A study on Hermite-Hadamard type inequalities for s-convex functions via conformable fractional

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1 Sud. Univ. Babeş-Bolyai Mah. 6(7), No. 3, DOI:.493/subbmah A sudy on Hermie-Hadamard ye inequaliies for s-convex funcions via conformable fracional inegrals Erhan Se and Abdurrahman Gözınar Absrac. In he resen noe, firsly we esablished a generalizaion of Hermie Hadamard s inequaliy for s-convex funcions via conformable fracional inegrals which generalized Riemann-Liouville fracional inegrals. Secondly, we roved new ideniy involving conformable fracional inegrals via bea and incomleed bea funcions.then, by using his ideniy, some Hermie Hadamard ye inegral inequaliies for s-convex funcions in he second sense are obained. Mahemaics Subjec Classificaion (): 6A33, 6A5, 6D, 6D5. Keywords: s-convex funcions, Hermie-Hadamard inequaliy, conformable fracional inegrals.. Inroducion One of he mos famous inequaliy for convex funcions is so called Hermie- Hadamard inequaliy as follows: Le f : I R R be a convex funcion and a, b I wih a < b, hen a + b f b a f(x)dx f(a) + f(b) (.) This famous inequaliy discovered by C. Hermie and J. Hadamard is imoran in he lieraure. For more sudies via Hermie Hadamard ye inequaliies see 3] in he references. Definiion.. Le f : I R R be a funcion and a, b I wih a < b, he funcion f : I R R is said o be convex if he inequaliy holds for all x, y I and, ]. f (x + ( ) y) f (x) + ( ) f (y)

2 3 Erhan Se and Abdurrahman Gözınar Definiion.. 7, 5] A funcion f : R + R is said o be s-convex in he second sense if f(αx + βy) α s f(x) + β s f(y) for all x, y R + and all α, β wih α + β. We denoe his by K s. I is obvious ha he s-convexiy means jus he convexiy when s. In ] Dragomir and Fizarick roved a varian of Hermie-Hadamard inequaliy which holds for s-convex funcions in he second sense. Theorem.3. Suose ha f :, ), ) is an s-convex funcion in he second sense, where s (, ] and le a, b, ), a < b. If f L a, b], hen he following inequaliy hold: a + b s f b a f (x) dx f (a) + f (b) s + (.) The consan k s+ is he bes ossible in he second inequaliy in (.). For more sudy relaed o s-convexiy in he second sense, see, e.g, (for examle) (3], 5], ]). Theory of convex funcions has grea imorance in various fields of ure and alied sciences. I is known ha heory of convex funcions is closely relaed o heory of inequaliies. Many ineresing convex funcions inequaliies esablished via Riemann-Liouville fracional inegrals. Now, les us give some necessary definiion and mahemaical reliminaries of fracional calculus heory as follows, which are used los of sudy. For more deails, one can consul (8]-], 4], 6]-3], 8]). Definiion.4. Le f L a, b]. The Riemann-Liouville inegrals Ja+f α and Jb α f of order α > wih a are defined by and J α a+f(x) Γ(α) J α bf(x) Γ(α) x a b x (x ) α f()d, x > a ( x) α f()d, x < b resecively. Here Γ() is he Gamma funcion and is definiion is Γ() e x x dx. I is o be noed ha Ja+f(x) Jb f(x) f(x) and in he case of α, he fracional inegral reduces o he classical inegral. The bea funcion defined as follows: B (a, b) Γ(a)Γ(b) Γ(a + b) a ( ) b d, a, b >,

