ON THE HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATOR

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1 Krgujevc ournl of Mthemtics Volume 44(3) (), Pges ON THE HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATOR H. YALDIZ AND M. Z. SARIKAYA Abstrct. In this er, using generl clss of frctionl integrl oertors, we estblish new frctionl integrl ineulities of Hermite-Hdmrd tye. The min results re used to derive Hermite-Hdmrd tye ineulities involving the fmilir Riemnn-Liouville frctionl integrl oertors.. Introduction Let f : I R R be convex ming defined on the intervl I of rel numbers nd, b I, with < b. The following double ineulity is well known in the literture s the Hermite-Hdmrd ineulity 5: ( ) b (.) f b f () f (b) f (x) dx. b The most well-known ineulities relted to the integrl men of convex function re the Hermite-Hdmrd ineulities. In, Drgomir nd Agrwl roved the following results connected with the right rt of (.). Lemm.. Let f : I R R be differentible ming on I,, b I with < b. If f L, b, then the following eulity holds: (.) f () f (b) b f(x)dx b ( t)f (t ( t)b)dt. b Key words nd hrses. Frctionl integrl oertor, convex function, Hermite-Hdmrd ineulity. Mthemtics Subject Clssifiction. Primry: 6A33. Secondry: 6D, 6D5, 33E. Received: Setember 4, 7. Acceted: My,. 369

2 37 H. YALDIZ AND M. Z. SARIKAYA Theorem.. Let f : I R R be differentible ming on I,, b I with < b. If f is convex on, b, then the following ineulity holds: f () f (b) (.3) b f(x)dx ( f () f (b) ). b Menwhile, in, Sriky et l. gve the following interesting Riemnn-Liouville integrl ineulities of Hermite-Hdmrd tye. Theorem.. Let f :, b R be ositive function with < b nd f L (, b). If f is convex function on, b, then the following ineulities for frctionl integrls hold: ( ) b (.4) f with α >. Γ(α ) α α f(b) α bf() f () f (b), Lemm.. Let f :, b R be differentible ming on (, b) with < b. If f L, b, then the following eulity for frctionl integrls holds: f() f(b) Γ(α ) α α (.5) f(b) bf() α b ( t) α t α f (t ( t)b) dt. Theorem.3. Let f :, b R be differentible ming on (, b) with < b. If f is convex on, b, then the following ineulity for frctionl integrls holds: f() f(b) Γ(α ) α α f(b) bf() α (.6) b (α ) ( α ) f () f (b). For some recent results connected with frctionl integrl ineulities see (-) In 7, Rin defined the following results connected with the generl clss of frctionl integrl oertors (.7) F ρ,λ (x) F (),(),... ρ,λ (x) k (k) Γ (ρk λ) xk, ρ, λ >, x < R, where the coefficents (k), k N N {}, is bounded seuence of ositive rel numbers nd R is the rel number. With the hel of (.7), Rin nd Agrwl et l. defined the following left-sided nd right-sided frctionl integrl oertors, resectively, s follows: (.) ρ,λ,;ωϕ(x) (.9) ρ,λ,b;ωϕ(x) x b x (x t) λ F ρ,λ ω (x t) ρ ϕ(t)dt, x >, (t x) λ F ρ,λ ω (t x) ρ ϕ(t)dt, x < b,

