GENERALIZED SOME HERMITE-HADAMARD-TYPE INEQUALITIES FOR CO-ORDINATED CONVEX FUNCTIONS

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1 Plestine Journl o Mthemtis Vol 7(2)(28), Plestine Polytehni University-PPU 28 GENERALIZED SOME HERMITE-HADAMARD-TYPE INEQUALITIES FOR CO-ORDINATED CONVEX FUNCTIONS Mehmet Zeki Srıky nd Tub Tunç Communited by Aymn Bdwi MSC 2 Clssiitions: Primry 26D7, 26D; Seondry 26D5, 26A33 Keywords nd phrses: Hermite-Hdmrd s ineulity, onvex untion, o-ordinted onvex untion Abstrt In this pper, severl ineulities o Hermite-Hdmrd type or untions onvex on the o-ordintes re given Obtined results in this work re the generliztion o the some Hermite-Hdmrd type ineulities or o-ordinted onvex untions Introdution Let : I R R be onvex untion deined on the intervl I o rel numbers nd, b I, with < b Then the ollowing double ineulity is known in the literture s the Hermite- Hdmrd s ineulity or onvex untions: ( ) b 2 (x) dx b () (b) 2 Let us onsider bidimensionl intervl : [, b] [, d] in R 2 with < b nd < d A untion : R 2 R is sid to be onvex on i or ll (x, y), (z, w) nd t [, ], it stisies the ollowing ineulity: (tx ( t) z, ty ( t) w) t (x, y) ( t) (z, w) A modiition or onvex untion on ws deined by Drgomir [6], s ollows: A untion : R is sid to be onvex on the o-ordintes on i the prtil mppings y : [, b] R, y (u) (u, y) nd x : [, d] R, x (v) (x, v) re onvex deined or ll x [, b] nd y [, d] A orml deinition or o-ordinted onvex untion my be stted s ollows: Deinition A untion : R is lled o-ordinted onvex on, i or ll (x, u), (y, v) nd t, s [, ], it stisies the ollowing ineulity: (tx ( t) y, su ( s) v) ts (x, u) t( s)(x, v) s( t)(y, u) ( t)( s)(y, v) Note tht every onvex untion : R is o-ordinted onvex but the onverse is not generlly true (see, [6]) In [6], Drgomir proved the ollowing ineulity whih is Hermite-Hdmrd type ineulity or o-ordinted onvex untions on the retngle rom the plne R 2 Theorem 2 Suppose tht : R is o-ordinted onvex, then we hve the ollowing

2 538 Mehmet Zeki Srıky nd Tub Tunç ineulities; ( b 2, d ) 2 2 b ( x, d ) dx 2 d ( ) b 2, y dy b (b )(d ) b d (x, y) dydx (x, )dx (x, d)dx b (, y)dy d (b, y)dy (, ) (, d) (b, ) (b, d) The bove ineulities re shrp Some new Hermite-Hdmrd type ineulities or o-ordinted onvex untions re proved by mny uthors In [2], Alomri nd Drus deined o-ordinted s-onvex untions nd proved some ineulities bsed on this deinition In [8], Lti nd Alomri proved similr results or h-onvex untions on the o-ordintes In [5], ineulities o Hdmrd type or o-ordinted log-onvex untions re deined in retngle rom the plne by Alomri nd Drus In [9], Sriky et l proved Hermite-Hdmrd type ineulities or dierentible o-ordinted onvex untions In [5], Özdemir et l proved Hdmrd s type ineulities or o-ordinted m onvex nd (α, m) onvex untions For reent developments bout Hermite-Hdmrd s ineulity or some onvex untions on the oordintes, plese reer to ([],[2], [5]-[], []-[8] nd [2]) Also severl ineulities or onvex untions on the o-ordintes see the reerenes [3], [], [2], [3], nd [9] The im o the this pper is to obtin generlized new Hermite-Hdmrd type ineulities o o-ordinted onvex untions o 2-vribles 2 Min Results We strt with the ollowing Lemm whih is importnt our min results Lemm 2 Suppose tht : [, b] [, d] R 2 R is prtil dierentible mpping on nd m, m 2, n, n 2 R I 2 L( ), then we hve the ollowing eulity; n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (2) (b )(d ) (m n )(b ) (x, y)dydx (m 2 n 2 )(d ) (n (x, ) m (x, d))dx (n 2 (, y) m 2 (b, y))dy

