Co-ordinated s-convex Function in the First Sense with Some Hadamard-Type Inequalities

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1 Int. J. Contemp. Mth. Sienes, Vol. 3, 008, no. 3, Co-ordinted s-convex Funtion in the First Sense with Some Hdmrd-Type Inequlities Mohmmd Alomri nd Mslin Drus Shool o Mthemtil Sienes Fulty o Siene nd Tehnology Universiti Kebngsn Mlysi Bngi Selngor, Mlysi Abstrt In this pper Hdmrd s type inequlity o s onvex untion in irst sense nd s onvex untion o vribles on the o ordintes re given. A monotoni nonderesing mpping onneted with the Hdmrd s inequlity or Lipshitzin s onvex mpping in the irst sense o one vrible is estblished. Keywords: Hdmrd s inequlity, s Convex untion, Co ordinted s onvex untion. Introdution Let : I R R be onvex mpping deined on the intervl I o rel numbers nd, b I, with <b. The ollowing double inequlity: ( ) + b b ()+ (b) (x) dx b () is known in the literture s Hdmrd s inequlity or onvex mppings. First uthor: lomri@mth.om Corresponding uthor: mslin@pkris..ukm.my

2 558 M. Alomri nd M. Drus In [9], Orliz introdued two deinitions o s onvexity o rel vlued untions. A untion : R + R, where R + =[0, ), is sid to be s onvex in the irst sense i (αx + βy) α s (x)+β s (y) () or ll x, y [0, ), α, β 0 with α s + β s = nd or some ixed s (0, ]. We denote this lss o untions by K s. Also, untion : R + R, where R + =[0, ), is sid to be s onvex in the seond sense i (αx + βy) α s (x)+β s (y) (3) or ll x, y [0, ), α, β 0 with α + β = nd or some ixed s (0, ]. We denote this lss o untions by K s. These deinitions o s onvexity, or so lled ϕ untions, ws introdued by Orliz in [9] nd ws used in the theory o Orliz spes (see [7], [8], [0]). A untion : R + R + is sid to be ϕ untion i (0) = 0 nd is non deresing nd ontinuous. Its esily to hek tht the both s onvexity men just the onvexity when s =. In [4], Hudzik nd Mligrd onsidered mong others the lss o untions whih re s onvex in the irst sense. This lss is deined in the ollowing wy: A untion :[0, ) R is sid to be s onvex in the irst sense i (αx + βy) α s (x)+β s (y) (4) holds or ll x, y [0, ), α, β 0 with α s + β s = nd or some ixed s (0, ]. It n be esily seen tht every onvex untion is onvex. Also, in [4], Hudzik nd Mligrd proved vrint properties o s onvex untion in the irst nd in the seond sense, let us tke the ollowing theorem. Theorem. Let 0 <s. I K s nd (0) = 0 then K s. In [5] Drgomir nd Fitzptrik proved vrint o Hdmrd s inequlity whih holds or s onvex untions in the irst sense.

3 Co-ordinted s-onvex untion 559 Theorem. Suppose tht :[0, ) [0, ) is n s onvex untion in the irst sense, where s (0, ) nd let, b [0, ), <b. I L [0, ], then the ollowing inequlities hold: ( ) + b The bove inequlities re shrp. b (x) dx b ()+s (b). (5) s + Also, in [5], Drgomir nd Fitzptrik proved vrint o Hdmrd s inequlity whih holds or s onvex untions in the seond sense. Theorem.3 Suppose tht :[0, ) [0, ) is n s onvex untion in the seond sense, where s (0, ) nd let, b [0, ), <b.i L [0, ], then the ollowing inequlities hold: s ( ) + b b (x) dx b ()+ (b) s + (6) the onstnt k = is the best possible in the seond inequlity in (.3). The s+ bove inequlities re shrp. Ater tht, in [6], Drgomir estblished the ollowing similr inequlity o Hdmrd s type or o-ordinted onvex mpping on retngle rom the plne R. Theorem.4 Suppose tht :Δ R is o-ordinted onvex on Δ. Then one hs the inequlities ( + b, + d ) The bove inequlities re shrp. b d (x, y) dydx (b )(d ) (, )+ (, d)+ (b, )+ (b, d) 4 (7) In this pper we will point out Hdmrd type inequlity o s onvex untion in irst sense nd s onvex untions o vribles on the o ordintes. A monotoni nonderesing mpping onneted with the Hdmrd s inequlity or Lipshitzin s onvex mpping in the irst sense o one vrible is given. For reinements, ounterprts, generliztions nd new Hdmrd s type inequlities see [ 6].