3 HERMITE-HADAMARD TYPE INEQUALITIES where λ, ρ >, ω R, nd ϕ (t) is such tht the integrls on the right side exists. It is esy to verify tht ρ,λ,;ωϕ(x) nd ρ,λ,b;ωϕ(x) re bounded integrl oertors on L (, b), if (.) M :F ρ,λ ω ρ <. In fct, for ϕ L (, b), we hve (.) ρ,λ,;ωϕ(x) M λ ϕ nd (.) ρ,λ,b ;ωϕ(x) M λ ϕ, where b ϕ : ϕ (t) dt The imortnce of these oertors stems indeed from their generlity. Mny useful frctionl integrl oertors cn be obtined by secilizing the coefficient (k). Here, we just oint out tht the clssicl Riemnn-Liouville frctionl integrls I α nd I α b of order α defined by (see, 3, 4, 6) (.3) (I α ϕ) (x) : Γ (α) nd (.4) (Ib α ϕ) (x) : Γ(α) follow esily by setting x b x (x t) α ϕ(t)dt, x >, α > (t x) α ϕ(t)dt, x < b, α >, (.5) λ α, () nd w in (.) nd (.9), nd the boundedness of (.3) nd (.4) on L (, b) is lso inherited from (.) nd (.), (see ). In this er, using generl clss of frctionl integrl oertors, we estblish new frctionl integrl ineulities of Hermite-Hdmrd tye. The min results re used to derive Hermite-Hdmrd tye ineulities involving the fmilir Riemnn- Liouville frctionl integrl oertors... Min Results In this section, using frctionl integrl oertors, we strt with stting nd roving the frctionl integrl counterrts of Lemm., Theorem. nd Theorem.. Then some other refinements will folllow. We begin by the following theorem.

4 37 H. YALDIZ AND M. Z. SARIKAYA Theorem.. Let ϕ :, b R be convex function on, b, with < b, then the following ineulities for frctionl integrl oertors hold: ( ) b ( ϕ (.) λ Fρ,λ ω (b ρ,λ,;ω ϕ ) (b) ( )ρ ρ,λ,b ;ωϕ ) () with λ >. ϕ () ϕ (b), Proof. For t,, let x t ( t)b, y ( t) tb. The convexity of ϕ yields ( ) ( ) b x y ϕ (x) ϕ (y) (.) ϕ ϕ, i.e., ( ) b (.3) ϕ ϕ (t ( t)b) ϕ (( t) tb). Multilying both sides of (.3) by t λ F ρ,λ ω ρ t ρ, then integrting the resulting ineulity with resect to t over,, we obtin ϕ ( b ) t λ F ρ,λ ω ρ t ρ dt t λ F ρ,λ ω ρ t ρ ϕ (t ( t)b) dt t λ F ρ,λ ω ρ t ρ ϕ (( t) tb) dt. Clculting the following integrls by using (.7), we hve nd t λ F ρ,λ ω ρ t ρ dt F ρ,λ ω ρ, t λ F ρ,λ ω ρ t ρ ϕ (t ( t)b) dt λ b (b x) λ F ρ,λ ω (b x) ρ ϕ (x) dx t λ F ρ,λ ω ρ t ρ ϕ (( t) tb) dt

5 HERMITE-HADAMARD TYPE INEQUALITIES λ As conseuence, we obtin ( ) b (.4) Fρ,λ ω ρ ϕ b (x ) λ F ρ,λ ω (x ) ρ ϕ (x) dx. ( λ ρ,λ,;ω ϕ ) (b) ( ρ,λ,b ;ωϕ ) () nd the first ineulity is roved. Now, we rove the other ineulity in (.), Since ϕ is convex, for every t,, we hve (.5) ϕ (t ( t)b) ϕ (( t) tb) ϕ () ϕ (b). Then multilying both hnd sides of (.5) by t λ F ρ,λ ω ρ t ρ nd integrting the resulting ineulity with resect to t over,, we obtin t λ F ρ,λ ω ρ t ρ ϕ (t ( t)b) dt t λ F ρ,λ ω ρ t ρ ϕ (( t) tb) dt ϕ () ϕ (b) t λ F ρ,λ ω ρ t ρ dt. Using the similr rguments s bove we cn show tht ( λ ρ,λ,;ω ϕ ) (b) ( ρ,λ,b ;ωϕ ) () F ρ,λ ω ρ ϕ () ϕ (b) nd the second ineulity is roved. Remrk.. If in Theorem. we set λ α, (), w, then the ineulities (.) become the ineulities (.4) of Theorem.. Remrk.. If in Theorem. we set λ, (), w, then the ineulities (.) become the ineulities (.). Before strting nd roving our next result, we need the following lemm. Lemm.. Let ϕ :, b R be differentible ming on (, b) with < b nd λ >. If ϕ L, b, then the following eulity for frctionl integrls holds: (.6) ϕ () ϕ (b) ( λ Fρ,λ ω (b ρ,λ,;ω ϕ ) (b) ( )ρ ρ,λ,b ;ωϕ ) ()