3 GENERALIZED HADAMARD-TYPE INEQUALITIES 539 (b )(d ) (m n )(m 2 n 2 ) [m (m n )s] [m 2 (m 2 n 2 )t] 2 (t ( t)b, s ( s)d)dtds Proo Tking prtil integrtion, we hve [m (m n )s] [m 2 (m 2 n 2 )t] 2 (t ( t)b, s ( s)d)dtds [m (m n )s] (m 2 n 2 ) b [m 2 (m 2 n 2 )t] b s (t ( t)b, s ( s)d) dt s ds [m (m n )s] n 2 b (m 2 n 2 ) b b (m 2 n 2 ) s (, s ( s)d) m 2 b (t ( t)b, s ( s)d) dt s ds ( [m (m n )s] n 2 s (, s ( s)d) m 2 (t ( t)b, s ( s)d) (b, s ( s)d) s (22) ) (b, s ( s)d) ds s [m (m n )s] (t ( t)b, s ( s)d) dtds s Agin tking prtil integrtion in the inl eulity o (22), it ollows tht; [m (m n )s] ( n 2 s (, s ( s)d) m 2 ) (b, s ( s)d) ds s (23) (m 2 n 2 ) [m (m n )s] (t ( t)b, s ( s)d) dtds s

4 5 Mehmet Zeki Srıky nd Tub Tunç [m (m n )s] (n 2 (, s ( s)d) m 2 (b, s ( s)d)) d (m n ) d (m 2 n 2 ) (m n ) d (n 2 (, s ( s)d) m 2 (b, s ( s)d)) ds [m (m n )s] n n 2 (, ) m 2 (b, ) d (m n ) d (m 2 n 2 ) (m n ) d (t ( t)b, s ( s)d) d (t ( t)b, s ( s)d) ds dt m n 2 (, d) m 2 (b, d) d (n 2 (, s ( s)d) m 2 (b, s ( s)d)) ds (t ( t)b, ) (t ( t)b, d) n m d d (t ( t)b, s ( s)d) ds dt Writing (23) in (22) nd then using hnge o vrible x t ( t)b nd y s ( s)d (b )(d ) or t, s [, ] nd inlly multiplying the both sides by (m n )(m 2 n 2 ), we get (2) This ompletes the proo Corollry 22 I we hoose m m 2 m nd n n 2 n in Lemm 2, it ollows tht; n 2 (, ) nm (b, ) mn (, d) m 2 (b, d) (m n) 2 (2) (b )(d ) (m n)(b ) (x, y)dydx (m n)(d ) (n (x, ) m (x, d))dx (n (, y) m (b, y))dy (b )(d ) (m n) 2 [m (m n)s] [m (m n)t] 2 (t ( t)b, s ( s)d)dtds