4 560 M. Alomri nd M. Drus Hdmrd s Inequlity In [], Alomri nd Drus estblished the deinition o s onvex untion in the seond sense on o ordintes. Similrly, one n deine the s onvex untion in the irst sense on o ordintes, s ollows: Deinition. Consider the bidimensionl intervl Δ:=[, b] [, d] in [0, ) with <bnd <d. The mpping :Δ R is s onvex in the irst sense on Δ i (αx + βz,αy + βw) α s (x, y)+β s (z, w), holds or ll (x, y), (z, w) Δ with α, β 0 with α s + β s =nd or some ixed s (0, ]. Thereore, one n tlk bout o ordinted s onvex untion in the irst sense, s ollows: A untion :Δ R is s onvex in the irst sense on Δ is lled o ordinted s onvex in the irst sense on Δ i the prtil mppings y :[, b] R, y (u) = (u, y) nd x :[, d] R, x (v) = (x, v), re s onvex in the irst sense or ll y [, d] nd x [, b] suh tht s (0, ], i.e, the prtil mppings y nd x s onvex with sme ixed s (0, ]. The ollowing inequlities is onsidered the Hdmrd type inequlities or s onvex untion in the irst sense on the o ordintes. Theorem. Suppose tht :Δ=[, b] [, d] [0, ) [0, ) is s onvex untion on the o ordintes in the irst sense on Δ. Then one hs the inequlities: ( + b, + d ) b b ( (b )(d ) (s +) b x, + d b d b + d (, y) dy + d ) dx + d ( ) + b d,y dy (x, y) dydx (8) (x, ) dx + s d d s b (x, d) dx b (b, y) dy (, )+s (b, )+s (, d)+s (b, d) (s +).

5 Co-ordinted s-onvex untion 56 The bove inequlities re shrp. Proo. Sine :Δ R is o ordinted s onvex in irst sense on Δ it ollows tht the mpping g x :[, d] [0, ), g x (y) = (x, y) iss onvex on [, d] or ll x [, b]. Then by s Hdmrd s inequlity (5) one hs: ( ) + d g x d d Tht is, ( x, + d ) d (x, y) dy d g x (y) dy g x ()+sg x (d), x [, b]. s + (x, )+s (x, d), x [, b]. s + Integrting this inequlity on [, b], we hve b ( x, + d ) b d dx (x, y) dydx (9) b (b )(d ) b (x, ) dx + s b (x, d) dx. s + b b A similr rguments pplied or the mpping g y :[, b] [0, ), g y (x) = (x, y), we get d ( ) + b d,y d b dy (x, y) dxdy (0) (d )(b ) d (, y) dy + s d (b, y) dy. s + d d Summing the inequlities (9) nd (0), we get the seond nd the third inequlities in (8). nd Thereore, by s Hdmrd s inequlity (5), we lso hve: ( + b, + d ) d ( ) + b d,y dy () ( + b, + d ) b ( x, + d ) dx () b

