WEIGHTED HARDY-TYPE INEQUALITIES

1 downloads 0 Views 416KB Size Report
Abstract. The aim of this paper is to establish some new weighted Hardy-type inequalities involving convex and monotone convex functions using Hilfer ...
WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTION FOR FRACTIONAL CALCULUS OPERATORS ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , JOSIP PECARI C

Abstract. The aim of this paper is to establish some new weighted Hardy-type inequalities involving convex and monotone convex functions using Hilfer fractional derivative and fractional integral operator with generalized Mittag-Leffler function in its kernel. We also discuss one dimensional cases of our related results. As a special case of our general results we obtain the results of [12]. Moreover, the refinement of Hardy-type inequalities for Hilfer fractional derivative are also included. 1. Introduction Fractional calculus deals with the study of fractional order integral and derivative operators calculus and have been of great importance during the last few decades. Oldham and Spanier [23] published their fundamental work in their book in 1974 and Podlubny [25] publication from 1999, which deals principally with fractional differential equations. For further details and literature about the fractional calculus we refer to [1], [2], [14] and the references cited therein. Numerous mathematicians obtained new Hardy-type inequalities for different fractional integrals and fractional derivatives. For details we refer to [3], [5], [7], [8], [13], [16], [24]. The general theory for the Hardy-type inequalities has attracted a lot of attention during a long time, see e.g. the books [15], [18], [20] and the reference therein. One reason is that such results are of special interest for technical sciences. Especially actions of kernels operators of type (1.2) and (1.3) below are important since the kernel k(x, y) represent unit impulse answers in systems which need not to be time invariant (f (y) and g(x) represent the ”insignals” and ”outsignals” respectively). Some current knowledge can be found in Section 7.5 of the new 2017 book [20] by Kufner, Persson and Samko, see also the related review article [19]. But still there are many open questions in this area, see e.g. those pointed out in [20, Section 7.5]. In this paper we present some new resuts concerning Hardy-type inequalities not covered by the literature mentioned above. The following definitions are presented in [22]. Definition 1.1. Let I be an interval in R. A function Φ : I → R is called convex if Φ(λx + (1 − λ)y) ≤ λΦ(x) + (1 − λ)Φ(y), (1.1) 2000 Mathematics Subject Classification. 26D15, 26D10, 26A33. Key words and phrases. Convex function, kernel, Hilfer fractional derivatives, fractional integral. 1

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

2

for all points x, y ∈ I and all λ ∈ [0, 1]. The function Φ is strictly convex if inequality (1.1) holds strictly for all distinct points in I and λ ∈ (0, 1). Definition 1.2. Let Φ : I −→ R be a convex function, then the sub-differential of Φ at x, denoted by ∂Φ(x), is defined as ∂Φ(x) = {α ∈ R : Φ(y) − Φ(x) − α(y − x) ≥ 0, y ∈ I}. Let (Σ1 , Ω1 , µ1 ) and (Σ2 , Ω2 , µ2 ) be measure spaces with positive σ-finite measures. Let U (f ) denote the class of functions g : Ω1 → R with the representation Z g(x) = k(x, y)f (y)dµ2 (y), (1.2) Ω2

and Ak be an integral operator defined by 1 g(x) = (Ak f )(x) := K(x) K(x)

Z k(x, y)f (y)dµ2 (y),

(1.3)

Ω2

where k : Ω1 × Ω2 → R is measurable and non-negative kernel, f : Ω2 → R is measurable function and Z 0 < K(x) := k(x, y)dµ2 (y), x ∈ Ω1 . (1.4) Ω2

The following theorem was given in [4] and [9] (see also [17]). Theorem 1.3. Let 0 < p ≤ q < ∞, or −∞ < q ≤ p < 0, (Ω1 , Σ1 , µ1 ) and (Ω2 , Σ2 , µ2 ) be measure spaces with σ-finite measures, u be a weight function on Ω1 , k be a non-negative measurable function on Ω1 × Ω2 , K be defined on Ω1 by   pq (1.4) and that the function x 7→ u(x) k(x,y) is integrable on Ω1 for each y ∈ Ω2 , K(x) and that v is defined on Ω2 by  pq   pq  Z k(x, y) dµ1 (x) < ∞. v(y) :=  u(x) K(x) Ω1

If Φ is a non-negative convex function on the interval I ⊆ R and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the inequality  pq

 Z

Z

q

v(y)Φ (f (y)) dµ2 (y) −

 Ω2

u(x)[Φ ((Ak f )(x))] p dµ1 (x) Ω1



q p

Z Ω1

u(x) Φ K(x)

q p −1

Z ((Ak f )(x))

k(x, y)r(x, y)dµ2 (y)dµ1 (x)

(1.5)

Ω2

holds for all measurable functions f : Ω2 → I, where Ak is defined by (1.3) and r : Ω1 × Ω2 → R is a non-negative function defined by r(x, y) = | |Φ(f (y)) − Φ((Ak f )(x))| − |ϕ((Ak f )(x))| |f (y) − (Ak f )(x)| |.

(1.6)

If Φ is a non-negative concave function, then the order of terms on the left hand side of (1.5) is reversed. If Φ is a non-negative monotone convex function on the

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTIONS...

3

interval I ⊆ R, and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ Int I, then the inequality  pq

 Z

Z v(y)Φ (f (y)) dµ2 (y) −

 Ω2

q

u(x)Φ p ((Ak f )(x)) dµ1 (x) Ω1

Z Z u(x) pq −1 q ≥ Φ ((Ak f )(x)) sgn(f (y)−(Ak f )(x))k(x, y)r1 (x, y) dµ2 (y) dµ1 (x) p K(x) Ω1

Ω2

(1.7) holds for all measurable functions f : Ω2 → I, where Ak f is defined by (1.3) and r1 : Ω1 × Ω2 → R is a non-negative function defined by r1 (x, y) = Φ(f (y)) − Φ((Ak f )(x)) − ϕ((Ak f )(x)) · (f (y) − (Ak f )(x)).

(1.8)

If Φ is a non-negative monotone concave function, then the order of terms on the left hand side of (1.7) is reversed. Remark 1.4. For p = q, Theorem 1.3 becomes [3, Theorem 2.1] (see also [17, Theorem 4.1]) and convex function Φ need not to be non-negative. Although the inequalities (1.5) and (1.7) hold for non-negative convex and monotone convex functions some choices of Φ are of our particular interest. Here, we consider the power weight function i.e. the function Φ : R+ → R be defined by Φ(x) = xs . It is a non-negative, convex and monotone function. Obviously, ϕ(x) = Φ0 (x) = sxs−1 , x ∈ R+ , so Φ is convex for s ∈ R \ [0, 1), concave for s ∈ (0, 1], and affine, that is, both convex and concave for s = 1. Corollary 1.5. Let Ω1 , Ω2 , µ1 , µ2 , u, k, K, p, q and v be as in Theorem 1.3. Let s ∈ R be such that s 6= 0, f : Ω2 → R be a non-negative measurable function (positive for s < 0), Ak f be defined by (1.3) and rs,k f (x, y) = |f s (y) − ((Ak f )(x))s | − |s| · ((Ak f )(x))s−1 |f (y) − (Ak f )(x)| , (1.9) for x ∈ Ω1 , y ∈ Ω2 . If s ≥ 1 or s < 0, then the following inequality   pq Z Z qs  v(y)f s (y)dµ2 (y) − u(x)A p f (x)dµ1 (x) k Ω2

Ω1



q p

Z

(q−p)s u(x) ((Ak f )(x)) p K(x)

Ω1

Z k(x, y)rs,k f (x, y) dµ2 (y) dµ1 (x)

(1.10)

Ω2

holds. Let qs

Ms,k f (x, y) = f s (y) − Akp f (x) − s · ((Ak f )(x))s−1 (f (y) − (Ak f )(x)) for x ∈ Ω1 , y ∈ Ω2 . If s ≥ 1 or s < 0, then the inequality

(1.11)

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

4

 pq

 Z 

v(y)f s (y)dµ2 (y) −

Ω2

Z

qs

u(x)Akp f (x)dµ1 (x)



1 Z Z (q−p)s u(x) q p ≥ ((Ak f )(x)) sgn(f (y)−(Ak f )(x))k(x, y)Ms,k f (x, y) dµ2 (y) dµ1 (x) p K(x)

Ω1

Ω2

(1.12) holds. If s ∈ (0, 1), then relations (1.10) and (1.12) hold with  pq

 Z

Z

qs p

u(x)Ak f (x)dµ1 (x) − 

v(y)f s (y)dµ2 (y)

Ω2

Ω1

on their left hand sides. Result for one dimensional settings, with intervals in R and Lebesgue measures was given in the following theorem (see [4] and c.f. also [17, Theorem 5.7]). Theorem 1.6. Let 0 < b ≤ ∞ and k : (0, b) × (0, b) → R, u : (0, b) → R be a non-negative measurable functions satisfying Zx K(x) =:

k(x, y)dy,

x ∈ (0, b),

(1.13)

0

and 

Zb 

w(y) = y 

k(x, y) K(x)

 pq

 pq u(x)

dx  < ∞, x

y ∈ (0, b).

