Sajid Iqbal, Josip Pe cari c, Muhammad Samraiz, Hassan Sajid Iqbal, Josip Pe cari c, Muhammad...

download Sajid Iqbal, Josip Pe cari c, Muhammad Samraiz, Hassan Sajid Iqbal, Josip Pe cari c, Muhammad Samraiz,

of 24

  • date post

    28-Jul-2020
  • Category

    Documents

  • view

    0
  • download

    0

Embed Size (px)

Transcript of Sajid Iqbal, Josip Pe cari c, Muhammad Samraiz, Hassan Sajid Iqbal, Josip Pe cari c, Muhammad...

  • Commun. Korean Math. Soc. 35 (2020), No. 1, pp. 161–184

    https://doi.org/10.4134/CKMS.c180458

    pISSN: 1225-1763 / eISSN: 2234-3024

    ON SOME WEIGHTED HARDY-TYPE INEQUALITIES

    INVOLVING EXTENDED RIEMANN-LIOUVILLE

    FRACTIONAL CALCULUS OPERATORS

    Sajid Iqbal, Josip Pečarić, Muhammad Samraiz, Hassan Tehmeena, and Živorad Tomovski

    Abstract. In this article, we establish some new weighted Hardy-type

    inequalities involving some variants of extended Riemann-Liouville frac-

    tional derivative operators, using convex and increasing functions. As special cases of the main results, we obtain the results of [18,19]. We also

    prove the boundedness of the k-fractional integral operator on Lp[a, b].

    1. Introduction

    Fractional calculus deals with the non integer order derivative and integral operators and draws increasing attention due to its applications in many fields see e.g. the books [30, 31]. The first application of fractional calculus was due to Abel in his solution to the Tautocrone problem [1]. It also has applications in biophysics, quantum mechanics, wave theory, polymers, continuum mechan- ics, Lie theory, field theory, spectroscopy and in group theory, among other applications [14–16,25].

    Many mathematicians originate the Hardy-type inequalities for different fractional integral and derivative operators. Because of the fundamental impor- tance of such inequalities in technical sciences, over the years much effort and time have been dedicated to the improvement and generalizations of Hardy- type inequalities. But still there are many open questions in this area, see e.g. those pointed out in [24, Section 7.5]. For further details and literature about the rich history of Hardy-type inequalities and fractional calculus, we refer the books [4,13,15] and the papers [2,6,7,10,18,20]. In the present work, we shall introduce some new results concerning Hardy-type inequalities not covered by the literature mentioned above.

    Received November 7, 2018; Revised February 19, 2019; Accepted March 11, 2019.

    2010 Mathematics Subject Classification. 26D15, 26D10, 26A33. Key words and phrases. Hardy-type inequalities, Riemann-Liouville, fractional integral

    operator, convex and increasing function. The research of second author was supported by the Ministry of Education and Science

    of the Russian Federation (Agreement number No. 02.a03.21.0008).

    c©2020 Korean Mathematical Society

    161

  • 162 S. IQBAL, J. PEČARIĆ, M. SAMRAIZ, H. TEHMEENA, AND Ž. TOMOVSKI

    2. Preliminaries and basic results

    We start with the definition of the Riemann-Liouville fractional integrals (see [22]).

    Definition 2.1. Let [a, b], (−∞ < a < b < ∞) be a finite interval on the real axis R. The Riemann-Liouville fractional integrals Iαa+f and I

    α b− f of order

    α > 0 are defined by

    (Iαa+f)(x) = 1

    Γ(α)

    ∫ x a

    f(t)(x− t)α−1dt, (x > a)

    and

    (Iαb−f)(x) = 1

    Γ(α)

    ∫ b x

    f(t)(t− x)α−1dt, (x < b),

    respectively. Here Γ is the usual gamma function.

