g(t)}.
(2.5)
Next, multiplying both sides of (2.5) by F(t,ρ) (ρ ∈ (0,t), t > 0), where F(t,ρ) is
given by (2.4), and integrating with respect to ρ from 0 to t, and using formula (1.7),
we arrive at the desired result (2.1).
?
THEOREM 2. Let f and g be two synchronous functions on [0,∞), then
Γ(1−β+η)
Γ(1−β)Γ(1+α+η) tβIγ,δ,ζ
? Iα,β,η
0,t
0,t
{f(t)g(t)}+
Γ(1−δ+ζ)
Γ(1−δ)Γ(1+γ+ζ) tδIα,β,η
{g(t)}+Iγ,δ,ζ
0,t
0,t
{f(t)g(t)}
{f(t)} Iγ,δ,ζ
0,t
{f(t)} Iα,β,η
0,t
{g(t)},
(2.6)
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S. D. PUROHIT AND R. K. RAINA
for all t > 0, α > max{0,−β}, γ > max{0,−δ}, β , δ <1, β −1< η < 0, δ −1 <
ζ < 0.
Proof. Multiplying both sides of (2.5) by
t−γ−δ(t −ρ)γ−1
Γ(γ)
2F1
?
γ +δ,−ζ;γ;1−ρ
t
?
(ρ ∈ (0,t); t > 0),
which (in view of the arguments mentioned above in the proof of Theorem 1) remains
positiveundertheconditionsstatedwith Theorem2. Integratingthe resultinginequality
so obtained with respect to ρ from 0 to t, we get
Iγ,δ,ζ
0,t
? Iα,β,η
0,t
{1}Iα,β,η
0,t
{f(t)} Iγ,δ,ζ
{f(t)g(t)}+Iα,β,η
0,t
{1}Iγ,δ,ζ
0,t
{f(t)g(t)}
0,t
{g(t)}+Iγ,δ,ζ
0,t
{f(t)} Iα,β,η
0,t
{g(t)},
which on using (1.7) readily yields the desired result (2.6).
?
REMARK 1. It may be noted that the inequalities (2.1) and (2.6) are reversed if
the functions are asynchronous on [0,∞), i.e.
{(f(x)− f(y))(g(x)−g(y))} ? 0,
(2.7)
for any x,y ∈ [0,∞).
REMARK 2. For α = γ, β = δ , η = ζ , Theorem 2 immediately reduces to The
orem 1.
THEOREM 3. Let (fi)i=1,···,nbe n positive increasing functions on [0,∞), then
Iα,β,η
0,t
?
n
∏
i=1
fi(t)
?
?
?
Γ(1−β)Γ(1+α+η) tβ
Γ(1−β +η)
?n−1
n
∏
i=1
Iα,β,η
0,t
{fi(t)},
(2.8)
for all t > 0, α > max{0,−β}, β < 1, β −1 < η < 0.
Proof. We prove this theorem by induction. Clearly, for n = 1 in (2.8), we have
Iα,β,η
0,t
{f1(t)} ? Iα,β,η
0,t
{f1(t)}(t > 0, α > 0).
Next, for n = 2, in (2.8), we get
Iα,β,η
0,t
{f1(t)f2(t)} ?Γ(1−β)Γ(1+α+η) tβ
Γ(1−β +η)
Iα,β,η
0,t
{f1(t)} Iα,β,η
0,t
{f2(t)}
(t > 0, α > 0),
which holds in view of (2.1) of Theorem 1.
By the induction principle, we suppose that the inequality
Iα,β,η
0,t
?n−1
∏
i=1
fi(t)
?
?
?
Γ(1−β)Γ(1+α+η) tβ
Γ(1−β +η)
?n−2n−1
∏
i=1
Iα,β,η
0,t
{fi(t)},
(2.9)
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CHEBYSHEV TYPE INEQUALITIES FOR THE SAIGO FRACTIONAL INTEGRALS
243
holds true for some positive integer n ? 2.
Now (fi)i=1,···,n are increasing functions implies that the function ∏n−1
also an increasing function. Therefore, we can apply inequality (2.1) of Theorem 1 to
the functions ∏n−1
i=1fi(t) = g and fn= f to get
i=1fi(t) is
Iα,β,η
0,t
?
n
∏
i=1
fi(t)
?
?Γ(1−β)Γ(1+α+η) tβ
Γ(1−β +η)
Iα,β,η
0,t
?n−1
∏
i=1
fi(t)
?
Iα,β,η
0,t
{fn(t)},
provided that t > 0, α > max{0,−β}, β < 1, β −1 < η < 0.
Making use of (2.9) now, this last inequality above leads to the result (2.8), which
proves Theorem 3.
