Int. Journl o Mth. Anlysis, Vol., 008, no. 3, 639-646 The Hdmrd s Inequlity or s-conve Function M. Alomri nd M. Drus School o Mthemticl Sciences Fculty o Science nd Technology Universiti Kebngsn Mlysi Bngi 43600 Selngor, Mlysi lomri@mth.com Corresponding uthor. mslin@pkrisc.cc.ukm.my Abstrct A monotone nondecresing mpping connected with Hdmrd type inequlity or s conve unction nd some pplictions re given. s Hdmrd s inequlity, s Conve unction, Jensen s in- Keywords: equlity Introduction Let : I R R be conve mpping deined on the intervl I o rel numbers nd, b I, with <b. The ollowing double inequlity: ( + b b (+ (b ( b is known in the literture s Hdmrd s inequlity or conve mppings. In [] Hudzik nd Mligrd considered mong others the clss o unctions which re s conve in the second sense. This clss is deined in the ollowing wy: unction :[0, R is sid to be s conve in the second sense i (λ +( λ y λ s (+( λ s (y ( holds or ll, y [0,, λ [0, ] nd or some ied s (0, ]. It cn be esily seen tht every s conve unction is conve when s. (
640 M. Alomri nd M. Drus In [] Drgomir nd Fitzptrick proved vrint o Hdmrd s inequlity which holds or s conve unctions in the second sense; which is so clled s Hdmrd type inequlity or s conve unction in nd sense. Theorem. Suppose tht :[0, [0, is n s conve unction in the second sense, where s (0, nd let, b [0,, <b.i L [0, ], then the ollowing inequlities hold: ( + b s b ( b (+ (b s + the constnt k is the best possible in the second inequlity in (.3. s+ The bove inequlities re shrp. In [6], Yng nd Hong estblished the ollowing theorem which is reinement o the second inequlity o (. Theorem. Suppose tht :[, b] R is conve on [, b] nd the mpping F :[0, ] R is deined by Then F (t b [ (b (( +t + ( t (i F is n conve on [0, ]. + (ii F is monotone incresing on [0, ]. (iii One hs the bounds (( +t b + ( t ] (3 nd in F (t F (0 t [0,] b (, (b sup F (t F ( t [0,] (+ (b.
Hdmrd s inequlity 64 For more reinements, counterprts nd generliztion see [3 6]. Hdmrd s Inequlity Lemm. Let :[, b] R be s conve unction nd let y y b with + y + y. Then Proo. ( + ( (y + (y (4 First we show tht ( + ( (y + (y. I y y then we re done. Suppose y y nd write y y y y + y y y y, y y y y + y y y y, since is s conve, we hve ( + ( y y y (y + y y y (y which completes the proo. + y (y + y (y y y y y y ( + (y + ( + y (y y y y y (y + (y. (5 The ollowing inequlity is considered the mpping connected with the inequlity (3. Theorem. Suppose tht :[, b] R is s conve on [, b] nd the mpping F :[0, ] R is deined by F (t Then b (s +(b [ (( +t + ( t + (( +t b + ( t ]
64 M. Alomri nd M. Drus (i F is n s conve on [0, ]. (ii F is monotone incresing on [0, ]. (iii One hs the bounds Proo. in F (t F (0 t [0,] (, (s +(b sup F (t F ( t [0,] (+ (b. s + (i For ll α, β 0 with α + β nd t,t [0, ], we hve: F (αt + βt b ( +(αt + βt (b ( +(αt + βt + b (b b ( α ( + t +( t (b + b ( α ( + t b +( t (b αs b [ ( ( + t (b + βs b [ ( ( + t (b α s F (t +β s F (t. + (αt + βt b + (αt + βt + β ( + t +( t d + ( t ( ( + t + Thereore, F is s conve unction on [0, ]. + ( t ( ( + t + (ii Let 0 t t, b. Since b ( ( + t b + ( t + β ( + t b +( t d b + ( t ] b + ( t ]
Hdmrd s inequlity 643 Thus, we hve nd since Thus, b ( ( + t F (t (b ( ( + t + ( + t [ ( + t [ ( + t + ( t + ( t + ( t b + ( t b [ (b + d. ( ( + t + ( t b + ( t ] (b + ( + t + ( t ( + t b + ( t (b + ( + t b + ( t (b + ] [ ( + t + ] [ ( + t + b + ( t b + ( t (b + (b + nd since is s conve on [, b], nd by Lemm., we hve: F (t (b ( ( + t + b [ ( ( + t + ( t b + ( t ] (b + ] ] b [ ( ( + t (b F (t. + ( t ( ( + t + This shows tht F (t is monotone incresing or ll t [0, ]. b + ( t ]
644 M. Alomri nd M. Drus (iii It ollows rom (ii, tht, or ll t [0, ] nd F (t F (0 b [ (s +(b ( ( ] + b + + d b (, (6 (s +(b F (t F ( b [ ( + (b] d (s +(b (+ (b s + (7 Remrk : In (6 nd (7, set s we get inequlity. Also, i we set s in (3 we get the sme result. 3 Hdmrd s Inequlity For Lipschitzin Mpping Theorem 3. Let :[, b] R stisy Lipschitzin conditions. Tht is, or t nd t [0, ], we hve where L is positive constnt. Then (t (t L t t Proo. F (t F (t L t t (b s + (8 For t,t [0, ], we hve F (t (s +(b b [ ( ( + t + ( t ( ( + t + ( t
Hdmrd s inequlity 645 ( ( + + t (s +(b b L L t t (b (s + This completes the proo. b + ( t ( ( + t [ ( t t + ( ( t t + t t b + ( t ] d b + Remrk : In (8 i we tke t 0 nd t, then (8 reduce to (+ (b s + b (s +(b ( ( ] t t d L (b (s +. (9 The inequlity (9 is the s Hdmrd type inequlity or Lipschitzin mpping o one vrible. Acknowledgement : GUP TMK 07 0 07. The work here is supported by the Grnt: UKM Reerences [] H. Hudzik, L. Mligrnd, Some remrks on s-conve unctions, Aequtiones Mth., 48 (994, 00 -. [] S.S. Drgomir, S. Fitzptrick, The Hdmrd s inequlity or s-conve unctions in the second sense, Demonstrtio Mth., 3 4 (999, 687-696. [3] S. S. Drgomir, On Hdmrd s inequlity or conve unctions on the co-ordintes in rectngle rom the plne, Tiwnese Journl o Mthemtics, 5 (00, 775-788. [4] S. S. Drgomir, A mpping in connection to Hdmrd s inequlity, An Ostro. Akd. Wiss. Mth. -Ntur (Wien 8 (99, 7-0. [5] S. S. Drgomir, Two mppings in connection to Hdmrd s inequlity, Mth. Anl. Appl., 67 (99, 49-56.
646 M. Alomri nd M. Drus [6] G. S. Yng nd M. C. Hong, A note on Hdmrd s inequlity, Tmkng J. Mth., 8 (997, 33-37. Received: Jnury 3, 008