TAMKANG JOURNAL OF MATHEMATICS Volume 41, Number 4, 353-359, Winter 1 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI, M. DARUS AND S. S. DRAGOMIR Abstrct. In this note we obtin some inequlities of Hermite-Hdmrd type for functions whose second derivtives bsolute vlues re qusi-convex. Applictions for specil mens re lso provided. 1. Introduction Let f : I R R be convex mpping defined on the intervl I of rel numbers nd, b I, with < b. The following two inequlities: ) + b b f f x) dx hold. This double inequlity is known in the literture s the Hermite Hdmrd inequlity for convex functions. In recent yers mny uthors estblished severl inequlities connected to this fct. For recent results, refinements, counterprts, generliztions nd new Hermite-Hdmrd stype inequlities see [1] [18]. We recll tht the notion of qusi-convex function generlizes the notion of convex function. More exctly, 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]. 1.1) Clerly, ny convex function is qusi-convex function. Furthermore, there exist qusiconvex functions which re not convex, see for instnce [1] [5] nd [1]). Recently, D.A. Ion [1] obtined two inequlities of the right hnd side of Hermite- Hdmrd s type for functions whose derivtives in bsolute vlues re qusi-convex functions, s follow: Corresponding uthor: M. Alomri. Received October, 9; revised Mrch, 1. Mthemtics Subject Clssifiction. 6A15, 6A51, 6D1. Key words nd phrses. Qusi-convex function, Hermite-Hdmrd s inequlity, mens. The first uthor cknowledges the finncil support of the Universiti Kebngsn Mlysi, Fculty of Science nd Technology, UKM GUP TMK 7 17). 353
354 M. ALOMARI, M. DARUS AND S. S. DRAGOMIR Theorem 1. Let f : I R R be differentible mpping on I,, b I with < b. If f is qusi-convex on [, b], then the following inequlity holds: f x) dx b b mx { f ), f b) }. 4 Theorem. Let f : I R R be differentible mpping on I,, b I with < b. If f p/p 1) is qusi-convex on [, b], then the following inequlity holds: f x)dx b b ) { mx f ) p/p 1), f b) p/p 1)}) p 1)/p. p + 1 The min im of this pper is to estblish new refined inequlities of the right-hnd side of Hermite-Hdmrd result for the clss of functions whose second derivtives t certin powers re qusi-convex functions.. Hermite-Hdmrd Type Inequlities In order to prove our min theorems, we need the following lemm [1], [16]. Lemm 1. Let f : I R R be twice differentible mpping on I,, b I with < b nd f is integrble on [, b], then the following equlity holds: b f x)dx b ) t 1 t) f t + 1 t)b)dt. A simple proof of this equlity cn be lso done integrting by prts twice in the right hnd side. The detils re left to the interested reder. The next theorem gives new result of the upper Hermite-Hdmrd inequlity for qusi-convex functions. Theorem 3. Let f : I R R be twice differentible mpping on I,, b I with < b nd f is integrble on [, b]. If f is n qusi-convex on [, b], then the following inequlity holds: b f x) dx b ) 1 mx { f ), f b) }. Proof. From Lemm 1, we hve b f x) dx
