FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE SECOND DERIVATIVE ARE

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1 Kagujevac Jounal of Mathematics Volume 4) 6) Pages 7 9. FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE SECOND DERIVATIVE ARE s )-CONVEX IN THE SECOND SENSE K. BOUKERRIOUA T. CHIHEB AND B. MEFTAH Abstact. In this pape the authos intoduce a new class of convex functions called s )-convex functions in the second sense and establish some new Hemite- Hadamad type ineualities involving Riemann-Liouville integal opeato.. Intoduction One of the most well-known ineualities in mathematics fo convex functions is so called Hemite-Hadamad integal ineuality ) a + b.) f b fx)dx b a a f a) + f b) whee f is a eal continuous convex function on the finite inteval a b]. If the function f is concave then.) holds in the evese diection see 4]). The Hemite-Hadamad ineuality play an impotant ole in nonlinea analysis and optimization. The above double ineuality has attacted many eseaches vaious genealizations efinements extensions and vaiants of.) have appeaed in the liteatue via classical integation and factional calculus we can mention the woks ] and the efeences cited theein. Recently Wang et al. 9] poved the following Hemite-Hadamad s ineualities whose powe of second deivatives ae -convex via factional integals. Key wods and phases. Hemite-Hadamad ineuality convex functions Riemann-Liouville integal opeato. Mathematics Subject Classification. Pimay 6D5 Seconday 6D 39A. Received: Novembe 9 5. Accepted: May 6. 7

2 FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES 73 Theoem. 9] Theoem 4.). Let f : b ] R be a twice diffeentiable mapping with b >. If f > ) is measuable and -convex on a b] fo some fixed < and a ; I = I = b a) α + ) b a) α + ) fo f a) f b) and I = fo f a) = f b) ; and p + =. ) p f a) + f b) ) ) pα + p + + ) p f a) + f b) ) ) pα + p + + ) p pα + p + f a) f b) ln f a) ln f b) ) b a) α + ) f p a) pα + p + ) Theoem. 9] Theoem 4.). Let f : b ] R be a twice diffeentiable mapping with b >. If f > ) is measuable and -convex on a b] fo some fixed < and a < b then the following ineuality holds fo factional integals: f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] I whee I = b a) f a) + f b) ) α + fo < and α I = ] b a) f a) + f b) ) α + ) α ] ) + β + α + + ) + β + α + +

3 74 K. BOUKERRIOUA T. CHIHEB AND B. MEFTAH fo > ; I = b a) f a) f b) α + ) ln f a) ln f b) ) ln f a) ln f b) ) i i f b) ) i f a) ] α + + ) i i= fo f a) f b) and I = fo f a) = f b) ; and p + =. ) b a) α + ) f a) α + + In 8] Lin et al. poved the following Hemite-Hadamad s ineualities whose second deivatives ae -convex via factional integals. Theoem.3 8] Poposition 4.). Let f : a b] R be a twice diffeentiable mapping with a < b. If f is integable and -convex on a b] fo some fixed < then the following ineuality fo factional integals holds Γ α + ) b a) α J α a +fb) + J α f a) + f b) bfa)] K whee K = b a) f a) + f b) α + fo < K = b a) f a) + f b) α + fo > K = b a) f b) α + ) k k ln k fo = and k = f a) / f b). )] + α + + β + α + )] + α + + β + α + i= ) i ln k ) i α + ) i i= ] ln k ) i α + ) i Theoem.4 8] Poposition 4.5). Let f : a b] R be a twice diffeentiable mapping with a < b. If f > is integable and -convex on a b] fo some fixed < then the following ineuality fo factional integals holds Γ α + ) b a) α J α a +fb) + J α f a) + f b) bfa)] K

4 FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES 75 whee K = b a) α + fo < fo > K = b a) α + K = b a) f b) α + fo = and k = f a) / f b). ] ) f a) + f b) ) α + ] ) f a) + f b) ) α + ] ) α + ] + ] + k ) ln k Motivated by the above esults in this pape we extend those esults fo a new class of convexity called s )-convex functions in the second sense.. Peliminaies In this section we ecall some concepts of convexity which ae well known in the liteatue. Let I be an inteval of R. Definition.. 4] A function f : I R is said to be convex if holds fo all x y I and all t ]. f tx + t) y) tf x) + t) fy) Definition.. 4] A positive function f : I R is said to be logaithmically convex if holds fo all x y I and all t ]. ftx + t)y) fx)] t fy)] t) Definition.3. 5] A nonnegative function f : I ) R is said to be s-convex in the second sense on I fo some fixed s ] if holds fo all x y I and t ]. ftx + t)y) t s fx) + t) s fy) Definition.4. ] A positive function f : I ) R is said to be s- logaithmically convex function in the second sense on I fo some fixed s ] if ftx + t)y) fx)] ts fy)] t)s holds fo all x y I and t ].

