Some generalized Hermite-Hadamard type integral inequalities for generalized s-convex functions on fractal sets

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1 Kılıçman Saleh Advances in Dierence Equations 05 05:30 DOI 086/s R E S E A R C H Open Access Some generalized Hermite-Hadamard type integral inequalities or generalized s-convex unctions on ractal sets Adem Kılıçman * Wedad Saleh * Correspondence: akilic@upmedumy Department o Mathematics Institute or Mathematical Research University Putra Malaysia Serdang Malaysia Full list o author inormation is available at the end o the article Abstract In this article some new integral inequalities o generalized Hermite-Hadamard type or generalized s-convex unctions in the second sense on ractal sets have been established MSC: 6A5; 6D07; 6D5; 53C Keywords: s-convex unctions; ractal space; local ractional derivative Introduction The convexity o unctions is an important concept in the class mathematical analysis course it plays a signiicant role in many ields or example in biological system economy optimization so on 7] Furthermore there are a lot o several inequalities related to the class o convex unctions For example Hermite-Hadamard s inequality is one o the well-known results in the literature which can be stated as ollows Theorem Hermite-Hadamard s inequality Let be a convex unction on a ] with < a I is integral on a ] then a + a a x dx + a In 8] Dragomir Fitzpatrick demonstrated a variation o Hadamard s inequality which holds or s-convex unctions in the second sense Theorem Let : R + R + be an s-convex unction in the second sense 0<s < a R + < a I L a ] then s a + a a x dx + a s + In recent years ractional calculus played an important part in ractal mathematics engineering In the sense o Melbrot a ractal set is the one whose Hausdor dimension strictly exceeds the topological dimension 9 5] Many researchers studied the properties o unctions on ractal space constructed many kinds o ractional calculus by 05 Kılıçman Saleh This article is distributed under the terms o the Creative Commons Attribution 40 International License which permits unrestricted use distribution reproduction in any medium provided you give appropriate credit to the original authors the source provide a link to the Creative Commons license indicate i changes were made

2 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page o 5 using dierent approaches 6 8] Particularly in 9] Yang stated the analysis o local ractional unctions on ractal space systematically which includes local ractional calculus the monotonicity o unction The outline o this article is as ollows In Section we state the operations with real line number ractal sets some deinitions are given Some integral inequalities o generalized Hermite-Hadamard type or generalized s-convex unctions in the second sense are studied in Section 3 Finally some applications are also illustrated in Section 4 The conclusions are in Section 5 Preliminaries Let R α be the real line numbers on ractal space Then by using Gao-Yang-Kang s concept one can explain the deinitions o the local ractional derivative local ractional integral as in 9 3] Now i r α rα rα 3 Rα 0 < α then r α + rα Rα r αrα Rα r α + rα rα + rα r + r α r + r α 3 r α +rα + rα 3 rα + rα +rα 3 4 r αrα rα rα r r α r r α 5 r αrα rα 3 rα rα rα 3 6 r αrα + rα 3 rα rα +rα rα 3 7 r α +0α 0 α + r α rα rα α α r α rα Let us state some deinitions about the local ractional calculus on R α Deinition 9] A non-dierentiable unction y: R R α is called local ractional continuous at x 0 i or any ε >0thereexistsδ >0suchthat yx yx0 < ε α holds or x x 0 < δ whereε δ R y C α a i it is local ractional continuous on the interval a Deinition 9] The local ractional derivative o ymoorderα at m m 0 is deined by y α m 0 dα ym dm α ym ym 0 lim m m mm0 0 m m 0 α where Ɣm 0 mz e m dm Ithereexistsy n+α md α m Dα mym n +timesor any m I Rtheny D n+α I n 0 Deinition 3 9] Thelocalractionalintegralounctionymoorderα is deined by where y C α a ] I a α ym lim t 0 ytdt α n yt i t i α i

