ON A NEW CLASS OF HARDY-TYPE INEQUALITIES WITH FRACTIONAL INTEGRALS AND FRACTIONAL DERIVATIVES

Transcription

1 RAD HAZU. MATEMATIČKE ZNANOSTI Vol. 8 = 59 (204): 9-06 ON A NEW CLASS OF HARDY-TYPE INEQUALITIES WITH FRACTIONAL INTEGRALS AND FRACTIONAL DERIVATIVES Sjid Ibl, Kristin Krulić Himmelreich nd Josi Pečrić Abstrct. This er is devoted to new clss of generl weighted Hrd-te ineulities for rbitrr convex functions with some lictions to different te of frctionl integrls nd frctionl derivtives.. Introduction In [5], A. Čižmešij et l. recentl introduced new clss of generl Hrdte ineulities. Let (Ω, Σ, µ ) nd (Ω 2, Σ 2, µ 2 ) be mesure sces with σ-finite mesures nd A k be n integrl oertor defined b (.) A k f(x) := k(x, )f()dµ 2 (), K(x) Ω 2 where k : Ω Ω 2 R is mesurble nd nonnegtive kernel, f is mesurble function on Ω 2, nd (.2) K(x) := k(x, )dµ 2 (), x Ω. Ω 2 Throughout the er, we consider tht K(x) > 0.e. on Ω. Theorem.. Let < <. Let (Ω, Σ, µ ) nd (Ω 2, Σ 2, µ 2 ) be mesure sces with σ-finite mesures, u be weight function on Ω, v be mesurble µ 2.e. ositive function on Ω 2, k be non-negtive mesurble function on Ω Ω 2, nd K be defined on Ω b (.2). Let K(x) > 0 for ll 200 Mthemtics Subject Clssifiction. 26D0, 26D5, 26A33. Ke words nd hrses. Hrd-te ineulities, convex function, kernel, frctionl derivtive, frctionl integrls. 9

2 92 S. IQBAL, K. KRULIĆ HIMMELREICH AND J. PEČARIĆ ( ) x Ω nd let the function x k(x,) K(x) be integrble on Ω for ech fixed Ω 2. Suose tht Φ : I [0, ) is bijective convex function on n intervl I R. If there exist rel rmeter s (, ) nd ositive mesurble function V : Ω 2 R such tht A(s, V ) = F(V, v) su V s Ω 2 () Ω ( k(x, ) K(x) ) dµ (x) where F(V, v) = V (s ) ()v ()dµ 2 (), Ω 2 then there is ositive rel constnt C, such tht the ineulit (.3) Φ (A k f(x)) dµ (x) C v()φ (f()) dµ 2 () Ω Ω 2 holds for ll mesurble functions f : Ω 2 R with vlues in I nd A k f be defined on Ω b (.). Moreover, if C is the smllest constnt for (.3) to hold, then <s< Here, our im is to construct new results relted to this new clss of Hrd-te ineulities for frctionl integrls of function with resect to nother incresing function, Riemnn-Liouville frctionl integrls, Hdmrd-te frctionl integrls, Cnvti-te frctionl derivtive, Cuto frctionl derivtive, Erdeli-Kóber frctionl integrls. Mn uthors gve imrovements nd generliztions of Hrd-te ineulities for convex functions s well s for suerudrtic functions, (see [6], [7], [0], [] [2], [3], [4], [6], [7]). We lso recll imortnt result of G. H. Hrd. Let [, b], < < b < be finite intervl on rel xis R, nd, then (.4) I α +f K f, I α b f K f holds, where K = (b )α Γ(α + ),