3 A sudy on Hermie-Hadamard ye inequaliies 3 where Γ (α) is Gamma funcion. The incomlee bea funcion is defined by B x (a, b) x a ( ) b d, x. For x, he incomlee bea funcion coincides wih he comlee bea funcion. For easy undersanding he comuaion in our heorems, le us give some roeries of bea and incomleed bea funcion: B(a, b) B (a, b) + B (b, a), i.e B(a, b) B (a, b) + B (b, a) B x (a +, b) ab x(a, b) (x) a ( x) b a + b B x (a, b + ) bb x(a, b) + (x) a ( x) b a + b B(a, b + ) + B(a +, b) B(a, b) In ] Sarıkaya e al. gave a remarkable inegral inequaliy of Hermie-Hadamard ye involving Riemann-Liouville fracional inegrals as follows: Theorem.5. Le f : a, b] R be a osiive funcion wih a < b and f L a, b]. If f is convex funcion on a, b], hen he following inequaliy for fracional inegrals hold: a + b f Γ(α + ) () α (J a α +f)(b) + (J b α f (a) + f (b) f)(a)] (.3) I is obviously seen ha, if we ake α in Theorem.5, hen he inequaliy (.3) reduces o well known Hermie-Hadamard inequaliy as (.). Hermie-Hadamard ye inequaliies for s-convex funcions via Riemann- Liouville fracional inegral is given in ] as follows: Theorem.6. Le f : a, b] R be a osiive funcion wih a < b and f L a, b]. If f is s-convex maing in he second sense on a, b], hen he following inequaliy for fracional inegral wih α > and s (, ] hold: a + b s f where B(a,b) is Euler bea funcion. Γ(α + ) () α (J a α +f)(b) + (J b α f)(a)] (.4) α (a) + f (b) + B(α, s + )]f α + s Sarikaya e al. esablished an ideniy which we will generalize for conformable fracional inegral in secion 3 for differeniable convex maings via Riemann- Liouville fracional inegral. Then hey gave some resuls by using his ideniy.

4 3 Erhan Se and Abdurrahman Gözınar Lemma.7. ] Le f : a, b] R be a differeniable maing on (a, b) wih a < b. If f La, b], hen he following equaliy for fracional inegrals holds: f(a) + f(b) Γ(α + ) () α J a α +f(b) + J b α f(a)] (.5) ( ) α α] f (a + ( )b)d. Theorem.8. ] Le f : a, b] R be a differeniable maing on (a, b) wih a < b. If f La, b], hen he following inequaliy for fracional inegrals holds: f(a) + f(b) Γ(α + ) () α J a α +f(b) + Iα b f(a)] (.6) ( (α + ) α ) f (a) + f (b) Recenly, some auhors sared o sudy on conformable fracional inegral. In 8], Khalil e al. defined he fracional inegral of order < α only. In ], Abdeljawad gave he definiion of lef and righ conformable fracional inegrals of any order α >. Definiion.9. Le α (n, n+] and se β αn hen he lef conformable fracional inegral saring a a if order α is defined by (Iαf)() a ( x) n (x a) β f(x)dx n! a Analogously, he righ conformable fracional inegral is defined by ( b I α f)() n! b (x ) n (b x) β f(x)dx. Noice ha if α n + hen β α n n + n where n,,, 3... and hence (I a αf)() (J a n+f)(). In 4] Se e.al. gave Hermie-Hadamard inequaliy for conformable fracional inegral as follows: Theorem.. Le f : a, b] R be a funcion wih a < b and f L a, b]. If f is a convex funcion on a, b], hen he following inequaliies for conformable fracional inegrals hold: a + b f Γ(α + ) () α Γ(α n) (Ia αf)(b) + ( b f (a) + f (b) I α f)(a)] wih α (n, n + ], where Γ is Euler Gamma funcion. (.7) For some sudies on conformable fracional inegral, see (], ], 4], 6]). In aers (5]-7]), Se e.al obained some Hermie-Hadamard, Osrowski, Chebyshev, Fejer ye inequaliies by using conformable fracional inegrals for various classes of funcions. The aim of his sudy is o esablish new Hermie-Hadamard inequaliies relaed o oher fracional inegral inequaliies for conformable fracional inegral.