(1.14)

y

If 0 < p ≤ q < ∞, or −∞ < q ≤ p < 0, Φ is a non-negative convex function on the interval I ⊆ R and ϕ : I → R satisfies that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the following inequality  b  pq Z Zb q dy  w(y)Φ (f (y))  − u(x)Φ p ((Ak f )(x)) dx y x 0

0

q ≥ p

Zb 0

u(x) pq −1 Φ ((Ak f )(x)) K(x)

Zx k(x, y) r(x, y)dy

dx x

(1.15)

0

holds for all measurable functions f : (0, b) → R with values in I, where r is defined by (1.6). If Φ is a non-negative monotone convex function on the interval I ⊆ R and ϕ : I → R satisfies that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the following inequality

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTIONS...

5

 b  pq Z Zb q dy  w(y)Φ (f (y))  − u(x)Φ p ((Ak f )(x)) dx y x 0

0

Zb Zx u(x) pq −1 q dx ≥ ((Ak f )(x)) sgn(f (y) − (Ak f )(x))k(x, y) r1 (x, y)dy Φ p K(x) x 0

0

(1.16) holds for all measurable functions f : (0, b) → R, where r1 is defined by (1.8) and Ak f is defined by 1 (Ak f )(x) := K(x)

Zx k(x, y)f (y)dy,

x ∈ (0, b).

(1.17)

0+

If 0 < p ≤ q < ∞, or −∞ < q ≤ p < 0, and Φ is a non-negative (monotone) concave function, then (1.15) and (1.16) hold with reverse order integral of their left hand sides. The paper is organized in the following way: After this introduction, in Section 2 we give the generalized Hardy-type inequalities involving generalized Mittag-Leffler function appearing in the kernel for convex and monotone convex functions and Hilfer fractional derivative. We give the related inequalities as an application for the power function. We also include the results for the one dimensional settings. In addition to this we construct inequalities in quotient for the generalized fractional integral operator. In Section 3 we derive the results for Hilfer fractional derivative. Results analogous to those in Section 2 given for Hilfer fractional derivative. We present some new inequalities of Hardy-type for Hilfer fractional derivative. Moreover, we deduce in particular the results of [10] and [12] from our general results. 2. Refined Hardy-type inequalities for fractional integral operator with generalized Mittag-Leffler function in its kernel In this section, we first give the definition of Mittag-Leffler function [21] and fractional integral operator involving generalized Mittag-Leffler function appearing in the kernel [28]. Let R(α) be a real part of complex number α. Definition 2.1. Let α, β, γ, δ ∈ C; min{R(α), R(β), R(γ), R(δ)} > 0; p, q > 0. Then the generalized Mittag-Leffler function defined in [28] is given by γ,δ,q Eα,β,p (z) =

∞ X

(γ)qn zn , Γ(αn + β) (δ)pn n=0

(2.1)

where (γ)n represents the Pochhammer symbol, defined by (γ)n = γ(γ − 1)(γ − 2) . . . (γ − n + 1). The function (2.1) represents all the previous generalizations of Mittag-Leffler function by setting ∞ P (γ)n γ,δ zn • p = q = 1, it reduces to Eα,β (z) = Γ(αn+β) (δ)n defined by Salim in [27]. n=0

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

6

γ,q • δ = p = 1, it represents Eα,β (z) =

∞ P n=0

(γ)qn z n Γ(αn+β) n! ,

which was introduced by

Shukla and Prajapati in [29]. In [30] Srivastava and Tomovski investigated the properties of this function and its existence for a wider set of parameters. • δ = p = q = 1, the operator (2.1) was defined by Prabhakar in [26] and was ∞ P (γ)n γ zn denoted as: Eα,β (z) = Γ(αn+β) n! . n=0

• γ = δ = p = q = 1, it reduces to Wiman’s function presented in [32]. Moreover, if β = 1, then the original Mittag-Leffler function Eα (z) will be the result (see [21]). We denote 

 β−1 γ,δ,q eγ,δ,q Eα,β,p (ωxα ) . α,β,p (x; ω) = x

Definition 2.2. Let α, β, γ, δ, ω ∈ C; min{R(α), R(β), R(γ), R(δ)} > 0; p, q > 0. For all f ∈ L(a, b) we introduce an integral operator 

Eγ,δ,q α,β,p,ω;a+ f



Zx (x) =

eγ,δ,q α,β,p (x − t; ω) f (t)dt,

(2.2)

a

which contains the generalized Mittag-Leffler function (2.1) in its kernel. Our first main result is given in the following theorem. Theorem 2.3. Let 0 < p ≤ q < ∞, or −∞ < q ≤ p < 0 and α, β, γ, δ, p, q be as in Definition 2.2 and let u be a weight function defined on (a, b). For each y ∈ (a, b), v˜ is defined on (a, b) by 

Zb

v˜(y) := 

u(x) y

eγ,δ,q α,β,p (x − y; ω) γ,δ,q eα,β+1,p (x − a; ω)

! pq

 pq dx < ∞.

If Φ is a non-negative convex function on the interval I ⊆ R and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the inequality 

Zb

 a

 pq

  Eγ,δ,q α,β,p,ω;a+ f (x)  dx v˜(y)Φ (f (y)) dy  − u(x)Φ  γ,δ,q eα,β+1,p (x − a; ω) a    γ,δ,q  Zb E f (x) + q α,β,p,ω;a u(x) q −1  Φ p  γ,δ,q ≥ p eγ,δ,q eα,β+1,p (x − a; ω) α,β+1,p (x − a; ω) Zb



q p

a

Zx ×

eγ,δ,q ˜(x, y)dydx (2.3) α,β,p (x − y; ω) r

a

holds for all measurable functions f : (a, b) → R and r˜ : (a, b) × (a, b) → R is a non-negative function defined by

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTIONS...

7

   γ,δ,q  E f (x) α,β,p,ω;a+  r˜(x, y) = Φ(f (y)) − Φ  γ,δ,q eα,β+1,p (x − a; ω)      Eγ,δ,q Eγ,δ,q α,β,p,ω;a+ f (x) α,β,p,ω;a+ f (x) . (2.4)  f (y) − − ϕ  γ,δ,q eα,β+1,p (x − a; ω) eγ,δ,q (x − a; ω) α,β+1,p If Φ is a non-negative monotone convex function on the interval I ⊆ R, and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ Int I, then the inequality   b  pq   γ,δ,q  Z Zb Eα,β,p,ω;a+ f (x) q  v˜(y)Φ (f (y)) dy  − u(x)Φ p   dx eγ,δ,q (x − a; ω) α,β+1,p a a     γ,δ,q Zb E f (x) + q α,β,p,ω;a q u(x)  ≥ Φ p −1  γ,δ,q p eγ,δ,q (x − a; ω) e (x − a; ω) α,β+1,p α,β+1,p a     x γ,δ,q Z Eα,β,p,ω;a+ f (x) γ,δ,q   × sgn f (y) − γ,δ,q eα,β,p (x − y; ω) r˜1 (x, y) dy dx (2.5) e (x − a; ω) α,β+1,p

