Hermite-Hadamard Inequality for Geometrically Quasiconvex Functions on a Rectangular Box

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Mth. Si. Ltt. 5, No., 59-69 (26 59 Mthmtil Sins Lttrs An Intrntionl Journl http://d.doi.or/.8576/msl/58 Hrmit-Hdmrd Inulit for Gomtrill Qusionv Funtions on Rtnulr Bo Ali Brni nd Ftmh Mlmir Dprtmnt of Mthmtis, Lorstn Univrsit, P. O. Bo 465, Khormbd, Irn Rivd: 27 Jul. 25, Rvisd: 2 Ot. 25, Aptd: 23 Ot. 25 Publishd onlin: Jn. 26 Abstrt: In this ppr, w introdu th onpt of omtrill usionv funtions on rtnulr bo in R 3. Thn, som Hrmit-Hdmrd tp inulitis for funtions whos third drivtivs in bsolut vlu r omtrill onv r ivn. Kwords: Hrmit-Hdmrd inulit, rtnulr bo, omtrill usionv funtions Introdution A funtion f : I R R, is sid to b onv if for vr, I nd t [,], f(t+( ttf(+( t f(. Lt f : I R b onv funtion nd,b I with <b, w hv th followin inulit ( +b f f(d 2 b f(+ f(b. 2 This rmrkbl rsult is wll known in th litrtur s Hrmit-Hdmrd inulit. Both inulitis hold in th rvrsd dirtion if f is onv. W not tht Hrmit-Hdmrd inulit m b rrdd s rfinmnt of th onpt of onvit nd it follows sil from Jnsn s inulit. Sin thn som rfinmnts of th Hrmit-Hdmrd inulit for onv funtions hv bn tnsivl invstitd b numbr of uthors, s for mpl [-,4-6]. Lt I R + : (, b n inrvl nd f : I R + b ontinuous funtion. Thn, f is sid to b omtrill onv on I, if for vr, I nd λ [,], f( λ λ f( λ f( λ. In [6], S.S. Dromir dfind onv funtions on th o-ordints (or o-ordintd onv funtions on th st :[,b][,d] in R 2 with <b nd <d s follows: Dfinition.. A funtion f : R is sid to b onv on th o-ordints on if for vr [,d] nd [,b], th prtil mppins, nd f :[,b] R, f :[,d] R, f (u f(u,, f (v f(,v, r onv. This mns tht for vr(,,(,w nd t,s [,], f(t+( t,s+(sw ts f(,+s( t f(, + t(s f(,w+( t(s f(,w. Clrl, vr onv funtion is o-ordintd onv. Furthrmor, thr ist o-ordintd onv funtions whih r not onv. Sin thn svrl importnt nrlitions introdud on this tor, s [, 8-2] nd rfrns thrin. Rll tht funtion f : I R R, is sid to b usionv if for vr, I nd λ [,], f(λ +(λ m f(, f(. In [3], M.E. Ödmir t l. introdud th notion of o-ordintd usionv funtions whih nrli th notion of o-ordintd onv funtions s follows: Dfinition.2. A funtion f : [,b][,d] Ris sid to b usionv on th o-ordints on if for vr [, d] nd [, b], th prtil mppin, f :[,b] R, f (u f(u,, Corrspondin uthor -mil: brni.@lu..ir 26 NSP Nturl Sins Publishin Cor.