5 GENERALIZED HADAMARD-TYPE INEQUALITIES 5 Remrk 23 I we tke m n in (2), we hve; (, ) (b, ) (, d) (b, d) 2 (d ) (b )(d ) ((, y) (b, y))dy (b ) whih is proved by Sriky etl in [9] b (b )(d ) (x, y)dydx ((x, ) (x, d))dx ( 2s) ( 2t) 2 (t ( t)b, s ( s)d)dtds Theorem 2 Suppose tht : [, b] [, d] R 2 R is prtil dierentible mpping on nd m, m 2, n, n 2 R I 2 is onvex untion on the o-ordintes on, then we hve the ollowing ineulity; n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (b )(d ) (b )(d ) (m n )(m 2 n 2 ) (x, y)dydx A ( 2 A B (, ) A 2 B 2 (, d) A 2 2B (b, ) A 2 ) 2B 2 (b, d) A (m 2 n 2 )(d ) (n 2 (, y)m 2 (b, y))dy (m n )(b ) A m3 2 3m 2n 2 2 2n3 2 6(m 2 n 2 ) 2, A 2 n3 2 3m2 2 n 2 2m 3 2 6(m 2 n 2 ) 2 B m3 3m n 2 2n3 6(m 2 n 2 ) 2, B 2 n3 3m2 n 2m 3 6(m 2 n 2 ) 2 Proo From Lemm 2, we know n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (b )(d ) (x, y)dydx A (n (x, )m (x, d))dx, (b )(d ) (m n )(m 2 n 2 ) m (m n )s m 2 (m 2 n 2 )t 2 (t ( t)b, s ( s)d) dtds

6 52 Mehmet Zeki Srıky nd Tub Tunç Sine 2 is o-ordinted onvex on, we n write n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (b )(d ) (x, y)dydx A (25) (b )(d ) (m n )(m 2 n 2 ) (m (m n )s)(m 2 (m 2 n 2 )t) t 2 (, s ( s)d) ( t) 2 } ] (b, s ( s)d) dt ds Firstly, we lulte the right-side integrl o (25), then we hve ; (m 2 (m 2 n 2 )t) t 2 (, s ( s)d) ( t) 2 } (b, s ( s)d) dt m 2 m 2 n 2 (m 2 (m 2 n 2 )t) t 2 (, s ( s)d) ( t) 2 } (b, s ( s)d) dt ((m 2 n 2 )t m 2 ) m 2 m 2 n 2 Thereore we get; 2 A (, s ( s)d) A 2 t 2 (, s ( s)d) ( t) 2 } (b, s ( s)d) dt 2 (b, s ( s)d) n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (b )(d ) (x, y)dydx A (26) (b )(d ) (m n )(m 2 n 2 ) } (m (m n )s) A 2 (, s ( s)d) A 2 2 (b, s ( s)d) ds Now, we lulte similr wy or other integrl Sine 2 is o-ordinted onvex on, we n write;

7 GENERALIZED HADAMARD-TYPE INEQUALITIES 53 (m (m 2 n )s) A (, s ( s)d) A 2 (27) 2 } (b, s ( s)d) ds A (m (m n )s) s 2 (, ) ( s) 2 } (, d) ds A 2 (m (m n )s) s 2 (b, ) ( s) 2 } (b, d) ds A m m n (m (m n )s) s 2 (, ) ( s) 2 } (, d) ds ((m n )s m ) s 2 (, ) ( s) 2 } (, d) ds m m n A 2 m m n (m (m n )s) s 2 (b, ) ( s) 2 ((m n )s m ) s 2 (b, ) ( s) 2 } (b, d) ds m m n } (b, d) ds 2 A B (, ) A 2 B 2 (, d) A 2 2B (b, ) A 2 2B 2 (b, d) Writing (27) in (26), we obtin the reuired ineulity Corollry 25 I we hoose m m 2 m nd n n 2 n in Theorem 2, it ollows tht; n 2 (, ) nm (b, ) mn (, d) m 2 (b, d) (m n) 2 (b )(d ) (x, y)dydx A (28) (b )(d ) (m n) 2 ( 2 A B (, ) A 2 B 2 (, d) A 2 2B (b, ) A 2 ) 2B 2 (b, d)