a

holds for all measurable functions f : (a, b) → R and r˜1 : (a, b) × (a, b) → R is a non-negative function defined by     γ,δ,q Eα,β,p,ω;a+ f (x)  r˜1 (x, y) = Φ(f (y)) − Φ  γ,δ,q eα,β+1,p (x − a; ω)      γ,δ,q    f (x) Eγ,δ,q Eα,β,p,ω;a+ f (x) + α,β,p,ω;a  · f (y) −  . (2.6) − ϕ  γ,δ,q γ,δ,q eα,β+1,p (x − a; ω) eα,β+1,p (x − a; ω) If Φ is a non-negative (monotone) concave, then the order of terms on the left hand side of inequalities (2.3) and (2.5) is reversed. Proof. Applying Theorem 1.3 with Ω1 = Ω2 = (a, b), dµ1 (x) = dx, dµ2 (y) = dy,  γ,δ,q eα,β,p (x − y; ω) , a ≤ y ≤ x ; ˜ y) = k(x, (2.7) 0, x < y ≤ b, where (see Lemma 3.2 in [11]), and Zx γ,δ,q ˜ K(x) = eγ,δ,q α,β,p (x − y; ω) dy = eα,β+1,p (x − a; ω) , a

and  (Ak f )(x) = we get inequalities (2.3) and (2.5).

 Eγ,δ,q f (x) α,β,p,ω;a+

eγ,δ,q α,β+1,p (x − a; ω)

,

(2.8) 

8

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

Remark 2.4. Theorem 2.3 generalizes the result of [12], i.e. for p = q, Theorem 2.3 becomes [12, Theorem 3.6] and convex function Φ need not to be non-negative. Particular to our interest, we consider the power function Φ : R+ → R defined by Φ(x) = xs . Obviously, ϕ(x) = Φ0 (x) = sxs−1 , x ∈ R+ . ˜ K, ˜ p, q and v˜ be as in Theorem 2.3. Let s ∈ R be such that Corollary 2.5. Let k, s 6= 0, f : (a, b) → R be a non-negative measurable function (positive for s < 0) and    γ,δ,q s Eα,β,p,ω;a+ f (x) s  r˜s,k f (x, y) = f (y) −  γ,δ,q eα,β+1,p (x − a; ω)      γ,δ,q s−1 γ,δ,q Eα,β,p,ω;a+ f (x) Eα,β,p,ω;a+ f (x) , (2.9) f (y) −  −|s| ·  γ,δ,q eα,β+1,p (x − a; ω) eγ,δ,q α,β+1,p (x − a; ω) for x, y ∈ (a, b). If s ≥ 1 or s < 0, then the inequality

  b  pq  qs   γ,δ,q p Z Zb Eα,β,p,ω;a+ f (x)  dx  v˜(y)f s (y)dy  − u(x)  eγ,δ,q (x − a; ω) α,β+1,p a

a



q p

Zb a

 u(x) eγ,δ,q α,β+1,p

(x − a; ω)

 Zx

×

Eγ,δ,q α,β,p,ω;a+ f

eγ,δ,q α,β+1,p



(x)

(x − a; ω)

 (q−p)s p 

eγ,δ,q ˜s,k f (x, y) dy dx α,β,p (x − y; ω) r

(2.10)

a

holds. Let  s Eγ,δ,q (x) +f α,β,p,ω;a ˜ s,k f (x, y) = f s (y) −   − M eγ,δ,q (x − a; ω) α,β+1,p      γ,δ,q s−1   Eα,β,p,ω;a+ f (x) Eγ,δ,q f (x) α,β,p,ω;a+  f (y) −  s  γ,δ,q eα,β+1,p (x − a; ω) eγ,δ,q (x − a; ω) α,β+1,p 

for x, y ∈ (a, b). If s ≥ 1 or s < 0, then the inequality

(2.11)

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTIONS...

9

  b   γ,δ,q  pq  qs p Z Zb Eα,β,p,ω;a+ f (x) s  v˜(y)f (y)dy  − u(x)   dx eγ,δ,q α,β+1,p (x − a; ω) a

a

   γ,δ,q  (q−p)s Zb p E f (x) α,β,p,ω;a+ u(x) q   ≥ p eγ,δ,q eγ,δ,q α,β+1,p (x − a; ω) α,β+1,p (x − a; ω) a     Zx Eγ,δ,q α,β,p,ω;a+ f (x) γ,δ,q ˜ s,k f (x, y) dy dx (2.12) e × sgn f (y) − γ,δ,q (x − y; ω) M α,β,p eα,β+1,p (x − a; ω) a

holds. If s ∈ (0, 1), then versions of the inequalities (2.10) and (2.12) hold with reverse order of terms on the left hand sides. Here we give the result for one dimensional settings, with intervals in R and Lebesgue measures in next theorem. Theorem 2.6. Let 0 < b ≤ ∞, α, β, γ, δ, p, q be as Definition 2.2 and u be a weight function. For each y ∈ (0, b) we let the function w ˜ : (0, b) → R be defined by  b  pq ! pq Z eγ,δ,q (x − y; ω) dx α,β,p w(y) ˜ = y (2.13) u(x)  . γ,δ,q x eα,β+1,p (x; ω) y

If Φ is a non-negative convex function on the interval I ⊆ R and ϕ : I → R satisfies that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the following inequality   b   γ,δ,q  pq  Zb Z E f (x) dx + q α,β,p,ω;a dy  w(y)Φ  ˜ (f (y))  − u(x)Φ p  y x eγ,δ,q (x; ω) α,β+1,p 0 0   x   γ,δ,q Zb Eα,β,p,ω;a+ f (x) Z q u(x) q  eγ,δ,q (x − y; ω) r˜(x, y)dy dx Φ p −1  ≥ α,β,p γ,δ,q γ,δ,q p x eα,β+1,p (x; ω) eα,β+1,p (x; ω) 0

0

(2.14) holds for all measurable functions f : (0, b) → R and r˜(x, y) is defined by (2.4). If Φ is a non-negative monotone convex function on the interval I ⊆ R and ϕ : I → R satisfies that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the following inequality   b  pq   γ,δ,q  Z Zb E f (x) + q α,β,p,ω;a dy  w(y)Φ  dx ˜ (f (y))  − u(x)Φ p  y x eγ,δ,q (x; ω) α,β+1,p 0 0    γ,δ,q  Zb Eα,β,p,ω;a+ f (x) q q u(x)  ≥ Φ p −1  γ,δ,q p eγ,δ,q (x; ω) e (x; ω) α,β+1,p α,β+1,p 0     x γ,δ,q Z Eα,β,p,ω;a+ f (x) dx γ,δ,q e × sgn f (y) − (x − y; ω) r ˜ (x, y)dy (2.15) 1 α,β,p x eγ,δ,q (x; ω) 0

α,β+1,p

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

10

holds for all measurable functions f : (0, b) → R and r˜1 (x, y) defined by (2.6). ˜ t) given by (2.7) and Proof. Applying Theorem 1.6 with k(x, (Ak f )(x) =

1

Zx

eγ,δ,q α,β+1,p (x; ω)

0

eγ,δ,q α,β,p (x − y; ω) f (y)dy,

then we obtain inequalities (2.14) and (2.15).