5 76 K. BOUKERRIOUA T. CHIHEB AND B. MEFTAH Definition.5. 3] A positive function f : I ) R is said to be -convex on I if { tf ftx + t)y) x) + t) f y)] if fx)] t fy)] tλ if = holds fo all x y I and t ]. Definition ] Let f L a b]. The Riemann-Liouville integals J α a + f and J α b f of ode α > with a ae defined by J α a +fx) = x x t) α ft)dt Γ α) x > a a J α b fx) = b Γ α) x t x) α ft)dt b > x espectively whee Γα) = e t t α dt is the Gamma function and J a fx) = + J b fx) = fx). Lemma.. 9] Fo α > and k > z > we have J α k) = H α k z) = whee α) i = i j= α + j). t) α k t dt = ln k) i < α) i i= t α k t dt = z α k z z ln k) i < α) i Lemma.. 9] Let f : a b] R be a twice diffeentiable mapping on a b) with a < b. If f La b] then the following euality fo factional integals holds: f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] b a) = t) α+ t α+] f ta + t)b) dt. α + ) Also we ecall that the Eule Beta function is defined as follows β x y) = i= t x t) y dt = Γ x) Γ y) Γ x + y).

6 FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES Main Results In ode to pove ou esults we fist intoduce a new concept of convexity called s )-convexity in the second sense. Definition 3.. A positive function f : I ) R is said to be s )-convex in the second sense if { t f tx + t)y) s f x) + t) s f y)] if fx)] ts fy)] t)s if = holds fo some fixed s ] t ] and all x y I. Example 3.. We define the function g as follows: { a if t = g t) = bt s + c) if t > whee a b c R and s ] such that b and c a and >. The function g is s )-convex in the second sense because { g a if t = t) = f t) = bt s + c if t > is s-convex in the second sense fo moe details see ]. Remak 3.. Obviously Definition 3. ecaptue all definitions cited above fo wellchosen values of s and. Theoem 3.. Let f : a b] ) be twice diffeentiable mapping with a 3.) whee c) b a) α + ) f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] T α + ) s )] s + ) s + α + )) β + α + f a) + f b) ] fo > b a) Ea b s) α + ) s f b) f a) s f b) s ln f a) ln f b) ln f a) f b) s ) i α + ) i f a) f b) s i= ln f a) ) i f b) s ) i α + ) i i=

7 78 K. BOUKERRIOUA T. CHIHEB AND B. MEFTAH fo = and f a) f b) and α b a) Nb s) α + ) α + ) fo = and f a) = f b) holds fo whee { if 3.) c) = if < f b) s if f a) f b) f b) if f a) f b) 3.3) Ea b s) = f a) s f b) s if f b) f a) f a) s f b) if f a) f b) 3.4) Nb s) = and α + ) i = i j= α + + j). { f b) s if f a) = f b) f b) s if f a) = f b) Poof. Fom Lemma. and popety of the modulus we have f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] b a) 3.5) t) α+ t α+] f ta + t)b) dt. α + ) Case : >. Since f is s )-convex in the second sense we get 3.6) f ta + t)b) t s f a) + t) s f b) ] it is easy to see that 3.7) f ta + t)b) c) whee c) is defined as in 3.). Substituting 3.7) into 3.5) we obtain b a) α + ) c) b a) α + ) ] t s f a) + t) s f b) t) α+ t α+] f ta + t)b) dt f a) t s t s t) α+ t s +α+] dt

8 FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES ) = = + c) b a) α + ) f b) f a) ] t) s s t) +α+ t α+ t) s c) b a) α + ) s )] α + ) s + ) s + α + )) β + α + c) b a) + f α + ) b) α + ) s + ) s + α + )) β α + s ) ] + c) b a) α + ) f a) + f b) ]. α + ) s )] s + ) s + α + )) β + α + dt Case : =. Since f is s )-convex we have 3.9) f ta + t)b) f a) ts f b) t)s using the fact that φ ts φ st and ψ ts ψ st+s is valid fo all < φ ψ and t s ] we get 3.) f ta + t)b) Ea b s) and ) f st a) if f a) f b) f b) 3.) f ta + t)b) Nb s) if f a) = f b) whee Ea b s) and Nb s) ae defined as in 3.3) and 3.4) espectively. Substituting 3.) into 3.5) we obtain f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] 3.) b a) Ea b s) α + ) = b a) Ea b s) α + ) t) α+ t α+] f a) f b) ) f st a) dt f b) ) f t) α+ st a) dt f b) t α+ f a) f b) ) st dt ) st dt.