8 5 Mehmet Zeki Srıky nd Tub Tunç A (m n)(d ) (n (, y)m (b, y))dy (m n)(b ) A m3 3mn 2 2n 3 6(m n) 2, A 2 n3 3m 2 n 2m 3 6(m n) 2 B m3 3mn 2 2n 3 6(m n) 2, B 2 n3 3m 2 n 2m 3 6(m n) 2 Remrk 26 I we tke m n in (28), we hve; (, ) (b, ) (, d) (b, d) (b )(d ) (b )(d ) 6 A 2 b (x, y)dydx A (n (x, )m (x, d))dx, 2 (, ) 2 (, d) 2 (b, ) 2 (b, d) whih is proved by Sriky etl in [9] ((x, ) (x, d))dx d ((, y) (b, y))dy Theorem 27 Suppose tht : [, b] [, d] R 2 R is prtil dierentible mpping on nd m, m 2, n, n 2 R I 2, > is onvex untion on the o-ordintes on, then we hve the ollowing ineulity; n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (b )(d ) (x, y)dydx A (b )(d ) (m n )(m 2 n 2 ) B 2 (, ) 2 (, d) 2 (b, ) 2 (b, d) A (m 2 n 2 )(d ) nd p B (n 2 (, y)m 2 (b, y))dy (m n )(b ) ( (m m 2 ) p (m 2 n ) p (m n 2 ) p (n n 2 ) p) p (m n ) p (m2 n 2 ) p (p ) 2 p (n (x, )m (x, d))dx,

9 GENERALIZED HADAMARD-TYPE INEQUALITIES 55 Proo From Lemm 2 nd using Hölder ineulity or double integrls, we get; n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (b )(d ) (x, y)dydx A (b )(d ) (m n )(m 2 n 2 ) (m (m n )s)(m 2 (m 2 n 2 )t) 2 (t ( t)b, s ( s)d) dtds (b )(d ) (m n )(m 2 n 2 ) (m (m n )s)(m 2 (m 2 n 2 )t) p dtds p 2 (t ( t)b, s ( s)d) dtds Sine 2 is o-ordinted onvex on, we n write the ollowing ineulities or t, s [, ] 2 (t ( t)b, s ( s)d) t 2 (, s ( s)d) ( t) 2 (b, s ( s)d) nd 2 (t ( t)b, s ( s)d) ts 2 (, ) ( t)s 2 (b, ) t( s) 2 (, d) ( t)( s) 2 (b, d) Thereore, it ollows tht;

10 56 Mehmet Zeki Srıky nd Tub Tunç n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (b )(d ) (x, y)dydx A (b )(d ) (m n )(m 2 n 2 ) B ts 2 (, ) ( t)s 2 (b, ) (b )(d ) (m n )(m 2 n 2 ) B t( s) 2 (, d) ( t)( s) 2 } ) (b, d) dtds 2 (, ) 2 (, d) 2 (b, ) 2 (b, d) Corollry 28 I we hoose m m 2 m nd n n 2 n in Theorem 27, it ollows tht; n 2 (, ) nm (b, ) mn (, d) m 2 (b, d) (m n) 2 (29) (b )(d ) (x, y)dydx A (b )(d ) (m n) 2 B 2 (, ) 2 (, d) 2 (b, ) 2 (b, d) A (m n)(d ) (n (, y)m (b, y))dy (m n)(b ) (n (x, )m (x, d))dx, B ( m 2(p) 2(mn) p n 2(p)) p (m n) 2 p (p ) 2 p

