The Hadamard s inequality for quasi-convex functions via fractional integrals

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1 Annls of the University of Criov, Mthemtics nd Computer Science Series Volume (), 3, Pges ISSN: The Hdmrd s ineulity for usi-convex functions vi frctionl integrls M E Özdemir nd Çetin Yildiz Abstrct In this pper, we give the Riemnn-Liouville frctionl integrls definitions We use these Riemnn-Liouville frctionl integrls to estblish some new integrl ineulities for usi-convex functions Also, some pplictions for specil mens of rel numbers re provided Mthemtics Subject Clssifiction 6D7, 6D, 6D5, 6A33 Key words nd phrses Qusi convex functions, Hdmrd s ineulity, Riemnn-Liouville frctionl integrl, Power-men ineulity Introduction Let rel function f be defined on some nonempty intervl I of rel line R The function f is sid to be usi-convex on I if ineulity f (tx + ( t)y) mx {f(x), f(y)} (QC) holds for ll x, y I nd t [, ] (see [5]) Let f : I R R be convex function on the intervl of I of rel numbers nd, b I with < b The following double ineulity ( ) + b f b b f(x)dx f() + f(b) is well-known in the literture s Hdmrd s ineulity For severl recent results concerning the ineulity () we refer the interested reder to ([], [], [3], [9], []- [5]) Clerly, ny convex function is usi-convex function Furthermore, there exist usi-convex functions which re not convex For exmple, consider the following: Let f : R + R, f(x) = ln x, x R + This function is usi-convex However f is not convex functions Definition for ll x, y I [5] The mpping f : I R is Jensen- or J-usi-convex if ( ) x + y f mx {f(x), f(y)} (JQC) Definition [5] For I R, the mpping f : I R is Wright-usi-convex if, for ll x, y I nd t [, ], one hs the ineulity [f(tx + ( t)y) + f(( t)x + ty)] mx {f(x), f(y)} (W QC) Received Mrch 5, 3 Accepted September, 3 () 67

2 68 M E ÖZDEMİR AND Ç YILDIZ or euivlently [f(y) + f(x + δ)] mx {f(x), f(y + δ)} for every x, y + δ I with x < y nd δ > In [5], Drgomir nd Perce proved the following results connected with the ineulity (): Theorem [5] Let f : I R be Wright-usi-convex mp on I nd suppose, b I R with < b nd f L [, b] Then we hve the ineulity b b f(t)dt mx {f(), f(b)} () Theorem [5] Let W QC(I) denote the clss of Wright-usi-convex functions on I R Then QC(I) W QC(I) JQC(I) In [], Ion proved the following results connected with usi-convex function: Theorem 3 [] Assume, b R with < b nd f : [, b] R is differentible function on (, b) If f is usi-convex on [, b] then the following ineulity holds true f() + f(b) b f(x)dx b b mx { f (), f (b) } (3) Theorem [] Assume, b R with < b nd f : [, b] R is differentible function on (, b) Assume p R with p > If f p/(p) is usi-convex on [, b] then the following ineulity holds true f() + f(b) b b f(x)dx b [ mx (p + ) /p { f p p (), f (b) In [], Alomri et l proved the following theorem for usi-convex function: p p }] p p Theorem 5 [] Let f : I R R be differentible mpping on I,, b I with < b If f is usi-convex on [, b],, then the following ineulity holds: f() + f(b) b f(x)dx b b ( { mx f (), f (b) }) (5) Now we give some necessry definitions nd mthemticl preliminries of frctionl clculus theory which re used throughout this pper Definition 3 [] Let f L [, b] The Riemnn-Liouville integrls J α f nd + Jb α f of order α > with re defined by J α +f(x) = x (x t) α f(t)dt, x > Γ(α) nd Jb α f(x) = b Γ(α) x (t x) α f(t)dt, x < b respectively where Γ(α) = e u u α du Here is J f(x) = J + b f(x) = f(x) ()

