Generalized Simpson-like Type Integral Inequalities for Differentiable Convex Functions via Riemann-Liouville Integrals

Transcription

1 International Journal of Mathematical Analysis Vol. 9, 15, no. 16, HIKARI Ltd, Generalized Simpson-like Type Integral Ineualities for Differentiable Convex Functions via Riemann-Liouville Integrals Jaekeun Park Department of Mathematics Hanseo University Seosan, Choongnam, , Korea Copyright c 15 Jaekeun Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, by setting up a generalized integral identity for differentiable functions via Riemann-Liouville fractional integrals, the author derive new estimates on generalization of Simpson-like types ineualities for functions whose derivatives in the absolute value at certain powers are convex and s-convex in the second sense. Mathematics Subject Classification: 6A51, 6A33, 6D1 Keywords: Convex functions, s-convex functions, Hermite-Hadamard type ineuality, Simpson type ineuality, Riemann-Liouville fractional integrals, Hölder s ineuality, mean-power integral ineuality 1 Introduction Let f : I R R be a function defined on the interval I of real numbers. Then f is called to be s-convex in the second sense on I if the following ineuality f(αx βy) α s f(x) β s f(y)

2 756 Jaekeun Park for all x, y I, α, β with α β = 1 and for some fixed s (, 1]. This class of s-convex functions in the second sense is usually denoted by K s. It can be easily seen that for s = 1, s-convexity reduces to ordinary convexity of functions defined on [, ). There are many results associated with convex functions in the area of ineualities, but some of those is the classical Hermite-Hadamard and Simpson type ineuality, respectively [11, 14]: Theorem 1.1. Let f : I R R be a convex function defined on the interval I of real numbers and a, b I with a < b. Then the following double ineuality holds: ( a b ) f 1 b a f(x)dx f(a) f(b). (1) Theorem 1.. [1] Let f : I R R be a differentiable function on the interior I of an interval I and a, b I with a < b. If f (x) M, x [a, b], then the following ineuality holds: f(x) 1 b a f(t)dt M [ (x a) (b x) ]. () Definition 1. The beta function, also called the Euler integral of the first kind, is a special function defined by and β(x, y) = β(a, x, y) = a is incomplete Beta function. t x 1 (1 t) y 1 dt, x, y >, t x 1 (1 t) y 1 dt, < a < 1, x, y >, We give some necessary definitions and mathematical preliminaries of fractional calculus theory which are used throughout this paper. Definition. Let f L([a, b]). The symbols J α a f and J α b f denote the leftside and right-side Riemann-Liouville integrals of the order α and are defined by J α a f(x) = 1 x (x t) α 1 f(t)dt ( a < x), Γ(α) and J α b f(x) = 1 Γ(α) a b x (t x) α 1 f(t)dt, ( < x < b), respectively, where Γ(α) is the Gamma function defined by Γ(α) = e t t α 1 dt and J a f(x) = J b f(x) = f(x).

3 Generalized Simpson-like type integral ineualities 757 In the caes of α = 1, the fractional integrals reduces to the classical integral. Recently, many authors have studied a number of ineualities by used the Riemann-Liouville fractional integrals, see [1-1,13,15-18] and the references cited therein. Especially, in [3, 13], Imdat Işcan, Noor, and Awan proved a variant of Hermite-Hadamard-like type and Ostrowski-like type ineualities which hold for the convex functions via Riemann-Liouville fractional integrals. Theorem 1.3. Let f : [a, b] R be twice differentiable function on (a, b) with a < b. If f L([a, b]) and f is convex on [a, b], then we have the following ineuality for fractional integrals: α 1 Γ(α 1) J α f(a) J α () α ( ab ) ( ab () 4 (α 1) α3 β( 1 f(b) ), α 1, ) Γ(α 3) Γ(α 4) f( a b ) [ f (a) f (b) ]. Theorem 1.4. Let f : I [, ) R be a differentiable function on the interior I of an interval I such that f L([a, b]), where a, b I with a < b. If f is s-convex on [a, b] for some fixed 1, x [a, b], µ [, 1] and α >, then the following ineuality for fractional integrals holds: where (1 µ) (x a) α (b x) α f(x) (x a) α f(a) (b x) α f(b) µ Γ(α 1) J α x f(a) J α x f(b) A (α, µ) [ (x a) α1 (b x)α1 1 f (x) A (α, µ, s) f (a) A 3 (α, µ, s) 1 ] f (x) A (α, µ, s) f (b) A 3 (α, µ, s), A 1 (α, µ) = αµ1 1 α 1 µ, α 1 s1 αµ1 α s 1 A (α, µ, s) = (s 1)(α s 1) µ s 1, ( 1 1 (1 µ α ) s1 ) A 3 (α, µ, s) = µ β(α 1, s 1) s 1 β(µ 1 α, α 1, s 1).