Remark 2.7. Some special cases of the above results are given below. • If we take m = k = 1 in Theorem 2.3, Corollary 2.5 and in Theorem 2.6, ∞ P (γ)n γ,δ zn then the inequalities reduces to the case Eα,β (z) = Γ(αn+β) (δ)n . n=0

• If we take δ = m = 1 in Theorem 2.3, Corollary 2.5 and in Theorem 2.6, ∞ P (γ)kn z n γ,k then the inequalities reduces to the case Eα,β (z) = Γ(αn+β) n! . n=0

• If we take δ = m = k = 1 in Theorem 2.3, Corollary 2.5 and in Theorem ∞ P (γ)n γ zn 2.6, then the inequalities reduces to the case Eα,β (z) = Γ(αn+β) n! . n=0

• If we take γ = δ = m = k = 1 in Theorem 2.3, Corollary 2.5 and in Theorem 2.6 then the inequalities reduces to Wiman’s function. Moreover, if β = 1, then the original Mittag-Leffler function Eα (z) will be the result. Next we will present some new generalized Hardy-type inequalities in quotient f1 form. For this if we substitute k(x, y) by k(x, y)f2 (y) and f by , where fi : f2 Ω2 → R, (i = 1, 2) are measurable functions in Theorem 1.6 we obtain the following result. Theorem 2.8. Let 0 < p ≤ q < ∞, or −∞ < q ≤ p < 0, (Ω1 , Σ1 , µ1 ) and (Ω2 , Σ2 , µ2 ) be measure spaces with σ-finite measures. Let fi : Ω2 → R be measurable functions, gi ∈ U (fi ), (i = 1, 2), where g2 (x) > 0 for every x ∈ Ω1 . Let u be a weight function on Ω1 , k be a non-negative measurable function on Ω1 × Ω2 and   pq be integrable on Ω1 . For each y ∈ Ω2 define let the function x 7→ u(x) k(x,y) g2 (x) s = s(y) on Ω2 by 



Z s(y) := f2 (y) 

u(x)

k(x, y) g2 (x)

 pq

 pq

dµ1 (x) < ∞.

Ω1

If Φ is a non-negative convex function on the interval I ⊆ R and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the inequality 



Z s(y)Φ



f1 (y) f2 (y)

 pq



Z dµ2 (y) −

Ω2

u(x)Φ

q p



g1 (x) g2 (x)

 dµ1 (x)

Ω1

q ≥ p

Z Ω1

u(x) pq −1 Φ g2 (x)



g1 (x) g2 (x)

Z k(x, y)f2 (y)d(x, y)dµ2 (y)dµ1 (x) Ω2

(2.16)

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTIONS...

11

(y) holds for all measurable functions fi : Ω2 → I, (i = 1, 2) such that ff21 (y) ∈ I, for all y ∈ Ω2 , and d : Ω1 × Ω2 → R is a non-negative function defined by       g1 (x) f1 (y) g1 (x) f1 (y) g1 (x) ϕ . − −Φ − d(x, y) = Φ f2 (y) g2 (x) g2 (x) f2 (y) g2 (x)

If Φ is a non-negative monotone convex function on the interval I ⊆ R, and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ Int I, then the inequality 



Z s(y)Φ



f1 (y) f2 (y)

 pq



Z dµ2 (y) −

Ω2

u(x)Φ

q p



g1 (x) g2 (x)

 dµ1 (x)

Ω1

Z   Z  u(x) pq −1 g1 (x) f1 (y) g1 (x) q sgn Φ − k(x, y)f2 (y)d1 (x, y) dµ2 (y) dµ1 (x) ≥ p g2 (x) g2 (x) f2 (y) g2 (x) Ω1

Ω2

(2.17) holds for all measurable functions fi : Ω2 → I, (i = 1, 2) such that ff21 (y) (y) ∈ I, for all y ∈ Ω2 and d1 : Ω1 × Ω2 → R is a non-negative function defined by         f1 (y) g1 (x) g1 (x) f1 (y) g1 (x) d1 (x, y) = Φ −Φ −ϕ · − f2 (y) g2 (x) g2 (x) f2 (y) g2 (x) If Φ is a non-negative (monotone) concave function, then the order of terms on the left hand sides of (2.16) and (2.17) are reversed. Remark 2.9. For p = q in Theorem 2.8 we get the result in [10, Theorem 1.3]. Our next result reads; Theorem 2.10. Let 0 < p ≤ q < ∞, or −∞ < q ≤ p < 0, α, β, γ, δ, p, q be as in Definition 2.2 and let u be a weight function defined on (a, b). For each y ∈ (a, b), define a function   pq  pq  Zb γ,δ,q eα,β,p (x − y; ω)   dx  s˜(y) := f2 (y)  u(x)    < ∞. γ,δ,q Eα,β,p,ω,a+ f2 (x) y If Φ is a non-negative convex function on the interval I ⊆ R and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the inequality 

Zb

 a

 pq

  γ,δ,q E f (x) + 1 α,β,p,ω,a f1 (y)  dx  s˜(y)Φ dy  − u(x)Φ   γ,δ,q f2 (y) E f (x) + 2 α,β,p,ω,a a    γ,δ,q  b Z Eα,β,p,ω,a+ f1 (x) q u(x) q     ≥ Φ p −1   γ,δ,q γ,δ,q p Eα,β,p,ω,a+ f2 (x) Eα,β,p,ω,a+ f2 (x) a 





Zb

q p

Zx × a

˜ eγ,δ,q α,β,p (x − y; ω) f2 (y)d(x, y)dydx (2.18)

12

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

holds for all measurable functions fi : Ω2 → I, (i = 1, 2) such that ff12 (y) (y) ∈ I, for all ˜ y ∈ (a, b), and d : (a, b) × (a, b) → R is a non-negative function defined by     γ,δ,q   Eα,β,p,ω,a+ f1 (x) f (y) 1 ˜ y) = Φ   d(x, − Φ  f 2 (y) Eγ,δ,q f (x) + 2 α,β,p,ω,a      γ,δ,q γ,δ,q E f (x) E f f1 (y) α,β,p,ω,a+ 1 α,β,p,ω,a+ 1 (x) .     − ϕ  − · Eγ,δ,q f (x) f2 (y) Eγ,δ,q f (x) α,β,p,ω,a+ 2

α,β,p,ω,a+ 2

If Φ is a non-negative monotone convex function on the interval I ⊆ R, and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ Int I, then the inequality   b   γ,δ,q  pq    Z Zb Eα,β,p,ω,a+ f1 (x) q f (y) 1  s˜(y)Φ  dx  dy  − u(x)Φ p   γ,δ,q f2 (y) E f (x) + 2 α,β,p,ω,a a a     γ,δ,q Zb E f (x) + 1 q α,β,p,ω,a u(x) q    Φ p −1   ≥  γ,δ,q γ,δ,q p E f (x) E f (x) α,β,p,ω,a+ 2 α,β,p,ω,a+ 2 a     Zx Eγ,δ,q α,β,p,ω,a+ f1 (x) f (y) 1 γ,δ,q e  × sgn  − (x − y; ω) f2 (y)d˜1 (x, y) dy dx α,β,p γ,δ,q f2 (y) f (x) E a

α,β,p,ω,a+ 2

(2.19) (y) holds for all measurable functions fi : Ω2 → I, (i = 1, 2) such that ff21 (y) ∈ I, for all ˜ y ∈ (a, b) and d1 : (a, b) × (a, b) → R is a non-negative function defined by    γ,δ,q    Eα,β,p,ω,a+ f1 (x) f (y) 1   − Φ  d˜1 (x, y) = Φ f2 (y) Eγ,δ,q f (x) α,β,p,ω,a+ 2      γ,δ,q    γ,δ,q E f Eα,β,p,ω,a+ f1 (x) (x) + 1 α,β,p,ω,a f (y) 1 ·  .   − ϕ  − f2 (y) Eγ,δ,q f (x) Eγ,δ,q f (x) α,β,p,ω,a+ 2

α,β,p,ω,a+ 2

If Φ is a non-negative (monotone) concave function, then the order of terms on the left hand sides of (2.18) and (2.19) are reversed. Proof. Applying Theorem 2.8 with Ω1 = Ω2 = (a, b), dµ1 (x) = dx, dµ2 (y) = dy, γ,δ,q γ,δ,q g1 (x) = (Eγ,δ,q α,β,p,ω,a+ f1 )(x), g2 (x) = (Eα,β,p,ω,a+ f2 )(x) and k(x, y) = eα,β,p (x − y; ω) , we obtain inequalities (2.18) and (2.19).  Remark 2.11. Since the right hand side of the inequalities (2.18) and (2.19) are non-negative, therefore we obtain the following inequality    γ,δ,q   b  pq   Z Zb E f (x) q α,β,p,ω,a+ 1  dx ≤  v(y)Φ f1 (y) dy  .  u(x)Φ p   (2.20) γ,δ,q f2 (y) Eα,β,p,ω,a+ f2 (x) a a Particularly for p = q, we obtain the inequality given in [12, Theorem 3.4].