5 A sudy on Hermie-Hadamard ye inequaliies 33. Hermie-Hadamard s inequaliies for conformable fracional inegrals In his secion, using he given roeries of conformable fracional inegrals, we will esablish a generalizaion of Hermie-Hadamard ye inequaliies for s-convex funcions. We will also noiced he relaion wih fracional and classical Hermie- Hadamard ye inegral inequaliies. Theorem.. Le f : a, b] R be a funcion wih a < b, s (, ] and f L a, b]. If f is an s-convex funcion on a, b], hen he following inequaliies for conformable fracional inegrals hold: Γ(α n) a + b Γ(α + ) f (.) () α s (Ia αf)(b) + ( b I α f)(a)] ] B(n + s +, α n) + B(n +, α n + s) f (a) + f (b) n! s wih α (n, n + ], n,,,... where Γ is Euler Gamma funcion and B(a, b) is a bea funcion. Proof. Le x, y a, b]. If f is a s-convex funcion on a,b], s s x + y f f(x) + f(y) if we change he variables wih x a + ( )b, y ( )a + b, a + b s f f(a + ( )b) + f(( )a + b). (.) Mulilying boh sides of above inequaliy wih n! n ( ) αn and inegraing he resuling inequaliy wih resec o over, ], we ge s a + b n! f n ( ) αn d n! + n! n! b a b n ( ) αn f(a + ( )b)d n ( ) αn f(( )a + b)d n αn b x x a f(x) dx n αn y a b y f(y) dy + n! a () α Ia αf(b) + b I α f(a)].

6 34 Erhan Se and Abdurrahman Gözınar Noe ha a + b Γ(α + ) f s () α Γ(α n) Ia αf(b) + b I α f(a)] (.3) where n ( ) αn d B(n +, α n) Γ(n + )Γ(α n) Γ(α + ) which means ha he lef side of (.) is roved. Since f is s-convex in he second sense, o rove he righ side of (.) we have he following inequaliies: Adding hese wo inequaliies, we ge f(a + ( )b) s f(a) + ( ) s f(b) f(( )a + b) ( ) s f(a) + s f(b). f(a + ( )b) + f(( )a + b) s + ( ) s ]f(a) + f(b)]. Mulilying boh sides of he resuling inequaliy wih n! n ( ) αn and inegraing wih resec o over, ], we have () α Ia αf(b) + b I α f(a)] (.4) n ( ) αn s + ( ) s ]f(a) + f(b)]d n! ] B(n + s +, α n) + B(n +, α n + s) f(a) + f(b)]. n! Combining (.3) and (.4) comlees he roof. Remark.. If we choose s in Theorem (.), by using relaion beween Γ and B funcions, he inequaliy (.) reduced o inequaliy (.7). Remark.3. If we choose α n + in Theorem., he inequaliy (.) reduced o inequaliy (.4). And also if we choose α, s in he inequaliy (.), hen we ge well-known Hermie-Hadamard inequaliy as (.). 3. Some new Hermie Hadamard ye inequaliies via conformable inegraion In order o achieve our aim, we will give an imoran ideniy for differeniable funcions involving conformable fracional inegrals as follows: Lemma 3.. Le f : a, b] R be a differeniable maing on (a, b) wih a < b. If f La, b], hen he following inequaliy for conformable fracional inegrals holds: f(a) + f(b) n! B(n +, α n) () α Ia αf(b) + b I α f(a)] (3.) { () B (n +, α n) B (n +, α n) ] } f (a + ( )b)d

7 A sudy on Hermie-Hadamard ye inequaliies 35 where B(a, b), B (a, b) is Euler bea and incomleed bea funcions resecively and α (n, n + ], n,,,.... Proof. Le I B (n +, α n) B (n +, α n) ] f (a + ( )b)d. Then, inegraing by ars and changing variables wih x a+()b, we can wrie I I B (n +, α n)f (a + ( )b)d (3.) ( ) x n ( x) αn dx f (a + ( )b)d ( ) f(a + ( )b)d x n ( x) αn dx a b + ( ) n αn f(a + ( )b) d a b ( ) f(b) x n ( x) αn dx + b a ( x a ) n ( b x ) αn f(x) dx a b B(n +, α n) f(b) n! () α+ (b I α f)(a) B (n +, α n)f (a + ( )b)d (3.3) B (n +, α n) f(a + ( )b) a b n ( ) αn f(a + ( )b) d a b B(n +, α n) f(a) + b n αn b x x a f(x) dx a B(n +, α n) f(a) + n! () α+ (Ia αf)(b). I means ha I I I. Thus, by mulilying boh sides by ba i.e we have desired resul. I I I Remark 3.. If we choose α n + in Lemma 3., he equaliy (3.) becomes he equaliy (.5).