3 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page 3 o 5 with t i t i+ t i t max{ t i : i n }wheret i t i+ ] i 0n t 0 < t < < t n < t n a is a partition o the interval a ] In 4] the authors introduced the generalized convex unction established the generalized Hermite-Hadamard s inequality on ractal space Let : I R R α or any x x I γ 0 ] i the ollowing inequality γ x + γ x γ α x + γ α x holds then is called a generalized convex unction on I Inα wehaveaconvex unction convexity is deined only in geometrical terms as being the property o a unction whose graph bears tangents only under it 5] Theorem Generalized Hermite-Hadamard s inequality Let a I a α be a generalized convex unction on a ] with < a Then a + a a α Iα a x + a α Note that it will be reduced to the class Hermite-Hadamard s inequality iα In 3] Mo Sui introduced the deinitions o two kinds o generalized s-convex unctions on ractal sets as ollows Deinition 4 i A unction : R + R α is called generalized s-convex 0<s <intheirstsensei γ x + γ x γ sα x +γ sα x 3 or all x x R + all γ γ 0 with γ s + γ s we denote this class o unctions by GKs ii A unction : R + R α is called generalized s-convex 0<s < in the second sense i inequality 3holdsorallx x R + all γ γ 0 with γ + γ we denote this class o unctions by GKs In the same paper 3] Mo Sui proved that all unctions rom GKs s 0 are non-negative 3 Main results In 6] the authors demonstrated a variation o generalized Hadamard s inequality which holds or a generalized s-convex unction in the second sense Now we will give another proo or generalized s-hadamard sinequality Theorem 3 Let : R + R α + be a generalized s-convex unction in the second sense 0< s < a R + with < a I L a ] then a + a a α Iα a x Ɣ + sα a +a 4 Ɣ + s +α

4 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page 4 o 5 Proo Since is generalized s-convex in the second sense then γ + γ a γ + γ a γ 0 ] Integrating the above inequality with respect to γ on 0 ] we have 0 I α γ + γ a a 0 I α γ Let x γ + γ a Thenwehave + a 0 I α γ Ɣ + sα Ɣ + s +α 0 I α γ + γ a a α a Iα x Now it ollows that a α Iα a x a α Iα a x Ɣ + sα a +a Ɣ + s +α a+ b Then the second inequality in 4isproved Inordertoprovetheirstinequalityin4 we use the ollowing inequality: x + x x + x x x I 5 Now assume that x γ + γ a x γ + γ a with γ 0 ] Then we get by inequality 5that a + a γ + γ a + γ + γ a γ 0 ] By integrating both sides o the above inequalities over 0 ] we have a + a dγ α a α Iα a x Then it ollows that 0 a + a a α Iα a x This completes the proo Remark 3 I we set c Ɣ+sαƔ+α Ɣ+s+α inequality o 4 or s 0 ] then it is best possible in the second

5 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page 5 o 5 As the unction :0] 0 α α ]givenbyx x sα is generalized s-convex in the second sense a α Iα a x Ɣ + sα Ɣ + s +α Ɣ + sα Ɣ + s +α 0 x dx α Ɣ + sα 0 + Ɣ + s +α Similarly i α then inequalities 4 reduce to inequalities Theorem 3 Let A: 0] R α be a unction such as Aγ a α Iα a γ x + γ a + a γ 0 ] where : a ] R α is a generalized s-convex unction in the second sense s 0 ] a R + < a L a ] Then i A GKs on 0 ] ii we have the inequality Aγ a + a γ 0 ] 6 iii the ollowing inequality also holds: A min { A γ A γ } γ 0 ] 7 where A γ γ a α Iα a x+ γ a + a Ɣ + A γ Ɣ + s +α + γ a + γ + a γ + γ + a or γ 0 ] iv I à max{a γ A γ } γ 0 ] then à { Ɣ + γ +a + α γ Ɣ + s +α a + a }