    These integrals are called the left-sided and the right-sided fractional inte- grals and are bounded in Lp(a, b), 1 ≤ p ≤ ∞, that is (2.1) ‖Iαa+f‖p ≤ K‖f‖p, ‖I

    α b−f‖p ≤ K‖f‖p,

    where

    K = (b− a)α

    Γ(α+ 1) .

    Inequalities given in (2.1) were proved by Hardy in one of his paper [12]. Next is the definition of convex function presented in [13].

    Definition 2.2. Let I be an interval in R. A function Φ : I → R is called convex if the following inequality

    (2.2) Φ(λx+ (1− λ)t) ≤ λΦ(x) + (1− λ)Φ(t) holds for all points x, y ∈ I and all λ ∈ [0, 1]. The function Φ is strictly convex if inequality (2.2) holds strictly for all distinct points in I and λ ∈ (0, 1).

    The generalized Lp space given in [28] defined as follows:

    Definition 2.3. A space Lp,r[a, b] is defined as a space of continuous real valued function h(t) on [a, b], such that(∫ b

    a

    |h(t)|ptrdt

    ) 1 p

    0,

  • ON SOME WEIGHTED HARDY-TYPE INEQUALITIES 163

    where (t)n,k = t(t+ k)(t+ 2k) · · · (t+ (n− 1)k), n ≥ 1, is called Pochhammer k symbol. The integral representation is given by

    (2.4) Γk(t) =

    ∫ ∞ 0

    xt−1e −xk k dx, R(t) > 0.

    Specially for k = 1, Γ1(t) = Γ(t).

    Next is the well known definition of Riemann-Liouvill fractional derivative (see [22], [32]) of order α defined by

    Dαx{f(x)} = 1

    Γ(−α)

    ∫ x 0

    f(t)(x− t)−α−1dt, R(α) < 0.

    Let (Σ1,Ω1, µ1) and (Σ2,Ω2, µ2) be measure spaces with positive σ-finite mea- sures. Let U(f) denote the class of functions g : Ω1 → R with the representation

    g(x) :=

    ∫ Ω2

    k(x, t)f(t)dµ2(t)

    and Ak be an integral operator defined by

    (2.5) Akf(x) := g(x)

    K(x) =

    1

    K(x)

    ∫ Ω2

    k(x, t)f(t)dµ2(t),

    where k : Ω1 × Ω2 → R is measurable and non-negative kernel, f : Ω2 → R is a measurable function and

    (2.6) 0 < K(x) :=

    ∫ Ω2

    k(x, t)dµ2(t), x ∈ Ω1.

    The following theorem is given in [18].

    Theorem 2.5. Let (Σ1,Ω1, µ1) and (Σ2,Ω2, µ2) be measure spaces with σ- finite measures, u be a weight function on Ω1, k be a non-negative measurable

    kernel on Ω1 × Ω2. Let 0 < p ≤ q < ∞ and the function x 7→ u(x)k(x,t)K(x) is integrable on Ω1. Then for each fixed t ∈ Ω2, v is known by

    v(t) :=

    ∫ Ω1

    u(x) k(x, t)

    K(x) dµ1(x) 0 for every x ∈ Ω1. Let u be a weight function on Ω1 and

  • 164 S. IQBAL, J. PEČARIĆ, M. SAMRAIZ, H. TEHMEENA, AND Ž. TOMOVSKI

    k a non-negative measurable kernel on Ω1 × Ω2. Assume that the function x 7→ u(x) f2(t)k(x,t)g2(x) is integrable on Ω1 for each fixed t ∈ Ω2. Define p on Ω2 by

    p(t) := f2(t)

    ∫ Ω1

    u(x)k(x, t)

    g2(x) dµ1(x)

  • ON SOME WEIGHTED HARDY-TYPE INEQUALITIES 165

    Definition 3.1. Let R(r) > 0, R(s) > 0 and R(α) < 0. Then the extended Riemann-Liouville fractional derivative Dαx{f(x); r, s} of order α, be such that

    (3.1) Dαx{f(x); r, s} = 1

    Γ(−α)

    ∫ x 0

    f(t)(x− t)−α−1 exp ( −rx t − sx

    (x− t)

    ) dt.