11 GENERALIZED HADAMARD-TYPE INEQUALITIES 57 Remrk 29 I we tke m n in (29), we hve; (, ) (b, ) (, d) (b, d) (b )(d ) (x, y)dydx A (b )(d ) (p ) 2 p 2 (, ) 2 (, d) 2 (b, ) 2 (b, d) A 2 b whih is proved by Srıky et l in [9] ((x, ) (x, d))dx d ((, y) (b, y))dy Theorem 2 Suppose tht : [, b] [, d] R 2 R is prtil dierentible mpping on nd m, m 2, n, n 2 R I 2, is onvex untion on the o-ordintes on, then we hve the ollowing ineulity; n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (b )(d ) (x, y)dydx A (b )(d ) (m n )(m 2 n 2 ) C ( 2 A B (, ) 2 A B 2 (, d) 2 A 2 B (b, ) 2 A 2 B 2 (b, d) ) A (m 2 n 2 )(d ) (n 2 (, y)m 2 (b, y))dy (m n )(b ) (n (x, )m (x, d))dx, A m3 2 3m 2n 2 2 2n3 2 6(m 2 n 2 ) 2, A 2 n3 2 3m2 2 n 2 2m 3 2 6(m 2 n 2 ) 2 B m3 3m n 2 2n3 6(m 2 n 2 ) 2, B 2 n3 3m2 n 2m 3 6(m 2 n 2 ) 2 C ( (m 2 n 2 )(m2 2 n2 2 ) ) (m n )(m 2 n 2 )

12 58 Mehmet Zeki Srıky nd Tub Tunç Proo By Lemm 2, nd power men ineulity or double integrls, we get; n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (b )(d ) (x, y)dydx A (b )(d ) (m n )(m 2 n 2 ) (m (m n )s)(m 2 (m 2 n 2 )t) 2 (t ( t)b, s ( s)d) dtds (b )(d ) (m n )(m 2 n 2 ) (m (m n )s)(m 2 (m 2 n 2 )t) dtds (m (m n )s)(m 2 (m 2 n 2 )t) 2 (t ( t)b, s ( s)d) dtds Sine 2 is o-ordinted onvex on, it ollows tht; n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (b )(d ) (x, y)dxdy A (2) (b )(d ) (m n )(m 2 n 2 ) C (m (m n )s)(m 2 (m 2 n 2 )t) ts 2 (, ) t( s) 2 (, d) ( t)s 2 (b, ) ( t)( s) 2 }) (b, d) Firstly, we lulte the right-side integrl o (2), then we hve ;

13 GENERALIZED HADAMARD-TYPE INEQUALITIES 59 (m 2 (m 2 n 2 )t) ts 2 (, ) ( t)s 2 (b, ) t( s) 2 (, d) ( t)( s) 2 } (b, d) dt m 2 m 2 n 2 (m 2 (m 2 n 2 )t) ts 2 (, ) ( t)s 2 (b, ) t( s) 2 (, d) ( t)( s) 2 } (b, d) dt ((m 2 n 2 )t m 2 ) ts 2 (, ) m 2 m 2 n 2 t( s) 2 (, d) ( t)s 2 (b, ) m 3 2 s 2 6(m 2 n 2 ) 2 (, ) s 2m3 2 3m2 2 n 2 6(m 2 n 2 ) 2 s 2n3 2 3m 2n 2 2 6(m 2 n 2 ) 2 ( t)( s) 2 } (b, d) dt 2 (b, ) 2 (, ) n 3 2 s 2 6(m 2 n 2 ) 2 (b, ) s m3 2 3m 2n 2 2 2n3 2 6(m 2 n 2 ) 2 s n3 2 3m2 2 n 2 2m 3 2 6(m 2 n 2 ) 2 2 sa (, ) m 3 2 ( s) 2 6(m 2 n 2 ) 2 (, d) 2 (, ) ( s) 2m3 2 3m2 2 n 2 6(m 2 n 2 ) 2 ( s) 2n3 2 3m 2n 2 2 6(m 2 n 2 ) 2 2 (b, d) 2 (, d) n 3 2 ( s) 2 6(m 2 n 2 ) 2 (b, d) 2 (b, ) ( 2 s)a (, d) ( s) m3 2 3m 2n 2 2 2n3 2 6(m 2 n 2 ) 2 ( s) n3 2 3m2 2 n 2 2m 3 2 6(m 2 n 2 ) 2 2 sa 2 (b, ) 2 (, d) 2 (b, d) ( 2 s)a 2 (b, d) Thus, we obtin;