8 36 Erhan Se and Abdurrahman Gözınar Now, using he obained ideniy, we will esablish some inequaliies conneced wih he lef ar of he inequaliy (.) Theorem 3.3. Le f : a, b] R be a differeniable maing on (a, b) wih a < b. If f La, b] and f is s-convex in he second sence wih s (, ], hen he following inequaliy for conformable fracional inegrals holds: f(a) + f(b) B(n +, α n) f (a) + f ] (b) s + { B (α n + s +, n + ) B (n +, α n + s + ) n! () α Ia αf(b) + b I α f(a)] (3.4) +B (n + s +, α n) B (α n, n + s + ) + B(n +, α n) } where B(a, b), B (a, b) is Euler bea and incomleed bea funcions resecively and α (n, n + ], n,,,.... Proof. Taking modulus on Lemma 3. and using s-convexiy of f we ge: f(a) + f(b) n! B(n +, α n) () α Ia αf(b) + b I α f(a)] (3.5) B (n +, α n) B (n +, α n) ] f (a + ( )b)d B (n +, α n) B (n +, α n) ] f (a + ( )b) d B (n +, α n) B (n +, α n) ] f (a + ( )b) d + B (n +, α n) B (n +, α n) ] f (a + ( )b) d { B (n +, α n) ( s f (a) + ( ) s f (b) ) d B (n +, α n) ( s f (a) + ( ) s f (b) ) d + B (n +, α n) ( s f (a) + ( ) s f (b) ) d B (n +, α n) ( s f (a) + ( ) s f (b) ) d }

9 A sudy on Hermie-Hadamard ye inequaliies 37 { f (a) B (n +, α n) B (n +, α n) ] s d + f (b) + f (a) + f (b) B (n +, α n) B (n +, α n) ] ( ) s d B (n +, α n) B (n +, α n) ] s) d B (n +, α n) B (n +, α n) ] ( ) s d. On he oher hand, using he roeries of incomleed bea funcion we have: B (n +, α n) B (n +, α n) (3.6) x n ( x) αn dx x n ( x) αn dx x n ( x) αn dx, where and B (n +, α n) B (n +, α n) (3.7) x n ( x) αn dx x n ( x) αn dx x n ( x) αn dx, where Using (3.6), (3.7) and Newon Leibniz formula and inegraing by ars we can wrie he following comuaion: Φ ( ( ) x n ( x) αn dx s d (3.8) ) ] x n ( x) αn s+ dx s + ( ( ) n αn n ( ) αn) s+ s + d s + s + αn+s ( ) n d + n+s+ ( ) αn d B (α n + s +, n + ) + B (n + s +, α n) ], ]

10 38 Erhan Se and Abdurrahman Gözınar Φ Φ 3 ( ( ) x n ( x) αn dx ( ) s d (3.9) ) ] ( ) x n ( x) αn s+ dx s + ( ( ) n αn n ( ) αn) ( ) s+ d s + x n ( x) αn dx s + s + B(n +, α n) B (α n, n + s + ) s + αn ( ) n+s+ ] d + n ( ) αn+s d B (n +, α n + s + ) ], ( ) x n ( x) αn dx s d (3.) ( ) ] x n ( x) αn s+ dx s + s + s + ( n ( ) αn + αn ( ) n) s+ d x n ( x) αn dx n+s+ ( ) αn d + s + B(n +, α n) B (α n, n + s + ) s + B (n +, α n + s + ) ] ] αn+s ( ) n d and Φ 4 ( ) x n ( x) αn dx ( ) s d (3.) ( ) ] ( ) x n ( x) αn s+ dx s + + ( n ( ) αn + αn ( ) n) ( ) s+ d s +

