Hadamard-Type Inequalities for s-convex Functions

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1 Interntionl Mthemtil Forum, 3, 008, no. 40, Hdmrd-Type Inequlitie or -Convex Funtion Mohmmd Alomri nd Mlin Dru Shool o Mthemtil Siene Fulty o Siene nd Tehnology Univeriti Kebngn Mlyi Bngi Selngor, Mlyi Abtrt In thi pper Hdmrd type inequlitie or onvex untion in both ene nd onvex untion on the o ordinte re given. Keyword: Hdmrd inequlity, onvex untion, o ordinted onvex untion Introdution Let : I R R be onvex mpping deined on the intervl I o rel number nd, b I, with <b. The ollowing double inequlity: ) + b b )+ b) x) dx b i known in the literture Hdmrd inequlity or onvex mpping. In [7] Drgomir nd Fitzptrik proved vrint o Hdmrd inequlity whih hold or onvex untion in the irt ene. ) Theorem. Suppoe tht :[0, ) [0, ) i n onvex untion in the irt ene, where 0, ) nd let, b [0, ), <b. I L [0, ], then the ollowing inequlitie hold: ) + b b b x) dx Firt uthor: lomri@mth.om Correponding uthor: mlin@pkri..ukm.my )+ b). ) +

2 966 M. Alomri nd M. Dru The bove inequlitie re hrp. Alo, In [7], Drgomir nd Fitzptrik proved vrint o Hdmrd inequlity whih hold or onvex untion in the eond ene. Theorem. Suppoe tht :[0, ) [0, ) i n onvex untion in the eond ene, where 0, ) nd let, b [0, ), <b.i L [0, ], then the ollowing inequlitie hold: ) + b b x) dx b )+ b) + the ontnt k = i the bet poible in the eond inequlity in 3). The + bove inequlitie re hrp. Ater tht, in [8], Drgomir etblihed the ollowing imilr inequlity o Hdmrd type or o ordinted onvex mpping on retngle rom the plne R. 3) Theorem.3 Suppoe tht :Δ R i o-ordinted onvex on Δ. Then one h the inequlitie + b, + d ) The bove inequlitie re hrp. b d x, y) dydx b )d ), )+, d)+ b, )+ b, d) 4 In [], M. Alomri nd M. Dru etblihed the ollowing imilr inequlity o Hdmrd type or o ordinted onvex mpping in the eond ene on retngle rom the plne R. 4) Theorem.4 Suppoe tht :Δ=[, b] [, d] [0, ) [0, ) i onvex untion on the o ordinte on Δ. Then one h the inequlitie: + b, + d ) b d x, y) dydx 5) b )d ), )+ b, )+, d)+ b, d) +).

3 Hdmrd-type inequlitie 967 Alo, In [5], M. Alomri nd M. Dru etblihed the ollowing imilr inequlity o Hdmrd type or o ordinted onvex mpping in the irt ene on retngle rom the plne R. Theorem.5 Suppoe tht :Δ=[, b] [, d] [0, ) [0, ) i onvex untion on the o ordinte in the irt ene on Δ. Then one h the inequlitie: + b, + d ) b d x, y) dydx 6) b )d ), )+ b, )+, d)+ b, d) +). The bove inequlitie re hrp. In thi pper we will point out Hdmrd type inequlitie o onvex untion on the o ordinte in the both ene. For reinement, ounterprt, generliztion nd new Hdmrd type inequlitie ee [ 8]. Remrk On A Previou Reult In thi etion we give ome remrk on previou reult or the uthor. The ollowing lemm oited with onvex untion o eond ene) w onidered by Alomri nd Dru in [3]. Lemm. Let :[, b] R be onvex untion o eond ene). Let y x x y b with x + x = y + y. Then x )+ x ) y )+ y ). 7) Atully, the proo w given in [3] o thi property i orret or onvex untion but not or onvex untion. The orretion o thi proo i given ollow: Proo. Firtly, we how tht x )+ x ) y )+ y ). I y = y then we re done. Suppoe y y nd ine i onvex untion o eond ene, then or α, β 0 with α + β = nd or ll 0 <, we hve x = y x y y ) y + x y y y ) y, x = y x y y ) y + x y y y ) y