3 ON A NEW CLASS OF HARDY-TYPE INEQUALITIES 93 I α f nd I α + b f of order α > 0 denote the Riemnn-Liouville frctionl integrls defined b nd I α +f(x) = Γ(α) I α b f(x) = Γ(α) x x (x ) α f()d, (x > ) ( x) α f()d, (x < b), where Γ is the Gmm function, i.e. Γ(α) = 0 e t t α dt. If 0 < α < nd < < α, then the oertors Iα nd I α + b re bounded from L (, b) into L (, b), where = α. This is known s Hrd- Littlewood theorem. Lter on, the ineulit (.4) is discussed nd roved b S. G. Smko et l. in [8, Theorem 2.6]. For detils we refer [5, Remrk 2.](lso see [8]). G. H. Hrd roved the ineulit (.4) involving left-sided frctionl integrl in one of his initil er, see [9]. The clcultion for the constnt K is hidden inside the roof. Throughout this er, ll mesures re ssumed to be ositive, ll functions re ssumed to be ositive nd mesurble nd exressions of the form 0, nd 0 0 re tken to be eul to zero. Moreover, b weight u = we men non-negtive mesurble function on the ctul intervl or more generl set. For rel rmeter 0, b we denote its conjugte exonent =, tht is + =. The er is orgnized in the following w: After introduction, in Section 2, we construct nd discuss new clss of generlized ineulities of Hrdte using different kinds of frctionl integrls nd frctionl derivtives. 2. The Min Results Let us recll some fcts bout frctionl derivtives needed in the seuel, for more detils see e.g. [], [8]. Let 0 < < b. B C m ([, b]) we denote the sce of ll functions on [, b] which hve continuous derivtives u to order m, nd AC([, b]) is the sce of ll bsolutel continuous functions on [, b]. B AC m ([, b]) we denote the sce of ll functions g C m ([, b]) with g (m ) AC([, b]). For n α R we denote b [α] the integrl rt of α (the integer k stisfing k α < k + ) nd α is the ceiling of α (min{n N, n α}). B L (, b)

4 94 S. IQBAL, K. KRULIĆ HIMMELREICH AND J. PEČARIĆ we denote the sce of ll functions integrble on the intervl (, b), nd b L (, b) the set of ll functions mesurble nd essentill bounded on (, b). Clerl, L (, b) L (, b). We strt with definitions nd some roerties of the frctionl integrls of function f with resect to given function g. For detils see e.g. [5,. 99]. Let (, b), < b be finite or inite intervl of the rel line R nd α > 0. Also let g be n incresing function on (, b) nd g be continuous function on (, b). The left- nd right-sided frctionl integrls of function f with resect to nother function g in [, b] re given b (I+;gf)(x) α = x g (t)f(t)dt Γ(α) [g(x) g(t)] α, x > nd (Ib ;gf)(x) α = g (t)f(t)dt, x < b, Γ(α) [g(t) g(x)] α x resectivel. Our first result dels with frctionl integrl of f with resect to nother incresing function g is given. Theorem 2.. Let < α > 0, u be weight function on (, b), v be.e. ositive function on (, b), g be incresing function on (, b] such tht g be continuous on (, b), I α +;gf denotes the left sided frctionl integrl of f with resect to nother incresing function g. Suose tht Φ : I [0, ) is bijective convex function on n intervl I R. If there exist rel rmeter s (, ) nd V : (, b) R is ositive mesurble function such tht (2.) A(s, V ) = V (s ) ()v ()d ( su V s () α g ()(g(x) g()) α ) (,b) (g(x) g()) α dx

5 ON A NEW CLASS OF HARDY-TYPE INEQUALITIES 95 then there exists ositive constnt C, such tht the ineulit (2.2) [ ( )] Γ(α + ) Φ (g(x) g()) α Iα + ;gf(x) dx C v()φ (f())d. holds. Moreover, if C is the smllest constnt for (2.2) to hold, then <s< Proof. Aling Theorem. with Ω = Ω 2 = (, b), dµ (x) = dx, dµ 2 () = d, { k(x, ) = g () Γ(α)(g(x) g()), α < x ; 0, x < b, we get tht K(x) = Γ(α+) (g(x) g())α Γ(α+), A k f(x) = (g(x) g()) I α α f(x), +;g nd the ineulit given in (.3) reduces to (2.2). This comlete the roof. If we choose the function Φ : R + R defined b Φ(x) = x t, t, then the following result is obtined. Corollr 2.2. Let < t, α > 0, u be weight function on (, b), v be.e. ositive function on (, b), g be incresing function on (, b] such tht g be continuous on (, b), I α +;gf denotes the left sided frctionl integrl of f with resect to nother incresing function g. If there exist rel rmeter s (, ) nd V : (, b) R is ositive mesurble function such tht A(s, V ) = V (s ) ()v ()d ( su V s () α g ()(g(x) g()) α ) (,b) (g(x) g()) α dx then there exists ositive constnt C, such tht the ineulit (2.3) ( ) t Γ(α + ) (g(x) g()) α Iα +;gf(x) dx C v()(f()) t d

