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}