4 968 M. Alomri nd M. Dru without lo o generlity, et k = y x y y ), k = x y y y ), k3 = y x y y ), k4 = x y y y ) uh tht, γ = k + k + k 3 + k 4 > 0;α = k +k 3 nd β = k +k 4. Thereore, γ γ α + β = nd by onvexity, we hve x )+ x ) = y x y y whih omplete the proo. + y x y y ) y + ) y + x y y y x y y y ) y ) y y x y )+ x y y ) 8) y y y y + y x y )+ x y y ) y y y y = y x + x ) y )+ x + x ) y y ) y y y y = y )+ y ). Alo, by looking deeply on Theorem.4, we ind the let ide o inequlity 5) i inorret. The orretion o Theorem.4. pointed out ollow : Theorem. Suppoe tht :Δ=[, b] [, d] [0, ) [0, ) i onvex untion on the o ordinte on Δ. Then one h the inequlitie: 4 + b, + d ) b d x, y) dydx 9) b )d ), )+ b, )+, d)+ b, d) +). Indeed, the dierene between 5) nd 9) i the let hnd ide, thereore we will give the proo o the let hnd ide only, to ee the proo o the right hnd ide ee []. Proo. Sine :Δ R i o ordinted onvex on Δ it ollow tht the mpping g x : [, d] [0, ), g x y) = x, y) i onvex on [, d] or ll

5 Hdmrd-type inequlitie 969 x [, b]. Then, by 3) one h: ) + d g x d d Tht i, x, + d ) d x, y) dy d Integrting thi inequlity on [, b], we hve b x, + d ) dx b b )d ) + b g x y) dy g x )+g x d), x [, b]. + x, )+ x, d), x [, b]. + b b d x, y) dydx 0) x, ) dx + b b x, d) dx. A imilr rgument pplied or the mpping g y :[, b] [0, ), g y x) = x, y), we get d ) d b + b d,y dy x, y) dxdy ) d )b ) + d d, y) dy + d d b, y) dy. Summing the inequlitie 0) nd ), we get the eond nd the third inequlitie in 9). Thereore, by 3), we hve 4 + b, + d ) d ) + b d,y dy ) nd + b 4, + d ) b x, + d ) dx 3) b whih give, by ddition the irt inequlity in 9). The deinition o onvex untion in both ene ) in retngle rom the plne, w deined by Alomri nd Dru in [4]. In the next etion ome Hdmrd type inequlitie re onidered.

6 970 M. Alomri nd M. Dru 3 Some Hdmrd Type Inequlitie Conider the bidimenionl intervl Δ := [, b] [, d] in[0, ) with <b nd <d. A mpping :Δ R i lled onvex o irt ene on Δ i there exit, 0, ] with = +, uh tht αx + βz,αy + βw) α x, y)+β z, w) 4) hold or ll x, y), z, w) Δ, α, β 0 with α + β = nd or ll ixed, 0, ]. We denote thi l o untion by MWO,. Let :Δ R be onvex on Δ, then i lled o ordinted onvex o irt ene on Δ i the prtil mpping y :[, b] R, y u) = u, y) nd x : [, d] R, x v) = x, v), re, onvex untion in the irt ene or ll, 0, ], y [, d] nd x [, b]; repetively, with = + 0, ]. Alo, mpping :Δ Ri lled onvex o eond ene on Δ i there exit, 0, ] with = +, uh tht 4) hold or ll x, y), z, w) Δ, α, β 0 with α + β = nd or ll ixed, 0, ]. We denote thi l o untion by MWO,. Similrly, we deine the onvex untion o eond ene on the o ordinte, i.e., untion i lled o ordinted onvex o eond ene on Δ i the prtil mpping y :[, b] R, y u) = u, y) nd x :[, d] R, x v) = x, v), re, onvex untion in the eond ene or ll, 0, ], y [, d] nd x [, b]; repetively, with = + 0, ]. The ollowing inequlitie i onidered Hdmrd type inequlitie onneted with inequlity 4) or onvex untion in the eond ene on the o ordinte. Theorem 3. Suppoe tht :Δ=[, b] [, d] [0, ) [0, ) i onvex untion o eond ene on the o ordinte on Δ. Then one h the inequlitie: 4 +4 ) b x, + d b + b, + d ) ) dx + d d ) + b,y dy b d x, y) dydx 5) b )d )

7 Hdmrd-type inequlitie 97 b [ x, )+ x, d)] dx +)b ) d + [, y)+ b, y)] dy +)d ) ) +) + +) [, )+, d)+ b, )+ b, d)] The bove inequlitie re hrp. Proo. Sine :Δ R i o ordinted onvex on Δ it ollow tht the mpping g x :[, d] [0, ), g x y) = x, y) i onvex on [, d] or ll x [, b] with 0, ]. Then by Hdmrd inequlity 3) one h: ) + d g x d d Tht i, x, + d ) d x, y) dy d Integrting thi inequlity on [, b], we hve b x, + d ) dx b b )d ) + b g x y) dy g x )+g x d), x [, b]. + x, )+ x, d), x [, b]. + b d b x, y) dydx 6) x, ) dx + b b x, d) dx. A imilr rgument pplied or the mpping g y :[, b] [0, ), g y x) = x, y), where, g y i onvex on [, b] or ll y [, d] with 0, ] d ) + b d,y d b dy x, y) dxdy 7) d )b ) + d d, y) dy + d d b, y) dy. Summing the inequlitie 6) nd 7), we get the eond nd the third inequlitie in 5).