6 96 S. IQBAL, K. KRULIĆ HIMMELREICH AND J. PEČARIĆ holds. Moreover, if C is the smllest constnt for (2.3) to hold, then <s< Here, we give secil cse for the Riemmn-Liouville frctionl integrl. If g(x) = x, then I α +;xf(x) reduces to I α + f(x) left-sided Riemnn Liouville frctionl integrl, so the following result follows. Corollr 2.3. Let < α > 0, t, u be weight function on (, b), v be.e. ositive function on (, b), nd I α + f denotes the left-sided Riemnn-Liouville frctionl integrl of f. If there exist rel rmeter s (, ) nd V : (, b) R is ositive mesurble function such tht (2.4) A(s, V ) = V (s ) ()v ()d ( ) su V s () α (x ) α (,b) (x ) α dx then there exists ositive constnt C, such tht the ineulit (2.5) ( ) t Γ(α + ) (x ) α Iα +f(x) dx C v()f t ()d holds. Moreover, if C is the smllest constnt for (2.5) to hold, then <s< Remrk 2.4. If we tke g(x) = log x, then I α +;gf(x) reduces to J α + f(x) left-sided Hdmrd-te frctionl integrl tht is defined for α > 0 b (J α + f)(x) = Γ(α) x ( log x ) α f()d, x >.

7 ON A NEW CLASS OF HARDY-TYPE INEQUALITIES 97 If there exist rel rmeter s (, ) nd ositive mesurble function V : (, b) R with < t, such tht A(s, V ) = V (s ) ()v ()d ( ) su V s () α (log x log ) α (,b) (log x log ) α dx then there exists ositive constnt C, such tht the ineulit (2.6) ( ) t Γ(α + ) (log x log ) α Jα + f(x) dx C v()f t ()d holds. Moreover, if C is the smllest constnt for (2.6) to hold, then <s< Next we give result with resect to the generlized Riemnn-Liouville frctionl derivtive. Let us recll the definition, for detils see [2]. Let α > 0 nd n = [α] + where [ ] is the integrl rt nd we define the generlized Riemnn-Liouville frctionl derivtive of f of order α b ( ) n x (D α f)(x) = d (x ) n α f() d. Γ(n α) dx In ddition, we stiulte D 0 f := f =: I0 f, I α f := Dα f if α > 0. If α N then D α f = dα f dx α, the ordinr α-order derivtive. The sce I α (L(, b)) is defined s the set of ll functions f on [, b] of the form f = I α ϕ for some ϕ L(, b), [8, Chter, Definition 2.3]. According to Theorem 2.3 in [8,. 43], the ltter chrcteriztion is euivlent to the condition (2.7) I n α f AC n [, b],