6 A. Brni, F. Mlmir: Hrmit-Hdmrd inulit nd f :[,d] R, f (v f(,v, r usionv. This mns tht for vr (,,(,w nd s,t [,], f(t+( t,s+(sw m f(,, f(,w, f(,, f(,w. Sin thn svrl importnt nrlitions on this tor provd b M.E. Ödmir t l. in [9, 2, 3]. On th othr hnd F. Qi nd B.A. Xi in [9] introdud th notion of omtrill usionv funtions nd stblishd som intrl inulitis of Hrmit-Hdmrd tp. Dfinition.3. A funtion f : I R : [, R, is sid to b omtrill usionv on I if for vr, I nd λ [,], f( λ λ m f(, f(. Not tht if f drsin nd omtrill usionv thn, it is usionv. If f inrsin nd usionv thn, it is omtrill usionv. W rll som rsults introdud in [9]. Lmm.. Lt f : I R + : (, R, b diffrntibl funtion on I nd,b I with < b. If f L([,b] thn, (i (lnb f(b(ln f( b f( lnbln lnbln d ( t b t ln( t b t f ( t b t dt. (ii M(,b : ln( t b t dt, (2 N(,b : t b t ln( t b t dt. (3 In [4], M. E. Ödmir dfind th onpt of omtrill onv funtions on th o-ordints s follows: Dfinition.4. Lt + :[,b][,d] b subst of R 2 + with < b nd < d. A funtion f : + R is sid to b omtrill onv on th o-ordints if for vr [,d] nd [,b] th prtil mppins, nd f :[,b] R, f :[,d] R, f (u f(u,, f (v f(,v, r omtrill onv funtion. This mns tht for vr(,,(,w + nd t,s [,], f( t t, s w s ts f(,+s( t f(, + t(s f(,w+( t(s f(,w. In [5], M.E. Ödmir dfind onv funtions on rtnulr bo s follows: Lt us onsidr th rtnulr bo Ω : [,b][,d] [, f] in R 3. Th mppin : Ω R is sid to b onv on Ω if for vr(,,,(u,v,w Ω nd λ [,], (λ +(λu,λ +(λv,λ +(λw λ (,,+(λ(u,v,w. A funtion : Ω R is sid to onv on th o-ordints on Ω if for vr(, [,b][,d],(, [,b][, f] nd(, [,d][, f], th prtil mppin, :[,b][,d] R, (u,v(u,v,, [, f], :[,b][, f] R, (u,w(u,,w, [,d], :[,d][, f] R, (v,w(,v,w, [,b], r onv. In [4], th notion of o-ordintd usionv funtions on rtnulr bo in R 3 whih nrli th notion of o-ordintd onv funtions is ivn s follows: Dfinition.5. A funtion : Ω R is sid to usionv on th o-ordints on Ω if for vr (, [,b] [,d], (, [,b] [, f] nd (, [,d][, f], th prtil mppin, :[,b][,d] R, (u,v(u,v,, [, f], :[,b][, f] R, (u,w(u,,w, [,d], :[,d][, f] R, (v,w(,v,w, [,b], r usionv. This mns tht for vr (,,,(u,v,w Ω nd t,s,r [,] (t+( tu,s+(sv,r+(rw m(,,,(,,w,(,v,,(u,,, (u,v,w,(u,v,,(u,,w,(,v,w. Motivtd b bov rsults in this ppr w introdu th notion of omtrill usionv funtions on rtnulr bo in R 3. Thn, som rsults in onntion to Hrmit-Hdmrd inulit r ivn. 2 Min rsults In this stion w introdu th notion; omtrill usionv funtions on rtnulr bo for funtions dfind on rtnulr in R 3 +, whih is nrlition of th notion omtrill onv funtions on rtnulr bo. Thn, w stblish som Hrmit-Hdmrd tp inulitis for this lss of funtions. In this stion, Ω + is rtnulr bo in R 3 dfind b Ω + :[,b][,d][, f], whr,b,,d R +. In [5] th notion of o-ordintd omtrill usionv funtions on th pln in R 2 is ivn s follows: 26 NSP Nturl Sins Publishin Cor.