9 8 K. BOUKERRIOUA T. CHIHEB AND B. MEFTAH Fom Lemma. with z = we get 3.3) 3.4) and 3.5) ) f t) α+ s a) t dt = f b) ) f t α+ s a) t dt = f b) ) f s a) t dt = f b) = f a) f b) ln f a) f b) ln i= s s f a) f b) s i= f a) f b) ) s ) i α + ) i ln f a) ) i f b) s ) i α + ) i s f b) f a) s f b) s ln f a) ln f b) using 3.3) 3.4) and 3.5) in 3.) we obtain f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] 3.6) b a) Ea b s) α + ) i= s f b) ln f a) f b) s ) i f a) α + ) i f b) f a) s f b) s ln f a) ln f b) s i= ln f a) ) i f b) s ) i. α + ) i Now substituting 3.) into 3.5) we get f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] α b a) 3.7) Nb s). α + ) α + ) Fom 3.8) 3.6) and 3.7) we obtain the desied ineuality in 3.). The poof is complete. Remak 3.. With the same assumptions of Theoem 3. if f x) M on a b] ineuality 3.) becomes 3.8) f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] T whee c) b a) α + ) α + ) s )] s + ) s + α + )) β + α + M

10 FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES 8 fo > and fo =. α b a) M α + ) α + ) Coollay 3.. Let f : a b] ) be twice diffeentiable mapping with a holds 3.9) f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] α b a) 4 α + ) α + ) f a) + f b) ]. Coollay 3.. Let f : a b] ) be twice diffeentiable mapping with a holds f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] 3.) b a) α + ) α + ) β s + α + ) s + ) s + α + ) ] f a) + f b) ]. Coollay 3.3. Let f : a b] ) be twice diffeentiable mapping with a holds 3.) whee c) b a) α + ) f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] T α + ) + ) + α + )) β )] + α + f a) + f b) ] fo > b a) f a) f b) ln α + ) ln f a) ln f b) f f a) f b) b) α + ) i= i ) ln f a) i f a) ) i f b) α + ) i i= fo = and f a) f b) α b a) f b) α + ) α + ) ) i

11 8 K. BOUKERRIOUA T. CHIHEB AND B. MEFTAH fo = and f a) = f b). Remak 3.3. Coollay 3.3 is simila to the Poposition 4. fom 8]. Coollay 3.4. Let f : a b] ) be twice diffeentiable mapping with a holds 3.) whee f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] T b a) f a) f b) ln α + ) ln f a) ln f b) f f a) f b) b) α + ) i f a) fo f a) f b) and fo f a) = f b). ln f a) ) i f b) α + ) i i= ) i α b a) f b) α + ) α + ) Theoem 3.. Let f : a b] ) be twice diffeentiable mapping with a . If f fo > is s )-convex function in the second sense fo some fixed s ] then the following ineuality fo factional integals holds fo and α > 3.3) whee fo > f a) + f b) c)] b a) α + ) i= Γ α + ) b a) α J α a +fb) + J α b fa)] T b a) ) α f a) + f b) ] α + ) α + α + ) s )] s + ) s + α + )) β + α + ) α Ea b s )) α + f a) s f b) s s ln f a) s ln f b) f b) s ln i= f a) f b) α + ) i ) i s ) i

12 FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES 83 f a) s i= ln f a) ) i f b) s ) i α + ) i fo = and f a) f b) and α b a) α + ) α + ) Nb s )) fo = and f a) = f b) whee f b) s if f a) f b) f b) if f a) f b) 3.4) Ea b s ) = f a) s) f b) s if f b) f a) f a) s) f b) if f a) f b) and 3.5) Nb s ) = { f b) s if f a) = f b) f b) s) if f a) = f b) c) is defined as in 3.) and α + ) i = i j= α + + j). Poof. Using Lemma. popety of the modulus and powe mean ineuality we get f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] b a) t) α+ t α+] dt α + ) t) α+ t α+] f ta + t)b) dt 3.6) = ) b a) α α + ) α + t) α+ t α+] f ta + t)b) dt. Case : >. Since f is s )-convex in the second sense we have 3.7) f ta + t)b) t s f a) + t) s f b) ]

