Punjb University Journl of Mthemtics (ISSN 116-56) Vol. 45 (13) pp. 33-38 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: sbiriub@yhoo.com S. Qisr College of Mthemtics nd Sttistics, Chongqing University, Chongqing, 41331, P. R. Chin. Emil: shhidqisr9@yhoo.com 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
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)
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.
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)
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 >
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