Some new Ostrowski type fractional integral inequalities for generalized (s, m, ϕ)-preinvex functions via Caputo k-fractional derivatives

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1 Proyecciones Journal of Mathematics Vol. 37, N o, pp. 33-5, March 28. Universidad Católica del Norte Antofagasta - Chile Some new Ostrowsi type fractional integral ineualities for generalized (s, m, ϕ)-preinvex functions via Caputo -fractional derivatives Artion Kashuri University Ismail Qemali, Albania and Rozana Lio University Ismail Qemali, Albania Received : June 27. Accepted : July 27 Abstract In the present paper, the notion of generalized (s, m, ϕ)-preinvex function is applied to establish some new generalizations of Ostrowsi type integral ineualities via Caputo -fractional derivatives. At the end, some applications to special means are given. 2 Mathematics Subject Classification : Primary: 26A5; Secondary: 26A33, 26D7, 26D, 26D5. Keywords : Ostrowsi type ineuality, Hölder s ineuality, power mean ineuality, s-convex function in the second sense, m-invex.

2 34 Artion Kashuri and Rozana Lio. Introduction and Preliminaries The following notations are used throughout this paper. We use I to denote an interval on the real line R =(, + ) andi to denote the interior of I. For any subset K R n,k is used to denote the interior of K. R n is used to denote a n-dimensional vector space. The set of continuous differentiable functions of order n on the interval [a, b] is denoted by C n [a, b]. The following result is nown in the literature as the Ostrowsi ineuality (see []), which gives an upper bound for the approximation of the integral Z b average f(t)dt by the value f(x) atpointx [a, b]. b a a Theorem.. Let f : I R, where I R is an interval, be a mapping differentiable in the interior I of I, and let a, b I with a<b.if f (x) M for all x [a, b], then (.) ³ f(x) Z b f(t)dt b a M(b a) a 4 + x a+b 2 2, (b a) 2 x [a, b]. For other recent results concerning Ostrowsi type ineualities (see []- [3],[],[6],[34]-[38],[42]). Ostrowsi ineuality is playing a very important role in all the fields of mathematics, especially in the theory of approximations. Thus such ineualities were studied extensively by many researches and numerous generalizations, extensions and variants of them for various ind of functions lie bounded variation, synchronous, Lipschitzian, monotonic, absolutely, continuous and n-times differentiable mappings etc. appeared in a number of papers (see [4]-[5],[7]-[2]). In recent years, one more dimension has been added to this studies, by introducing a number of integral ineualities involving various fractional operators lie Riemann- Liouville, Erdelyi-Kober, Katugampola, conformable fractional integral operators etc. by many authors (see [2]-[3]). Riemann-Liouville fractional integral operators are the most central between these fractional operators.

3 Some new Ostrowsi type fractional integral ineualities for In numerical analysis many uadrature rules have been established to approximate the definite integrals. Ostrowsi ineuality provides the bounds of many numerical uadrature rules (see [8]). In recent decades Ostrowsi ineuality is studied in fractional calculus point of view by many mathematicians (see [9]-[]). Definition.2. (see [33]) A function f : [, + ) R is said to be s-convex in the second sense, if (.2) f(λx +( λ)y) λ s f(x)+( λ) s f(y) for all x, y, λ [, ] and s (, ]. It is clear that a -convex function must be convex on [, + ) asusual. The s-convex functions in the second sense have been investigated in (see [33]). Definition.3. (see [39]) A set K R n is said to be invex with respect to the mapping η : K K R n, if x + tη(y,x) K for every x, y K and t [, ]. Notice that every convex set is invex with respect to the mapping η(y, x) =y x, but the converse is not necessarily true. For more details (see [39],[4]). Definition.4. (see [4]) The function f defined on the invex set K R n is said to be preinvex with respect η, if for every x, y K and t [, ], we have that f(x + tη(y,x)) ( t)f(x)+tf(y). The concept of preinvexity is more general than convexity since every convex function is preinvex with respect to the mapping η(y,x) = y x, but the converse is not true. Definition.5. For R + and x C, the -gamma function is defined by n! n n x (.3) Γ (x) = lim. n (x) n, Its integral representation is given by (.4) Γ (α) = Z t α e t dt.

