Some new Hardy-type inequalities for Riemann-Liouville fractional q-integral operator
Persson and Shaimardan Journal of Inequalities and Applications
Some new Hardy-type inequalities for Riemann-Liouville fractional q-integral operator
Lars-Erik Persson 0 1 3
Serikbol Shaimardan 2
0 Narvik University College , P.O. Box 385, Narvik, 8505 , Norway
1 Luleå University of Technology , Luleå, 971 87 , Sweden
2 L.N. Gumilyov Eurasian National University , Munaytpasov St. 5, Astana, 010008 , Kazakhstan
3 Luleå University of Technology , Luleå, 971 87 , Sweden
We consider the q-analog of the Riemann-Liouville fractional q-integral operator of order n ∈ N. Some new Hardy-type inequalities for this operator are proved and discussed. MSC: Primary 26D10; 26D15; secondary 33D05; 39A13
2 Preliminaries
f (x)Dq g(x) =
f qj g qj – g qj+ .
– qα
[α]q := – q ,
[n]q! :=
(x – a)qn :=
The q-gamma function q is defined by
q(n + ) := [n]q!, n ∈ N .
Dqf (x) :=
f (x) dqx := ( – q)a
f (x) dqx := ( – q)
k=–∞
X(,z](t)f (t) dqt = ( – q)
X[z,∞)(t)f (t) dqt = ( – q)
kn–=j
kn–=kn–
An–,(i, j) =
k=k
Iq,nf (x) :=
Kn–(x, s)f (s) dqs,
≤ C
∀f ≥
(Snf )j =
fi =
An–(i, j)fi,
kn–=
k=
qi≤z
qi≥z
k=
i=–∞
i=–∞
k=j
fiAn–,(i, j),
j=–∞
i=–∞
≤ C
uir Sn∗f i
≤ C∗
Apm,(j, k)vj–p
An–,m+(k, i)uir
r
, n ∈ N.
Arm,(i, k)uir
, n ∈ N.
Moreover, A(n) ≈ C, where C is the best constant in ().
(ii) Let < p ≤ r < ∞ and n ≥ . Then inequality () holds if and only if
A∗(n) = max≤m≤n– A∗m(n) < ∞, where
A∗m(n) = sup
k∈Z
Bm(n) =
i=–∞ j=i
Apm,(j, i)vj–p
Apm,(j, i)vj–p
An–,m+(i, k)urk
r
+Ei,j = Ei,j – Ei,j+, n ∈ N.
i=–∞
j=–∞
k=–∞
k=–∞
Moreover, B(n) ≈ C, where C is the best constant in ().
(ii) Let < r < p < ∞ and n ≥ . Then inequality () holds if and only if
B∗(n) = max≤m≤n– B∗m(n) < ∞, where
B∗m(n) =
Arm,(j, i)ujr
i=–∞ j=i
Arm,(j, i)ujr
+Ei,j = Ei,j – Ei,j+, ∀n ∈ N.
for all ∞ > i ≥ k ≥ j > –∞.
ai(,nj) ≈
(γ ) n,γ
ai,k dk,j , γ = , , . . . , n – , n ∈ N
j=–∞
j=–∞
3 The main results
ur(x) Iq,nf (x) r dqx
≤ C
∞ vp(x)f p(x) dqx
Therefore, inequality () can be rewritten as
ur(x) Iq,nf (x) r dqx =
ur(x) Iq,nf (x) r dqx.
ur(x) Iq,nf (x) r dqx
≤ C
∞ vp(x)f p(x) dqx
Iq∗,nf (s) :=
X[s,∞)(x)Kn–(x, s)f (x) dqx,
Iq,nf (x) =
X(,x](s)Kn–(x, s)f (s) dqs,
Qnm– =
ur(x) Iq∗,nf (x) r dqx
≤ C∗
∞ vp(x)f p(x) dqx
Qnm– =
X(,z](s)K mp (z, s)v–p (s) dqs
× Dq
× Dq
X(,z](s)K mp (z, s)v–p (s) dqs
X(,z](s)K mr(z, s)ur(s) dqs
X(,z](s)K mr(z, s)ur(x) dqs
A+(z) =
A–(z) =
A+(z) =
A–(z) =
Hn– =
Aq+ = sup A+(z),
z>
Qn– = ≤mk≤anx– Qkn–
X[z,∞)(x)ur(x)
X(,z](t)Knp–(x, t)v–p (t) dqt
X(,z](t)v–p (t)
X[z,∞)(x)Knr–(x, t)ur(x) dqx
X[z,∞)(t)v–p (t)
X(,z](x)Knr–(t, x)ur(x) dqx
X(,z](x)ur(x)
Hn– = ≤mk≤anx– Hkn–,
X(,z](s)K mp (z, s)v–p (s) dqs
X(,z](x)K mr(z, x)ur(x) dqx
Qn– = ≤mk≤anx– Qkn–.