4 758 Jaekeun Park In this paper, we give some generalized integral ineualities connected with the Simpson-like type for differentiable functions whose derivatives in the absolute value at certain powers are convex s-convex in the second sense via fractional integrals. Lemmas Now we turn our attention to establish integral ineualities of Simpson-like type ineuality for convex functions via Riemann-Liouville fractional integrals, we need the lemmas below: Lemma 1. Let f : I R R be a differentiable function on the interior I of an interval I such that f L([a, b]), where a, b I with a < b. Then, for any λ 1 and n, the following identity holds: I(f; α; λ, n) 1 λf(a) (1 λ)f(b) ( 1 1 f(λa (1 λ)b) n n) Γ(α 1) λ α1 J α λ α (1 λ) α () α x f(a) (1 λ)α1 J α x f(b) [ = λ(1 λ)() (t α 1 n )f ( tx (1 t)a ) dt where x = λa (1 λ)b. ( 1 n tα )f ( tx (1 t)b ) ] dt, (3) Proof. Integrating by parts and changing variable of definite integral, we have: = (t α 1 n )f ( tx (1 t)a ) dt 1 1 (1 λ)() n f(a) ( 1 1 ) Γ(α 1) f(x) n (1 λ) α () J α α x f(a). (4) Similarly, we have = ( 1 n tα )f ( tx (1 t)b ) dt 1 1 λ() n f(b) ( 1 1 ) Γ(α 1) f(x) n λ α () J α α x f(b). (5) Multiplying both sides of (5) and (6) by λ(1 λ)(b a), respectively, and adding the resulting two eualities we obtain the desired λ 1 λ result. and λ(1 λ)(b a)

5 Generalized Simpson-like type integral ineualities 759 Note that, if we choose n = 3 and λ = 1, then we have 1 f(a) 4f( a b 6 ) f(b) ( 1) 1 α Γ(α 1) J () α ( ab = [ 4 (t α 1 3 )f ( t a b ( 1 3 tα )f ( t a b Lemma. For ξ 1, one has (a) (b) ξ t α dt δ 1 (α, ξ, ) = ξ 1 α α 1 ) α f(a) J α ( ab (1 t)a ) dt f(b) ) (1 t)b ) ] dt. (6) β( 1 α, 1 ) β( 1 α, 1 ) β(ξ, 1 α, 1 ), ξ t α tdt δ (α, ξ, ) = ξ α α β( α, 1 ) β( α, 1 ) β(ξ, α, 1 ). (7) Proof. These eualities follows from a straightfoward computation of definite integrals. 3 Some ineualities of Simpson-like type Now we turn our attention to establish new integral ineualities of Hermite- Hadamard and Ostrowski type for convex functions via fractional integrals. Theorem 3.1. Let f : I R R be a differentiable function on the interior I of an interval I and f L([a, b]), where a, b I with a < b and λ [, 1] and n. If f is convex on [a, b], then the following ineuality holds: I(f; α; λ, n) λ(1 λ)() [ δ 1 (α, 1 n, 1) (λ 1)δ (α, 1 f n, 1) (a) δ 1 (α, 1 n, 1) (1 λ)δ (α, 1 f n, 1) (b) ],

6 76 Jaekeun Park Proof. From Lemma 1, the convexity of f on [a, b], and the noted power-mean integral ineuality, we have I(f; α; λ, n) [ λ(1 λ)() t α 1 ( f tx (1 t)a ) dt n 1 n tα ( f tx (1 t)b ) ] dt [( λ(1 λ)() t α 1 ) f (1 t)dt (a) f (b) n ( t α 1 ) f ( tdt λa (1 λ)b ) ] n [ = λ(1 λ)() δ (α, 1 n, 1) ( f λa (1 λ)b ) δ 1 (α, 1 n, 1) δ (α, 1 f n, 1) (a) f (b) ]. (8) By Theorem 1.1 and the ineuality (8), we have I(f; α; λ, n) [ λ(1 λ)() δ 1 (α, 1 n, 1) (λ 1)δ (α, 1 f n, 1) (a) δ 1 (α, 1 n, 1) (1 λ)δ (α, 1 f n, 1) (b) ]. Theorem 3.. Let f : I R R be a differentiable function on the interior I of an interval I and f L([a, b]), where a, b I with a < b, λ [, 1], and n. If f is convex on [a, b] for > 1 with 1 1 = 1, then the following p ineuality holds: I(f; α; λ, n) where x = λa (1 λ)b. λ(1 λ)()δ 1 p 1 (α, 1 n, p) [ (1 λ) f (a) (1 λ) f (b) λf (a) ( λ) f (b) 1 ], Proof. From Lemma 1, the convexity of f on [a, b] for > 1 with 1

7 Generalized Simpson-like type integral ineualities p 1 = 1, and Hölder integral ineuality, we have I(f; α; λ, n) λ(1 λ)() t α 1 p dt n [ ( f tx (1 t)a ) dt 1 ( f tx (1 t)b ) dt 1 p By the convexity of f on [a, b] for > 1 with 1 p 1 1 ]. (9) = 1, we have ( f tx (1 t)a ) f (a) f (x) dt, (1) ( f tx (1 t)b ) f (x) f (b) dt. (11) By substituting (1) and (11) in (9), we get I(f; α; λ, n) λ(1 λ)()δ 1 p 1 (α, 1 n, 1) [ f (a) f (x) 1 f (x) f (b) 1 ]. (1) By the ineuality (8), we have f (a) f (x) (1 λ) f (a) (1 λ) f (b), (13) f (x) f (b) λ f (a) ( λ) f (b). (14) By substituting (13) and (14) in (1), we get the second ineuality. Theorem 3.3. Let f : I R R be a differentiable function on the interior I of an interval I and f L([a, b]), where a, b I with a < b, λ [, 1], and n. If f is convex on [a, b] for > 1 with 1 1 = 1, then the following p ineuality holds: I(f; α; λ, n) [( λ(1 λ)() δ 1 (α, 1 n, ) (1 λ)δ (α, 1 ) f n, ) (a) (1 λ)δ (α, 1 n, ) f (b) 1 ( λδ (α, 1 n, ) f (a) ( δ 1 (α, 1 n, ) λδ (α, 1 ) f n, ) (b) 1 ].

8 76 Jaekeun Park 1 p 1 Proof. From Lemma 1, the convexity of f on [a, b] for > 1 with = 1, and Hölder integral ineuality, we have I(f; α; λ, n) [ λ(1 λ)() t α 1 ( f tx (1 t)a ) 1 dt n t α 1 ( f tx (1 t)b ) 1 ] dt n [( λ(1 λ)() t α 1 f tdt) ( λa (1 λ)b ) n ( t α 1 f (1 t)dt) ( a ) 1 n ( t α 1 f tdt) ( λa (1 λ)b ) n ( t α 1 ) (1 t)dt f ( b ) 1 ] n λ(1 λ)() [ δ (α, 1 n, ) ( f x ) ( δ 1 (α, 1 n, ) δ (α, 1 ) f n, ) ( a ) 1 δ (α, 1 n, ) ( f x ) ( δ 1 (α, 1 n, ) δ (α, 1 ) f n, ) ( b ) 1 ]. (15) By the convexity of f on [a, b] for > 1 with 1 p 1 = 1, we have f (x) = f ( λa (1 λ)b ) λ f (a) (1 λ) f (b). (16) By substituting (16) in (15), we get the desired result. Theorem 3.4. Let f : I R R be a differentiable function on the interior I of an interval I and f L([a, b]), where a, b I with a < b, λ [, 1], and n. If f is convex on [a, b] for 1 with 1 1 = 1, then the following p ineuality holds: I(f; α; λ, n) λ(1 λ)()δ 1 p 1 (α, 1 n, 1) [( δ 1 (α, 1 n, 1) (1 λ)δ (α, 1 ) f n, 1) ( a )

9 Generalized Simpson-like type integral ineualities 763 (1 λ)δ (α, 1 ) n, 1) f ( b ) 1 λδ (α, 1 ) n, 1) f ( a ) ( δ 1 (α, 1 n, 1) λδ (α, 1 ) n, 1) f ( b ) 1 ]. Proof. Suppose that 1. By Lemma 1, the convexity of f on [a, b], and the power-mean integral ineuality, it follows that I(f; α; λ, n) λ(1 λ)() [ t α 1 1 p dt t α 1 ( f tx (1 t)a ) dt n n t α 1 1 p dt t α 1 ( f tx (1 t)b ) dt n n 1 1 ] = λ(1 λ)()δ 1 p 1 (α, 1 n, 1) [ t α 1 ( f t(λa (1 λ)b ) (1 t)a ) 1 dt n t α 1 ( f t(λa (1 λ)b ) (1 t)b ) 1 ] dt. (17) n By the convexity of f on [a, b], we have (i) t α 1 f ( t(λa (1 λ)b ) (1 t)a ) dt n ( δ 1 (α, 1 n, 1) (1 λ)δ (α, 1 ) f n, 1) ( a ) (1 λ)δ (α, 1 n, 1) ) f ( b ), (18) (ii) t α 1 ( f t(λa (1 λ)b ) (1 t)b ) dt n λδ (α, 1 n, 1) f ( a ) ( δ 1 (α, 1 n, 1) λδ (α, 1 ) n, 1) f ( b ). (19) By substituting (18) and (19) in (17), we get the desired result. Theorem 3.5. Let f : I R R be a differentiable function on the interior I of an interval I and f L([a, b]), where a, b I with a < b and λ [, 1] and n. If f is s-convex in the second sense on [a, b], then the following

10 764 Jaekeun Park ineuality holds: I(f; α; λ, n) λ(1 λ)()[ λ s δ 3 (α, n, s) δ 4 (α, n, s) f (a) (1 λ) s δ 3 (α, n, s) δ 4 (α, n, s) f (b) ], where δ 3 (α, n, s) = αn sα1 α s 1 (s 1)(s α 1) 1 n(s 1), δ 4 (α, n, s) = 1 (1 ( 1 n ) 1 α n(s 1) β(α 1, s 1) β(( 1 n ) 1 α, α 1, s 1). Proof. From Lemma 1, the s-convexity of f on [a, b], we have I(f; α; λ, n) λ(1 λ)() [( t α 1 f t dt) ( s λa (1 λ)b ) n ( t α 1 f (1 t) dt) s (a) n ( t α 1 f (1 t) dt) s (b) n ( t α 1 ) f t s dt ( λa (1 λ)b ) ] n = λ(1 λ)() [( t α 1 ) f (1 t) s dt (a) f (b) n ( t α 1 ) t s dt f ( λa (1 λ)b ) ] n λ(1 λ)()[ λ s δ 3 (α, n, s) δ 4 (α, n, s) f (a) (1 λ) s δ 3 (α, n, s) δ 4 (α, n, s) f (b) ].

11 Generalized Simpson-like type integral ineualities 765 References [1] Z. Dahmani, On Minkowski and Hermite-Hadamard integral ineualities for s-convex functions via fractional integration, Ann. Funct. Anal.,1(1) (1) [] S. S. Dragomir, Muhammad Ibal Bhatti, Muhammad Ibal, Some new Hermite-Hadamard s type fractional integral ineualities, J. Comput. Anal. and Appl.,18(4) (15) 1-7. [3] Imdat Işcan, Generalization of different type integral ineualities for (α, m)-convex functions via fractional integrals, arxiv: v1[math.ca]5 Mar 13, (13). [4] Imdat Işcan, On generalization of different type integral ineualities for s-convex functions via fractional integrals, Math. Sci. and Appl. E-Notes, (1) (14) [5] Imdat Işcan, New general integral ineualities for uasi-geometrically convex functions via fractional integrals, J. Ineual. and Appl., 13(1)(13) [6] Imdat Işcan, Hermite-Hadamard type ineualities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 38 (14) [7] Imdat Işcan, On generalization of different type integral ineualities for some convex functions via fractional integrals, arxiv: , (1) 1. [8] Imdat Işcan, Generalization of different type integral ineualities for s-convex functions via fractional integrals, Appl. Anal., (13) [9] U. N. Katugampola, New approach to a generalized fractional integrals, Appl. Math. Comput., 18(3) (11) [1] YuMei Liao, JianHua Deng, JinRong Wang, Riemann-Liouville fractional Hermite-Hadamard ineualities. Part II: for twice differentiable geometricarithmetically s-convex functions, J. Ineual. Appl., 13:517 (13) [11] Meihui Qu, Wenjun Liu, J. Park, Some new Hermite-Hadamard-type ineualities for geometric-arithmetically s-convex functions, WSEAS Trans. on Math., 13 (14)

12 766 Jaekeun Park [1] D. S. Mitrinović, J. E. Pečarić, A. M. Fink, Ineualities involving functions and their integrals and derivatives, Kluwer Academic Pub., Dortrecht, (1991). [13] Muhammad Aslam Noor, Muhammad Uzair Awan, Some integral ineualities for two kind of convexities via fractional integrals, TJMM, 5() (13) [14] J. Park, Generalization of some Simpson-like type ineualities via differentiable s-convex mappings in the second sense, Inter. J. Math. Math. Sci., 11 (11) 13 Pages, Art ID: [15] M. Z. Sarikaya, H. Ogunmez, On new ineualities via Riemann-Liouville fractional integration, Abst. Appl. Anal., 1 (1) 1 Pages, Art ID: [16] M. Z. Sarikaya, E. Set, H. Yaldiz, N. Başak, Hermite-Hadamard s ineualities for fractional integrals and related fractional ineualities, Math. Comput. Modelling, 57(9) (13) [17] E. Set, New ineualities of Ostrowski type for mapping whose derivatives are s-convex in the second sense via fractional integrals, Comput. and Math. with Appl., 63 (1) [18] J. Wang, X. Li, M. Feckan, Y. Zhou, Hermite-Hadamard-type ineualities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal., (1). Received: February 1, 15; Published: March 1, 15