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTIONS...

Remark 2.12. If Φ is strictly convex on I and inequality given in (2.20) is strict.

f1 (x) f2 (x)

13

is non-constant, then the

3. Hardy-type inequalities for Hilfer fractional derivative operator Let x > a > 0. By L1 (a, x) we denote the space of all Lebesgue integrable functions on the interval (a, x). For any f ∈ L1 (a, x) the Riemann-Liouville fractional integral of f of order ν is defined by (Iaν+ f )(s)

1 = Γ(α)

Zs

(x − y)ν−1 f (t)dt = (f ∗ Kν )(s), s ∈ [a, x], (ν > 0),

(3.1)

a ν−1

where Kν (s) = sΓ(ν) . The integral on the right side of (3.1) exists for almost s ∈ [a, x] and Iaν+ f ∈ L1 (a, x). The Riemann-Liouville fractional derivative of f ∈ L1 (a, x) of order ν is defined by  n d ν (Ian−ν f )(x), (ν > 0, n = [ν + 1]). (Da+ f )(s) = + dx By C m [a, x] we denote the space of all functions on [a, x] which have continuous derivatives up to order m, and AC[a, x] is the space of all absolutely continuous functions on [a, x]. By AC m [a, x] we denote the space of all functions f ∈ C m [a, x] with f (m−1) ∈ AC[a, x]. By L∞ (a, x) we denote the space of all measurable functions essentially bounded on [a, x]. Let µ > 0, m = [µ] + 1 and f ∈ AC m [a, b]. The Caputo derivative of order µ > 0 is defined as C

(

Daµ+ f )(x)

 =

Iam−µ +

dm f dxm



1 (x) = Γ(m − µ)

Zx

(x − s)m−µ−1

dm f (s)ds. dxm

a

Let us recall the definition of Hilfer fractional derivative presented in [31]. Definition 3.1. Let f ∈ L1 [a, b], f ∗ K(1−ν)(1−µ) ∈ AC 1 [a, b]. The fractional derivµ,ν ative operator Da+ of order 0 < µ < 1 and type 0 < ν ≤ 1 with respect to x ∈ [a, b] is defined by   ν(1−µ) d (1−ν)(1−µ) µ,ν  Da+ f (x) := Ia+ Ia+ f (x) , (3.2) dx whenever the right hand side exists. The derivative (3.2) is usually called Hilfer fractional derivative. The more general integral representation of equation (3.2) given in [6] is defined by: Let f ∈ L1 [a, b] , f ∗ K(1−ν)(n−µ) ∈ AC n [a, b] , n − 1 < µ < n, 0 < ν ≤ 1, n ∈ N. Then   n  ν(n−µ) d (1−ν)(n−µ) µ,ν  f (x) , (3.3) Da+ f (x) = Ia+ I dxn a+ which coincide with (3.2) for n = 1. µ,0 µ Specially for ν = 0, Da+ f = Da+ f is a Riemann-Liouville fractional derivative of µ,1 µ order µ, and for ν = 1 it is a Caputo fractional derivative Da+ f = C Da+ f of order

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

14

µ. Applying the properties of Riemann-Liouville integral the relation (3.3) can be rewritten in the form: µ,ν  Da+ f (x)



=

ν(n−µ)

Ia+



n−(1−ν)(n−µ)

Da+ Zx

1 Γ (ν (n − µ))

=

(x − y)

  f (x)

ν(n−µ)−1

   µ+ν(n−µ) Da+ f (t) dt.

(3.4)

a

Our first result of this section is given in next theorem. Theorem 3.2. Let 0 < p ≤ q < ∞, or −∞ < q ≤ p < 0, f ∈ L1 [a, b] and the µ,ν fractional derivative operator Da+ of order n − 1 < µ < n and type 0 < ν ≤ 1, and let u be a weight function on (a, b). For each y ∈ (a, b), vˆ = vˆ(y) is defined on (a, b) by 

Zb

vˆ(y) := (ν (n − µ)) 

ν(n−µ)−1



(x − y) (x − a)ν(n−µ)

u(x)

 pq

 pq

dx < ∞.

(3.5)

y

If Φ is a non-negative convex function on the interval I ⊆ R and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the inequality 

Zb



 pq    µ+ν(n−µ) f (y) dy  vˆ(y)Φ Da+

a

Zb −

u(x)Φ

Γ (ν (n − µ) + 1)

q p

ν(n−µ)

(x − a)

a

q ≥ (ν (n − µ)) p

Zb

u(x) ν(n−µ)

a

(x − a)

Φ

! µ,ν  Da+ f (x)

dx

Γ (ν (n − µ) + 1)

q p −1

ν(n−µ)

(x − a) Zx ×

(x − y)

ν(n−µ)−1

! µ,ν  Da+ f (x)

rˆ(x, y)dydx (3.6)

a µ+ν(n−µ)

holds for all measurable functions Da+ is a non-negative function defined by

  µ+ν(n−µ) rˆ(x, y) = Φ Da+ f (y) − Φ !  Γ (ν (n − µ) + 1) µ,ν − ϕ D f (x) a+ ν(n−µ) (x − a)

f : (a, b) → R and rˆ : (a, b)×(a, b) → R

Γ (ν (n − µ) + 1) ν(n−µ)

µ,ν  Da+ f (x)

!

(x − a)  Γ (ν (n − µ) + 1) µ,ν D f (x) f (y) − . a+ ν(n−µ) (x − a) (3.7)

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTIONS...

15

If Φ is a non-negative monotone convex function on the interval I ⊆ R, and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ Int I, then the inequality  b  pq Z    µ+ν(n−µ)  vˆ(y)Φ Da+ f (y) dy  a

Zb −

u(x)Φ

q p

a

Γ (ν (n − µ) + 1) (x − a)

ν(n−µ)

! µ,ν  Da+ f (x)

Zb q u(x) q ≥ (ν (n − µ)) Φ p −1 p (x − a)ν(n−µ)

dx

Γ (ν (n − µ) + 1) ν(n−µ)

(x − a)

a

Zx

! µ,ν  Da+ f (x)

  Γ (ν (n − µ) + 1) µ+ν(n−µ) µ,ν  f (x) Da+ × sgn f (y) − Da+ ν(n−µ) (x − a) a ν(n−µ)−1 × (x − y) rˆ1 (x, y) dy dx µ+ν(n−µ)

holds for all measurable functions Da+ R is a non-negative function defined by rˆ1 (x, y) = Φ



µ+ν(n−µ) f Da+

−ϕ

×



ν(n−µ)

(x − a)

Γ (ν (n − µ) + 1) ν(n−µ)

(3.8)

f : (a, b) → R and rˆ1 : (a, b) × (a, b) →

Γ (ν (n − µ) + 1)

 (y) − Φ

!

! µ,ν  Da+ f (x)

! µ,ν  Da+ f (x)

(x − a) !   Γ (ν (n − µ) + 1) µ+ν(n−µ) µ,ν  Da+ f (y) − Da+ f (x) . (3.9) ν(n−µ) (x − a)

If Φ is a non-negative (monotone) concave function, then the order of terms on the left hand sides of (3.6) and (3.8) are reversed. Proof. Applying Theorem 1.3 with Ω1 = Ω2 = (a, b), dµ1 (x) = dx, dµ2 (y) = dy, ( (x−y)ν(n−µ)−1 ˆ y) = Γ(ν(n−µ)) , a ≤ y ≤ x; k(x, (3.10) 0, x < y ≤ b, we get ν(n−µ)

(x − a) Γ (ν (n − µ) + 1) and the integral operator Ak f (x) takes the form ˆ (x) = K

(Ak f )(x) =

Γ (ν (n − µ) + 1) ν(n−µ)

(x − a)

µ,ν  Da+ f (x)

and vˆ as in (3.5), we get inequalities (3.6) and (3.8).

(3.11)

(3.12) 

Remark 3.3. For p = q, Theorem 1.3 becomes [12, Theorem 3.6] and the convex function Φ need not to be non-negative.