13 84 K. BOUKERRIOUA T. CHIHEB AND B. MEFTAH the ineuality 3.7) can be estimate as 3.8) f ta + t)b) c) whee c) is defined as in 3.). Now substituting 3.8) into 3.6) we obtain f a) + f b) c)] b a) α + ) f a) t s f a) + t) s f b) ] Γ α + ) b a) α J α a +fb) + J α b fa)] α ) α + t s t s t) α+ t s +α+] t s dt + f b) ] t) s s t) +α+ t α+ t) s dt 3.9) = c)] b a) α + ) Case : =. Since f is s )-convex we have ) α f a) + f b) ] α + α + ) s )] s + ) s + α + )) β + α + 3.3) f ta + t)b) f a) ) t s f b) ) t) s. Repeating a simila agument in Theoem 3. we obtain 3.3) f ta + t)b) f a) ) st Ea b s ) f b) if f a) f b) and 3.3) f ta + t)b) Nb s ) if f a) = f b) using 3.3) into 3.6) it yields f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] ) b a) α 3.33) Nb s )). α + ) α +.

14 FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES 85 Now substituting 3.3) into 3.6) and using the fact that t) n n t n we get f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] ) b a) α Ea b s )) f a) t α + ) α + f b) s) dt t) α+ f a) f b) t s) dt t α+ f a) f b) s) t dt 3.34) ) b a) α f a) s f b) s = Ea b s )) α + ) α + s ln f a) s ln f b) ln f a) f b) s f b) s ) i ln f a) f a) s ) i f b) s ) i α + ) i α + ) i i= i=. Making as 3.34) 3.33) and 3.9) we get the euied ineuality in 3.3). Theoem 3.3. Let f : a b] ) be twice diffeentiable mapping with a 3.35) f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] T whee fo > b a) α + ) b a) α + ) ) ) p α + ) p c)] f a) + f b) ] p α + ) + s + ) p α + ) p Ea b s )) p α + ) + fo = and f a) f b) and b a) α + ) p α + ) p α + ) + f a) s f b) s ] s ln f a) s ln f b) ) p Nb s )) fo = and f a) = f b) holds fo whee c) Ea b s ) and Nb s ) ae defined as in 3.) 3.4) and 3.5) espectively.

15 86 K. BOUKERRIOUA T. CHIHEB AND B. MEFTAH Poof. Using Lemma. popety of the modulus and Hölde s ineuality we get f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] b p a) t) α+ t α+] p dt f ta + t)b) dt α + ) 3.36) = b a) α + ) b a) α + ) t) pα+) t pα+)] dt f ta + t)b) dt ) p α + ) p p α + ) + p f ta + t)b) dt noting that fo any t ] α > and fixed p t) α+ t α+] p = t) α+ ) t α+] p t) α+] p t pα+) t) pα+) t pα+) whee we have use the fact that fo all t n ] t) n n t n. Case : >. Since f is s )-convex in the second sense we have 3.37) f ta + t)b) t s f a) + t) s f b) ] and 3.38) f ta + t)b) c) t s f a) + t) s f b) ] whee c) is defined as in 3.). Using 3.38) into 3.36) we obtain f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] ) b a) p α + ) p c)] α + ) p α + ) + t s f a) dt + t) s f b) dt

16 FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES ) = b a) α + ) ) ) p α + ) p c)] f a) + f b) ]. p α + ) + s + Case : =. Since f is s )-convex by a simila agument to Theoem 3. we have 3.4) f ta + t)b) f a) ) st Ea b s ) f b) if f a) f b) and 3.4) f ta + t)b) Nb s ) if f a) = f b) using 3.4) into 3.36) we obtain f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] ) b a) p α + ) p 3.4) Nb s )). α + ) p α + ) + Now substituting 3.4) into 3.36) it yields f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] ) p α + ) p 3.43) Ea b s )) p α + ) + b a) α + ) f a) s f b) s s ln f a) s ln f b) Fom 3.43) 3.4) and 3.39) we get the desied ineuality in 3.35). This completes the poof. Remak 3.4. Theoem 3.3 will be educed to Theoem 4. fom 9] also to the Poposition 4.5 fom 8] in the case s =. Theoem 3.4. Suppose that all the assumptions of Theoem 3.3 ae satisfied then the following ineuality fo factional integals with α > holds 3.44) f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] T whee ]. c)] b a) f a) + f b) ] α + ) α + ) s + ) s + α + ) + ) β α + ) + s ) ] +