4 36 Artion Kashuri and Rozana Lio One can note that Γ (α + ) =αγ (α). For =, (.4) gives integral representation of gamma function. Definition.6. For R + and x C, the -beta function with two parameters x and y is defined as (.5) β (x, y) = t x ( t) y dt. For =, (.5) gives integral representation of beta function. Z Theorem.7. Let x, y >, then for -gamma and -beta function the following euality holds (.6) β (x, y) = Γ (x)γ (y) Γ (x + y). Definition.8. (see [2]) Let α>and α/ {, 2, 3,...}, n=[α]+,f C n [a, b] such that f (n) exists and are continuous on [a, b]. The Caputo fractional derivatives of order α are defined as follows: (.7) and (.8) c D α a+f(x) = c D α b f(x) = Z x f (n) (t) dt, x > a Γ(n α) a (x t) α n+ Z ( )n b f (n) (t) dt, x < b. Γ(n α) x (t x) α n+ If α = n {, 2, 3,...} and usual derivative of order n exists, then Caputo fractional derivative c Da+f α (x) coincides with f (n) (x). In particular we have ³ (.9) c D (x) =³ c Db f (x) =f(x) a+f where n =andα =. In the following we recall Caputo -fractional derivatives. Definition.9. (see [3]) Let α>, and α / {, 2, 3,...}, n= [α] +,f C n [a, b]. The Caputo -fractional derivatives of order α are defined as follows: (.) c D α, a+ f(x) = Γ n α Z x f (n) (t) dt, x > a a (x t) α n+

5 Some new Ostrowsi type fractional integral ineualities for and (.) c D α, b f(x) = ( ) n Γ n α Z b f (n) (t) dt, x < b. x (t x) α n+ The aim of this paper is to establish some generalizations of Ostrowsi type ineualities using new identity given in Section 2 to generalized (s, m, ϕ)- preinvex functions via Caputo -fractional derivatives. In Section 3, some applications to special means are given. In Section 4, some conclusions and future research are also given. 2. Main results Definition 2.. (see [32]) A set K R n is said to be m-invex with respect to the mapping η : K K (, ] R n for some fixed m (, ], if mx + tη(y,x, m) K holds for each x, y K and any t [, ]. Remar 2.2. In Definition 2., under certain conditions, the mapping η(y, x, m) could reduce to η(y, x). For example when m =, then the m-invex set degenerates an invex set on K. Definition 2.3. (see [2]) Let K R be an open m-invex set with respect to η : K K (, ] R and ϕ : I K a continuous function. For f : K R and any fixed s, m (, ], if (2.) f (mϕ(x)+tη(ϕ(y),ϕ(x),m)) m( t) s f(ϕ(x)) + t s f(ϕ(y)) is valid for all x, y I,t [, ], then we say that f(x) is a generalized (s, m, ϕ)-preinvex function with respect to η. Remar 2.4. In Definition 2.3, it is worthwhile to note that the class of generalized (s, m, ϕ)-preinvex function is a generalization of the class of s-convex in the second sense function given in Definition.2. In this section, in order to present some new Ostrowsi type integral ineualities to generalized (s, m, ϕ)-preinvex functions via Caputo - fractional derivatives, we need the following new interesting lemma to obtain our results. Lemma 2.5. Let α>, and α / {, 2, 3,...}, n=[α] +. Let ϕ : I K be a continuous function. Suppose K R be an open