14 55 Mehmet Zeki Srıky nd Tub Tunç n n 2 (, ) n m 2 (b, ) m n 2 (, d) m m 2 (b, d) (m n )(m 2 n 2 ) (b )(d ) (x, y)dydx A (2) (b )(d ) (m n )(m 2 n 2 ) C (m (m n )s) 2 sa (, ) ( 2 s)a (, d) 2 sa 2 (b, ) ( 2 s)a 2 (b, d) } ds ) Now, we lulte similr wy or other integrl Sine 2 is o-ordinted onvex on, we n write; (m (m n )s) 2 sa 2 (b, ) 2 sa (, ) ( 2 } s)a 2 (b, d) ds ( 2 s)a (, d) (22) m m n (m (m n )s) 2 sa 2 (b, ) 2 sa (, ) ( 2 } s)a 2 (b, d) ds ( 2 s)a (, d) m m n ((m n )s m ) 2 sa 2 (b, ) 2 sa (, ) ( 2 } s)a 2 (b, d) ds ( 2 s)a (, d)

15 GENERALIZED HADAMARD-TYPE INEQUALITIES 55 A m 3 6(m n ) 2 A 2 m 3 6(m n ) 2 A 2n 3 3m n 2 6(m n ) 2 A 2 2n 3 3m n 2 6(m n ) 2 A m 3 3m n 2 2n3 6(m n ) 2 2 (, ) 2 (b, ) A 2 m 3 3m n 2 2n3 6(m n ) 2 2 A B (, ) A 2m 3 3m2 n 6(m n ) 2 2 (, ) 2 (b, ) 2 (, ) A 2 2m 3 3m2 n 6(m n ) 2 A n 3 6(m n ) 2 A 2 n 3 6(m n ) 2 2 (b, ) 2 A B 2 (, d) 2 (, d) 2 (b, d) 2 (, d) 2 (b, d) A n 3 3m2 n 2m 3 6(m n ) 2 Writing (22) in (2) we obtin the reuired ineulity A 2 n 3 3m2 n 2m 3 6(m n ) 2 2 A 2 B (b, ) 2 (, d) 2 (b, d) 2 A 2 B 2 (b, d) Corollry 2 I we hoose m m 2 m nd n n 2 n in Theorem 2, it ollows tht; n 2 (, ) nm (b, ) mn (, d) m 2 (b, d) (m n) 2 (b )(d ) (x, y)dydx A (23) (b )(d ) (m n) 2 C ( 2 A B (, ) 2 A B 2 (, d) 2 A 2 B (b, ) 2 A 2 B 2 (b, d) ) A (m n)(d ) (n (, y)m (b, y))dy (m n)(b ) A m3 3mn 2 2n 3 6(m n) 2, A 2 n3 3m 2 n 2m 3 6(m n) 2 B m3 3mn 2 2n 3 6(m n) 2, B 2 n3 3m 2 n 2m 3 6(m n) 2 ( (m C 2 n 2 ) 2 ) (m n) 2 (n (x, )m (x, d))dx,