8 98 S. IQBAL, K. KRULIĆ HIMMELREICH AND J. PEČARIĆ d j dx j In α f() = 0, j = 0,,..., n. A function f L(, b) stisfing (2.7) is sid to hve n integrble frctionl derivtive D α f, [8, Chter, Definition 2.4]. The following lemm hel us to rove the next result. For detils see [2]. Lemm 2.5. Let β > α 0, n = [β] +, m = [α] +. Identit D α f(x) = Γ(β α) is vlid if one of the following conditions holds: x (x ) β α D β f() d, x [, b]. (i) f I β (L(, b)). (ii) I n β f AC n [, b] nd D β k f() = 0 for k =,... n. (iii) D β k f C[, b] for k =,..., n, D β f AC[, b] nd D β k f() = 0 for k =,... n. (iv) f AC n [, b], D βf L(, b), Dα f L(, b), β α / N, Dβ k f() = 0 for k =,..., n nd D α k f() = 0 for k =,..., m. (v) f AC n [, b], Df β L(, b), D α f L(, b), β α = l N, D β k f() = 0 for k =,..., l. (vi) f AC n [, b], Df β L(, b), D α f L(, b) nd f() = f () = = f (n 2) () = 0. (vii) f AC n [, b], D βf L(, b), Dα f L(, b), β / N nd Dβ f is bounded in neighborhood of t =. Corollr 2.6. Let < t, β > α 0, u be weight function on (, b), v be.e. ositive function on (, b), D α f denotes the generlized Riemnn-Liouville frctionl derivtive of f nd let the ssumtions of the Lemm 2.5 be stisfied. If there exist rel rmeter s (, ) nd ositive mesurble function V : (, b) R such tht A(s, V ) = V (s ) ()v ()d ( ) su V s () (β α)(x ) β α (,b) (x ) β α dx then there exists ositive constnt C, such tht the ineulit (2.8) ( ) t Γ(β α + ) (x ) β α Dα f(x) dx C v()(df()) β t d

9 ON A NEW CLASS OF HARDY-TYPE INEQUALITIES 99 holds. Moreover, if C is the smllest constnt for (2.8) to hold, then <s< Proof. Aling Theorem. with Ω = Ω 2 = (, b), dµ (x) = dx, dµ 2 () = d, { (x ) β α k(x, ) = Γ(β α), < x ; 0, x < b, we get tht K(x) = (x )β α Γ(β α+) nd A kf(x) = Γ(β α+) (x ) D α β α f(x). Relce f b Df. β Then the ineulit (.3) becomes (2.9) [ ( )] Γ(β α + ) Φ (x ) β α Dα f(x) dx C v()φ ( Df() ) β d. For t, the function Φ : R + R be defined b Φ(x) = x t, then (2.9) becomes (2.8). Now we define Cnvti-te frctionl derivtive (ν frctionl derivtive of f), for detils see [] nd [3]. We consider C ν ([, b]) = {f Cn ([, b]) : I ν f (n) C ([, b])}, ν > 0, n = [ν], [.] is the integrl rt, nd ν = ν n, 0 ν <. For f C ν ([, b]), the Cnvti-ν frctionl derivtive of f is defined b where D = d/dx. D ν f = DI ν f (n), Lemm 2.7. Let ν > γ > 0, n = [ν], m = [γ]. Let f C ν ([, b]), be such tht f (i) () = 0, i = m, m +,..., n. Then (i) f C γ ([, b]) (ii) (D γ f)(x) = Γ(ν γ) for ever x [, b]. x (x t) ν γ (Df)(t)dt, ν In the following Corollr, we construct new ineulit for the Cnvtite frctionl derivtive. Corollr 2.8. Let < t, u be weight function on (, b), v be.e. ositive function on (, b), nd let the ssumtions in Lemm

10 00 S. IQBAL, K. KRULIĆ HIMMELREICH AND J. PEČARIĆ 2.7 be stisfied. If there exist rel rmeter s (, ) nd V : (, b) R is ositive mesurble function such tht A(s, V ) = V (s ) ()v ()d ( ) su V s () (ν γ)(x ) ν γ (,b) (x ) ν γ dx then there exists ositive constnt C, such tht the ineulit (2.0) ( ) t Γ(ν γ + ) (x ) ν γ Dγ f(x) dx C v()(df()) ν t d holds. Moreover, if C is the smllest constnt for (2.0) to hold, then <s< Proof. Aling Theorem. with Ω = Ω 2 = (, b), dµ (x) = dx, dµ 2 () = d, { (x ) ν γ k(x, ) = Γ(ν γ), < x ; 0, x < b, we get tht K(x) = (x )ν γ Γ(ν γ+) nd A kf(x) = Γ(ν γ+) (x ) Df(x). γ Relce f b ν γ D ν f. Then the ineulit given in (.3) become (2.) [ ( )] Γ(ν γ + ) Φ (x ) ν γ Dγ f(x) dx C v()φ (Df()) ν d. If we tke Φ(x) = x t, t, x R +, then (2.) becomes (2.0). Next, we give the result for Cuto frctionl derivtive, for detils see [,. 449]. The Cuto frctionl derivtive is defined s: Let α 0, n = α, g AC n ([, b]). The Cuto frctionl derivtive is given b D g(t) α x g (n) () = d, Γ(n α) (x ) α n+