3 THE HADAMARD S INEQUALITY FOR QUASI-CONVEX FUNCTIONS 69 In the cse of α =, the frctionl integrl reduces to the clssicl integrl For some recent results connected with frctionl integrl ineulities see ([]-[8]) In [3], Srıky et l proved the following Lemm nd estblished some ineulities for frctionl integrls Lemm 6 [3] Let f : [, b] R, be differentible mpping on (, b) with < b If f L[, b], then the following eulity for frctionl integrls holds: f() + f(b) Γ(α + ) (b ) α [J α +f(b) + J α b f()] = b [( t) α t α ] f (t + ( t)b)dt The im of this pper is to estblish Hdmrd type ineulities for usi-convex functions vi Riemnn-Liouville frctionl integrl MAIN RESULTS Theorem Let f : [, b] R, be positive function with < b nd f L [, b] If f is usi-convex function on [, b], then the following ineulity for frctionl integrls holds: with α > Γ(α + ) (b ) α [J α +f(b) + J b α f()] mx {f(), f(b)} Proof Since f is usi-convex function on [, b], we hve f (t + ( t)b) mx {f(), f(b)} nd f (( t) + tb) mx {f(), f(b)} By dding these ineulities we get [f (t + ( t)b) + f (( t) + tb)] mx {f(), f(b)} (6) Then multiplying both sides of (6) by t α nd integrting the resulting ineulity with respect to t over [, ], we obtin t α f (t + ( t)b) dt + = b t α f (( t) + tb) dt ( ) α b u f(u) du b b b + mx {f(), f(b)}, α ie Γ(α + ) (b ) α [J α +f(b) + J b α f()] mx {f(), f(b)} The proof is complete ( ) α v f(v) dv b b Remrk If we choose α = in Theorem, we hve the ineulity ()

4 7 M E ÖZDEMİR AND Ç YILDIZ Theorem Let f : [, b] R, be differentible mpping on (, b) with < b If f is usi-convex on [, b] nd α >, then the following ineulity for frctionl integrls holds: f() + f(b) Γ(α + ) (b ) α [J α +f(b) + J b α f()] (7) b ( ) α + α mx { f (), f (b) } Proof Using Lemm 6 nd the usi-convex of f with properties of modulus, we hve f() + f(b) Γ(α + ) (b ) α [J α +f(b) + J b α f()] b ( t) α t α f (t + ( t)b) dt b = b ( t) α t α mx { f (), f (b) } dt mx { f (), f (b) } = b ( ) α + α mx { f (), f (b) } where we use the fct tht [( t) α t α ] dt + [t α ( t) α ] dt ( t) α t α dt = [( t) α t α ] dt + [t α ( t) α ] dt = ( ) α + α which completes the proof Remrk If we choose α = in (7), then the ineulity (7) reduces to the ineulity (3) of Theorem 3 Theorem 3 Let f : [, b] R, be differentible mpping on (, b) with < b such tht f L [, b] If f is usi-convex on [, b], nd p >, then the following ineulity for frctionl integrls holds: f() + f(b) Γ(α + ) (b ) α [J α +f(b) + J b α f()] (8) b ( { mx f (), f (b) }) (αp + ) p where p + = nd α [, ]

5 THE HADAMARD S INEQUALITY FOR QUASI-CONVEX FUNCTIONS 7 Proof From Lemm 6 nd using Hölder ineulity with properties of modulus, we hve f() + f(b) Γ(α + ) (b ) α [J α +f(b) + J b α f()] b ( t) α t α f (t + ( t)b) dt b p ( t) α t α p dt f (t + ( t)b) dt We know tht for α [, ] nd t, t [, ], hence t α t α t t α, ( t) α t α p dt = = t αp dt [ t] αp dt + αp + [t ] αp dt Since f is usi-convex on [, b], we get f() + f(b) Γ(α + ) (b ) α [J α +f(b) + J b α f()] b ( { mx f (), f (b) }) (αp + ) p which completes the proof Remrk 3 If in Theorem 3, we choose α =, then the ineulity (8) becomes the ineulity () of Theorem Theorem Let f : [, b] R, be differentible mpping on (, b) with < b such tht f L [, b] If f is usi-convex on [, b] nd, then the following ineulity for frctionl integrls holds: f() + f(b) Γ(α + ) (b ) α [J α +f(b) + J b α f()] (9) b ( ) (mx { f α + α (), f (b) }) with α >