17 88 K. BOUKERRIOUA T. CHIHEB AND B. MEFTAH fo > b a) α + ) Ea b s )) f b) s i= fa) s fb) s s ln f a) s ln f b) ln f a) f b) s ) i f a) s α + ) + ) i fo = and f a) f b) and b a) α + ) Nb s )) ln f a) ) i f b) s ) i α + ) + ) i i= ] α + ) α + ) + fo = and f a) = f b) whee c) Ea b s ) and Nb s ) ae defined as in 3.) 3.4) and 3.5) espectively. Poof. Using Lemma. popety of the modulus and Hölde s ineuality we get f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] b p a) dt t) α+ t α+] f ta + t)b) dt α + ) 3.45) b a) α + ) t) α+) t α+)] f ta + t)b) dt Case : >. Since f is s )-convex in the second sense we have 3.46) f ta + t)b) t s f a) + t) s f b) ] the ineuality 3.46) can be efomulated as 3.47) f ta + t)b) c) t s f a) + t) s f b) ] whee c) is defined as in 3.). Using 3.47) into 3.45) we obtain f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] c)] b a) α + ) f a) t s t s t) α+) t s +α+)] dt.

18 FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES 89 + f b) ] t) s s t) +α+) t α+) t) s dt 3.48) = c)] b a) α + ) f a) α + ) s )] s + ) s + α + ) + ) β + α + ) + + f b) α + ) s + ) s + α + ) + ) β α + ) + s ) ]] + = c)] b a) α + ) α + ) s + ) s + α + ) + ) β α + ) + s ) ] + f a) + f b) ]. Case : =. Since f is s )-convex by a simila agument to Theoem 3.3 we have 3.49) f ta + t)b) f a) ) st Ea b s ) f b) if f a) f b) and 3.5) f ta + t)b) Nb s ) if f a) = f b) using 3.5) into 3.45) we obtain f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] ] b a) α + ) Nb s )) α + ) 3.5). α + ) + Now substituting 3.49) into 3.45) using Lemma. we get f a) + f b) Γ α + ) b a) α J α a +fb) + J α b fa)] 3.5) b a) α + ) Ea b s )) f b) s i= ln f a) f b) f a) s f b) s s ln f a) s ln f b) s ) i α + ) + ) i

19 9 K. BOUKERRIOUA T. CHIHEB AND B. MEFTAH f a) s i= ln f a) ) i f b) s ) i. α + ) + ) i Fom 3.48) 3.5) and 3.5) we obtain the desied ineuality in 3.44). completes the poof. This Remak 3.5. Theoem 3.4 will be educed to Theoem 4. fom 9] in the case s =. Refeences ] A. O. Akdemi and M. Tunc On some integal ineualities fo s-logaithmically convex functions and thei applications axiv pepint axiv:.584. ] R.-F. Bai F. Qi and B.-Y. Xi Hemite-Hadamad type ineualities fo the m- and α m)- logaithmically convex functions Filomat 7 3) 7. 3] A. Baani S. Baani and S. Dagomi Hemite-Hadamad ineualities fo functions whose second deivatives absolute values ae p-convex RGMIA Res. Rep. Coll. 4 ) Aticle 73 8 pp. 4] A. Baani S. Baani and S. S. Dagomi Refinements of Hemite-Hadamad ineualities fo functions when a powe of the absolute value of the second deivative is p-convex J. Appl. Math. ) Aticle pp. 5] W. W. Beckne Stetigkeitsaussagen fü eine Klasse Veallgemeinete Konvexe Funktionen in Topologischen Lineaen Räumen Publ. Inst. Math. Beogad) N.S.) 337) 978) 3. 6] Z. Dahmani On Minkowski and Hemite-Hadamad integal ineualities via factional integation Ann. Funct. Anal. ) ] J. Deng and J. Wang Factional Hemite-Hadamad ineualities fo α m)-logaithmically convex functions J. Ineual. Appl. 3) Aticle 364 pp. 8] S. Dagomi and C. Peace Selected Topics on Hemite-Hadamad Ineualities and Applications RGMIA Monogaphs Victoia Univesity. 9] S. S. Dagomi and R. P. Agawal Two ineualities fo diffeentiable mappings and applications to special means of eal numbes and to tapezoidal fomula Appl. Math. Lett. 998) ] S. S. Dagomi J. Pečaić and L. E. Pesson Some ineualities of Hadamad type Soochow J. Math. 995) ] H. Hudzik and L. Maliganda Some emaks on s-convex functions Aeuationes Math ). ] İ. İşcan and S. Wu Hemite-Hadamad type ineualities fo hamonically convex functions via factional integals Appl. Math. Comput. 38 4) ] U. N. Katugampola New appoach to a genealized factional integal Appl. Math. Comput. 8 ) ] A. A. Kilbas H. M. Sivastava and J. J. Tujillo Theoy and applications of factional diffeential euations vol. 4 of Noth-Holland Mathematics Studies Elsevie Science B.V. Amstedam 6. 5] U. S. Kimaci M. Klaičić Bakula M. E. Özdemi and J. Pečaić Hadamad-type ineualities fo s-convex functions Appl. Math. Comput. 93 7) ] W.-H. Li and F. Qi Some Hemite-Hadamad type ineualities fo functions whose n-th deivatives ae α m)-convex Filomat 7 3) ] Y. Liao J. Deng and J. Wang Riemann-Liouville factional Hemite-Hadamad ineualities. Pat I: fo once diffeentiable geometic-aithmetically s-convex functions J. Ineual. Appl. 3) Aticle pp.