6 374 H. YALDIZ AND M. Z. SARIKAYA Fρ,λ ω (b )ρ ( t) λ F ρ,λ ω ρ ( t) ρ ϕ (t ( t)b) dt t λ Fρ,λ ω ρ t ρ ϕ (t ( t)b) dt. Proof. Here, we ly integrtion by rts in integrls of right hnd side of (.6), then we hve (.7) ( t) λ F ρ,λ ω ρ ( t) ρ ϕ (t ( t)b) dt t λ F ρ,λ ω ρ t ρ ϕ (t ( t)b) dt ( t) λ Fρ,λ ω ρ ( t) ρ ϕ (t ( t)b) b ( t) λ Fρ,λ ω ρ ( t) ρ ϕ (t ( t)b) dt b t λ Fρ,λ ω ρ t ρ ϕ (t ( t)b) b t λ Fρ,λ ω ρ t ρ ϕ (t ( t)b) dt. b Now we use the substitution rule lst integrls in (.7), fter by using definition of left nd right-sided frctionl integrl oertor, we obtin roof of this lemm. Remrk.3. If in Lemm. we set λ α, (), nd w, then the ineulities (.6) become the eulity (.5) of Lemm.. Remrk.4. If in Lemm. we set λ, (), nd w, then the ineulities (.6) become the eulity (.) of Lemm.. We hve the following results. Theorem.. Let ϕ :, b R be differentible ming on (, b) with < b nd λ >. If ϕ is convex on, b, then the following ineulity for frctionl integrls holds: (.) ϕ () ϕ (b) ( λ Fρ,λ ω (b ρ,λ,;ω ϕ ) (b) ( )ρ ρ,λ,b ;ωϕ ) ()

7 HERMITE-HADAMARD TYPE INEQUALITIES F ρ,λ ω ρ Fρ,λ ω ρ where ϕ () ϕ (b), ( (k) : (k) ). ρkλ Proof. Using Lemm. nd the convexity of ϕ, we find tht ϕ () ϕ (b) ( λ Fρ,λ ω (b ρ,λ,;ω ϕ ) (b) ( )ρ ρ,λ,b ;ωϕ ) () (k) ω Fρ,λ ω (b k ρk )ρ k Γ (ρk λ ) ( t) ρkλ t ρkλ t ϕ () ( t) ϕ (b) dt (k) ω k ρk Fρ,λ ω ρ k Γ (ρk λ ) ( t) ρkλ t ρkλ t ϕ () ( t) ϕ (b) dt t ρkλ ( t) ρkλ t ϕ () ( t) ϕ (b) dt Fρ,λ ω ρ This comletes the roof. ( F ρ,λ ω ρ ) ( ϕ () ϕ (b) ). Remrk.5. If in Theorem. we set λ α, (), nd w, then the ineulity (.) become the ineulities (.6) of Theorem.3. Remrk.6. If in Theorem. we set λ, (), nd w, then, the ineulity (.) become the ineulities (.3) of Theorem.. Theorem.3. Let ϕ :, b R be differentible ming on (, b) with < b. If ϕ is convex on, b for some >, then the following ineulity for frctionl integrls holds: ϕ () ϕ (b) ( λ Fρ,λ w (b ρ,λ,;w ϕ ) (b) ( )ρ ρ,λ,b ;wϕ ) () Fρ,λ w ρ F ρ,λ w ρ