6 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page 6 o 5 Proo i Let γ γ 0 ] μ μ 0withμ + μ then Aμ γ + μ γ a α Iα a μ γ + μ γ x + μ γ + μ γ a + a { a α Iα a μ γ x + γ a + a + μ γ x + γ a } + a μ Aγ +μ Aγ which implies that A GKs on 0 ] ii Let γ 0 ] by the change o variable m γ x + γ +a wehave Aγ γ α a α γ + γ +a I α γ a + γ +a m b b α b Iα b m By using the irst generalized Hermite-Hadamard inequality we have b b α b Iα b m inequality 6isobtained I γ 0theinequality a + a a + a b + b a + a also holds iii By using the second part o generalized Hadamard s inequality we get b b α b Iα b m Ɣ + sα b +b Ɣ + s +α Ɣ + sα Ɣ + s +α + γ a + γ + a A γ γ 0 ] γ + γ + a I γ 0 then the inequality a + a holdsasitisequivalentto A0 A 0 α Ɣ + sα Ɣ + s +α Ɣ + s +α Ɣ + sα α a + a 0 α a + a

7 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page 7 o 5 we know that or s 0 Since a + a 0 α γ x + γ a + a γ x+ γ a + a or γ 0 ] x a ] then we obtain Aγ a α Iα a γ x + γ a + a γ a α Iα a x+ γ A γ a + a Then the proo o inequality 7is complete iv We have A γ Ɣ + sα γ + γ + a Ɣ + s +α Ɣ + sα Ɣ + s +α γ + γ + γ a + γ + a a + a ] + γ a + γ a + a Ɣ + sα γ +a + α γ Ɣ + s +α γ 0 ] a + a ] ] Since a α Iα a x Ɣ + sα a +a Ɣ + s +α γ a + a α Ɣ + sα a + a γ Ɣ + s +α then Ɣ + sα A γ γ a +a Ɣ + s +α + α Ɣ + sα a + a γ Ɣ + s +α the proo o Theorem 3 is complete

8 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page 8 o 5 Remark 3 In particular: I we choose s intheorem3thenweget: a a α Iα a γ x + γ a + a { α min γ a α Iα a x+ γ α a + a γ + γ a + a b Since we have + γ a + γ a } + a { α Ã max γ a α Iα a x+ γ α Ãγ γ + γ + a a + a γ α +a + α γ α + γ a + γ + a a + a Now i one chooses α intheorem3 then we can easily obtain: a a { min γ s a s + b Similarly we have { Ã max γ s s + γ x + γ + a γ + γ + a a γ + γ + a dx x dx + γ s a + a + γ a + γ + a x dx + γ s a + a + γ a + γ + a ] } } } Ãγ γ s +a ] + γ s a + a s + or γ 0 ] 3 I one considers α s intheorem3thenweget:

9 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page 9 o 5 a b a { min γ γ x + γ a + a dx a γ + γ + a { Ã max γ a a γ + γ + a a + a x dx + γ + γ a + γ + a a + a x dx + γ + γ a + γ + a } } Ãγ γ +a ] a + a + γ or γ 0 ] Theorem 33 Let g :0] R α be a unction such as gγ a α γ x + γ x dx α dx α γ 0 ] where : a ] R α + is a generalized s-convex unction in the second sense s 0 ] a R + which < a L a ] Then: i g GKs in 0 ] I GKs then g GK s ii gγ + g γ or all γ 0 ] gγ is symmetric about γ iii We have the inequality gγ Aγ 4 a + a or γ 0 ] 8 iv We have the inequality gγ min { g γ g γ } 9 where g γ γ + γ ] a α Iα a x ] Ɣ + sα g γ a + γ + γ a + γ a +γ a + a ] Ɣ + s +α or γ 0 ]

10 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page 0 o 5 Proo i Take {γ γ } 0 ] γ + γ t t D GK s thenwehave gγ t + γ t a α a α γ t + γ t x + γ t + γ t x dx α dx α γ t x + x t x + γ t x + x t x ] dx α dx α γ + γ a α a α γ gt +γ gt which implies that g GKs in 0 ] ii Let γ 0 ] then g γ + a α γ + a α γ g γ gγ is symmetric about γ because gγ g γ iii Let us observe that gγ a α Now since x is ixed in a ] then the unction A x :0] R α can be given by A x γ a α t x + x t x dx α dx α t x + x t x dx α dx α x + γ x dx α dx α x + x + γ dx α dx α a γ x a α + γ x dx α dx α γ x + γ x dx α a α Iα a γ x + γ x