16 552 Mehmet Zeki Srıky nd Tub Tunç Remrk 22 I we tke m n in (23), we hve; (, ) (b, ) (, d) (b, d) (b )(d ) (x, y)dydx A (b )(d ) 6 A 2 2 (, ) 2 (, d) 2 (b, ) 2 (b, d) b whih is proved by Srıky etl in [9] ((x, ) (x, d))dx d ((, y) (b, y))dy Reerenes [] M Alomri nd M Drus, Co-ordinted s-onvex untion in the irst sense with some Hdmrd-type ineulities, Int J Contemp Mth Sienes, 3(32), 28, [2] M Alomri nd M Drus, The Hdmrd s ineulity or s -onvex untions o 2-vribles on the oordintes, Int J Mth Anl, 2, 3 (28), [3] M Alomri nd M Drus, Grüss-type ineulities or Lipshitzin onvex mppings on the o-ordintes, Leture series on geometri untion theory I, in onjuntion with the Workshop or geometri untion theory, April, Puri Pujngg-UKM: (29), [] M Alomri nd M Drus, Fejer ineulity or double integrls, Ft Universittis (NI S): Ser Mth Inorm 2(29), 5-28 [5] M Alomri nd M Drus, On the Hdmrd s ineulity or log-onvex untions on the oordintes, J o Ineul nd Appl, Artile ID 2837, 29, 3 pges [6] S S Drgomir, On Hdmrd s ineulity or onvex untions on the o-ordintes in retngle rom the plne, Tiwnese Journl o Mthemtis, (2), [7] M A Lti nd M Alomri, Hdmrd-type ineulities or produt two onvex untions on the oordintes, Int Mth Forum, (7), 29, [8] M A Lti nd M Alomri, On the Hdmrd-type ineulities or h-onvex untions on the oordintes, Int J o Mth Anlysis, 3(33), 29, [9] M Z Sriky, E Set, ME Ozdemir, S S Drgomir, New some Hdmrd s type ineulities or oordinted onvex untions, Tmsui Oxord J o Inormtion nd Mth Sienes, 28(2) (2), [] M ZSriky, On the Hermite-Hdmrd-type ineulities or o-ordinted onvex untion vi rtionl integrls, Integrl Trnsorms nd Speil Funtions, Vol 25, No 2, 3-7, 2 [] M Z Sriky nd H Yldiz, On the Hdmrd s type ineulities or L-Lipshitzin mpping, Konurlp Journl o Mthemtis, Volume, No 2, pp 33- (23) [2] M Z Sriky, H Budk nd H Yldiz, Cebysev type ineulities or o-ordinted onvex untions, Pure nd Applied Mthemtis Letters, 2(2), 36- [3] M Z Sriky, H Budk nd H Yldiz, Some new Ostrowski type ineulities or o-ordinted onvex untions, Turkish Journl o Anlysis nd Number Theory, 2, Vol 2, No 5, [] E Set, M Z Sriky nd H Ogulmus, Some new ineulities o Hermite-Hdmrd type or h-onvex untions on the o-ordintes vi rtionl integrls, Ft Universittis, Series: Mthemtis nd Inormtis, Vol 29, No (2), 397 [5] M E Ozdemir, E Set nd M Z Sriky, New some Hdmrd s type ineulities or oordinted m- onvex nd (α, m)-onvex untions, RGMIA, Res Rep Coll, 3 (2), Supplement, Artile

17 GENERALIZED HADAMARD-TYPE INEQUALITIES 553 [6] M E Ozdemir, ÇYıldız And A O Akdemir, On some new Hdmrd-Type ineulities or o-ordinted usi-onvex untions, Het J Mth Stt, 5 (22), [7] M E Ozdemir, H Kvurmı, A O Akdemir nd M Avı, Ineulities or onvex nd s-onvex untions on : [, b] [, d], Jo Ineul nd Appl, 22, 22:2 [8] F Chen, A note on the Hermite-Hdmrd ineulity or onvex untions on the o-ordintes, Jo Mth Ineulities, 8(), 2, [9] M K Bkul nd J Peri, On the Jensen s ineulity or onvex untions on the o-ordintes in retngle rom the plne, Tiwnese Journl o Mthemtis, (5), 26, [2] D Y Hwng, K L Tseng, nd G S Yng, Some Hdmrd s ineulities or o-ordinted onvex untions in retngle rom the plne, Tiwnese Journl o Mthemtis, (27), Author inormtion Mehmet Zeki Srıky, Deprtment o Mthemtis, Fulty o Siene nd Arts, Düze University, Düze, TURKEY E-mil: srikymz@gmilom Tub Tunç, Deprtment o Mthemtis, Fulty o Siene nd Arts, Düze University, Düze, TURKEY E-mil: tubtun3@gmilom Reeived: April, 26 Aepted: April 22, 27