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

16

Especially for the power function we obtain the next corollary. ˆ K ˆ and vˆ be as in Theorem 3.2. Let s ∈ R be such that Corollary 3.4. Let u, k, µ+ν(n−µ) s 6= 0, Da+ f : (a, b) → R be a non-negative measurable function (positive for s < 0) and

!s   s Γ (ν (n − µ) + 1) µ+ν(n−µ) µ,ν  f (y) − hs,k f (x, y) = Da+ Da+ f (x) ν(n−µ) (x − a) !s−1 Γ (ν (n − µ) + 1) Γ (ν (n − µ) + 1) µ,ν  µ,ν  −|s| · f (y) − D f (x) D f (x) , a+ a+ ν(n−µ) ν(n−µ) (x − a) (x − a) (3.13)

for x, y ∈ (a, b). If s ≥ 1 or s < 0, then the inequality



 pq

Zb vˆ(y)





µ+ν(n−µ)

Da+



f (y)

s

dy 

a

Zb −

u(x)

Γ (ν (n − µ) + 1) (x − a)

a

q ≥ (ν (n − µ)) p

Zb a

ν(n−µ)

Γ (ν (n − µ) + 1)

u(x) (x − a)

! qs p µ,ν  f (x) Da+

ν(n−µ)

(x − a) Zx

×

(x − y)

ν(n−µ)

ν(n−µ)−1

dx ! (q−p)s p µ,ν  Da+ f (x)

hs,k f (x, y) dy dx

(3.14)

a

holds. Let

Ns,k f (x, y) =



µ+ν(n−µ) Da+ f

−s

×



s (y) −

Γ (ν (n − µ) + 1)

Γ (ν (n − µ) + 1) ν(n−µ)

ν(n−µ)

(x − a)

!s µ,ν  Da+ f (x)

!s−1 µ,ν  Da+ f (x)

(x − a) !    Γ (ν (n − µ) + 1) µ+ν(n−µ) µ,ν  Da+ f (y) − Da+ f (x) (3.15) ν(n−µ) (x − a)

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTIONS...

17

for x, y ∈ (a, b). If s ≥ 1 or s < 0, then the inequality 

Zb



 pq   s µ+ν(n−µ) v(y) Da+ f (y) dy 

a

Zb −

u(x)

(x − a)

a

Γ (ν (n − µ) + 1) ν(n−µ)

(x − a)

a

×

sgn



µ+ν(n−µ) Da+ f



µ,ν  Da+ f (x)

ν(n−µ)

Zb u(x) q ≥ (ν (n − µ)) p (x − a)ν(n−µ) Zx

! qs p

Γ (ν (n − µ) + 1)



(y) −

dx ! (q−p)s p µ,ν  Da+ f (x)

!

Γ (ν (n − µ) + 1)

µ,ν  Da+ f (x)

ν(n−µ)

(x − a)

a

ν(n−µ)−1

× (x − y)

Ns,k f (x, y) dy dx (3.16)

holds. If s ∈ (0, 1), then inequalities corresponding to (3.14) and (3.16) hold with reverse order of terms on the left hand sides. Next we give the results for one dimensional settings involving Hilfer fractional derivative. Theorem 3.5. Let 0 < p ≤ q < ∞, or −∞ < q ≤ p < 0, f ∈ L1 [a, b] and the µ,ν fractional derivative operator Da+ of order n − 1 < µ < n and type 0 < ν ≤ 1, and let u be a weight function. For each y ∈ (0, b) let the function w ˆ : (0, b) → R be defined by  b  pq ! pq Z ν(n−µ)−1 (x − y) dx w(y) ˆ = y (ν (n − µ))  (3.17) u(x)  . ν(n−µ) x (x − a) y

If Φ is a non-negative convex function on the interval I ⊆ R and ϕ : I → R satisfies that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the following inequality  b  pq Z    dy µ+ν(n−µ)  w(y)Φ  ˆ Da+ f (y) y 0

Zb −

u(x)Φ

Γ (ν (n − µ) + 1)

q p

(x − a)

0

q ≥ (ν (n − µ)) p

Zb

u(x) ν(n−µ)

0

(x − a)

Φ

ν(n−µ)

q p −1

! µ,ν  Da+ f (x)

Γ (ν (n − µ) + 1) ν(n−µ)

(x − a) Zx ×

ν(n−µ)−1

(x − y) 0

dx x ! µ,ν  Da+ f (x)

rˆ(x, y)dy

dx x

(3.18)

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

18

µ+ν(n−µ)

holds for all measurable functions Da+ f : (0, b) → R and rˆ(x, y) is defined by (3.7). If Φ is a non-negative monotone convex function on the interval I ⊆ R and ϕ : I → R satisfies that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the following inequality  b  pq Z    dy µ+ν(n−µ)   w(y)Φ ˆ Da+ f (y) y 0

Zb −

u(x)Φ

q p

Γ (ν (n − µ) + 1) ν(n−µ)

(x − a) Zb q u(x) q Φ p −1 ≥ (ν (n − µ)) p (x − a)ν(n−µ)

! µ,ν  Da+ f (x)

dx x

0

Γ (ν (n − µ) + 1) (x − a)

0

Zx sgn



µ+ν(n−µ) Da+ f



0

ν(n−µ)

! µ,ν  Da+ f (x)

!  Γ (ν (n − µ) + 1) µ,ν  Da+ f (x) (y) − ν(n−µ) (x − a) ν(n−µ)−1

× (x − y) µ+ν(n−µ)

holds for all measurable functions Da+ by (3.9).

dx rˆ1 (x, y)dy (3.19) x

f : (0, b) → R and rˆ1 (x, y) is defined

Our next result reads: Theorem 3.6. Let 0 < p ≤ q < ∞, or −∞ < q ≤ p < 0, α, β, γ, δ, p, q be as in Definition 2.2 and let u be a weight function defined on (a, b). For each y ∈ (a, b), define a function p  b ! pq  q Z µ+ν(n−µ) ν(n−µ)−1 f2 )(y)  (Da+ (x − y) sˆ(y) := dx < ∞. u(x) (ν (n − µ)) (Daµ,ν + f2 ) (x) y

If Φ is a non-negative convex function on the interval I ⊆ R and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the inequality 

Zb sˆ(y)Φ

 a

  pq Daµ,ν f (x) 1 +  dx dy  − u(x)Φ  µ,ν (Da+ f2 ) (x) a    b µ,ν Z D f (x) 1 a q + q u(x)  ≥ Φ p −1  p (Daµ,ν (Daµ,ν + f2 ) (x) + f2 ) (x)

µ+ν(n−µ) (Da+ f1 )(y) µ+ν(n−µ) (Da+ f2 )(y)

!

 pq

Zb



a

Zx ×

ν(n−µ)−1

(x − y) ˆ y)dydx (3.20) (Daµ+ν(n−µ) f2 )(y)d(x, + Γ (ν (n − µ))

a µ+ν(n−µ)

holds for all measurable functions Da+ µ+ν(n−µ) (Da f1 )(y) + µ+ν(n−µ) (Da+ f2 )(y)

fi : (a, b) → R, (i = 1, 2) such that

∈ I for all y ∈ (a, b), and dˆ : (a, b) × (a, b) → R is a non-negative

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTIONS...