?
By setting β = 0 (and δ = 0 additionally for Theorem 2), and using the relation
(1.6), Theorems 1 to 3 yield the following integral inequalities involving the Erd´ elyi
Kober type fractional integral operator defined by (1.6):
COROLLARY 1. Let f and g be two synchronous functions on [0,∞), then
Iα,η{f(t)g(t)} ?Γ(1+α +η)
Γ(1+η)
Iα,η{f(t)} Iα,η{g(t)},
(2.10)
for all t > 0, α > 0, −1 < η < 0.
COROLLARY 2. Let f and g be two synchronous on [0,∞), then
Γ(1+η)
Γ(1+α +η)Iγ,ζ{f(t)g(t)}+
? Iα,η{f(t)} Iγ,ζ{g(t)}+Iγ,ζ{f(t)} Iα,η{g(t)},
Γ(1+ζ)
Γ(1+γ +ζ)Iα,η{f(t)g(t)}
(2.11)
for all t > 0, α, γ > 0, −1 < max(η,ζ) < 0.
COROLLARY 3. Let (fi)i=1,···,nbe n positive increasing functions on [0,∞), then
Iα,η
?
n
∏
i=1
fi(t)
?
?
?Γ(1+α +η)
Γ(1+η)
?n−1
n
∏
i=1
Iα,η{fi(t)},
(2.12)
for all t > 0, α > 0, −1 < η < 0.
Next, if we replace β by −α (and δ by −γ additionally for Theorem 2), and
make use of the relation (1.5), then Theorems 1 to 3 corresponds to the known results
due to Belarbi and Dahmani [3, pp. 2–4, Theorems 3.1 to 3.3].
3. Fractional qintegral inequalities
In this section, we establish some fractional qintegral inequalities which may
be regarded as qextensions of the results derived in the previous section. For the
convenience of the reader, we deem it proper to give here basic definitions and related
details of the qcalculus.
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S. D. PUROHIT AND R. K. RAINA
The qshifted factorial is defined for α, q ∈ C as a product of n factors by
(α;q)n=
?
1;
n = 0
(1−α)(1−αq) ···(1−αqn−1) ; n ∈ N ,
(3.1)
and in terms of the basic analogue of the gamma function
(qα;q)n=Γq(α +n)(1−q)n
Γq(α)
(n > 0),
(3.2)
where the qgamma function is defined by ([6, p. 16, eqn. (1.10.1)])
Γq(t) =(q;q)∞(1−q)1−t
(qt;q)∞
(0 < q < 1).
(3.3)
We note that
Γq(1+t) =(1−qt)Γq(t)
1−q
,
(3.4)
and if q < 1, the definition (3.1) remains meaningful for n = ∞, as a convergent
infinite product given by
∞
∏
j=0
(α;q)∞=
(1−αqj).
(3.5)
Also, the qbinomial expansion is given by
(x−y)ν= xν(−y/x;q)ν= xν
∞
∏
n=0
?
1−(y/x)qn
1−(y/x)qν+n
?
.
(3.6)
Let t0∈ R, then we define a specific time scale
Tt0= {t;t =t0qn,n a nonnegativeinteger}∪{0}, 0 < q < 1,
(3.7)
and for convenience sake, we denote Tt0by T throughout this paper.
The Jackson’s qderivative and qintegral of a function f defined on T are, re
spectively, given by (see [6, pp. 19, 22])
Dq,tf(t) =f(t)− f(tq)
t(1−q)
(t ?= 0, q ?= 1)
(3.8)
and
?t
0
f(τ)dqτ =t (1−q)
∞
∑
k=0
qkf(tqk).
(3.9)
DEFINITION 3. The RiemannLiouville fractional qintegral operator of a func
tion f(t) of order α (due to Agarwal [1]) is given by
Iα
q{f(t)} =
tα−1
Γq(α)
?t
0(qτ/t;q)α−1f(τ) dqτ
(α > 0, 0 < q < 1),
(3.10)
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CHEBYSHEV TYPE INEQUALITIES FOR THE SAIGO FRACTIONAL INTEGRALS
245
where
(a;q)α=
(a;q)∞
(aqα;q)∞
(α ∈ R).
(3.11)
DEFINITION 4. For α >0, η ∈R and 0<q<1, the basic analogueof theKober
fractional integral operator (cf. [2]) is given by
Iα,η
q
{f(t)} =t−η−1
Γq(α)
?t
0(qτ/t;q)α−1τηf(τ) dqτ.