11 ON A NEW CLASS OF HARDY-TYPE INEQUALITIES 0 for ll x [, b]. The bove function exists lmost everwhere for x [, b]. The following result cover the cse of the Cuto frctionl derivtive. Corollr 2.9. Let < t, u be weight function on (, b), v be.e. ositive function on (, b), nd D α f denotes the Cuto frctionl derivtive of f, f AC n ([, b]). If there exist rel rmeter s (, ) nd V : (, b) R is ositive mesurble function such tht (2.2) A(s, V ) = V (s ) ()v ()d ( ) su V s () (n α)(x ) n α (,b) (x ) n α dx then there exists ositive constnt C, such tht the ineulit (2.3) ( ) t Γ(n α + ) (x ) n α Dα f(x) dx C v()(f (n) ()) t d holds. Moreover, if C is the smllest constnt for (2.3) to hold, then <s< Proof. Aling Theorem. with Ω = Ω 2 = (, b), dµ (x) = dx, dµ 2 () = d, { (x ) n α k(x, ) = Γ(n α), < x ; 0, x < b we get tht K(x) = (x )n α Γ(n α+) nd A kf(x) = Γ(n α+) (x ) D α n α f(x). Relce f b f (n). Then the ineulit given in (.3) tkes the form (2.4) [ ( )] Γ(n α + ) Φ (x ) n α Dα f(x) dx C v()φ ( f (n) () ) d. If we choose Φ(x) = x t, t, x R +, then (2.4) becomes (2.3). We continue with the following lemm tht is given [4].

12 02 S. IQBAL, K. KRULIĆ HIMMELREICH AND J. PEČARIĆ Lemm 2.0. Let ν > γ 0, n = [ν]+, m = [γ]+ nd f AC n ([, b]). Suose tht one of the following conditions hold: () ν, γ N 0 nd f i (0) = 0 for i = m,..., n. (b) ν N 0, γ N 0 nd f i (0) = 0 for i = m,..., n 2. (c) ν N 0, γ N 0 nd f i (0) = 0 for i = m,..., n. (d) ν N 0, γ N 0 nd f i (0) = 0 for i = m,..., n 2. Then for ll x b. D γ + f(x) = Γ(ν γ) x (x ) ν γ D ν +f()d Corollr 2.. Let < t, u be weight function on (, b), v be.e. ositive function on (, b), nd let the ssumtions in Lemm 2.0 be stisfied. Let D γ + f denotes the Cuto frctionl derivtive of f, f AC n ([, b]). If there exist rel rmeter s (, ) nd V : (, b) R is ositive mesurble function such tht A(s, V ) = V (s ) ()v ()d ( ) su V s () (ν γ)(x ) ν γ (,b) (x ) ν γ dx then there exists ositive constnt C, such tht the ineulit (2.5) ( ) t Γ(n α + ) (x ) ν γ Dγ f(x) dx C v()(d ν + +f())t d holds. Moreover, if C is the smllest constnt for (2.5) to hold, then <s< Proof. Aling Theorem. with Ω = Ω 2 = (, b), dµ (x) = dx, dµ 2 () = d, { (x ) ν γ k(x, ) = Γ(ν γ), < x ; 0, x < b,