19

function defined by    ! µ,ν µ+ν(n−µ) D f (x) 1 a + (D f )(y) a+ 1 ˆ   d(x, y) = Φ −Φ µ,ν µ+ν(n−µ) (Da+ f2 ) (x) (Da+ f2 )(y)      µ+ν(n−µ) µ,ν Daµ,ν f (x) D f (x) 1 1 a + + (Da+ f1 )(y) . (3.21)  − ϕ  − µ,ν µ,ν µ+ν(n−µ) (Da+ f2 ) (x) (Da+ (Da+ f2 ) (x) f2 )(y) If Φ is a non-negative monotone convex function on the interval I ⊆ R, and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ Int I, then the inequality q   b   !  p Zb µ,ν Z µ+ν(n−µ) D f (x) 1 a q + (D f )(y) a+ 1  sˆ(y)Φ  dx dy  − u(x)Φ p  µ+ν(n−µ) (Daµ,ν + f2 ) (x) (Da+ f2 )(y) a a     x    Zb µ,ν µ,ν Z µ+ν(n−µ) D f (x) D f (x) 1 1 a a q + + f1 )(y) q u(x)  sgn  (Da+  − ≥ Φ p −1  µ,ν µ+ν(n−µ) p (Daµ,ν (Daµ,ν (D f2 ) (x) a + f2 ) (x) + f2 ) (x) + f2 )(y) (Da+ a a ν(n−µ)−1 (x − y) µ+ν(n−µ) (Da+ f2 )(y)dˆ1 (x, y) dy dx (3.22) Γ (ν (n − µ)) µ+ν(n−µ)

holds for all measurable functions Da+ µ+ν(n−µ) (Da f1 )(y) + µ+ν(n−µ) (Da+ f2 )(y)

fi : (a, b) → R, (i = 1, 2) such that

∈ I, for all y ∈ (a, b) and dˆ1 : (a, b) × (a, b) → R is a non-negative

function defined by   f Daµ,ν 1 (x) +  − Φ (Daµ,ν + f2 ) (x)        µ,ν µ+ν(n−µ) i D f (x) Daµ,ν f1 (x) 1 a + + f )(y) (D a+ 1    − · − ϕ . (3.23) µ+ν(n−µ) (Daµ,ν (Daµ,ν + f2 ) (x) + f2 ) (x) f2 )(y) (Da+

h dˆ1 (x, y) = Φ

µ+ν(n−µ) (Da+ f1 )(y) µ+ν(n−µ) f2 )(y) (Da+



!

If Φ is a non-negative (monotone) concave function, then the order of terms on the left hand sides of (3.20) and (3.22) are reversed. Proof. Applying Theorem 2.8 with Ω1 = Ω2 = (a, b), dµ1 (x) = dx, dµ2 (t) =   µ+ν(n−µ) µ,ν dt, g1 (x) = Daµ,ν f (x), g f1 )(y), and 1 2 (x) = Da+ f2 (x), f1 (y) = (Da+ + µ+ν(n−µ)

f2 (y) = (Da+ and (3.22).

f2 )(y), k(x, y) defined by (3.10), we obtain inequalities (3.20) 

Remark 3.7. Since the right hand sides of the inequalities (3.20) and (3.22) are non-negative, we obtain the following inequality q     b ! p µ,ν Z Zb µ+ν(n−µ) D f (x) a+ 1 q f1 )(y)  dx ≤  v(y)Φ (Da+ u(x)Φ p  dy  µ,ν µ+ν(n−µ) (Da+ f2 ) (x) (Da+ f2 )(y) a

a

(3.24) Particularly for p = q, we obtain the inequality given in [12, Theorem 3.4].

20

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

Remark 3.8. If Φ is strictly convex on I and

µ+ν(n−µ) (Da f1 )(y) + µ+ν(n−µ)

(Da+

f2 )(y)

is non-constant, then

the inequality given in (3.24) is strict. Now we present some Hardy-type inequalities for Hilfer fractional derivative. We continue our analysis about improvements by taking the non-negative difference of the left hand side and the right hand side of the inequalities given in (1.10) and (1.12) as:  pq  b Z Zb sq s   v(y)f (y)dµ2 (y) ρ(s) = − u(x)((Ak f )(x)) p dµ1 (x) a

q − p

a

Zb

q u(x) ((Ak f )(x))s( p −1) K(x)

a

Zb k(x, y)rp,k f (x, y)dydx,

(3.25)

a

where rp,k f (x, y) is defined by (1.9) and 

Zb

 pq

Zb

s

v(y)f (y)dy  −

π(s) =  a

sq

u(x) ((Ak f )(x)) p dx a

Zb Zb q u(x) pq −1 − Φ ((Ak f )(x)) sgn(f (y) − (Ak f )(x))k(x, y) Mp,k f (x, y) dy dx , p K(x) a

a

(3.26) where Mp,k f (x, y) is defined by (1.11). Theorem 3.9. Let 0 < p ≤ q < ∞, s ≥ 1, ν(n − µ) ≥ 1 − pq , f ∈ L1 [a, b] and the µ,ν fractional derivative operator Da+ of order n − 1 < µ < n and type 0 < ν ≤ 1. µ+ν(n−µ) µ,ν f and Da+ f the following inequality Then for non-negative functions Da+ holds true: 0 ≤ ρ(s) ≤ H(s) − M (s) ≤ H(s), where q p

ρ(s) =

(ν (n − µ)) (ν (n − µ) − 1) pq + 1

− (Γ(ν (n − µ) + 1))

sq p

 b  pq Z p  (b − y)ν(n−µ)−1+ q ((Daµ+ν(n−µ) f )(y))s dy  +

a

Zb (x − a)

(ν(n−µ))q(1−s) p

 sq µ,ν (Da+ f )(x) p dx − M (s),

a

q

q(ν (n − µ))(Γ(ν (n − µ) + 1))s( p −1) M (s) = p

Zb (x − a)

(ν(n−µ))(q−p)(1−s) p

s( q −1) µ,ν (Da+ f )(x) p

a

Zx × a

hp,k f (x, y)(x − y)ν(n−µ)−1 dy dx,

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTIONS...

21

where hp,k f (x, y) is defined by (3.13) and q

H(s) = (b − a)(ν(n−µ)) p (1−s) 

q((ν(n−µ))s−1)+p p

q p

 (ν (n − µ)) (b − a) × (ν (n − µ) − 1) pq + 1

 b  pq Z  (Daµ+ν(n−µ) f (y))s dy  +

a

− (Γ(ν (n − µ) + 1))

sq p

Zb

 sq p

µ,ν (Da+ f (x)) dx . (3.27)

a

Moreover, 0 ≤ π(s) ≤ H(s) − B(s) ≤ H(s), where 

q p

(ν (n − µ))  (ν (n − µ) − 1) pq + 1

π(s) =

sq

Zb

 pq p

(b − y)ν(n−µ)−1+ q ((Daµ+ν(n−µ) f )(y))s dy  +

a

Zb

− (Γ(ν (n − µ) + 1)) p

(x − a)

 sq µ,ν (Da+ f )(x) p dx − B(s),

(ν(n−µ))q(1−s) p

a q

B(s) =

q(ν (n − µ))(Γ(ν (n − µ) + 1))s( p −1) p Zb Zx   Γ(ν (n − µ) + 1) µ,ν µ+ν(n−µ) × (Da+ f )(x) sgn (Da+ f )(y) − (x − a)ν(n−µ) a

× (x − a)

a

(ν(n−µ))(q−p)(1−s) p

q

s( −1) µ,ν (Da+ f )(x) p

(x − y)

ν(n−µ)−1

Np,k f (x, y)dy dx

and Np,k f (x, y) is defined by (3.15) and H(s) is defined by (3.27). Proof. Applying Theorem 1.3 with Ω1 = Ω2 = (a, b), dµ2 (x) = dx, dµ2 (y) = dy, ( (x−y)ν(n−µ)−1 a < y ≤ x; Γ(ν(n−µ)) , k(x, y) = 0, x < y ≤ b, we get that K(x) =

(x−a)ν(n−µ) Γ(ν(n−µ)+1)

and (Ak f )(x) =

Γ(ν(n−µ)+1) µ,ν (Da+ f )(x). (x−a)ν(n−µ)

Replace f

(ν(n−µ))q p

µ+ν(n−µ) Da+ f.

by For the particular weight function u(x) = (x−a) , x ∈ (a, b) p q ν(n−µ)−1+ p q q we get v(y) = ((ν (n − µ))(b − y) )/(((ν (n − µ) − 1) p + 1) ) and then (3.25) takes the form  b  pq q Z p (ν (n − µ)) p  (b − y)ν(n−µ)−1+ q ((Daµ+ν(n−µ) f )(y))s dy  ρ(s) = q + (ν (n − µ) − 1) p + 1 a

sq

Zb

− (Γ(ν (n − µ) + 1)) p

(x − a) a

(ν(n−µ))q(1−s) p

 sq µ,ν (Da+ f )(x) p dx − M (s).