11 A sudy on Hermie-Hadamard ye inequaliies 39 n ( ) αn+s d + s + ] αn ( ) n+s+ d B (α n + s +, n + ) + B (n + s +, α n) ], Using he fac ha B(a, b) B (a, b) + B (b, a) and combining (3.8), (3.9), (3.), (3.) wih (3.5) comlees he roof. Corollary 3.4. Taking s in Theorem 3.3 i.e f is convex, we ge he following resul: f(a) + f(b) B(n +, α n) ( f (a) + f ) (b) { B (α n +, n + ) B (n +, α n + ) n! () α Ia αf(b) + b I α f(a)] +B (n + 3, α n) B (α n, n + 3) + B(n +, α n) } (3.) Remark 3.5. Taking α n + in Corollary 3.4, he inequaliy (3.) reduces o (.6). Theorem 3.6. Le f : a, b] R be a differeniable maing on (a, b), a < b and > wih + q. If f La, b] and f q is s-convex in he second sense, hen he following inequaliy for conformable fracional inegrals holds: ( f(a) + f(b) B(n +, α n) Ψ f (a) q + f (b) q s + ) ] n! () α Ia αf(b) + b I α f(a)] q. (3.3) where B(a, b) is Euler bea funcion, α (n, n + ], n,,,... and. ( ) Ψ x n ( x) αn dx

12 3 Erhan Se and Abdurrahman Gözınar Proof. Taking modulus and using Hölder inequaliy wih a funcion of f q convexiy we ge inequaliies as follow: f(a) + f(b) n! B(n +, α n) () α Ia αf(b) + b I α f(a)] (3.4) B (n +, α n) B (n +, α n) ] f (a + ( )b)d B (n +, α n) B (n +, α n) f (a + ( )b) d B (n +, α n) B (n +, α n) ] d f (a + ( )b) ] q q d. I follows ha: and Ψ B (n +, α n) B (n +, α n) d (3.5) + + ( B (n +, α n) B (n +, α n)) d ( B (n +, α n) B (n +, α n)) d ( x n ( x) dx) αn d ( x n ( x) dx) αn d ( x n ( x) dx) αn d f (a + ( )b) q d f (a) q s d + f (b) q ( ) s d ( f (a) q + f (b) q) (3.6) s + which comlees he roof. Corollary 3.7. If we ake s in Theorem 3.6, he inequaliy (3.3) reduces o following inequaliy: f(a) + f(b) B(n +, α n) n! () α Ia αf(b) + b I α f(a)] Ψ f (a) q + f (b) q ] q (3.7)

13 A sudy on Hermie-Hadamard ye inequaliies 3 where B(a, b) is Euler bea funcion and ( Ψ x n ( x) dx) αn. Corollary 3.8. If we ake α n + in corollary 3.7, he inequaliy (3.7) reduces o following inequaliy: f(a) + f(b) B(α, ) Γ(α) () α J a α +f(b) + J b α f(a)] (3.8) Ψ f (a) q + f (b) q ] q, ( ( ) α α ) where Ψ d. α Remark 3.9. If we ake α in Corollary 3.8, he inequaliy (3.8) reduces o following inequaliy: f(a) + f(b) b f(x)dx () (3.9) a f (a) q + f (b) q ] q, + which is he same as Theorem.3 in ]. Remark 3.. If we ake α (, ] in Corollary 3.8, hen he inequaliy (3.8) reduces o secial case of Corollary for s in 9], which is he same as f(a) + f(b) Γ(α + ) () α J a α +f(b) + J b α f(a)] (3.) f (a) q + f (b) q ] q. α + References ] Abdeljawad, T., On conformable fracional calculus, J. Comu. Al. Mah., 79(5), ] Abdeljawad, T., On mulilicaive fracional calculus, arxiv:5, o476v mah.ca], 6 Oc 5. 3] Alomari, M., Darus, M., Dragomir, S.S., Cerone, P., Osrowski ye inequaliies for funcions whose derivaives are s-convex in he second sense, Al. Mah. Le., 3(9), ] Anderson, O.R., Taylor s formula and inegral inequaliies for conformable fracional derivaives, arxiv: v mah. CA], Se 4. 5] Avci, M., Kavurmaci, H., Özdemir, M.E., New inequaliies of Hermie-Hadamard ye via s-convex funcions in he second sense wih alicaions, Al. Mah. Comu., 7(),