13 ON A NEW CLASS OF HARDY-TYPE INEQUALITIES 03 we get tht K(x) = (x )ν γ Γ(ν γ+) nd A kf(x) = Γ(ν γ+) (x ) D γ ν γ f(x). Relce f b + D ν f. Then the ineulit given in (.3) tkes the form + (2.6) [ ( )] Γ(ν γ + ) Φ (x ) ν γ Dγ f(x) dx C + v()φ (D ν +f()) d. For t, the function Φ : R + R is defined b Φ(x) = x t, then (2.6) becomes (2.5). Now we resent definitions nd some roerties of the Erdéli-Kober te frctionl integrls. Some of these definitions nd results were resented in Smko et l. in [8]. Let (, b), (0 < b ) be finite or inite intervl of the hlf-xis R +. Also let α > 0, σ > 0, nd η R. We consider the left- nd right-sided integrls of order α R defined b (2.7) (I α +;σ;ηf)(x) = σx σ(α+η) Γ(α) nd (2.8) (Ib α σxση ;σ;ηf)(x) = Γ(α) x x t ση+σ f(t)dt (x σ t σ ) α, x > t σ( η α) f(t)dt (t σ x σ ) α, x < b, resectivel. Integrls (2.7) nd (2.8) re clled the Erdéli-Kober te frctionl integrls. Now, we give the following result for Erdéli-Kober te frctionl integrls. Corollr 2.2. Let < t, α > 0, u be weight function on (, b), v be.e. ositive function on (, b), I α f denotes +;σ;η the Erdéli-Kober te frctionl integrls of f, nd 2 F (, b; c; z) denotes the hergeometric function. If there exist rel rmeter s (, ) nd V : (, b) R is ositive mesurble function such tht A(s, V ) = V (s ) ()v ()d ( su V s () α σ x ση ση+σ (x σ σ ) α ) (,b) (x σ σ ) α dx 2F (x)

14 04 S. IQBAL, K. KRULIĆ HIMMELREICH AND J. PEČARIĆ then there exists ositive constnt C, such tht the ineulit (2.9) ( t Γ(α + ) ( ( σ ) α x) 2F Iα + ;σ;ηf(x)) b dx C v()f t ()d. (x) holds. Moreover, if C is the smllest constnt for (2.9) to hold, then where nd <s< ( σ ) 2F (x) = 2 F ( η, α; α + ; x) 2F () = 2 F (η, α; α + ; ( ) σ ) b. Proof. Aling Theorem. with Ω = Ω 2 = (, b), dµ (x) = dx, dµ 2 () = d, { k(x, ) = σx σ(α+η) Γ(α) (x σ σ ) ση+σ, α < x ; 0, x < b, we get tht K(x) = ( ( Γ(α+) ) σ ) α x 2F ( η, α; α + ; ( σ) x) nd Γ(α+) A k f(x) = ( ( x) σ ) α 2F Iα (x) +;σ;ηf(x), then ineulit (.3) becomes (2.20) [ ( )] Γ(α + ) Φ ( ( σ ) α x) 2F (x) Iα + ;σ;η f(x) dx C v()φ (f()) d. If we choose Φ(x) = x t, t, x R +, then (2.20) becomes (2.9). Remrk 2.3. Similr result cn be obtined for the right sided frctionl integrl of f with resect to nother incresing function g, right sided Riemnn-Liouville frctionl integrl, right sided Hdmrd-te frctionl integrls, right sided Erdéli-Kober te frctionl integrls, but here we omit the detils. Acknowledgements. This reserch is rtill funded b Higher Eduction Commission of Pkistn. This work hs been suorted in rt b Crotin Science Foundtion under the roject 5435.