Our main results read as follows.
I(z) :=
X(,z](t)f (t) dqt
X[z,∞)(x)g(x) dqx
β
.
Kn–m–(x, t)Km(t, s)
sup I(z) = ( – q)α+β sup
z> k∈Z
i=–∞
I+(z) :=
I–(z) :=
X[z,∞)(x)g(x)
X(,z](t)K (x, t)f (t) dqt
X(,z](t)f (t)
X[z,∞)(x)K (x, t)g(x) dqx
sup I+(z) = sup ( – q)
z> k∈Z
sup I–(z) = sup ( – q)
z> k∈Z
j=–∞
( – q)
( – q)
qjK qj, qi g qj
Qnm– =
( – q)
qtK mp qi, qt v–p qt
× ( – q)
qjKnr–m– qj, qi ur qj
( – q)qnK mr qi, qn v–p qn
i=–∞
i=–∞
j=–∞
j=–∞
j=–∞
Qnm– =
( – q)
qtK mr qi, qt ur qt
× ( – q)
( – q)qnK mr qi, qn ur qn
+En,i = En,i – En,i+.
4 Proofs
≤ (x – qs)qn–m–(x – qs)qm
= (x – qs)qn– = Kn–(x, s)
K–m–(x, t)Km(t, s)
for n = .
This means that the inequality
Kn–(x, s) <
Kn–m–(x, t)Km(t, s)
Kl–m–(x, t)Km(t, s) x – ql–s
Kl–m–(x, t)Km(t, s) x – ql–m–t + ql–m–t – ql–s
Kl–m–(x, t)Km(t, s) x – ql–m–t
Kl–m–(x, t)Km(t, s)ql–m– t – qm+s
Kl–(x, t)K(t, s)
K(x, t)Kl(t, s).
Kl–(x, t)K(t, s) +
Kl–m–(x, t)Km(t, s)
Kl–m–(x, t)Km(t, s) + l –
l – q
K(x, t)Kl–(t, s)
Kl–(x, s) <
Kl–m–(x, t)Km(t, s).
Proof of Lemma . From () and () it follows that
i=–∞
i=–∞
β
.
β
.
If z = qk , then, for k ∈ Z,
qj≤z
sup I(z) = sup sup
z> k∈Z qk≤z<qk–
= ( – q)α+β sup
k∈Z
We have proved that () holds wherever β > .
Next we assume that α > . Let qk+ < z < qk , k ∈ Z. Then we get that
I(z) = ( – q)α+β sup
k∈Z j=k+
i=–∞
and analogously as above we find that
i=–∞
I+(z) = ( – q)
qjg qj ( – q)
I+(z) = ( – q)
qjg qj ( – q)
I+(z) = ( – q)
qjg qj ( – q)
j=–∞
j=–∞
j=–∞
Qnm– =
i=–∞
X(,qi](s)K mp qi, s v–p (s) dqs
X(,qi](s)K mp qi, s v–p (s) dqs
respectively.
Hence, we conclude that
I+(z) = ( – q)
j=–∞
qjg qj ( – q)
i=–∞
X(,qi+](s)K mp qi+, s v–p (s) dqs
( – q)
qtK mp qi, qt v–p qt
× ( – q)
qjKnr–m– qj, qi ur qj
( – q)qnK mr qi, qn v–p qn
j=–∞
and the first equality in Lemma . is proved.
The second inequality can be proved in a similar way, so we leave out the details. The
proof is complete.
Proof of Theorem . By using formulas () and () we find that inequality () can be
rewritten as
j=–∞
≤ C
i=–∞
( – q)r+qjur qj
qif qi Kn– qj, qi
( – q)qif p qi vp qi
ujr = ( – q)r+qjur qj ,
vip = ( – q)qi(–p)vp qi , (...truncated)