14 3 Erhan Se and Abdurrahman Gözınar 6] Benkheou, N., Hassani, S., Torres, D.E.M., A conformable fracional calculus on arbirary ime, J. King Saud Univ. Sci., 8(6), ] Breckner, W.W., Seigkeisaussagen fr eine Klasse verallgemeinerer konvexer funkionen in oologischen linearen Raumen, Publ. Ins. Mah., 3(978), 3-. 8] Chen, F., Exensions of he Hermie-Hadamard Inequaliy for convex funcions via fracional inegrals, J. Mah. Ineq., ()(6), ] Dahmani, Z., New inequaliies in fracional inegrals, In. J. Nonlinear Sci., 9(4)(), ] Dahmani, Z., Tabhari, L., Taf, S., New generalizaions of Gruss inequaliy using RiemannLiouville fracional inegrals, Bull. Mah. Anal. Al., (3)(), ] Dragomir, S.S., Agarval, R.R., The Hadamard s inequaliy for s-convex funcions in he second sense, Demonsraio Mah., 3(4)(999), ] Dragomir, S.S., Fizarik, S., Two inequaliies for differeniable maings and alicaions o secial means of real numbers and o raezoidal formula, Al. Mah. Le., (5)(998), ] Dragomir, S.S., Pearce, C.E.M., Seleced Toics on Hermie-Hadamard Inequaliies and Alicaions, RGMIA Monograhs, Vicoria Universiy,. 4] Gorenflo, R., Mainardi, F., Fracional Calculus: Inegral and Differenial Equaions of Fracional Order, 8, arxiv rerin arxiv: ] Hudzik, H., Maligranda, L., Some remarks on s-convex funcions, Aequaiones Mah., 48(994), -. 6] 7] İşcan, İ., Generalizaion of differen ye inegral inequaliies for s-convex funcions via fracional inegrals, Al. Anal., 93(9)(4), İşcan, İ., Hermie Hadamard ye inequaliies for harmonically convex funcions via fracional inegral, Al. Mah. Comu., 38(4), ] Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definiion of fracional derivaive, J. Comu. Al. Mah., 64(4), ] Özdemir, M.E., Kavurmacı, H., Yıldız, Ç., Fracional inegral inequaliies via s-convex funcions, Turkish J. Anal. Number Theory, 5()(7), 8-. ] Podlubni, I., Fracional Differenial Equaions, Academic Press, San Diego, 999. ] Sarıkaya, M.Z., Se, E., Yaldız, H., Başak, N., Hermie-Hadamard s inequaliies for fracional inegrals and relaed fracional inequaliies, Mah. Comu. Model., 57(3), ] Se, E., Sarıkaya, M.Z., Özdemir, M.E., Yıldırım, H., The Hermie-Hadamard s inequaliy for some convex funcions via fracional inegrals and relaed resuls, J. Al. Mah. Sa. Inform., ()(4), ] Se, E., New inequaliies of Osrowski ye for maings whose derivaives are s-convex in he second sense via fracional inegrals, Comu. Mah. Al., 63(7)(), ] Se, E., Akdemir, A.O., Mumcu, İ., The Hermie-Hadamard s inequaliy and is exenions for conformable fracioanal inegrals of any order α >, Submied. 5] Se, E., Akdemir, A.O., Mumcu, İ., Osrowski ye inequaliies for funcions whoose derivaives are convex via conformable fracional inegrals, J. Adv. Mah. Sud., (3)(7),

15 A sudy on Hermie-Hadamard ye inequaliies 33 6] Se, E., Akdemir, A.O., Mumcu, İ., Chebyshev ye inequaliies for conformable fracional inegrals, Submied. 7] Se, E., Mumcu, İ., Hermie-Hadamard-Fejer ye inequalies for conformable fracional inegrals, Submied. 8] Zhu, C., Feckan, M., Wang, J., Fracional inegral inequaliies for differeniable convex maings and alicaions o secial means and a midoin formula, J. Mah. Sa. Inform., 8()(), -8. Erhan Se Dearmen of Mahemaics, Faculy of Ars and Sciences, Ordu Universiy 5 Ordu, Turkey erhanse@yahoo.com Abdurrahman Gözınar Dearmen of Mahemaics, Faculy of Ars and Sciences, Ordu Universiy 5 Ordu, Turkey abdurrahmangozinar79@gmail.com