6 7 M E ÖZDEMİR AND Ç YILDIZ Proof From Lemm 6 nd using power-men ineulity with properties of modulus, we cn write f() + f(b) Γ(α + ) (b ) α [J α +f(b) + J b α f()] b ( t) α t α f (t + ( t)b) dt b ( t) α t α dt ( t) α t α f (t + ( t)b) dt Since f is usi-convex on [, b], we hve f() + f(b) Γ(α + ) (b ) α [J α +f(b) + J b α f()] b ( { mx f (), f (b) }) ( t) α t α dt = b ( { mx f (), f (b) }) [( t) α t α ] dt + [t α ( t) α ] dt = b ( ) (mx { f α + α (), f (b) }), which completes the proof Remrk We note tht the obtined ineulity (9) is better thn the ineulity (8) mening tht the pproch vi the power men ineulity is better pproch thn tht through Hölder s ineulity Remrk 5 If in Theorem, we choose α =, then the ineulity (9) becomes the ineulity (5) of Theorem 5 3 APPLICATIONS TO SPECIAL MEANS We now consider the mens for rbitrry rel numbers α, β (α β) We tke () Arithmetic men : () Logrithmic men: L(α, β) = A(α, β) = α + β, α, β R + α β ln α ln β, α = β, α, β, α, β R+ (3) Generlized log men: [ β n+ α n+ ] n L n (α, β) =, n Z\{, }, α, β + (n + )(β α) Now using the results of Section, we give some pplictions for specil mens of rel numbers

7 THE HADAMARD S INEQUALITY FOR QUASI-CONVEX FUNCTIONS 73 Proposition 3 Let, b R +, < b nd n Z Then, we hve A( n, b n ) L n n(, b) b mx { n, b n } Proof The ssertion follows from Theorem pplied to the usi-convex mpping f(x) = x n, x R nd α = Proposition 3 Let, b R +, < b nd n Z Then, for ll, we hve A( n, b n ) L n n(, b) b ( { mx ( n ), ( b n ) }) Proof The ssertion follows from Theorem pplied to the m-convex mpping f(x) = x n, x R nd α = References [] M Alomri nd M Drus, On the Hdmrd s ineulity for log-convex functions on the coordintes, Journl of Ineulities nd Applictions (9), Article ID 837, 3 pges [] M Alomri, M Drus nd SS Drgomir, Ineulities of Hermite-Hdmrd s type for functions whose derivtives bsolute vlues re usi-convex, RGMIA Res Rep Coll, Supplement, Article [3] MK Bkul, ME Özdemir, J Pečrić, Hdmrd type ineulities for m convex nd (α m) convex functions, J Ine Pure nd Appl Mth 9 (8), no, Article 96 [] S Belrbi nd Z Dhmni, On some new frctionl integrl ineulities, J Ine Pure nd Appl Mth (9), no 3, Article 86 [5] Z Dhmni, New ineulities in frctionl integrls, Interntionl Journl of Nonliner Science 9 (), no, [6] Z Dhmni, On Minkowski nd Hermite-Hdmrd integrl ineulities vi frctionl integrtion, Ann Funct Anl (), no, 5 58 [7] Z Dhmni, L Tbhrit, S Tf, Some frctionl integrl ineulities, Nonl Sci Lett A (), no, 55 6 [8] Z Dhmni, L Tbhrit, S Tf, New generliztions of Gruss ineulity using Riemnn- Liouville frctionl integrls, Bull Mth Anl Appl (), no 3, [9] SS Drgomir nd CEM Perce, Selected Topics on Hermite-Hdmrd Ineulities nd Applictions, RGMIA Monogrphs, Victori University, [] SS Drgomir nd CEM Perce, Qusi-convex functions nd Hdmrd s ineulity, Bull Austrl Mth Soc 57 (998), [] R Gorenflo, F Minrdi, Frctionl clculus: integrl nd differentil eutions of frctionl order, Springer Verlg, Wien, 997, 3 76 [] DA Ion, Some estimtes on the Hermite-Hdmrd ineulity through usi-convex functions, Annls of University of Criov, Mth Comp Sci Ser 3 (7), 8 87 [3] MZ Srıky, E Set, H Yldız nd N Bşk, Hermite-Hdmrd s ineulities for frctionl integrls nd relted frctionl ineulities,, 57 (3), no 9-, 3 7 [] E Set, ME Özdemir nd SS Drgomir, On the Hermite-Hdmrd ineulity nd other integrl ineulities involving two functions, Journl of Ineulities nd Applictions (), Article ID 8, 9 pges [5] ME Özdemir, SS Drgomir nd Ç YILDIZ, The Hdmrd Ineulity for convex function vi frctionl integrls, Act Mth Sci 33B (3), no 5, (M E Özdemir, Çetin Yildiz) Deprtment of Mthemtics, Attürk University, KK Eduction Fculty, 5, Cmpus, Erzurum, TURKEY E-mil ddress: emos@tuniedutr, yildizc@tuniedutr