11 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page o 5 As it was shown in the proo o Theorem 3orγ 0 ] we have equality A x γ b b α b Iα b m where b γ a + γ x b γ + γ x By using the generalized Hermite- Hadamard inequality we have b b α b Iα b m b + b γ a + a + γ x or all γ 0 x a ] Integrating on a ]overx wehave gγ A γ or γ 0 Further since gγ g γ then the proo o inequality 8 is done or γ 0 I γ 0 or γ theninequality8alsoholds iv Since γ x + γ x γ x + γ x or all x x a ]γ 0 ] integrating the above inequality on a ] wehave a α γ a α γ x + γ x dx α dx α + γ a α x dx α dx α x dx α dx α γ a α Iα a x + γ a α Iα a x γ + γ a α Iα a x The proo o the irst part in 9isdone By the second part o the generalized Hermite-Hadamard inequality we obtain A x γ b b α b Iα b m Ɣ + sα γ a + γ x + γ a + γ x Ɣ + s +α where b γ a + γ x b γ + γ x γ 0 ] Integrating this inequality on a ]overx then gγ Ɣ + sα Ɣ + s +α a α Iα a γ + γ x + a α Iα a γ a + γ x ]

12 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page o 5 A simple calculation shows that a α Iα a γ a + γ x c c α c Iα c m Ɣ + sα c +c ] Ɣ + s +α Ɣ + sα a + ] γ a + γ Ɣ + s +α where c a c γ a + γ γ 0 Similarly or γ 0 Then a α Iα a γ + γ x Ɣ + sα a + ] γ + γ a Ɣ + s +α ] Ɣ + sα gγ a + γ + γ a + γ a + γ a + a ] Ɣ + s +α I γ 0orγ then this inequality also holds Remark 33 I α in the above theorem then gγ a γ x + γ x dx dx γ 0 ] { γ gγ min s + γ s] a x dx a a + γ a + s + γ a + γ a + γ a + a ] } Theorem 34 Let us consider that a sum o A belongs to GK s A n a i γ i where a i γ then i supa α n i a i0 α n i a i ii A is symmetric about γ iii A GK s i γ x + γ x dx α dx α

13 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page 3 o 5 Proo i a i γ γ + γ i x dx α dx α Since i are generalized s-convex unctions we get a i γ γ + γ α α a i 0 α a i i x dx α dx α i i x dx α dx α i x dx α dx α ii a i γ is symmetric about γ since a iγ a i γ i Then A γ Aγ A also is iii Since a i γ x + γ x γ a i x + γ a i x then A γ x + γ x n a i γ x + γ x i γ n a i x + γ i γ Ax + γ Ax n a i x i that is A GK s 4 Applications to special means We now consider the applications o our theorems to the ollowing generalized means: A a aα + aα a α a 0 a α K a + a α a a 0 α G a a α aα a 0 In 3] the ollowing example is given Let 0 < s <a α aα aα 3 Rα Deineorx R + a α n n 0 a α nsα + a α 3 n >0 I a α 0α 0 α a α 3 aα then GK s

14 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page 4 o 5 Proposition 4 Let a R + < a a then the ollowing inequalities hold: K a + ] ] Ɣ + 3α α G a α A a 0 γ + γ α K a +γ α γ α G a ] γ α + γ α α Ɣ + 3α ] K a + ] α G a α γ α γ α A a Proo I GKs on a ]orsomeγ 0 ] s 0 ] then in Theorem 33 i :0] 0 α α ] xx α wherex a ]s so a α γ α + γ α Ɣ + 3α + α γ α γ α γ x + γ x α dx α dx α ] α a α + aα a α + a α aα + aα Then by Theorem 33weget ] a α Ɣ + 3α + a α + a α aα a α + aα Then we obtain inequality 0 By applying Theorem 33 we obtain inequality as ollows: γ + γ α a α + a α ] + γ α γ α a α aα α γ α + γ α α Ɣ + 3α α γ α γ α a α + a α α ] a α + a α + ] α α aα aα 5 Conclusion In this article we have established some new integral inequalities o generalized Hermite- Hadamard type or generalized s-convex unctions in the second sense on ractal sets R α 0<α < In particular our results extend some important inequalities in a classical situation; when α some relationships between these inequalities the classical inequalities have been established Finally we have also given some applications or these inequalities on ractal sets Competing interests The authors declare that they have no competing interests Authors contributions All authors jointly worked on deriving the results approved the inal manuscript

