Some new Hardy-type inequalities for Riemann-Liouville fractional q-integral operator

Journal of Inequalities and Applications, Sep 2015

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: 26D10, 26D15, 33D05, 39A13.

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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)


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Lars-Erik Persson, Serikbol Shaimardan. Some new Hardy-type inequalities for Riemann-Liouville fractional q-integral operator, Journal of Inequalities and Applications, 2015, pp. 296, 2015, DOI: 10.1186/s13660-015-0816-z