6 38 Artion Kashuri and Rozana Lio m-invex subset with respect to η : K K (, ] R for any fixed m (, ]. Assume that f : K R is a function on K such that f C n+ [mϕ(a),mϕ(a) +], where >. Then we have the following euality for Caputo -fractional derivatives η n α (ϕ(x),ϕ(a),m)f (n) (mϕ(a)+η(ϕ(x),ϕ(a),m)) ηn α (ϕ(x),ϕ(b),m)f (n) (mϕ(b)+η(ϕ(x),ϕ(b),m)) +( ) n+ (n α)γ n α f(mϕ(b)) c D α, (mϕ(a)+η(ϕ(x),ϕ(a),m)) f(mϕ(a)) c D α, (mϕ(b)+η(ϕ(x),ϕ(b),m)) = ηn α + (ϕ(x),ϕ(a),m) ηn α + (ϕ(x),ϕ(b),m) Z Z t n α f (n+) (mϕ(a)+tη(ϕ(x),ϕ(a),m))dt t n α f (n+) (mϕ(b)+tη(ϕ(x),ϕ(b),m))dt. Proof. Throughout this paper we denote I f,η,ϕ (x; α,, n, m, a, b) = ηn α + Z (ϕ(x),ϕ(a),m) t n α f (n+) (mϕ(a)+tη(ϕ(x),ϕ(a),m))dt ηn α + (ϕ(x),ϕ(b),m) (2.2) Z t n α f (n+) (mϕ(b)+tη(ϕ(x),ϕ(b),m))dt. Integrating by parts, we get I f,η,ϕ (x; α,, n, m, a, b) = ηn α + (ϕ(x),ϕ(a),m) tn α f (n) (mϕ(a)+tη(ϕ(x),ϕ(a),m)) η(ϕ(x),ϕ(a),m)

7 Some new Ostrowsi type fractional integral ineualities for n α η(ϕ(x),ϕ(a),m) Z t n α f (n) (mϕ(a)+tη(ϕ(x),ϕ(a),m))dt ηn α + (ϕ(x),ϕ(b),m) tn α f (n) (mϕ(b)+tη(ϕ(x),ϕ(b),m)) η(ϕ(x),ϕ(b),m) n α Z t n α f (n) (mϕ(b)+tη(ϕ(x),ϕ(b),m))dt η(ϕ(x),ϕ(b),m) = ηn α (ϕ(x),ϕ(a),m)f (n) (mϕ(a)+η(ϕ(x),ϕ(a),m)) ηn α (ϕ(x),ϕ(b),m)f (n) (mϕ(b)+η(ϕ(x),ϕ(b),m)) +( ) n+ (n α)γ n α c D α, (mϕ(a)+η(ϕ(x),ϕ(a),m)) f(mϕ(a)) c D α, (mϕ(b)+η(ϕ(x),ϕ(b),m)) f(mϕ(b)). 2 By using Lemma 2.5, one can extend to the following results. Theorem 2.6. Let α>, and α / {, 2, 3,...}, n=[α] +. Let ϕ : I K be a continuous function. Suppose K R be an open m-invex subset with respect to η : K K (, ] R for any fixed s, m (, ]. Assume that f : K R is a function on K such that f C n+ [mϕ(a),mϕ(a) +], where >. If f (n+) is generalized (s, m, ϕ)-preinvex function on [mϕ(a),mϕ(a) + ] and f (n+) M, then the following ineuality for Caputo -fractional derivatives holds: I f,η,ϕ (x; α,, n, m, a, b) M (2.3) Ã mβ µn α! +,s+ + n + s α + η(ϕ(x),ϕ(a),m) n α + + η(ϕ(x),ϕ(b),m) n α +

8 4 Artion Kashuri and Rozana Lio Proof. From Lemma 2.5, generalized (s, m, ϕ)-preinvexity of upper bound of f (n+) and properties of the modulus, we have η(ϕ(x),ϕ(a),m) n α + + η(ϕ(x),ϕ(b),m) n α + η(ϕ(x),ϕ(a),m) n α + + η(ϕ(x),ϕ(b),m) n α + 2 M Ã mβ I f,η,ϕ (x; α,, n, m, a, b) Z t n α Z t n α Z h t n α Z h t n α f (n+), f (n+) (mϕ(a)+tη(ϕ(x),ϕ(a),m)) dt f (n+) (mϕ(b)+tη(ϕ(x),ϕ(b),m)) dt m( t) sf (n+) (ϕ(a)) +t sf i (n+) (ϕ(x)) dt m( t) sf (n+) (ϕ(b)) +t sf (n+) i (ϕ(x)) dt µn α! +,s+ + n + s α + η(ϕ(x),ϕ(a),m) n α + + η(ϕ(x),ϕ(b),m) n α + Corollary 2.7. Under the same conditions as in Theorem 2.6, if we choose m = =,η(ϕ(y),ϕ(x),m)=ϕ(y) mϕ(x) and ϕ(x) =x, x I, we get the following ineuality for Caputo fractional derivatives: (x a)n α (x b) n α f (n) (x) b a n+ Γ (n α +) +( ) c D α b a x f(a) c Dx f(b) α µ M β (n α +,s+)+ (x a)n α+ +(b x) n α+ n + s α + b a (2.4)

9 Some new Ostrowsi type fractional integral ineualities for... 4 Theorem 2.8. Let α>, and α / {, 2, 3,...}, n=[α] +. Let ϕ : I K be a continuous function. Suppose K R be an open m-invex subset with respect to η : K K (, ] R for any fixed s, m (, ]. Assume that f : K R is a function on K such that f C n+ [mϕ(a),mϕ(a)+], where + >. If f (n+) is generalized (s, m, ϕ)-preinvex function on [mϕ(a),mϕ(a) + ] >, p + =and f (n+) M, then the following ineuality for Caputo -fractional derivatives holds: (2.5) M I f,η,ϕ (x; α,, n, m, a, b) µ n p α p + µ m + s + η(ϕ(x),ϕ(a),m) n α + + η(ϕ(x),ϕ(b),m) n α + Proof. Suppose that >. From Lemma 2.5, generalized (s, m, ϕ)- preinvexity of f (n+), Hölder ineuality, upper bound of f (n+) and properties of the modulus, we have η(ϕ(x),ϕ(a),m) n α + + η(ϕ(x),ϕ(b),m) n α + I f,η,ϕ (x; α,, n, m, a, b) Z t n α Z t n α η(ϕ(x),ϕ(a),m) n α + f (n+) (mϕ(a)+tη(ϕ(x),ϕ(a),m)) dt f (n+) (mϕ(b)+tη(ϕ(x),ϕ(b),m)) dt µz t (n α )p p dt µz f (n+) (mϕ(a)+tη(ϕ(x),ϕ(a),m)) dt + η(ϕ(x),ϕ(b),m) n α + µz t (n α )p p dt µz f (n+) (mϕ(b)+tη(ϕ(x),ϕ(b),m)) dt

10 42 Artion Kashuri and Rozana Lio 2 η(ϕ(x),ϕ(a),m) n α + µz t (n α )p p dt µz h m( t) sf (n+) (ϕ(a)) + t sf (n+) (ϕ(x)) i dt + η(ϕ(x),ϕ(b),m) n α + µz t (n α )p p dt µz h m( t) sf (n+) (ϕ(b)) + t sf (n+) (ϕ(x)) i dt µ M m + µ n α p s + p + η(ϕ(x),ϕ(a),m) n α + + η(ϕ(x),ϕ(b),m) n α + Corollary 2.9. Under the same conditions as in Theorem 2.8, if we choose m = =,η(ϕ(y),ϕ(x),m)=ϕ(y) mϕ(x) and ϕ(x) =x, x I, we get the following ineuality for Caputo fractional derivatives: (x a)n α (x b) n α f (n) (x) b a n+ Γ (n α +) +( ) c D α b a x f(a) c Dx f(b) α (2.6) M µ (n α) p + p µ 2 (x a)n α+ +(b x) n α+ s + b a Theorem 2.. Let α >, and α / {, 2, 3,...}, n=[α] +. Let ϕ : I K be a continuous function. Suppose K R be an open m-invex subset with respect to η : K K (, ] R for any fixed s, m (, ]. Assume that f : K R is a function on K such that f C n+ [mϕ(a),mϕ(a)+], where + >.