6 56 M. Alomri nd M. Drus whih give, by ddition the irst inequlity in (8). nd Finlly, by the sme inequlity we n lso stte: b (, )+s (b, ) (x, ) dx b s + b (, d)+s (b, d) (x, d) dx b s + d (, )+s (, d) (, y) dy d s + d (b, )+s (b, d) (b, y) dy d s + whih give, by ddition the lst inequlity in (8). Remrk.3 In (8) i s =then the inequlity redued to inequlity (7). Corollry.4 Suppose tht :Δ=[, b] [, b] [0, ) [0, ) is s onvex untion on the o ordintes in the irst sense on Δ. Then one hs the inequlities: ( + b, + b ) b (b ) { ( x, + b ) + ( )} + b,x dx b (b ) (x, y) dydx (3) b { (x, )+ (, x)+s [ (x, b)+ (b, x)]} dx (s +)(b ) (, )+s (b, )+s (, b)+s (b, b) (s +). The bove inequlities re shrp.

7 Co-ordinted s-onvex untion 563 Corollry.5 In Corollry.4 i in ddition is symmetri, i.e, (x, y) = (y, x) or ll (x, y) [, b] [, b], we hve ( + b, + b ) b ( x, + b ) dx (b ) b b (b ) (x, y) dydx (4) { (x, )+s (x, b)} dx (s +)(b ) (, )+s (, b)+s (b, b) (s +). The bove inequlities re shrp. Now, the ollowing inequlity is onsidered the mpping onneted with the inequlities in (5) nd (6), s ollows: Theorem.6 Let :[, b] [0, ) [0, ) be s onvex untion in the seond sense on [, b] nd (0) = 0. Deine untion H :[0, ] R be suh tht H (t) = Then, [ (tb +( t) )+s (t +( t) b)], 0 t s s+ [ (tb +( t) )+ ()], s t. s+ () H is s onvex in the irst sense on [0, ]. () H is non deresing untion on [0, ]. (3) We hve the bounds: ()+s (b) in H (t) = = H (0) t [0,] s + H (t) H () = ()+ (b) s + = sup H (t). t [0,] Proo. Suppose tht :[, b] [0, ) [0, ) bes onvex untion in the seond sense on [, b] nd (0) = 0. Then by Theorem. is s onvex untion in the irst sense on [, b].

8 564 M. Alomri nd M. Drus. Let t,t [0, ] nd α, β 0 with α s + β s =. To show tht H is s onvex we hve three non-trivil ses: () For t,t [0,s] H (αt + βt ) = s + [ ((αt + βt ) b +( αt + βt ) ) +s ((αt + βt ) +( αt + βt ) b)] = s + [ (α (t b +( t ) )+β (t b +( t ) )) +s (α (t +( t ) b)+β (t +( t ) b))] s + [αs (t b +( t ) )+β s (t b +( t ) ) +s (α s (t +( t ) b)+β s (t +( t ) b))] = α s (t b +( t ) )+s (t +( t ) b) s + +β s (t b +( t ) )+s (t +( t ) b) s + = α s H (t )+β s H (t ) (b) For t,t [s, ] H (αt + βt ) = s + [ ((αt + βt ) b +( αt + βt ) ) + ()] = s + [ (α (t b +( t ) )+β (t b +( t ) )) + ()] s + [αs (t b +( t ) )+β s (t b +( t ) ) + ()] = α s (t b +( t ) )+ () + β s (t b +( t ) )+ () s + s + = α s H (t )+β s H (t ) () Without loss o generlity, ssume tht t [0,s] nd t [s, ]. Now, sine 0 t s nd s t, then 0 αt αs nd βs βt β, thereore, βs s αt + βt αs + β. Hene, αt + βt [s, ] nd by se (b) bove we obtin H (αt + βt ) α s H (t )+β s H (t ) whih shows tht H is s onvex in the irst sense on [0, ].