15 ON A NEW CLASS OF HARDY-TYPE INEQUALITIES 05 References [] G. A. Anstssiou, Frctionl Differentition Ineulities, Sringer Science-Businness Medi, LLC, Dordrecht, [2] M. Andrić, J. Pečrić nd I. Perić, A multile Oil te ineulit for the Riemnn- Liouville frctionl derivtives, J. Mth. Ineul. 7 (203), [3] M. Andrić, J. Pečrić nd I. Perić, Imrovements of comosition rule for Cnvti frctionl derivtive nd lictions to Oil-te ineulities, Dnmic Sstems Al. 20 (20), [4] M. Andrić, J. Pečrić nd I. Perić, Comosition identities for the Cuto frctionl derivtives nd lictions to Oil-te ineulities, Mth. Ineul. Al. 6 (203), [5] A. Čižmešij, K. Krulić nd J. Pečrić, A new clss of generl refined Hrd te ineulities with kernels, Rd Hrvt. Akd. Znn. Umjet. Mt. Znn. 7 (203), [6] A. Čižmešij, K. Krulić nd J. Pečrić, Some new refined Hrd-te ineulities with kernels, J. Mth. Ineul. 4 (200), [7] N. Elezović, K. Krulić nd J. Pečrić, Bounds for Hrd te differences, Act Mth. Sin. (Engl. Ser.) 27 (20), [8] G. D. Hndle, J. J. Kolih nd J. Pečrić, Hilbert-Pchtte te integrl ineulities for frctionl derivtives, Frct. Clc. Al. Anl. 4 (200), [9] G. H. Hrd, Notes on some oints in the integrrl clculus, Messenger. Mth, 47 (98), [0] S. Ibl, K. Krulić nd J. Pečrić, On n ineulit of H. G. Hrd, J. Ineul. Al. 200, Art. ID , 23. [] S. Ibl, K. Krulić nd J. Pečrić, Imrovement of n ineulit of G. H. Hrd, Tmkng J. Mth. 43 (202), [2] S. Ibl, K. Krulić Himmelreich nd J. Pečrić, Imrovement of n ineulit of G. H. Hrd vi suerudrtic functions, Pnmer. Mth. J. 22 (202), [3] S. Ibl, K. Krulić Himmelreich nd J. Pečrić, On n ineulit of G. H. Hrd for convex functions with frctionl integrls nd frctonl derivtives, Tbil. Mth. J. 6 (203), 2. [4] S. Kijser, L. Nikolov, L.-E. Persson, nd A. Wedestig, Hrd te ineulities vi convexit, Mth. Ineul. Al. 8 (2005), [5] A. A. Kilbs, H. M. Srivstv nd J. J. Trujillo, Theor nd Aliction of Frctinl Differentil Eutions, North-Hollnd Mthemtics Studies 204, Elsevier, New York- London, [6] K. Krulić, J. Pečrić nd L.-E. Persson, Some new Hrd te ineulities with generl kernels, Mth. Ineul. Al. 2 (2009), [7] J. Pečrić, F. Proschn nd Y. L. Tong, Convex Functions, Prtil Orderings nd Sttisticl Alictions, Acdemic Press, Inc [8] S. G. Smko, A. A. Kilbs nd O. J. Mrichev, Frctionl Integrl nd Derivtives : Theor nd Alictions, Gordon nd Brech Science Publishers, Switzerlnd, 993.

16 06 S. IQBAL, K. KRULIĆ HIMMELREICH AND J. PEČARIĆ O novoj klsi nejednkosti Hrdjev ti s rzlomljenim integrlim i rzlomljenim derivcijm Sjid Ibl, Kristin Krulić Himmelreich i Josi Pečrić Sžetk. Ovj rd osvećen je novoj klsi oćenitih težinskih nejednkosti Hrdjev ti z roizvoljnu konveksnu funkciju s rimjenm n rzne vrste rzlomljenih integrl i rzlomljenih derivcij. Sjid Ibl Dertment of Mthemtics Universit of Srgodh Sub-Cmus Bhkkr, Bhkkr Pkistn E-mil: sjid_uos2000@hoo.com Kristin Krulić Himmelreich Fcult of Textile Technolog Universit of Zgreb Prilz brun Filiović Zgreb, Croti E-mil: kkrulic@ttf.hr Josi Pečrić Fcult of Textile Technolog Universit of Zgreb Prilz brun Filiović Zgreb, Croti E-mil: ecric@element.hr Received: