New estimates on generalization of some integral inequalities for (α, m)-convex functions

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1 Contemporary Analysis Applied M athetics Vol. No New estites on generalization of some integral ineualities for α m)-convex functions İmdat İşcan Giresun University Faculty of Arts Sciences Department of Mathetics Giresun Turkey e-il: imdat.iscan@giresun.edu.tr imdati@yahoo.com Abstract. In this paper it has been studied a unified approach to establish midpoint trapezoid Simpson s ineualities for functions whose derivatives in absolute value at certain power are α m)-convex. Key words. α m)-convex function Hermite-Hadard s ineuality Simpson type ineualities trapezoid ineuality midpoint ineuality. Introduction Let f : I R R be a convex function defined on the interval I of real numbers a b I with a < b. The following double ineuality is well known in the literature as Hermite-Hadard integral ineuality have ) a + b f b b a a fa) + fb)..) The class of α m)-convex functions was first introduced in [4] it is defined as follows: The function f : [ b] R b > is said to be α m)-convex α m) [ ] if we for all x y [ b] t [ ]. f tx + m t)y) t α fx) + m t α )fy) It can be easily that for α m) ) α ) ) m) ) α ) one obtains the following classes of functions: increasing α-starshaped starshaped m-convex convex α-convex. Denote by K α mb) the set of all α m)-convex functions on [ b] for which f). For recent results generalizations concerning α m)-convex functions see [ 8 ]. The following ineuality is well known in the literature as Simpson s ineuality. 53

2 New estites on generalization of some integral ineualities Let f : [a b] R be a four times continuously differentiable pping on a b) f 4) = f 4) x) <. Then the following ineuality holds: sup x ab) [ )] fa) + fb) a + b + f b 3 b a f 4) b a) a In recent years ny authors have studied error estitions for Simpson s ineuality; for refinements counterparts generalizations new Simpson s type ineualities see [8 ]. In this paper in order to provide a unified approach to establish midpoint ineuality trapezoid ineuality Simpson s ineuality for functions whose derivatives in absolute value at certain power are α m)-convex we derive a general integral identity for differentiable functions. Main results Throughout this section we will assume that I is an interval. In order to generalize the classical trapezoid midpoint Simpson type ineualities prove them we need the following lem. Lem. Let f : I R R be a differentiable pping on I such that f L[ mb] m ] mb I with a < b then for θ λ [ ] the following euality holds: θ) λf) + λ) fmb)) + θf λ) + λmb) m b a) = m b a) λ t θ) f t + t) [ λ) + λmb]) dt + λ) mb t θ) f tmb + t) [ λ) + λmb]) dt. A simple proof of the euality can be done by performing an integration by parts in the integrals from the right side changing the variable. The details are left to the interested reader. Theorem. Let f : I [ ) R be a differentiable pping on I such that f L[ mb] m ] b I with a < b θ λ [ ]. If f is α m)-convex on 54

3 İmdat İşcan [ b] for α [ ] then the following ineuality holds: θ) λf) + λ) fmb)) + θfmc) m b a) mb m b a) A θ) min B θ λ α m) B θ λ α m).) B θ λ α m) = B θ λ α m) = λ f ) A θ α) + m f C) A 3 θ α) ) + λ) f mb) A θ α) + m f C) A 3 θ α) ) λ f mc) A 4 θ α) + m f a) A 5 θ α) ) + λ) f mc) A 4 θ α) + m f b) A 5 θ α) ) A θ) = θ θ + A θ α) = A 3 θ α) = θ A 4 θ α) = C = λ) a + λb. A 5 θ α) = θ) θ α+ α + ) α + ) θ α + + α + θ α+ α + ) α + ) αθ α + + α α + ) θ) α+ α + ) α + ) θ α + + α + θ)α+ α θ) α + α + ) α + ) α + α + ) Proof. Suppose that C = λ) a + λb. From Lem. using the properties of modulus the well known power mean ineuality we have θ) λf) + λ) fmb)) + θf λ) + λmb) m b a) m b a) λ t θ f t + t) mc) dt + λ) m b a) λ t θ f tmb + t) mc) dt t θ dt mb t θ f t + t) mc) dt 55

4 New estites on generalization of some integral ineualities + λ) t θ dt Since f is α m)-convex on [ b] we know that for t [ ] t θ f tmb + t) mc) dt..) f t + t) mc) t α f ) + m t α ) f C) f tmb + t) mc) t α f mb) + m t α ) f C). Hence by simple computation t θ t α f ) + m t α ) f C) ) dt = f ) A θ α) + m f C) A 3 θ α).3) t θ t α f mb) + m t α ) f C) ) dt = f mb) A θ α) + m f C) A 3 θ α).4) t θ dt = θ θ +..5) Thus using.3)-.5) in.) we obtain the following ineuality: θ) λf) + λ) fmb)) + θfmc) m b a) m b a) A θ) λ f ) A θ α) + m f C) A 3 θ α) ) + λ) f mb) A θ α) + m f C) A 3 θ α) ) In ineuality.) if we use eualities t θ f t + t) mc) dt =..6) θ t f tmc + t) ) dt t θ f tmb + t) mc) dt = θ t f tmc + t) mb) dt 56

5 İmdat İşcan by the similar process since f is α m) convex on [ b] for t [ ] we get f tmc + t) ) t α f mc) + m t α ) f a) f tmc + t) mb) t α f mc) + m t α ) f b). Similarly by simple computation θ t t α f mc) + m t α ) f a) ) dt = f mc) A 4 θ α) + m f a) A 5 θ α).7) θ t t α f mc) + m t α ) f b) ) dt = f mc) A 4 θ α) + m f b) A 5 θ α)..8) Thus using.5).7).8) in.) we have the following ineuality: θ) λf) + λ) fmb)) + θfmc) m b a) m b a) A θ) λ f mc) A 4 θ α) + m f a) A 5 θ α) ) + λ) f mc) A 4 θ α) + m f b) A 5 θ α) )..9) From ineualities.6).9) ineuality.) is obtained. This completes the proof. Corollary.3 Under the assumptions of Theorem. with = θ) λf) + λ) fmb)) + θfmc) m b a) m b a) min B θ λ α m) B θ λ α m). Rerk.4 In Corollary.3 i) If we choose λ = θ = 3 α = we have [ + mb f) + 4f 6 m b a) min mb ) ] + fmb) m b a) 3 B ) m B 3 ) m 57

6 New estites on generalization of some integral ineualities B 3 m ) B 3 m ) = 8 [ f ) + f mb) ] + 9m 34 f a + b ) = 9 34 f + mb ) + m 8 [ f a) + f b) ]. ii) If we choose λ = θ = α = we have f) + fmb) m b a) m b a) min B ) m B ) m B m ) B m ) = [ f ) + f mb) ] + m f a + b ) = f + mb ) + m [ f a) + f b) ]. iii) If we choose λ = θ = α = we have ) + mb f m b a) m b a) min B ) m B ) m B m ) B m ) = 4 [ f ) + f mb) ] + m 6 f C) = 6 f mc) + m 4 [ f a) + f b) ]. Corollary.5 Under the assumptions of Theorem. with λ = θ = 3 we have θ) λf) + λ) fmb)) + θfmc) m b a) ) 5 m b a) min B 8 3 ) α m B 3 ) α m. Particularly for α = m = we have [ a + b fa) + 4f 6 8 f a) 34 ) ] + fb) b b a) a + 9 f a+b ) ) 8 f b) ) 5 b a) f a+b ) 648 ) 58

7 İmdat İşcan which slightly improves ineuality in [ Theorem ] for s =. Corollary.6 Under the assumptions of Theorem. with λ = θ = we have ) f) + fmb) m b a) m b a) min B ) α m B ) α m B α m ) = ) f ) + αm 4 α + f a + b ) ) + f mb) + αm f a + b ) ) B α m ) = 4 α + ) α + ) + f + mb ) ) f + α α + 3) m + mb ) f b) ) α α + 3) m +. f a) ) Corollary.7 Under the assumptions of Theorem. with λ = θ = we have ) ) + mb f m b a) m b a) min B ) α m B ) α m B α m ) = ) f ) α α + 3) m + 4 α + ) α + ) f a + b ) ) + f mb) α α + 3) m + f a + b ) ) B α m ) = 4 ) α + f + mb + f + mb ) + αm ) f a) ) + αm ) f b). 59

8 New estites on generalization of some integral ineualities Theorem.8 Let f : I [ ) R be a differentiable pping on I such that f L[ mb] m ] b I with a < b θ λ [ ]. If f is α m)-convex on [ b] for α [ ] > then the following ineuality holds: θ) λf) + λ) fmb)) + θfmc) m b a) ) θ p+ + θ) p+ p m b a) min B 3 λ α m) B 4 λ α m) p +.) B 3 λ α m) = B 4 λ α m) = λ E λ α m) + λ) E λ α m) λ E 3 λ α m) + λ) E 4 λ α m) E λ α m) = f ) + αm f C) α + E λ α m) = f mb) + αm f C) α + E 3 λ α m) = f mc) + αm f a) α + E 4 λ α m) = f mc) + αm f b) α + C = λ) a + λb p + =. Proof. Suppose that C = λ) a + λb. From Lem. by Hölder s integral ineuality we have θ) λf) + λ) fmb)) + θfmc) m b a) m b a) λ t θ f t + m t) C) dt + λ) t θ f tmb + m t) C) dt m b a) λ + λ) p t θ p dt p t θ p dt f t + m t) C) dt f tmb + m t) C) dt..) 6

9 İmdat İşcan Since f is α m)-convex on [ b] we know that for t [ ] f t + m t) C) t α f ) + m t α ) f C) f tmb + m t) C) t α f mb) + m t α ) f C). Hence by simple computation t α f ) + m t α ) f C) dt = f ) + αm f C).) α + t α f mb) + m t α ) f C) dt = f mb) + αm f C).3) α + t θ p dt = θp+ + θ) p+..4) p + Thus using.)-.4) in.) we obtain the following ineuality θ) λf) + λ) fmb)) + θfmc) m b a) ) θ p+ + θ) p+ p f m b a) λ ) + αm f C) ) p + α + Similarly λ) f mb) + αm f C) α + )..5) f t + t) mc) dt = f tmc + t) ) dt t α f mc) + m t α ) f a) ) dt = f mc) + αm f a).6) α + f tmb + t) mc) dt = f tmc + t) mb) dt t α f mc) + m t α ) f b) ) dt = f mc) + αm f b)..7) α + 6

10 New estites on generalization of some integral ineualities By using.).6).7 ) in.) we get the following ineuality θ) λf) + λ) fmb)) + θfmc) m b a) ) θ p+ + θ) p+ p f m b a) λ mc) + αm f a) ) p + α + λ) f mc) + αm f b) α + )..8) From ineualities.5).8) ineuality.) is obtained. This completes the proof. Corollary.9 Under the assumptions of Theorem.8 with λ = θ = 3 we have [ ) ] + mb f) + 4f + fmb) m b a) p+ ) p + 6 m b a) 3 p + ) min E α m) + E α m) E 3 α m) + E 4 α m) E α m) = f ) + αm f ) a+b α + E α m) = f mb) + αm f ) a+b α + ) + αm f a) E 3 α m) = E 4 α m) = f ma+b) α + ) f ma+b) + αm f b) α + Rerk. In Corollary.9 if we take α = m = then we obtain the following ineuality [ ) ] a + b fa) + 4f + fb) b ) ) b a + p+ p 6 b a 3 p + ) a f ) a+b ) + f a) f ) a+b ) + f b) + which is the same of the ineuality in [ Corollary 3] Corollary. Under the assumptions of Theorem.8 with λ = θ = we have ) f) + fmb) m b a) p m b a) 4 p + min E α m) + E α m) E 3 α m) + E 4 α m).. 6

11 İmdat İşcan Corollary. Under the assumptions of Theorem.8 with λ = θ = we have ) ) m a + b) f m b a) p m b a) 4 p + min E α m) + E α m) E 3 α m) + E 4 α m). References [] M.K. Bakula M.E. Ozdemir J. Pecaric Hadard type ineualities for m-convex α m)-convex functions J. Ineual. Pure Appl. Math. 94) 8) -. [] I. Iscan A new generalization of some integral ineualities for α m)-convex functions Mathetical Sciences 7) 3) -8 doi:.86/ [3] I.Iscan Hermite-Hadard type ineualities for functions whose derivatives are α m)- convex International Journal of Engineering Applied Sciences 3) 3) [4] V.G. Miheşan A generalization of the convexityseminer on Functional Euations Approxition Convexity Cluj-Napoca Ronia 993. [5] M.E. Ozdemir M. Avcı H. Kavurcı Hermite-Hadard-type ineualities via α m)- convexity Comput. Math. Appl. 6 ) [6] M.E. Ozdemir H. Kavurcı E. Set Ostrowski s type ineualities for α m)-convex functions Kyungpook Math. J. 5 ) [7] M.E. Ozdemir E. Set M.Z. Sarıkaya Some new Hadard s type ineualities for coordinated m-convex α m)-convex functions Hacet. J. Math. Stat. 4) ) 9 9. [8] J. Park Hermite-Hadard Simpson-like type ineualities for differentiable α m)- convex ppings Int. J. Math. Math. Sci. ) Article ID pages. [9] M.Z. Sarikaya N. Aktan On the generalization of some integral ineualities their applications Math. Comput. Modelling 54 ) [] M.Z. Sarikaya E. Set M.E. Özdemir On new ineualities of Simpson s type for s convex functions Comput. Math. Appl. 6 )

12 New estites on generalization of some integral ineualities [] M.Z. Sarikaya E. Set M.E. Özdemir On new ineualities of Simpson s type for convex functions RGMIA Res. Rep. Coll. 3) ) Article. [] E. Set M. Sardari M.E. Ozdemir J. Rooin On generalizations of the Hadard ineuality for α m)-convex functions RGMIA Res. Rep. Coll. 4) 9) -. 64