9 Co-ordinted s-onvex untion 565. Let t,t [0, ] nd without loss o generlity ssume tht 0 t t. Sine is s onvex in the irst sense then is non deresing on (0, ). Now, I 0 t t s, then it s esy to see tht H (t ) H (t ). Also, i s t t, then one n see tht H (t ) H (t ). It remins to hek t s t, to get tht, it suies to show tht the untion g (s) = () +s (s +( s) b) is non deresing on [t,t ]. Thereore, g (t )= ()+t (t +( t ) b) ()+t (t +( t ) b) =g (t ), with g (t =0)= () nd g (t =)= (), whih shows tht H is non deresing on [0, ]. 3. It ollows rom () tht, or ll t [0, ], ()+s (b) H (0) = s + This ompletes the proo. H (t) ()+ (b) s + = H (). Corollry.7 I :[, b] [0, ) [0, ) be s onvex untion in the irst sense on [, b]. Then, the result bove in Theorem.6 holds. Theorem.8 Let :[, b] R stisy Lipshitzin onditions. Tht is, or ll t,t [0, ], we hve (t ) (t ) L t t where, L is positive onstnt. Then L (b ) t t, 0 t t s H (t ) H (t ) Proo. For t,t [0, ], we hve two ses:. I 0 t t s, then L(b ) s+ t t, 0 <s t t L(b ) s+ ( t t + t ), 0 t s t (5) H (t ) H (t ) = s + (t b +( t ) )+s (t +( t ) b) [ (t b +( t ) )+s (t +( t ) b)] s + (t b +( t ) ) (t b +( t ) ) + s s + (t +( t ) b) (t +( t ) b) L (b ) t t

10 566 M. Alomri nd M. Drus. I 0 <s t t, then H (t ) H (t ) = s + (t b +( t ) )+ () [ (t b +( t ) )+ ()] s + (t b +( t ) ) (t b +( t ) ) L s + (b ) t t 3. Without loss o generlity, i 0 t s nd s t, then H (t ) H (t ) = s + (t b +( t ) )+s (t +( t ) b) [ (t b +( t ) )+ ()] s + (t b +( t ) ) (t b +( t ) ) + s + s (t +( t ) b) () L s + (b ) t t + s + (t +( t ) b) () L s + (b ) t t + L s + (b )( t t + t ) L s + (b ) t This ompletes the proo. Remrk.9 In (5) i we tke t =nd t =0, then (5) redue to H () H (0) = (b) L (b ) s (6) where, 0 <s<. The inequlity (6) is the s Hdmrd type inequlity or Lipshitzin s onvex mpping in the irst sense o one vrible. Aknowledgement. The work here is supported by the Grnt: UKM GUP TMK

11 Co-ordinted s-onvex untion 567 Reerenes [] M. Alomri nd M. Drus, A mpping onneted with Hdmrd type inequlties in 4 vribles, Int. Journl o Mth. Anlysis, (3) (008), [] M. Alomri nd M. Drus, The Hdmrd s inequlity or s onvex untion o vribles On The o ordintes, Int. Journl o Mth. Anlysis, (3) (008), [3] M. Alomri nd M. Drus, The Hdmrd s inequlity or s onvex untion, Int. Journl o Mth. Anlysis, (3) (008), [4] H. Hudzik, L. Mligrnd, Some remrks on s onvex untions, Aequtiones Mth., 48 (994), 00-. [5] S.S. Drgomir, S. Fitzptrik, The Hdmrd s inequlity or s-onvex untions in the seond sense, Demonstrtio Mth., 3 (4) (999), [6] S. S. Drgomir, On Hdmrd s inequlity or onvex untions on the o-ordintes in retngle rom the plne, Tiwnese Journl o Mthemtis, 5 (00), [7] W. Mtuszewsk nd W. Orliz, A note on the theorey o s normed spes o ϕ integrble untions, studi Mth., (98), [8] J. Musielk, Orliz soes nd modulr spes, Leture Notes in Mthemtis, Vol. 034, Springer Verlg, New York / Berlin, 983. [9] W. Orliz, A note on modulr spes, I, Bull. Ad. Polon. Si. Mth. Astronom. Phys., 9 (96), [0] S. Rolewiz, Metri Liner Spes, nd ed., PW N, Wrsw, 984. Reeived: My 5, 008