8 376 H. YALDIZ AND M. Z. SARIKAYA ( ϕ () 3 ϕ (b) ) ( 3 ϕ () ϕ (b) ), where with, λ >. (k) : (k) ( (ρk λ) ) ( ), (ρkλ) Proof. Using Lemm. nd the convexity of ϕ, nd Hölder s ineulity, we obtin ϕ () ϕ (b) ( λ Fρ,λ ω (b ρ,λ,;ω ϕ ) (b) ( )ρ ρ,λ,b ;ωϕ ) () (k) ω k ρk Fρ,λ ω ρ k Γ (ρk λ ) ( t) ρkλ t ρkλ dt t ϕ () ( t) ϕ (b) dt t ρkλ ( t) ρkλ dt t ϕ () ( t) ϕ (b) dt (k) ω k ρk Fρ,λ ω ρ k Γ (ρk λ ) ( t) (ρkλ) t (ρkλ) dt t ϕ () ( t) ϕ (b) dt t (ρkλ) ( t) (ρkλ) dt t ϕ () ( t) ϕ (b) dt Fρ,λ w ρ F ρ,λ w ρ ( ϕ () 3 ϕ (b) ) ( 3 ϕ () ϕ (b) ). Here, we use (A B) A B for ny A > B nd. This comletes the roof.

9 HERMITE-HADAMARD TYPE INEQUALITIES Corollry.. Under the ssumtion of Theorem.3 with λ α, () nd w, we hve ϕ() ϕ(b) Γ(α ) α α ϕ(b) bϕ() α ( ) ( ) α α ( ϕ () 3 ϕ (b) ) ( 3 ϕ () ϕ (b) ). Corollry.. If we tke α in Corollry., we hve ϕ() ϕ(b) b ϕ (t) dt ( ) ( b ( ϕ () 3 ϕ (b) ) ( ) ) ( 3 ϕ () ϕ (b) ). References R. P. Agrwl, M.-. Luo nd R. K. Rin, On Ostrowski tye ineulities, Fsc. Mth. 56() (6) DOI.55/fscmth-6-. S. S. Drgomir nd R. P. Agrwl, Two ineulities for differentible mings nd lictions to secil mens of rel numbers nd to trezoidl formul, Al. Mth. Lett. (5) (99), A. A. Kilbs, H. M. Srivstv nd.. Trujillo, Theory nd Alictions of Frctionl Differentil Eutions, North-Hollnd Mthemtics Studies 4, Elsevier, Amsterdm, 6. 4 R. Gorenflo nd F. Minrdi, Frctionl Clculus: Integrl nd Differentil Eutions of Frctionl Order, Sringer Verlg, Wien, 997, Hdmrd, Etude sur les rorietes des fonctions entieres et en rticulier d une fonction considree r, ornl de Mthemtiués Pures et Alouées 5 (93), S. Miller nd B. Ross, An Introduction to the Frctionl Clculus nd Frctionl Differentil Eutions, ohn Wiley & Sons, USA, New York, R. K. Rin, On generlized Wright s hyergeometric functions nd frctionl clculus oertors, Est Asin Mthemticl ournl () (5), 9 3. M. Z. Sriky, E. Set, H. Yldiz nd N. Bsk, Hermite-Hdmrd s ineulities for frctionl integrls nd relted frctionl ineulities, Mth. Comut. Modelling 57 (3), M. Z. Sriky nd H. Yldiz, On generliztion integrl ineulities for frctionl integrls, Nihonki Mth.. 5 (4), M. Z. Sriky, H. Yldiz nd N. Bsk, New frctionl ineulities of Ostrowski-Grüss tye, Le Mtemtiche 69() (4), M. Z. Sriky nd H. Yldiz, On Hermite-Hdmrd tye ineulities for ϕ-convex functions vi frctionl integrls, Mlys.. Mth. Sci. 9() (5), 43 5.

10 37 H. YALDIZ AND M. Z. SARIKAYA Dertment of Mthemtics, University of Krmnoğlu Mehmetbey, Krmnoğlu Mehmetbey University, Kmil Özdğ Science Fculty, Dertment of Mthemtics, Yunus Emre Cmus, 7 Krmn-TURKEY Emil ddress: yldizhtice@gmil.com Dertment of Mthemtics, University of Duzce, Duzce University, Fculty of Science nd Arts, Dertment of Mthemtics, Duzce-TURKEY Emil ddress: srikymz@gmil.com