Mth. Si. Ltt. 5, No., 59-69 (26 / www.nturlspublishin.om/journls.sp 6 Dfinition 2.. A funtion f : + R is sid omtrill usionv funtion on + R 2 + if for vr(,,(,w + nd λ [,], f( λ λ, λ w λ m f(,, f(,w. Dfinition 2.2. Lt + :[,b][,d] b subst of R 2 + with < b nd < d. A funtion f : + R is sid to b omtrill usionv on th o-ordints on + R 2 + if for vr [,d] nd [,b] th prtil mppins nd f :[,b] R, f :[,d] R, f (u f(u,, f (v f(,v, r omtrill usionv. This mns tht for vr (,,(,w + nd s,t [,], f( t t, s w s m f(,, f(,w, f(,, f(,w. Dfinition 2.3 A funtion : Ω + R will b lld omtrill usionv on th o-ordints on Ω + R 3 if for vr (, [,b] [,d], (, [,b][, f] nd (, [,d][, f], th prtil mppin, :[,b][,d] R, (u,v(u,v,, [, f], :[,b][, f] R, (u,w(u,,w, [,d], :[,d][, f] R, (v,w(,v,w, [,b], r omtrill usionv. This mns tht for vr (,,,(u,v,w Ω + nd t,s,r [,] f( t u t, s v s, r w r m(,,,(,,w,(,v,,(u,, (u,v,w,(u,v,,(u,,w,(,v,w. Dfinition 2.4. A funtion : Ω + R is sid omtrill usionv funtion on Ω + R 3 + if for vr(,,,(u,v,w Ω + nd λ [,], ( λ u λ, λ v λ, λ w λ m (,,,(u,v,w. Th followin lmm holds. Lmm 2.. Evr omtrill usionv mppin : Ω + R is omtrill usionv on th o-ordints. Proof. Suppos tht : Ω + R is omtrill usionv on Ω +. Thn for vr (, [,b][,d], (, [,b][, f] nd (, [,d][, f], th prtil mppin, :[,b][,d] R, (u,v(u,v,, [, f], :[,b][, f] R, (u,w(u,,w, [,d], :[,d][, f] R, (v,w(,v,w, [,b], r omtrill usionv on Ω +. For λ [,] nd v,v 2 [,d], w,w 2 [, f], on hs (v λ vλ 2,w λ wλ 2 (,v λ vλ 2,w λ wλ 2 ( λ λ,v λ vλ 2,w λ wλ 2 m(,v,w,(,v 2,w 2 m (v,w, (v 2,w 2, whih omplts th proof of omtrill usionvit of on [,d][, f]. Thrfor nd is lso omtrill usionv on [,b][, f] nd [,b][,d] for ll [,d] nd [, f] os likwis nd w shll omit th dtils.. Now w introdu th followin nw lmm whih w nd to rh our ol. Lmm 2.2. Suppos tht : Ω + R b prtil diffrntibl mppin on int(ω +. If 3 tsr L(Ω + thn, (lnbln(lnd ln(ln f ln ( (,, +E F Ω t b t s d s r f r ln( t b t ln( s d s ln( r f r tsr (t b t, s d s, r f r dtdsdr, whr C :(ln f ((lnd [ (lnb(b,d, f(ln(,d, f ] (ln[(lnb(b,, f(ln(,, f ], ( D :(ln (lnd [ (ln(,d,(lnb(b,d,] (ln [ (ln(b,,(lnb(,, ], (,, f E : (ln f + (,, (ln dd + +(ln d (,d, d +2 (,, (ln d +2 (b,, +(ln b d +3 (,, (ln d, +3 (4 26 NSP Nturl Sins Publishin Cor.