(3.12)
DEFINITION 5. For α > 0, β ∈ R, a basic analogue of the Saigo’s fractional
integral operator (introduced by Purohit and Yadav [10]) is given for τ/t < 1 by
Iα,β,η
q
{f(t)} =t−β−1q−η(α+β)
Γq(α)
×
?t
0(qτ/t;q)α−1Tq,qα+1τ
t
?
2Φ1
?
qα+β, q−η;qα;q,q
??
f(τ) dqτ,
(3.13)
where η is any nonnegative integer, and the function2Φ1(−) (see [6]) and the q
translation operator occurring in the righthand side of (3.13) are, respectively, defined
by
∞
∑
n=0
(c;q)n(q;q)n
2Φ1[a,b;c;q,t] =
(a;q)n(b;q)n
tn
(q < 1, t < 1)
(3.14)
and
Tq,τ(f(t)) =
+∞
∑
n=−∞
Antn(τ/t;q)n,
(3.15)
where (An)n∈Z(Z = 0,±1,±2,···) is any bounded sequence of real or complex num
bers.
Following [10], when f(t) =tµ, we obtain
Iα,β,η
q
{tµ} =
Γq(µ +1)Γq(µ +1−β +η)
Γq(µ +1−β)Γq(µ +1+α+η)tµ−β,
(3.16)
(0 < q < 1,
min(µ,µ −β +η) > −1,
t > 0).
We now state and prove the qintegral inequalities which can be treated as the
qanalogues of the inequalities (2.1), (2.6) and (2.8).
THEOREM 4. Let f and g be two synchronous functions on T, then
Iα,β,η
q
{f(t)g(t)} ?Γq(1−β)Γq(1+α +η) tβ
Γq(1−β +η)
Iα,β,η
q
{f(t)} Iα,β,η
q
{g(t)},
(3.17)
for all t > 0, 0 < q < 1, α > max{0,−β}, β < 1, η −β > −1.
Proof. Since the functions f and g are synchronousfunctionson T forall τ, ρ ?
0, thereforethe inequality (1.2)is satisfied. Now, on multiplyingboth sides of (1.2) (or,
Page 8
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S. D. PUROHIT AND R. K. RAINA
equivalently (2.3)) by
t−β−1q−η(α+β)
Γq(α)
(qτ/t;q)α−1Tq,qα+1τ
t
?
2Φ1
?
qα+β, q−η;qα;q,q
??
,
(3.18)
(τ ∈ (0,t), t > 0),
and taking qintegration with respect to τ from 0 to t, then on using Definition 5, we
get
Iα,β,η
q
{f(t)g(t)}+ f(ρ)g(ρ) Iα,β,η
q
{1} ? g(ρ) Iα,β,η
q
{f(t)}+ f(ρ) Iα,β,η
q
{g(t)}.
(3.19)
It may be observed that the function (3.18) remains positive for all values of τ ∈
(0,t) (t > 0) and under the conditions imposed with Theorem 4.
Next, multiplying both sides of (3.19) by
t−β−1q−η(α+β)
Γq(α)
(qρ/t;q)α−1Tq,qα+1ρ
t
?
2Φ1
?
qα+β, q−η;qα;q,q
??
(3.20)
(ρ ∈ (0,t), t > 0),
and noting that the function (3.20) is also positive for all ρ ∈ (0,t) (t > 0) and un
der the conditions imposed with Theorem 4, we perform qintegration in the resulting
inequality with respect to ρ from 0 to t, using the formula (3.16), the desired result
(3.17) is thus easily arrived at.
?
THEOREM 5. Let f and g be two synchronous functions on T, then
Γq(1−β +η)
Γq(1−β)Γq(1+α +η) tβIγ,δ,ζ
q
{f(t)g(t)}
+
Γq(1−δ +ζ)
Γq(1−δ)Γq(1+γ +ζ) tδIα,β,η
? Iα,β,η
q
{f(t)} Iγ,δ,ζ
q
q
{f(t)g(t)}
{g(t)}+Iγ,δ,ζ
q
{f(t)} Iα,β,η
q
{g(t)},
(3.21)
for all t > 0, 0 < q < 1, α > max{0,−β}, γ > max{0,−δ}, β , δ < 1, η −β ,
ζ −δ > −1.
Proof. To prove the above theorem, we start with the inequality (3.19). On multi
plying both sides of the inequality (3.19) by
t−δ−1q−ζ(γ+δ)
Γq(γ)
(qρ/t;q)γ−1Tq,qγ+1ρ
t
?
2Φ1
?
qγ+δ, q−ζ;qγ;q,q
??