    Specially for r = s we arrive at the extended Riemann-Liouville fractional derivative of order α given in [3] and is defined by

    (3.2) Dαx{f(x); r} = 1

    Γ(−α)

    ∫ x 0

    f(t)(x− t)−α−1 exp ( − rx

    2

    t(x− t)

    ) dt.

    Example 3.2. Consider the derivative given by (3.1) of xν corresponding to x = 1, we get

    Dαx{xν ; r, s}x=1 = Br,s(ν + 1,−α)

    Γ(−α) ,

    where Br,s(ν + 1,−α) is the extended beta functions (see [26]) defined by

    (3.3) Br,s(x, y) =

    ∫ 1 0

    tx−1(1− t)y−1e− r t−

    s 1−t dt,

    where x, y, r, s ∈ C, R(r) > 0, R(s) > 0. For r = s, Br,s becomes Br and for r = s = 0, we get the classical beta function defined by

    B(x, y) =

    ∫ 1 0

    tx−1(1− t)y−1dt, R(x) > 0, R(y) > 0.

    In [29] the following inequality is proved for the extended beta function (3.3):

    Br,s(x, y) ≤ (2r) 2x−1

    2 (2s) 2y−1

    2

    √ Γ(−2x+ 1, 2r)Γ(−2y + 1, 2s),

    where r, s > 0, 0 < x, y < 12 and Γ(x, y) is the incomplete gamma function.

    Lemma 3.3. Let R(r) > 0, R(s) > 0 and R(α) < 0. Then the following relation holds:

    K̆(x) = x−αBr,s(1,−α)

    Γ(−α) .(3.4)

    Proof. Since

    (3.5) k̆(x, t) =

    { 1

    Γ(−α) (x− t) −α−1 exp

    ( − rxt −

    sx (x−t)

    ) , 0 ≤ t ≤ x ;

    0, x < t ≤ b . Therefore

    K̆(x) = 1

    Γ(−α)

    ∫ x 0

    (x− t)−α−1 exp ( −rx t − sx

    (x− t)

    ) dt,

    which can be written as

    = x−α−1

    Γ(−α)

    ∫ x 0

    (1− t x

    )−α−1 exp

    ( −rx t − s

    (1− tx )

    ) dt.

    By substituting tx = y and using the simple calculation, we arrive at (3.4). �

  • 166 S. IQBAL, J. PEČARIĆ, M. SAMRAIZ, H. TEHMEENA, AND Ž. TOMOVSKI

    The first result for the operator (3.1) is as follows:

    Theorem 3.4. Let R(r) > 0, R(s) > 0 and R(α) < 0. Let Dαx{f(x); r, s} denote the extension of Riemann-Liouville fractional derivative of order α and let u be a weight function defined on (0, b). For each fixed t ∈ (0, b), define a function v̆ by

    v̆(t) =

    ∫ b t

    u(x) (x− t)−α−1 exp

    ( − rxt −

    sx (x−t)

    ) x−αBr,s(1,−α)

    dx 0, R(s) > 0 and R(α) < 0. Let Dαx{f(x); r, s} denote the extension of Riemann-Liouville fractional derivative of order α and let u be a weight function defined on (0, b). For each fixed t ∈ (0, b), define a function

    p̆(t) := f2(t)

    Γ(−α)

    ∫ b t

    u(x) (x− t)−α−1 exp

    ( − rxt −

    sx (x−t)

    ) Dαx{f2(x); r, s}

    dx

  • ON SOME WEIGHTED HARDY-TYPE INEQUALITIES 167

    Theorem 3.6. Let the assumption of Theorem 3.4 be satisfied, 0 < p ≤ q 0 and R(α) < 0. Let Dαx{f(x); r, s} denote the extension of Riemann-Liouville fractional derivative of order α and Dαx{f2(x); r, s}