62 A. Brni, F. Mlmir: Hrmit-Hdmrd inulit whr + [,b][,d], +2 [,b][, f] nd +3 [,d][, f] r substs ofr 2 + with <b, <d nd < f, nd F : (lnb (ln (ln (ln +(ln f (ln (ln (ln +(ln d (lnd (b,, (lnd (,, (lnb (,, f d (lnb (ln (ln (ln (,, d (ln f (,d, d (ln f (,, d (b,d, (,d, (b,, f d (b,, d (,d,, f d (,,, f d Proof. W dnot th riht hnd sid of (4 b I. Lttin t b t, s d s nd r f r for t,s,r [,]. B. intrtion b prts on Ω +, w hv (lnbln(lnd ln(ln f lni (lnbln(lnd ln(ln f ln t b t s d s r f r ln( t b t ln( s d s ln( r f r tsr (t b t, s d s, r f r dtdsdr (ln(ln(ln (,, (ln(ln (ln 2 (,, b 2 (,,d d (ln(ln (lnb 2 (b,, (ln 2 b (,, 2 (,,d d (lnb (ln(ln 2 (b,,d (ln (ln(ln 2 (,,d (ln(ln 2 (,,. (5 Similrl intrtion b prts in th riht sid of (5 ddu tht (lnbln(lnd ln(ln f lni (lnb (ln (ln (b,, d (b,,d (ln (ln (ln (ln (,, (,,d (ln (,, d (,,d d d (6 26 NSP Nturl Sins Publishin Cor.

Mth. Si. Ltt. 5, No., 59-69 (26 / www.nturlspublishin.om/journls.sp 63 (lnb (ln [(lnd (b,d,(ln (b,,] (b,,d (ln (ln [(lnd (,d, (ln (,,] (,,d (ln (lnd (,d,d (ln +(ln + (ln (,,d (,,. Ain intrtion b prts in th riht sid of (6 ddu tht (ln bln(lnd ln(ln f lni f (ln b(lnd (ln(b,d, (b,d, f (lnb(ln (ln(b,, (b,, (lnb f (ln(b,, (b,, d f (ln(lnd (ln(,d, (,d, f +(ln(ln (ln(,, (,, (ln f + (ln(,, (,, d (lnd f (ln(,d, (,d, d (ln f + (ln(,, (,, d f + (ln(,, (,, dd [(ln ] f(b,d, f(ln(b,d, (ln b(lnd (b,d, (7 (lnb(ln (b,, (lnb [(ln f(b,, f(ln(b,, ] (ln f(b,, f(ln (b,, (b,, d [(ln ] f(,d, f(ln(,d, (ln(lnd (,d, [(ln ] f(,, f(ln(,, +(ln(ln + + + (,, (ln (ln f(,, f(ln (,, (,, d (lnd (ln f(,d, f(ln(,d, (,d, d (ln (ln f(,, f(ln(,, (,, d (,, (ln f(,, f(ln (,, dd. Dividin both sids of (7 b (lnbln(lnd ln(ln f ln implis tht th ution (4 holds nd proof is ompltd.. Thorm 2.. Lt : Ω + R b prtil diffrntibl mppin on int(ω + nd tsr L(Ω +. If tsr is omtrill usionv funtion on th o-ordints on 26 NSP Nturl Sins Publishin Cor.

64 A. Brni, F. Mlmir: Hrmit-Hdmrd inulit Ω + thn th followin inulit holds: (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F N(,b N(,d N(, f m 3 tsr (,, (,, f 3 tsr (,d, (,d, f 3 tsr (b,, (b,, f 3 tsr (b,d, 3 (b,d, f whr, C, D, nd N(, b r dfind, rsptivl, in Lmm 2.2 nd Lmm., nd E Ẽ (lnbln(lnd ln(ln f ln, F F (lnbln(lnd ln(ln f ln, whr E, F r dfind, in Lmm 2.2. Proof. From Lmm 2.2, it follows tht (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F t b t s d s r f r ln( t b t ln( s d s ln( r f r tsr (t b t, s d s, r f r dtdsdr. Sin tsr is omtrill usionv on th o-ordints on Ω +, w hv tsr (t b t, s d s, r f r m 3 tsr (,, (,, f 3 tsr (,d, (,d, f 3 (b,, f tsr (b,, 3 tsr (b,d, 3 (b,d, f (8 whr t, s, r [, ]. From this inulit nd Lmm., it follows tht t b t s d s r f r ln( t b t ln( s d s ln( r f r tsr (t b t, s d s, r f r dtdsdr m 3 tsr (,, (,, f 3 tsr (,d, (,d, f 3 (b,, f tsr (b,, 3 tsr (b,d, 3 (b,d, f tsr t b t s d s r f r ln( t b t ln( s d s ln( r f r N(,b N(,d N(, f m 3 tsr (,, (,, f 3 tsr (,d, (,d, f 3 tsr (b,, (b,, f 3 tsr (b,d, 3 (b,d, f whih is th ruird inulit (8, sin t b t s d s r f r ln( t b t ln( s d s ln( r f r dtdsdr (t b t ln( t b t dt (s d s ln( s d s ds r f r ln( r f r dt N(,b N(,d N(, f. Th proof of thorm is ompltd.. Th followin orollr is n immdit onsun of thorm 2.. Corollr 2.. Suppos th onditions of th Thorm 2. r stisfid. Additionll, if 26 NSP Nturl Sins Publishin Cor.

Mth. Si. Ltt. 5, No., 59-69 (26 / www.nturlspublishin.om/journls.sp 65 ( tsr is inrsin on th o-ordints on Ω +, thn (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F N(,b N(,d N(, f (b,d, f tsr. (9 (2 tsr is drsin on th o-ordints on Ω +, thn. (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F N(,b N(,d N(, f tsr (,,. ( Thorm 2.2. Lt : Ω + R b prtil diffrntibl mppin on int(ω + nd tsr L(Ω +. If tsr is omtrill usionv funtion on th o-ordints on Ω + nd p,>, p +, thn th followin inulit holds: (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F [ N( p,b p N( p,d p N( p, f p ] /p [ m 3 tsr (,, (,, f tsr (,d, (,d, f tsr (b,, (b,, f tsr (b,d, ] / (b,d, f ( whr, C, D, Ẽ, F nd N(,b r dfind rsptivl, in Lmm 2., Thorm 2. nd Lmm.. Proof. Suppos tht p >. From Lmm 2.2 nd wllknown Höldr inulit for tripl intrls, w obtin (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F t b t s d s r f r ln( t b t ln( s d s ln( r f r tsr (t b t, s d s, r f r dtdsdr ( p(t b pt p(s d ps p(r f pr ln( t b t ln( s d s ln( p(r f pr p dtdsdr ( 3 tsr (t b t, s d s, r f r dtdsdr. (2 Sin tsr is omtrill usionv on th o-ordints on Ω +, w obtin tsr (t b t, s d s, r f r dtdsdr m 3 tsr (,, (,, f tsr (,d, (,d, f tsr (b,, (b,, f tsr (b,d, (b,d, f. tsr (3 W lso noti tht p(t b pt p(s d ps p(r f pr p ln( t b t ln( s d s ln( p(r f pr p dtdsdr p(t b pt ln( t b t p dt p(s d ps ln( s d s p ds p(r f pr ln( p(r f pr dr N( p,b p N( p,d p N( p, f p. (4 Combintion of (2, (3 nd (4, ivs th dsird inulit (. Hn th proof of th thorm is ompltd.. 26 NSP Nturl Sins Publishin Cor.

66 A. Brni, F. Mlmir: Hrmit-Hdmrd inulit Th followin orollr is n immdit onsun of thorm 2.2. Corollr 2.2. Suppos th onditions of th Thorm 2.2 r stisfid. Additionll, if ( tsr thn is inrsin on th o-ordints on Ω +, (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F [ N( p,b p N( p,d p N( p, f p ] /p (b,d, f tsr. (5 (2 tsr is drsin on th o-ordints on Ω +, thn. (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F [ N( p,b p N( p,d p N( p, f p ] /p tsr (,,. (6 Thorm 2.3. Lt : Ω + R b prtil diffrntibl mppin on int(ω + nd tsr L(Ω +. If tsr is omtrill usionv funtion on th o-ordints on Ω + for >, thn th followin inulit holds: (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F [M(,b M(,d M(, f] / [ ( 3 N ( /(,b /( N ( /(,d /( N ( /(, f /(] / (7 [ m 3 tsr (,, (,, f tsr (,d, (,d, f tsr (b,, (b,, f tsr (b,d, ] / (b,d, f whr, C, D, Ẽ, F nd M(, b, N(, b r dfind, rsptivl, in Lmm 2.2, Thorm 2. nd Lmm.. Proof. B Lmm 2.2, Höldr s inulit, nd th omtri usinvit of on Ω +, w hv tsr (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F t b t s d s r f r ln( t b t ln( s d s ln( r f r tsr (t b t, s d s, r f r dtdsdr [ (t( b t/( (s/( d s/( (r/( f r/( ] / ln( t b t ln( s d s ln( r f r dtdsdr [ ln( t b t ln( s d s ln( r f r tsr (t b t, s d s, r f r ] / dtdsdr [ (t/( b t/( (s/( d s/( (r/( f r/( ln( t b t ln( s d s ln( r f r ] / dtdsdr [ m 3 tsr (,, (,, f tsr (,d, (,d, f tsr (b,, (b,, f tsr (b,d, ] / (b,d, f tsr. 26 NSP Nturl Sins Publishin Cor.

Mth. Si. Ltt. 5, No., 59-69 (26 / www.nturlspublishin.om/journls.sp 67 Not tht b Lmm. w ddu tht, nd (t/( b t/( (s/( d s/( (r/( f r/( ln( t b t ln( s d s ln( r f r dtdsdr (t/( b t/( ln( t b t dt (s/( d s/( ln( s d s ds (r/( f r/( ln( r f r dr (3 3 N ( /(,b /( N ( /(,d /( N ( /(, f /(, ln( t b t ln( s d s ln( r f r dtds M(,b M(,d M(, f. Th proof of Thorm 2.3 is ompltd.. Thorm 2.4. Lt : Ω + R b prtil diffrntibl mppin on Ω nd 3 tsr L(Ω +. If tsr is omtrill usionv funtion on th o-ordints on Ω + nd >l>, thn (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F ( 3(/ ( 3/ l l [N ( l,b l N ( l,d l N ( l, f l] / [N ( (l/(,b (l/( N ( (l/(,d (l/( N ( (l/(, f (l/(] (/ [ m 3 tsr (,, (,, f tsr (,d, (,d, f tsr (b,, (b,, f tsr (b,d, ] / (b,d, f (8 whr, C, D, Ẽ, F nd N(, b r dfind, rsptivl, in Lmm 2.2, Thorm 2. nd Lmm.. Proof. From Lmm 2.2, Höldr s inulit, nd th omtri usionvit of tsr on th o-ordints on Ω + w t, (lnbln(lnd ln(ln f ln (,, Ω (lnbln(lnd ln(ln f ln + Ẽ F t b t s d s r f r ln( t b t ln( s d s ln( r f r tsr (t b t, s d s, r f r dtdsdr [ (l(t/( b (lt/( (l(s/( d (ls/( (l(r/( f (lr/( ln( t b t ln( s d s ln( r f r ] / dtdsdr [ ln( l(t b lt ln( l(s d ls ln( l(r f lr tsr (t b t, s d s, r f r ] / dtdsdr [ (l(t/( b (lt/( (l(s/( d (ls/( (l(r/( f (lr/( ln( t b t ln( s d s ln( r f r ] / dtdsdr [ l(t b lt l(s d ls l(r f lr ln( t b t ln( s d s ln( r f r ] / dtdsdr [ m 3 tsr (,, (,, f tsr (,d, (,d, f tsr (b,, (b,, f tsr (b,d, ] / (b,d, f tsr 26 NSP Nturl Sins Publishin Cor.