15 Kılıçman Saleh Advances in Dierence Equations 05 05:30 Page 5 o 5 Author details Department o Mathematics Institute or Mathematical Research University Putra Malaysia Serdang Malaysia Department o Mathematics University Putra Malaysia Serdang Malaysia Acknowledgements The authors would like to thank the reerees or valuable suggestions comments which helped the authors to improve this article substantially Received: 9 June 05 Accepted: 5 September 05 Reerences Kılıçman A Saleh W: Some inequalities or generalized s-convex unctions JP J Geom Topol Ruel JJ Ayres MP: Jensen s inequality predicts eects o environmental variation Trends Ecol Evol Li M Wang J Wei W: Some ractional Hermite-Hadamard inequalities or convex Godunova-Levin unctions Facta Univ Ser Math Inorm Liu W: Ostrowski type ractional integral inequalities or MT-convex unctions Miskolc Math Notes Lin Z Wang JR Wei W: Fractional Hermite-Hadamard inequalities through r-convexunctions viapowermeans Facta Univ Ser Math Inorm Liu W: New integral inequalities involving beta unction via P-convexity Miskolc Math Notes Budak H Sarikaya MZ: Some new generalized Hermite-Hadamard inequality or generalized convex unction applications RGMIA Res Rep Collect Dragomir SS Fitzpatrick S: The Hadamard inequalities or s-convex unctions in the second sense Demonstr Math Golmankhaneh AK Baleanu D: On a new measure on ractals J Inequal Appl Kolwankar KM Gangal AD: Local ractional calculus: a calculus or ractal space-time In: Fractals: Theory Applications in Engineering pp 7-8 Springer London 999 Yang Y-J Baleanu D Yang X-J: Analysis o ractal wave equations by local ractional Fourier series method Adv Math Phys 03 Article ID Noor MA Noor KI Awan MU Khan S: Fractional Hermite-Hadamard inequalities or some new classes o Godunova-Levin unctions Appl Math In Sci Liu W: Some Ostrowski type inequalities via Riemann-Liouville ractional integrals or h-convex unctions J Comput Anal Appl Erden S Sarikaya MZ: Generalized Bullen type inequalities or local ractional integrals its applications RGMIA Res Rep Collect Budak H Sarikaya MZ Yildirim H: New inequalities or local ractional integrals RGMIA Res Rep Collect Babakhani A Datardar-Gejji V: On calculus o local ractional derivatives J Math Anal Appl Carpinteri A Chiaia B Cornetti P: Static-kinematic duality the principle o virtual work in the mechanics o ractal media Comput Methods Appl Mech Eng Zhao Y Cheng DF Yang XJ: Approximation solutions or local ractional Schrödinger equation in the one-dimensional Cantorian system Adv Math Phys 03 ArticleID Yang XJ: Advanced Local Fractional Calculus Its Applications World Science Publisher New York 0 0 Yang XJ Baleanu D Machado JAT: Mathematical aspects o Heisenberg uncertainty principle within local ractional Fourier analysis Bound Value Probl Yang AM Chen ZS Srivastava HM Yang XJ: Application o the local ractional series expansion method the variational iteration method to the Helmholtz equation involving local ractional derivative operators Abstr Appl Anal 03 Article ID Yang XJ Baleanu D Khan Y Mohyud-Din ST: Local ractional variational iteration method or diusion wave equations on Cantor sets Rom J Phys Mo H Sui X: Generalized s-convex unctions on ractal sets Abstr Appl Anal 04 Article ID Mo H Sui X Yu D: Generalized convex unctions on ractal sets two related inequalities Abstr Appl Anal 04 Article ID Hörmer L: Notions o Convexity Birkhäuser Basel Mo H Sui X: Hermite-Hadamard type inequalities or generalized s-convex unctions on real linear ractal set R α 0 < α < 05 arxiv: