New Integral Inequalities of the Type of Hermite-Hadamard Through Quasi Convexity

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1 Punjb University Journl of Mthemtics (ISSN ) Vol. 45 (13) pp New Integrl Inequlities of the Type of Hermite-Hdmrd Through Qusi Convexity S. Hussin Deprtment of Mthemtics, College of Science, Qssim University, P. O. Box 6644, Burydh 5148, Sudi Arbi. And Deprtment of Mthemtics, Islmi University Bhwlpur, Pkistn. Emil: S. Qisr College of Mthemtics nd Sttistics, Chongqing University, Chongqing, 41331, P. R. Chin. Emil: Abstrct. In this pper, we estblish some new integrl inequlities of Hermite Hdmrd type for twice differentible functions through qusi convexity by using Riemnn-Liouville frctionl integrls. Applictions to specil mens of rel numbers re lso given. AMS (MOS) Subject Clssifiction Codes: 6D7; 6A33; 6D15; 6D1 Key Words: Hermite-Hdmrd-type inequlity, Qusi-convex function, Holder s integrl inequlity, Riemnn-Liouville frctionl integrls. 1. INTRODUCTION Let f : Φ I R R be function defined on the intervl I of rel numbers. Then f is clled convex, if f (λx + (1 λ) y) λf (x) + (1 λ) f (y), for ll x, y I nd λ [, 1]. Geometriclly, this mens tht if P, Q nd R re three distinct points on grph of f with Q between P nd R, then Q is on or below chord PR. There re mny results ssocited with convex functions in the re of inequlities. The notion of qusi-convex functions generlized the notion of convex functions. More precisely, function f : [, b] R is sid to be qusi-convex on [, b], if f (λx + (1 λ) y) mx f (x), f (y)}, x, y [, b]. Any convex function is qusi-convex function but the converse is not true. Becuse there exist qusi-convex functions which is not convex, (see []). For exmple, the function f : R + R, defined by f (x) = ln x, x R + is qusi-convex. However f is not convex function. 33

2 34 S. Hussin, S. Qisr There re mny results ssocited with convex functions in the re of inequlities, but one of those is the clssicl Hermite Hdmrd inequlity. This inequlity is defined s: Let f : I R R be convex function defined on the intervl Iof rel numbers with, b I nd < b. Then f stisfies the following well-known Hermite Hdmrd inequlity ( ) + b f 1 b b f (x)dx f () + f (b), (1. 1) for, b I, with < b. For severl recent results concerning the bove inequlity (1.1) we refer the interested reder to [1, 4, 8, 9, 1, 11, 1]. Recently, D. A. Ion [14] obtined the following two inequlities of the right hnd side of Hermite-Hdmrd s type functions whose derivtives in bsolute vlues re qusi-convex. Theorem 1. Let f : I R R be differentible function on I with, b I nd < b. If f is qusi-convex function on [, b], then we hve: f() + f(b) b b mx [ f (), f (b) ]. (1. ) 4 Theorem. Let f : I R R be differentible function on I with, b I nd < b. If f p is qusi-convex function on [, b] for some fixed p > 1, then we hve: f() + f(b) b (p + 1) 1/p b g(x)dx b f(x)g (x) dx [ mx f () p/(p 1), f (b) p/(p 1)}] (p 1)/p. (1. 3) In [], Alomri, Drus nd Kirmci estblished the following Hermite-Hdmrd inequlities for qusi-convex functions which give refinements of bove Theorems 1 nd. Theorem 3. Let f : I R R be differentible function on I with, b I nd < b. If f is qusi-convex on [, b], then we hve f() + f(b) b b [ ( ) } mx f (), + b 8 f ( ) + b + mx f, f (b) }] (1. 4)

3 New Integrl Inequlities of the Type of Hermite-HdmrdThrough Qusi Convexity 35 Theorem 4. Let f : I R R be differentible function on I with, b I nd < b. If f q is qusi-convex on [, b] nd p > 1, then we hve f() + f(b) b ( ) ( 1/p ( ) p }) p 1 (b ) 1 + b p mx 4 (1 + p) f p 1, f p p 1 () + ( ( ) p + b mx f p 1, f (b) p p 1 }) p 1 p (1. 5) Theorem 5. Let f : I R R be differentible function on I with, b I nd < b. If f (x) is qusi-convex on [, b], then we hve f() + f(b) [ (b ) b mx( f () q, f ( + b } 1 q 8 ) q ) + mx( f ( + b } 1 ] ) q, f (b) q q ) (1. 6) Alomri, Drus nd Drgomir in [3] introduced the following theorems for twice differentible qusi-convex functions. Theorem 6. Let f : I R R be differentible function on I with, b I nd < b. If f is qusi-convex on [, b], then we hve f() + f(b) (b ) b mx f (), f (b) }. (1. 7) 1 Theorem 7. Let f : I R R be differentible function on I with, b I nd < b. If f q is qusi-convex on [, b] nd q 1, then we hve f() + f(b) (b ) ( b mx f () q, f (b) q}) 1/q (1. 8) 1 In [13], R. Gorenflo, F. Minrdi defined the Riemnn-Liouville frctionl integrls s: Let f L 1 [, b]. The Riemnn-Liouville frctionl integrls J α f nd J α + b f of order α > with α re defined by nd J α 1 +f (x) = Γ (α) x b (x t) α 1 f (t) dt, ( < x), Jb f α (x) = 1 (t x) α 1 f (t) dt, (x < b), Γ (α) x respectively. Here Γ (α) = e u u α 1 du nd J+f (x) = Jb f (x) = f (x). Note tht if α = 1, the frctionl integrl reduces to the clssicl integrl.

4 36 S. Hussin, S. Qisr In this pper, we estblish some new integrl inequlities of Hermite Hdmrd type for twice differentible functions through qusi convexity by using Riemnn-Liouville frctionl integrls. Applictions to specil mens of rel numbers re lso given.. MAIN RESULTS In order to prove our min results, we use the following Lemm of [17]. Lemm 8. Suppose f : [, b] R be twice differentible mpping on (, b) with < b. If f L [, b], then we hve the following frctionl integrl inequlity: f()+f(b) (b ) J α α +f (b) + Jb α f () ] (b ) 1 (1 λ) λ f (λ + (1 λ) b) dλ. (.1) Theorem 9. Suppose f : [, b] R be differentible mpping on (, b) with < b nd f L 1 [, b]. If f is qusi-convex on [, b], for α >, then we hve the following frctionl integrl inequlity: f()+f(b) (b ) J α α +f (b) + Jb α f () ] α(b ) ()(α+) mx f (), f (b) }. (.) Proof. Using Lemm 8 nd qusi convexity of f on [, b], we get f()+f(b) (b ) J α α +f (b) + Jb α f () ] (b ) (b ) The proof is completed. 1 (1 λ) λ f (λ + (1 λ) b) dλ. 1 (1 λ) λ = (b ) mx f (), f (b) } mx f (), f (b) }dλ = α(b ) ()(α+) mx f (), f (b) }. 1 (1 λ) λ dλ Note tht, If we tke α = 1, in bove Theorem 9 with the properties of gmm functions, we get inequlity (1.7). Theorem 1. Suppose f : [, b] R be differentible mpping on (, b) with < b such tht f L 1 [, b]. If f q is qusi-convex on [, b], nd p > 1, then we hve the following frctionl integrl inequlity: f()+f(b) (b ) () Γ() where 1 p + 1 q = 1. [ (b ) J α α +f (b) + Jb α f () ] ( 1/p (mx 1 p()+1) f () q, f (b) q}) 1 q, Proof. Using Lemm 8, we get f()+f(b) (b ) J α α +f (b) + Jb α f () ] (b ) 1 (1 λ) λ f (λ + (1 λ) b) dλ. (.3)

5 New Integrl Inequlities of the Type of Hermite-HdmrdThrough Qusi Convexity 37 By Holders inequlity nd qusi convexity of f on [, b] with λ [, 1], we hve f()+f(b) (b ) J α α +f (b) + Jb α f () ] ( ( ) (b ) 1 pdλ ) 1/p ( () 1 (1 λ) λ 1 f λ + (1 λ) b q dλ ( ( ) (b ) 1 pdλ ) 1/p () 1 (1 λ) ( λ mx f () q, f (b) q}) 1 q ( ) 1/p = (b ) (mx () 1 p()+1 f () q, f (b) q}) 1 q The proof is completed. ) 1/q Theorem 11. Suppose f : [, b] R be differentible mpping on (, b) with < b such tht f L 1 [, b]. If f q is qusi-convex on [, b] nd q 1, then we hve the following frctionl integrl inequlity: f()+f(b) (b ) J α α +f (b) + Jb α f () ] α(b ) ()(α+) ( mx f () q, f (b) q}) 1 q. Proof. Using Lemm 8 nd Holder s inequlity, we get f()+f(b) (b ) J α α +f (b) + Jb α f () ] (b ) 1 (1 λ) λ (.4) f (λ + (1 λ) b) dλ ) p ) 1 1 (1 ( (1 λ) λ q 1/q dλ f (λ + (1 λ) b) dλ) q ( (b ) 1 () Using qusi-convexity of f on [, b] nd λ [, 1], we get f()+f(b) (b ) () Γ() ( = α(b ) ()(α+) The proof is completed. [ (b ) J α α +f (b) + Jb α f () ] (1 ) p ) 1 1 (1 λ) λ q ( dλ mx f () q, f (b) q}) 1 q ( mx f () q, f (b) q}) 1 q. Note tht, If we tke α = 1, in bove Theorem 11 with the properties of gmm functions, we get inequlity (1.8). 3. APPLICATIONS TO SOME SPECIAL MEANS First we recll the rithmetic men A(, b), logrithmic men L (, b)nd p-logrithmic men L p (, b) for rbitrry rel numbers nd b s follows: A = A (, b): = +b,, b >,, if = b L = L (, b) = b ln b ln, if b,, b >, if = b L p L p (, b) = b p+1 p+1 (p+1)(b ), if b, p Re \ 1, }:, b >