68 A. Brni, F. Mlmir: Hrmit-Hdmrd inulit ( 3(/ [N ( (l/(,b (l/( l N ( (l/(,d (l/( N ( (l/(, f (l/(] / ( 3/[ N ( l,b l N ( l,d l N ( l, f l] / l [ m 3 tsr (,, (,, f tsr (,d, (,d, f tsr (b,, (b,, f tsr (b,d, ] / (b,d, f Th proof of thorm is ompltd.. Thorm 2.5. Lt f : Ω + R b omtrill usionv funtion on th o-ordints on Ω +. If L(Ω +, thn f ( (b /2,(d /2,( f /2 (lnbln(lnd ln(ln f ln f(,, m (,,,(,, f,(b,,,(b,, f, (,d,,(,d, f,(b,d,,(b,d, f. (9 Proof. B omtri usionvit of f on o-ordints on Ω +, w hv f ( (b /2,(d /2,( f /2 m f( t b t, s d s, r f r, f( t b t, s d s, r f r m (,,,(,, f,(b,,,(b,, f, Sin (,d,,(,d, f,(b,d,,(b,d, f. f( t b t, s d s, r f r dtdsdr f( t b t, s d s, r f r dtdsdr (lnbln(lnd ln(ln f ln f(,,, (2 b intrtin in (2 w t f ( (b /2,(d /2,( f /2 m f( t b t, s d s, r f r dtdsdr, f( t b t, s d s, r f r dtdsdr, f( t b t, s d s, r f r dtdsdr (lnbln(lnd ln(ln f ln f(,, m (,,,(,, f,(b,,,(b,, f, (,d,,(,d, f,(b,d,,(b,d, f, nd proof is ompltd.. Thorm 2.6. Suppos tht,h : Ω + R r omtrill usionv funtions on th o-ordints on Ω +. If h L(Ω +. Thn, (lnbln(lnd ln(ln f ln (,, m (,, h(u,v,w,u [,b],,v [,d], w, [, f]. Proof. Lt t b t, s b s, r f r, t,s,r [,] nd usin th omtri usionvit of,h on th o-ordints on Ω + ilds (lnbln(lnd ln(ln f ln (,, f( t b t, s d s, r f r ( t b t, s d s, r f r dtdsdr m(,,,(,, f,(b,,,(b,, f, (,d,,(,d, f,(b,d,,(b,d, f m (,,,(,, f,(b,,,(b,, f, (,d,,(,d, f,(b,d,,(b,d, f, nd proof is ompltd.. 3 Conlusion This ppr is onrnd with th lbrt Hrmit-Hdmrd inulit for funtions of thr 26 NSP Nturl Sins Publishin Cor.

Mth. Si. Ltt. 5, No., 59-69 (26 / www.nturlspublishin.om/journls.sp 69 vribls. W obtind som rsults in onntion to Hrmit-Hdmrd inulit busin th notion of omtrill usionv funtions on rtnulr bo. In our opinion, dditionl rsrh should not onl fous on how to wkn th prtil diffrntibilit ondition in Lmm 2.2, but lso how to nrli th onvit. Aknowldmnt Th uthors r indbtd to rfrs for rfull rdin th ppr nd for mkin usful sustions. Rfrns [] A. Brni, S. Brni, Hrmit-Hdmrd inulitis for funtions whn powr of th bsolut vlu of th first drivtiv is P-onv, Bull. Aust. Mth. So. 86 (22, 26-34. [2] A. Brni, A. G. Ghnfri nd S. S. Dromir, Hrmit- Hdmrd inulit for funtions whos drivtivs bsolut vlus r prinv, J. Inul. Appl 22, 22:247. [3] A. Brni nd F. Mlmir, Nw som Hdmrd s tp inulitis for onv funtions on rtnulr bo, submittd. [4] A. Brni nd F. Mlmir, on som nw inulitis for usionv funtions on rtnulr bo, submittd. [5] A. Brni nd F. Mlmir, Hrmit-Hdmrd inulit for omtrill