22

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

Since

(ν(n−µ))q (1 p

− s) ≤ 0 and M (s) ≥ 0, we obtain that q (ν(n−µ)−1) p +1

q p

(ν (n − µ)) (b − a) (ν (n − µ) − 1) pq + 1

ρ(s) ≤

−(b − a)

(ν(n−µ))q (1−s) p

 b  pq Z  ((Daµ+ν(n−µ) f )(y))s dy  +

a

(Γ(ν (n − µ) + 1))

sq p

Zb

 sq µ,ν (Da+ f )(x) p dx − M (s)

a

=

H(s) − M (s)



H(s).

Moreover, (3.26) takes the form q p



(ν (n − µ))  (ν (n − µ) − 1) pq + 1

π(s) =

− (Γ(ν (n − µ) + 1))

sq p

Zb

 pq p

f )(y))s dy  (b − y)ν(n−µ)−1+ q ((Daµ+ν(n−µ) +

a

Zb (x − a)

 sq µ,ν (Da+ f )(x) p dx − B(s).

(ν(n−µ))q(1−s) p

a

Since

(ν(n−µ))q (1 p

− s) ≤ 0 and B(s) ≥ 0, we obtain that q p

π(s) ≤

q (ν(n−µ)−1) p +1

(ν (n − µ)) (b − a) (ν (n − µ) − 1) pq + 1 −(b − a)

(ν(n−µ))q (1−s) p

 b  pq Z  ((Daµ+ν(n−µ) f )(y))s dy  +

a

(Γ(ν (n − µ) + 1))

sq p

Zb

 sq µ,ν f )(x) p dx − B(s) (Da+

a

= H(s) − B(s) ≤ H(s). The proof is complete.



Remark 3.10. Since H(s) > 0 in Theorem 3.9, then after an elementary calculation we get the inequality given in [11, Remark 2.6]. Acknowledgements. We thank the careful referee for several good suggestions which have improved the final version of this paper. The research of author Josip Peˇcari´c has been fully supported by Croatian Science Foundation under the project 5435. The author Zivorad Tomovski acknowledges NWO grant-No. 040.11.629, Department of Applied Mathematics, TU Delft. References [1] D. Baleanu, O. G. Mustafa and R. P. Agarwal, Asymptotically linear solutions for some linear fractional differential equations, Abstr. Appl. Anal., Vol.(2010), Article ID 865139, 8. [2] D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos, Boston: World Scientific, 2012.

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING CONVEX FUNCTIONS...

23

ˇ zmeˇsija, K. Kruli´ [3] A. Ciˇ c and J. Peˇ cari´ c, Some new refined Hardy-type inequalities with kernels, J. Math. Inequal., 4, (2010), no. 4, 481-503. ˇ zmeˇsija, K. Kruli´ [4] A. Ciˇ c and J. Peˇ cari´ c, A new class of general refined Hardy type inequalities with kernels, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., 17(515) (2013), 53-80. [5] N. Elezovi´ c, K. Kruli´ c and J. Peˇ cari´ c, Bounds for Hardy type differences, Acta Math. Sinica(Eng Ser), 27 (2011), no.4, 671-684. ˇ Tomovski, Operational method for solution of fractional differential [6] R. Hilfer, Y. Luchko and Z. equations with generalized Riemann-Liouville fractional derivative, Frac. Calc. Appl. Anal., 12 (2009), no. 3, 299-318. [7] S. Iqbal, K. Kruli´ c and J. Peˇ cari´ c, On an inequality of H. G. Hardy, J. Inequal. Appl., vol. 2010. Artical ID 264347, 2010. [8] S. Iqbal, K. Kruli´ c and J. Peˇ cari´ c, On an inequality for convex function with some applications on fractional derivatives and fractional integrals, J. Math. Inequal., 5 (2011), no. 2, 219-230. [9] S. Iqbal, K. Kruli´ c Himmelreich and J. Peˇ cari´ c, Weighted Hardy type inequalities for monotone convex functions with some applications, Frac. Differ. Calc., 3, (2013), no.1, 31-53. [10] S. Iqbal, K. Kruli´ c and J. Peˇ cari´ c, Refinements of Hardy-type integral inequalities with kernels, Punjab Univ. J. Math., 48 (2016), no.1, pp. 19-28. [11] S. Iqbal, J. Peˇ cari´ c, M. Samraiz and Z. Tomovski, Hardy-type inequalities for generalized fractional integral operators, Tbilisi Math. J., 10 (2017), no.1, 1-16.. [12] S. Iqbal, J. Peˇ cari´ c, M. Samraiz and Z. Tomovski On some Hardy-type inequalities for fractional calculus operator, Banach J. Math. Anal. 11 (2017), no. 2, 438-457. [13] S. Kaijser, L. Nikolova, L-E. Persson, and A. Wedestig, Hardy type inequalities via convexity, Math. Inequal. Appl. 8 (2005), no.3, 403–417. [14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, New York-London, 2006. [15] V. Kokilashvili, A. Meskhi and L. E. Persson, Weighted Norm Inequalities for Integral Transformations with Product Kernels, Nova Science Publishers, Inc., New York, 2009. [16] K. Kruli´ c, J. Peˇ cari´ c and L. E. Persson, Some new Hardy type inequalities with general kernels, Math. Inequal. Appl., 12, (2009), 473-485. [17] K. Kruli´ c, J. Peˇ cari´ c and D. Pokaz, Inequalities of Hardy and Jensen, Zagreb, Element, (2013). [18] A. Kufner, L. Maligranda and L. E. Persson, The Hardy Inequality-About its History and Related Results, Vydavatelsky Servis Publishing House, Pilsen, 2007. [19] A. Kufner, L. E. Persson and N. Samko, Hary-type iequalities with kernels: the current status and some new results, Math. Nachr. 290 (2017), no. 1, 57-65. [20] A. Kufner, L. E. Persson and N. Samko, Weighted Inequalities of Hardy Type, Second Edition, World Scientific, 2017. [21] G. M. Mittag-Leffler, Sur la nouvelle fonction, C.R. Acad. Sci. Paris. 137 (1903), 554-558. [22] C. Niculescu and L. E. Persson, Convex Functions and their Applications. A Contemporary Approach, CMC Books in Mathematics Springer, New York, 2006. [23] K. B. Oldham, and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [24] J. Peˇ cari´ c, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Inc. 1992. [25] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [26] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7-15. [27] T. O. Salim, Some properties relating to the generalized Mittag-Leffler function, Adv. Appl. Math. Anal. 4 (2009), 21-30. [28] T. O. Salim and A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with fractional calculus, JFCA. 5 (2012), 1-13. [29] A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl. 336 (2007), 797-811. [30] H. M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernal, Appl. Math. Comput. 211 (2009), no.1, 198-210.

24

ˇ ´ 2 , LARS-ERIK PERSSON3 , AND ZIVORAD TOMOVSKI4 SAJID IQBAL1 , J. PECARI C

ˇ Tomovski, R. Hilfer and H. M. Srivastava, Fractional and Operational Calculus with Gener[31] Z. alized Fractional Derivative Operators and Mittag-Leffler Type Functions, Integral Transforms Spec. Funct. 21 (2010), no. 11, 797-814. [32] A. Wiman, Uber den fundamental satz in der theori der functionen, Acta Math. 29 (1905), 191-201. 1-Department of Mathematics, University of Sargodha (Sub-Campus Mianwali), Mianwali, Pakistan E-mail address: sajid [email protected]; [email protected] ´a 2-Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovic 28a, 10000 Zagreb, Croatia E-mail address: [email protected] 3-Department of Mathematics, Lule˚ a University of Technology, SE-971 87, Lule˚ a, Sweden and UIT The Artic University of Norway, P. O. Box 385, N-8505, Narvik, Norway E-mail address: [email protected] 4-Faculty of Mathematics and Natural Sciences, Gazi Baba bb, 1000 Skopje, Macedonia E-mail address: [email protected]