20 FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES 9 8] Z. Lin J. Wang and W. Wei Factional Hemite-Hadamad ineualities though -convex functions via powe means Facta Univ. Se. Math. Infom. 3 5) ] M. A. Noo K. I. Noo M. U. Awan and S. Khan Factional Hemite-Hadamad ineualities fo some new classes of Godunova-Levin functions Appl. Math. Inf. Sci. 8 4) ] M. A. Noo F. Qi and M. U. Awan Some Hemite-Hadamad type ineualities fo log-h-convex functions Analysis Belin) 33 3) ] M. E. Özdemi and Ç. Yıldız New ineualities fo Hemite-Hadamad and Simpson type with applications Tamkang J. Math. 44 3) 9 6. ] J. Pak Hemite-Hadamad-like type ineualities fo s-convex functions and s-godunova-levin functions of two kinds Appl. Math. Sci. 9 5) ] C. E. M. Peace J. Pečaić and V. Šimić Stolasky means and Hadamad s ineuality J. Math. Anal. Appl. 998) ] J. E. Pečaić F. Poschan and Y. L. Tong Convex functions patial odeings and statistical applications vol. 87 of Mathematics in Science and Engineeing Academic Pess Inc. Boston MA 99. 5] M. Z. Saıkaya On the Hemite-Hadamad-type ineualities fo co-odinated convex function via factional integals Integal Tansfoms Spec. Funct. 5 4) ] M. Z. Saikaya and N. Aktan On the genealization of some integal ineualities and thei applications Math. Comput. Modelling 54 ) ] M. Z. Saikaya E. Set H. Yaldiz and N. Başak Hemite-Hadamad s ineualities fo factional integals and elated factional ineualities Mathematical and Compute Modelling 579) 3) ] M. Tunç and H. Kavumaci-Önalan On some ineualities fo s-logaithmically convex functions in the second sense via factional integals Ceat. Math. Infom. 3 4) ] J. Wang J. Deng and M. Fečkan Hemite-Hadamad-type ineualities fo -convex functions based on the use of Riemann-Liouville factional integals Ukainian Math. J. 65) 3) 93. 3] J. Wang X. Li M. Fečkan and Y. Zhou Hemite-Hadamad-type ineualities fo Riemann- Liouville factional integals via two kinds of convexity Appl. Anal. 9 3) ] Y. Zhang and J. Wang On some new Hemite-Hadamad ineualities involving Riemann- Liouville factional integals J. Ineual. Appl. 3) Aticle 7 pp. 3] C. Zhu M. Fečkan and J. Wang Factional integal ineualities fo diffeentiable convex mappings and applications to special means and a midpoint fomula J. Appl. Math. Stat. Infom. 8 ) 8. Univesity of 8 May 945 Guelma Algeia. addess: khaledv4@yahoo.f tchiheb@yahoo.f Univesity of Badji-Mokhta Annaba Algeia. addess: badimeftah@yahoo.f

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