8 97 M. Alomri nd M. Dru nd Thereore, by 3), we hve 4 + b, + d ) d ) + b d,y dy 8) 4 + b, + d ) b x, + d ) dx 9) b whih give, by ddition the irt inequlity in 5). nd Finlly, by the me inequlity we n lo tte: b, )+ b, ) x, ) dx b + b, d)+ b, d) x, d) dx b + d, )+, d), y) dy d + d b, )+ b, d) b, y) dy d + whih give, by ddition the lt inequlity in 5). Remrk 3. In 5), i = =, then 5) redued to inequlity 4). Alo, in 5), i =, then 5) redued to inequlity 9). The ollowing inequlitie i onidered Hdmrd type inequlitie onneted with inequlity 4) or onvex untion in the irt ene on the o ordinte. Theorem 3.3 Suppoe tht :Δ=[, b] [, d] [0, ) [0, ) i onvex untion on the o ordinte in the irt ene on Δ. Then one h the inequlitie: + b, + d )

9 Hdmrd-type inequlitie 973 b x, + d b ) dx + d ) + b d,y dy b d x, y) dydx 0) b )d ) +)b ) b + +)d ) d [ x, )+ x, d)] dx [, y)+ b, y)] dy, )+, d)+ b, )+ b, d) +) +, )+, d)+ b, )+ b, d) +) The bove inequlitie re hrp. Proo. Sine :Δ R i o ordinted onvex in irt ene on Δ it ollow tht the mpping g x :[, d] [0, ), g x y) = x, y) i onvex on [, d] or ll x [, b]. Then by Hdmrd inequlity ) one h: ) + d g x d d Tht i, x, + d ) d d Integrting thi inequlity on [, b], we hve b x, + d ) dx b b )d ) + b g x y) dy g x )+ g x d), x [, b]. + x, y) dy x, )+ x, d), x [, b]. + b d b x, y) dydx ) x, ) dx + b b x, d) dx. A imilr rgument pplied or the mpping g y :[, b] [0, ), g y x) = x, y), we get d ) + b d,y dy d b x, y) dxdy ) d )b )

10 974 M. Alomri nd M. Dru d, y) dy + d + d d b, y) dy. Summing the inequlitie ) nd ), we get the eond nd the third inequlitie in 0). nd Thereore, by Hdmrd inequlity ), we lo hve: + b, + d ) d ) + b d,y dy 3) + b, + d ) b x, + d ) dx 4) b whih give, by ddition the irt inequlity in 0). nd Finlly, by the me inequlity we n lo tte: b x, ) dx, )+ b, ) b + b x, d) dx, d)+ b, d) b + d, y) dy, )+, d) d + d b, y) dy b, )+ b, d) d + whih give, by ddition the lt inequlity in 0). Remrk 3.4 In 0), i = =, then 0) redued to inequlity 4). Alo, in 0), i =, then 0) redued to inequlity 6). ACKNOWLEDGEMENT. The work here i upported by the Grnt: UKM GUP TMK

11 Hdmrd-type inequlitie 975 Reerene [] M. Alomri nd M. Dru, A mpping onneted with Hdmrd type inequltie in 4 vrible, Int. Journl o Mth. Anlyi, 3) 008), [] M. Alomri nd M. Dru, The Hdmrd inequlity or onvex untion o vrible On The o ordinte, Int. Journl o Mth. Anlyi, 3) 008), [3] M. Alomri nd M. Dru, The Hdmrd inequlity or onvex untion, Int. Journl o Mth. Anlyi, 3) 008), [4] M. Alomri nd M. Dru, On o ordinted onvex untion, Interntionl Mthemtil Forum, ubmitted. [5] M. Alomri nd M. Dru, Co ordinte onvex untion in the irt ene with ome Hdmrd type inequlitie, Int. J. Contemp. Mth. Si., ubmitted. [6] H. Hudzik, L. Mligrnd, Some remrk on onvex untion, Aequtione Mth., ), 00. [7] S.S. Drgomir, S. Fitzptrik, The Hdmrd inequlity or -onvex untion in the eond ene, Demontrtio Mth., 3 4) 999), [8] S. S. Drgomir, On Hdmrd inequlity or onvex untion on the o ordinte in retngle rom the plne, Tiwnee Journl o Mthemti, 5 00), Reeived: My, 008