11 Some new Ostrowsi type fractional integral ineualities for If f (n+) is generalized (s, m, ϕ)-preinvex function on [mϕ(a),mϕ(a) + ] and f (n+) M, then the following ineuality for Caputo -fractional derivatives holds: I f,η,ϕ (x; α,, n, m, a, b) Ã! M n α + mβ µn α +,s+ + n + s α + η(ϕ(x),ϕ(a),m) n α + + η(ϕ(x),ϕ(b),m) n α + (2.7) Proof. Suppose that. From Lemma 2.5, generalized (s, m, ϕ)- preinvexity of f (n+), the well-nown power mean ineuality, upper bound of f (n+) and properties of the modulus, we have I f,η,ϕ (x; α,, n, m, a, b) η(ϕ(x),ϕ(a),m) n α + Z t n α f (n+) (mϕ(a)+tη(ϕ(x),ϕ(a),m)) dt + η(ϕ(x),ϕ(b),m) n α + µz µz Z t n α η(ϕ(x),ϕ(a),m) n α + t n α f (n+) (mϕ(b)+tη(ϕ(x),ϕ(b),m)) dt µz t n α dt f (n+) (mϕ(a)+tη(ϕ(x),ϕ(a),m)) dt + η(ϕ(x),ϕ(b),m) n α + t n α µz t n α dt f (n+) (mϕ(b)+tη(ϕ(x),ϕ(b),m)) dt η(ϕ(x),ϕ(a),m) n α + µz t n α dt

12 44 Artion Kashuri and Rozana Lio µz h t n α m( t) sf (n+) (ϕ(a)) + t sf (n+) (ϕ(x)) i dt + η(ϕ(x),ϕ(b),m) n α + µz t n α dt µz h t n α m( t) sf (n+) (ϕ(b)) + t sf (n+) (ϕ(x)) i dt 2 Ã! M n α + mβ µn α +,s+ + n + s α + η(ϕ(x),ϕ(a),m) n α + + η(ϕ(x),ϕ(b),m) n α + Corollary 2.. Under the same conditions as in Theorem 2., if we choose m = =,η(ϕ(y),ϕ(x),m)=ϕ(y) mϕ(x) and ϕ(x) =x, x I, we get the following ineuality for Caputo fractional derivatives: (x a)n α (x b) n α f (n) (x) b a µ M n+ Γ (n α +) +( ) c D α b a x f(a) c Dx f(b) α n α + β (n α +,s+)+ n + s α + (2.8) (x a)n α+ +(b x) n α+ b a Corollary 2.2. Under the same conditions as in Theorem 2. for =, we get Theorem 2.6.

13 Some new Ostrowsi type fractional integral ineualities for Applications to special means In the following we give certain generalizations of some notions for a positive valued function of a positive variable. Definition 3.. (see [43]) A function M : R 2 + R +, is called a Mean function if it has the following properties:. Homogeneity: M(ax, ay) =am(x, y), for all a>, 2. Symmetry: M(x, y) =M(y,x), 3. Reflexivity: M(x, x) =x, 4. Monotonicity: If x x and y y, then M(x, y) M(x,y ), 5. Internality: min{x, y} M(x, y) max{x, y}. We consider some means for arbitrary positive real numbers α, β (α 6= β).. The arithmetic mean: A:=A(α, β) = α+β 2 2. The geometric mean: G:=G(α, β) = αβ 3. The harmonic mean: H:=H(α, β) = 2 α + β 4. The power mean: P r := P r (α, β) = ³ α r +β r 2 ³ β β r, r. 5. Theidentricmean: I:=I(α, β) =( e, α 6= β; α, α = β. 6. The logarithmic mean: L:=L(α, β) = α α β α ln(β) ln(α). p 7. The generalized log-mean: L p := L p (α, β) = βp+ α p+ (p+)(β α) ; p R \{, }. Ã! α, α 8. The weighted p-power mean: M 2,,α n p =( P n u, u 2,,u i= α i u p i ) p n where α i, u i > (i =, 2,...,n)with P n i= α i =.