6 38 S. Hussin, S. Qisr Now we present some new inequlities for the bove mens by using the results of sections. The following proposition follows from Theorem 9 pplied to qusi-convex mpping f (x) = x n, x N nd α = 1. Proposition 1. Let, b R +, < < b, nd n N. Then, we hve A ( n, b n ) L n n (, b) n(n 1) 1 (b ) mx n, b n }. The following proposition follows from Theorem 11 pplied to the mpping f (x) = x n, x N nd α = 1. Proposition 13. Let, b R +, < < b, ( nd n N. Then for ll q 1, we hve A ( n, b n ) L n n (, b) n(n 1) 1 (b ) mx (n )q, b (n )q}) 1 q. REFERENCES [1] M. Alomri nd M. Drus, On the Hdmrd s inequlity for log-convex functions on the coordintes, J. Ineq. Appl., Volume 9, Article ID 83147, 13 pges. Doi:1.1155/9/ [] M. Alomri, M. Drus nd U.S. Kirmci, Refinements of Hdmrd-type inequlities for qusi-convex functions with pplictions to trpezoidl formul nd to specil mens, Comp. Mth. Appl., 59 (1), 5-3. [3] M. Alomri, M. Drus nd S.S. Drgomir, New inequlities of Hermite-Hdmrd s type for functions whose second derivtives bsolute vlues re qusiconvex, Tmk. J. Mth., 41 (1) [4] M. K. Bkul, M Emin Ozdemir nd J. Pecric, Hdmrd type inequlities for m-convex nd(α, m)- convex, J. Inequl. Pure nd Appl. Mth., 9(8), Article 96. [ONLINE: [5] S. Belrbi nd Z. Dhmni, On some new frctionl integrl inequlities, J. Ineq. Pure nd Appl. Mth., 1(3) (9), Article 86. [6] L. Chun nd F. Qi, Integrl inequlities for Hermite-Hdmrd type for functions whose 3rd derivtives re s-convex, Applied Mthemtics, 3 (1), [7] S. S. Drgomir, R. P. Agrwl nd P. Cerone, On Simpson s inequlity nd pplictions, J. Ineq. Appl., 5(), [8] S. S. Drgomir nd C. E. M. Perce, Selected Topics on Hermite-Hdmrd Inequlities nd Applictions, RGMIA Monogrphs, Victori University,. Online: hdmd. [9] Z. Dhmni, On Minkowski nd Hermite-Hdmrd integrl inequlities vi frctionl integrtion, Ann. Funct. Anl. 1(1) (1), [1] Z. Dhmni, L. Tbhrit, S. Tf, New generliztions of Gruss inequlity using Riemnn-Liouville frctionl integrls, Bull. Mth. Anl. Appl., (3) (1), [11] Z. Dhmni, L. Tbhrit, S. Tf, Some frctionl integrl inequlities, Non l. Sci. Lett. A.,1() (1), [1] Z. Dhmni, New inequlities in frctionl integrls, Interntionl Journl of Nonliner Science, 9(4) (1), [13] R. Gorenflo, F. Minrdi, Frctionl clculus: integrl nd differentil equtions of frctionl order, Springer Verlg, Wien (1997), [14] D. A. Ion, Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions, Annls of University of Criov Mth. Comp. Sci. Ser., 34(7), [15] J. Pecric, F. Proschn nd Y.L. Tong, Convex functions, prtil ordering nd sttisticl pplictions, Acdemic Press, New York, [16] M. Z. Sriky, E. Set, H. Yldiz, nd N. Bsk, Hermite Hdmrd s inequlities for frctionl integrls nd relted frctionl inequlities, Mth. Comput. Model. (1), Online, doi:1.116/j.mcm [17] JinRong Wng, Xuezhu Li, Michl Fe, Hermite Hdmrd-type inequlities for Riemnn Liouville frctionl integrls vi two kinds of convexity, Applicble Anlysis, DOI:1.18/