,
(3.22)
(ρ ∈ (0,t), t > 0),
and taking basic integration with respect to ρ from 0 to t, we get
Iγ,δ,ζ
q
(1)Iα,β,η
q
? Iα,β,η
q
{f(t)g(t)}+Iα,β,η
q
(1)Iγ,δ,ζ
q
f(t) Iα,β,η
{f(t)g(t)}
f(t) Iγ,δ,ζ
q
g(t)+Iγ,δ,ζ
qq
g(t),
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CHEBYSHEV TYPE INEQUALITIES FOR THE SAIGO FRACTIONAL INTEGRALS
247
which yields the desired result by taking into account (3.16).
?
REMARK 3. The inequalities (3.17) and (3.21) are reversed if the functions are
asynchronous on T.
REMARK 4. Evidently, when α = γ, β = δ , η = ζ , then Theorem 5 leads to
Theorem 4.
THEOREM 6. Let (fi)i=1,···,nbe n positive increasing functions on T, then
Iα,β,η
q
?
n
∏
i=1
fi(t)
?
?
?
Γq(1−β)Γq(1+α +η) tβ
Γq(1−β +η)
?n−1
n
∏
i=1
Iα,β,η
q
{fi(t)},
(3.23)
for all t > 0, 0 < q < 1, α > max{0,−β}, β < 1, η −β > −1.
Proof. By applyingthe induction method and Theorem 4, one can easily establish
the above theorem. Therefore, we omit the further details of the proof of this theo
rem.
?
We now, briefly consider some consequences of the theorems derived in this sec
tion. If we set β = 0 (and additionally δ = 0 for Theorem 5), and make use of the
known result [10, p. 38, eqn. (3.7)], namely
Iα,0,η
q
{f(t)} = Iα,η
q
{f(t)},
(3.24)
(with suitable changes for the parameters in Theorem 5) then Theorems 4 to 6 yield
the following qintegral inequalities involving Erd´ elyiKober type fractional integral
operators:
COROLLARY 4. Let f and g be two synchronous functions on T, then
Iα,η
q
{f(t)g(t)} ?Γq(1+α +η)
Γq(1+η)
Iα,η
q
{f(t)} Iα,η
q
{g(t)},
(3.25)
for all t > 0, 0 < q < 1, α > 0 and η is any nonnegative integer.
COROLLARY 5. Let f and g be two synchronous functions on T, then
Γq(1+η)
Γq(1+α +η)Iγ,ζ
? Iα,η
q
q
{f(t)g(t)}+
Γq(1+ζ)
Γq(1+γ +ζ)Iα,η
{g(t)}+Iγ,ζ
q
{f(t)} Iα,η
q
{f(t)g(t)}
{f(t)} Iγ,ζ
qq
{g(t)},
(3.26)
where t > 0, 0 < q < 1, α, γ > 0, η, ζ are any nonnegative integers.
COROLLARY 6. Let (fi)i=1,···,nbe n positive increasing functions on T, then
Iα,η
q
?
n
∏
i=1
fi(t)
?
?
?Γq(1+α +η)
Γq(1+η)
?n−1
n
∏
i=1
Iα,η
q
fi(t),
(3.27)
Page 10
248
S. D. PUROHIT AND R. K. RAINA
where t > 0, 0 < q < 1, α > 0 and η is any nonnegativeinteger.
We observe that, if we replace β by −α and δ by −γ, and make use of the
relation [10, p. 38, eqn. (3.8)], and note the following relations:
Iα,−α,η
q
{f(t)} = Iα
q{f(t)}
(3.28)
and
Iγ,−γ,ζ
q
{f(t)} = Iγ
q{f(t)},
(3.29)
then, Theorems 4 to 6 reduce to the known qintegral inequalities due to¨Oˇ g¨ unmez
and¨Ozkan [9, pp. 4–6, Theorems 3.1 to 3.3], involving the RiemannLiouville type of
fractional qintegral operator.
Finally, it is interesting to observe that, if we let q → 1−, and use the limit formu
las:
(qα;q)n
(1−q)n= (α)n
Lim
q→1−
(3.29)
and
Lim
q→1−Γq(α) = Γ(α),
(3.30)
the results of Section 3 then correspond to the results obtained in Section 2.
Acknowledgement. Theauthorsexpresstheir sincerethanksto therefereeforsome
valuable suggestions.
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(Received March 15, 2012)
S. D. Purohit
Department of BasicSciences (Mathematics)
College of Technology and Engineering
M. P. University of Agriculture and Technology
Udaipur313001, Rajasthan, India
email: sunil_a_purohit@yahoo.com
R. K. Raina
M. P. University of Agriculture and Technology
Udaipur 313001, Rajasthan, India
Present address:
10/11, Ganpati Vihar, Opposite Sector5
Udaipur313002, Rajasthan, India
email: rkraina_7@hotmail.com
Journal of Mathematical Inequalities
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