INEQUALITIES OF HERMITE-HADAMARD TYPE 355 b ) b ) b ) b ) 1 which completes the proof. t 1 t) f t + 1 t)b) dt t 1 t)mx { f ), f b) }dt mx { f ), f b) } mx { f ), f b) } t 1 t)dt The corresponding version for powers of the bsolute vlue of the second derivtive is incorported in the following result: Theorem 4. Let f : I R R be twice differentible mpping on I,, b I with < b nd f is integrble on [, b]. If f p/p 1) is qusi-convex on [, b], for p > 1, then the following inequlity holds: b ) 8 where q p/p ). f x) dx π Γ 1 + p) Γ 3 + p) b mx { f ) q, f b) q}) 1/q Proof. From Lemm 1 nd using the well known Hölder integrl inequlity, we hve successively f x)dx b b ) b ) b ) b ) 8 t 1 t) f t + 1 t)b) dt ) )1/p p dt f t + 1 t)b) q dt π 1 p π Γ 1 + p) Γ 3 + p) Γ 1 + p) Γ 3 + p) )1/q mx { f ) q, f b) q}) 1/q mx { f ) q, f b) q}) 1/q,
356 M. ALOMARI, M. DARUS AND S. S. DRAGOMIR where 1/p+1/q 1. We note tht, the Bet nd Gmm functions see [7], pp 98 91), re defined respectively, s follows: nd β x, y) re used to evlute the integrl Γ x) t x 1 1 t) y 1 dt, x, y > e t t x 1 dt, x > ) p dt t p 1 t) p dt β p + 1, p + 1) Using the properties of Bet function, tht is, β x, x) 1 x β 1, x) nd β x, y) Γx)Γy) Γx+y), we cn obtin tht ) 1 β p + 1, p + 1) 1 p+1) β, p + 1 p 1 Γ 1 ) Γ p + 1) Γ 3 + p), where Γ 1 ) π, which completes the proof. A more generl inequlity is given using Lemm 1, s follows: Theorem 5. Let f : I R R be twice differentible mpping on I,, b I with < b nd f is integrble on [, b]. If f q is n qusi-convex on [, b], q 1, then the following inequlity holds: b f x)dx b ) { mx f ) q, f b) q}) 1/q 1 Proof. From Lemm 1 nd using well known power men inequlity, we hve f x)dx b b ) b ) b ) b ) 1 t 1 t) f t + 1 t) b) dt ) 1 1/q dt) ) 1 1/q 1 1 6 6 mx { f ) q, f b) q} mx { f ) q, f b) q}) 1/q ) ) 1/q f t + 1 t)b) q dt ) 1/q
INEQUALITIES OF HERMITE-HADAMARD TYPE 357 which completes the proof. 3. Applictions to specil mens We consider the mens for rbitrry rel numbers α, β α β). We tke 1. Arithmetic men:. Logrithmic men: L α, β) 3. Generlized log-men: Aα, β) α + β, α, β R. α β, α β, α, β, α, β R. ln α ln β [ β n+1 α n+1 ] 1 n L n α, β), n Z\ { 1, }, α, β R, α β. n + 1)β α) Now, using the results of Section, we give some pplictions for specil mens of rel numbers. Proposition 1. Let, b R, < b nd n N, n. Then, we hve L n n, b) An, b n ) n n ) 1 b ) mx { n, b n }. Proof. The ssertion follows from Theorem 3 pplied to the qusi-convex mpping f x) x n, x R. Proposition. Let, b R, < b nd / [, b]. Then, for ll p > 1, we hve L 1, b) A 1, b 1) b ) π Γ 1 + p) { 4 Γ 3 + p) mx 3q, b 3q}) 1/q. Proof. The ssertion follows from Theorem 4 pplied to the qusi-convex mpping f x) 1/x, x [, b]. Proposition 3. Let, b R, < b nd n N, n. Then, for ll q 1, we hve L n n, b) A n n n ) {, b) b ) mx n )q, b n )q}) 1/q. 1 Proof. The ssertion follows from Theorem 5 pplied to the qusi-convex mpping f x) x n, x R.
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INEQUALITIES OF HERMITE-HADAMARD TYPE 359 School Of Mthemticl Sciences, Universiti Kebngsn Mlysi, UKM, Bngi, 436, Selngor, Mlysi. E-mil: mwomth@gmil.com School Of Mthemticl Sciences, Universiti Kebngsn Mlysi, UKM, Bngi, 436, Selngor, Mlysi. E-mil: mslin@ukm.my Reserch Group in Mthemticl Inequlities & Applictions, School of Engineering & Science, Victori University, PO Box 1448, Melbourne City, MC 81, Austrli. E-mil: sever.drgomir@vu.edu.u