usionv funtions on o-ordints, RGMIA Rsrh Rport Colltion, 8 (25, Artil 37, 8 pp. [6] S. S. Dromir, On th Hdmrd s inulit for onv funtions on th o-ordints in rtnl from th pln, Tiwn. J. Mth 5 (2 775-778. [7] S. S. Dromir nd C.E.M. Pr, Sltd Topis on Hrmit-Hdmrd Tp Inulits nd Applitions, RGMIA (2, Monorphs, [ONLINE: http://jm.or//rgmia/monorphs/hrmit hdmrd.htmi]. [8] U. S. Kirmi, Inulits for diffrntibl mppins nd pplitions to spil mns of rl numbrs to midpoint formul, App. Mth. Comput 47 (24 37-46. [9] M. A. Ltif, S. Hussin nd S.S. Dromir, On Som nw inulit for o-ordintd usi-onv funtions, http://jm.or/rgmia/pprs/ v455.pdf, 2. [] M. S. Moslhin, Mtri Hrmit-Hdmrd tp inulitis, Houston. J. Mth 39 (23 77-89. [] M. E. Ödmir, A. O. Akdmir, Ç. Yıldı, On th oordintd onv funtions, Appl. Mth. Info. Si 8 (24 85-9. [2] M. E. Ödmir,Ç. Yıldı nd A.O. Akdmir, On som nw Hdmrd-tp inulitis for o-ordintd usi-onv funtions, Http. J. Mth. St 4 (22 697-77. [3] M. E. Ödmir, A.O. Akdmir, Ç. Yıldı, On o-ordintd usi-onv funtions, Ch. Mth. J 62 (22 889-9. [4] M. E. Ödmir, On th o-ordintd omtrill onv funtions, Abstrts of MMA23 nd AMOE23, M 27-3, 23, Trtu, Estoni. [5] M. E. Ödmir, A.O. Akdmir, On som Hdmrd-tp inulitis for onv funtions on rtnulr bo, volum 2, r 2 rtil ID jn-, ps doi:.5899/2/jn-. [6] M. A. Noor, K. I. Noor, M. U. Awn nd J. Lib, On Hrmit- Hdmrd inulitis for h-prinv funtions, Filomt 28 (24 463-474. [7] M. A. Noor, K. I. Noor, M. U. Awn, Hrmit- Hdmrd inulitis for rltiv smi-onv funtions nd pplitions, Filomt 28 (24 22-23 [8] J. Pčrić, F. Proshn nd Y. L. Ton, Conv Funtion, Prtil Ordrins nd Sttistil Applitions, Admi Prss (992, In. [9] F. Qi nd B.Y. Xi, Som Hrmit-Hdmrd tp inulitis for omtrill usi-onv funtions, Prodins of th Indin Adm of Sin 24 (24 33-342. [2] D. Y. Wn, K. L. Tsn, G. S. Yn, Som Hdmrd s inulit for o-ordintd onv funtions in rtnl from th pln, Tiwn J. Mth (27 63-73. [2] B-Y. Xi, J. Hu, F. Qi, Hrmit-Hdmrd tp inulitis for tndd s-onv funtions on th o-ordints in rtnl. J. Appl. Anl 2 (24 29-39. Ali Brni rivd th PhD dr in Mthmtis for Isfhn Univrsit, Irn. His rsrh intrsts r in th rs of inulitis nd nrlid onvit on Rimnnin mnifolds nd linr sps. H hs publishd rsrh rtils in rputd intrntionl journls of mthmtil sins. H is rfr of mthmtil journls. Ftmh Mlmir rivd th mstr dr in Mthmtis for Lorstn univrsit, Irn. His rsrh intrsts r in th rs of nrlid onvit. Now sh is PhD studnt of Mthmtis. 26 NSP Nturl Sins Publishin Cor.

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