14 46 Artion Kashuri and Rozana Lio It is well nown that L p is monotonic nondecreasing over p R with L := L and L := I. In particular, we have the following ineuality H G L I A. Now, let a and b be positive real numbers such that a<b. Consider the function M := M(ϕ(x),ϕ(y)) : [ϕ(x),ϕ(x)+η(ϕ(y),ϕ(x))] [ϕ(x),ϕ(x)+η(ϕ(y),ϕ(x))] R +, which is one of the above mentioned means and ϕ : I K be a continuous function. Therefore one can obtain various ineualities using the results of Section 2 for these means as follows: Replace η(ϕ(y),ϕ(x),m) with η(ϕ(y),ϕ(x)) and setting η(ϕ(y),ϕ(x)) = M(ϕ(x),ϕ(y)) for value m =and x, y I in (2.5) and (2.7), one can obtain the following interesting ineualities involving means: I f,m(, ),ϕ (x; α,, n,,a,b) M µ n α p p + µ 2 s + M n α + (ϕ(a),ϕ(x)) + M n α + (ϕ(b),ϕ(x)) (3.), M(ϕ(a),ϕ(b)) I f,m(, ),ϕ (x; α,, n,,a,b) ³ M + n α β n α +,s+ + n+s α + M n α + (ϕ(a),ϕ(x)) + M n α + (ϕ(b),ϕ(x)) (3.2) M(ϕ(a),ϕ(b)) Letting M(ϕ(x),ϕ(y)) = A, G, H, P r,i,l,l p,m p, x, y I, in (3.) and (3.2), we get the ineualities involving means for a particular choice of a generalized (s,,ϕ)-preinvex functions f (n+). The details are left to the interested reader. 4. Conclusions In the present paper, the notion of generalized (s, m, ϕ)-preinvex function was applied to established some new generalizations of Ostrowsi type ineualities via Caputo -fractional derivatives.

15 Some new Ostrowsi type fractional integral ineualities for Motivated by this new interesting class of generalized (s, m, ϕ)-preinvex functions we can indeed see to be vital for fellow researchers and scientists woring in the same domain. We conclude that our methods considered here may be a stimulant for further investigations concerning Ostrowsi and Hermite-Hadamard type integral ineualities for various inds of preinvex functions involving classical integrals, Riemann-Liouville fractional integrals, -fractional integrals, local fractional integrals, fractional integral operators, Caputo -fractional derivatives, -calculus, (p, )-calculus, time scale calculus and conformable fractional integrals. References [] Ç. Yildiz, M. E. Özdemir and M. Z. Sariaya, New generalizations of Ostrowsi-lie type ineualities for fractional integrals, Kyungpoo Math. J., 56, pp. 6-72, (26). [2] A. Kashuri and R. Lio, Ostrowsi type fractional integral ineualities for generalized (s, m, ϕ)-preinvex functions, Aust. J. Math. Anal. Appl., 3, (), Article 6, pp. -, (26). [3] A. Kashuri and R. Lio, Generalizations of Hermite-Hadamard and Ostrowsi type ineualities for MT m -preinvex functions, Proyecciones, 36, (), pp. 45-8, (27). [4] M.Alomari,M.Darus,S.S.DragomirandP.Cerone,Ostrowsitype ineualities for functions whose derivatives are s-convex in the second sense, Appl. Math. Lett., 23, pp. 7-76, (2). [5] R. P. Agarwal, M. J. Luo and R. K. Raina, On Ostrowsi type ineualities, Fasc. Math., 24, pp. 5-27, (26). [6] S. S. Dragomir, On the Ostrowsi s integral ineuality for mappings with bounded variation and applications, Math. Ine. & Appl.,, (2) (998). [7] S. S. Dragomir, The Ostrowsi integral ineuality for Lipschitzian mappings and applications, Comput. Math. Appl., 38, pp , (999).

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19 Some new Ostrowsi type fractional integral ineualities for... 5 Artion Kashuri Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, Vlora, Albania artionashuri@gmail.com and Rozana Lio Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, Vlora, Albania rozanalio86@gmail.com