Hermite-Hadamard inequality for geometrically quasiconvex functions on co-ordinates

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1 Int. J. Nonliner Anl. Appl No ISSN: eletroni Hermite-Hdmrd ineulity for geometrilly usionvex funtions on o-ordintes Ali Brni Ftemeh Mlmir Deprtment of Mthemtis Lorestn University P. O. Box 465 Khormbd Irn Communited by A. Ebdin Abstrt In this pper we introdue the onept of geometrilly usionvex funtions on the o-ordintes nd estblish some Hermite-Hdmrd type integrl ineulities for funtions defined on retngles in the plne. Some ineulities for produt of two geometrilly usionvex funtions on the o-ordintes re onsidered. Keywords: Hermite-Hdmrd ineulity; onvex funtions on o-ordintes; geometrilly usionvex funtions. 2 MSC: Primry 26A5; Seondry 26D5.. Introdution nd preliminries Let I R be rel intervl. A funtion f : I R is sid to be onvex if for every x y I nd t ] ftx + ty tfx + tfy. Let f : I R be onvex funtion nd b I with < b then we hve the following ineulity + b f b f + fb fxdx. 2 b 2 This remrkble result is well known in the literture s Hermite-Hdmrd ineulity. Both ineulities hold in the reversed diretion if f is onve. We note tht Hermite-Hdmrd ineulity my be regrded s refinement of the onept of onvexity nd it follows esily from Jensen s Corresponding uthor Emil ddresses: brni.@lu..ir Ali Brni mlmir.fteme@ymil.om Ftemeh Mlmir Reeived: April 25 Revised: August 25

2 48 Brni Mlmir ineulity. There hve been severl works in the literture whih re devoted to investigting refinements nd generliztions of the Hermite-Hdmrd ineulity for onvex funtions see for exmple ] nd referenes therein. In 4] S.S. Drgomir defined onvex funtions on the o-ordintes or o-ordinted onvex funtions on the set : b] d] in R 2 with < b nd < d s follows: Definition.. A funtion f : R is sid to be onvex on the o-ordintes on if for every y d] nd x b] the prtil mppings f y : b] R f y u fu y nd f x : d] R f x v fx v re onvex. This mens tht for every x y z w nd t s ] ftx + tz sy + sw tsfx y + s tfz y + t sfx w + t sfz w. Clerly every onvex funtion is o-ordinted onvex. Furthermore there exist o-ordinted onvex funtions whih re not onvex. The following Hermit-Hdmrd type ineulity for oordinted onvex funtions ws lso proved in 4]. Theorem.2. Suppose tht f : R is onvex on o-ordintes on. Then + b f 2 + d 2 b f x + d dx + d ] + b f 2 b 2 d 2 y dy 4 b d b b d The bove ineulities re shrp. b d fx y fx dx + b b d + f ydy + d d f + f d + fb + fb d. 4 fx ddx ] f ydy Sine then severl importnt generliztions introdued on this tegory see ] nd referenes therein. Rell tht funtion f : I R R is sid to be usionvex if for every x y I nd λ ] fλx + λy mx{fx fy}. In 5] M.E. Özdemir et l. introdued the notion of o-ordinted usionvex funtions whih generlize the notion of o-ordinted onvex funtions s follows:

3 Hermite-Hdmrd ineulity for geometrilly usionvex No Definition.3. A funtion f : b] d] R is sid to be usionvex on the o-ordintes on if for every y d] nd x b] the prtil mpping f y : b] R f y u fu y nd f x : d] R f x v fx v re usionvex. This mens tht for every x y z w nd s t ] ftx + tz sy + sw mx{fx y fx w fz y fz w}. Sine then severl importnt generliztions on this tegory introdued in 4]. On the other hnd the notion of geometrilly usionvex funtions is introdued by İ. İşn in 7] nd F. Qi nd B.A. Xi in 8] s follows: Definition.4. A funtion f : I R : R is sid to be geometrilly usionvex on I if for every x y I nd t ] fx t y t mx{fx fy}. Note tht if f deresing nd geometrilly usionvex then it is usionvex. If f inresing nd usionvex then it is geometrilly usionvex. We rell some results introdued in 8]. Lemm.5. Let f : I R + : R be differentible funtion on I nd b I with < b. If f L b] then ln bfb ln f ln b ln t b t ln t b t f t b t dt. b ln b ln fx x dx. Theorem.6. Let f : I R + R be differentible funtion on I nd f L b] for b I with < b. If f is geometrilly usionvex on b] then ln bfb ln f b fx ln b ln ln b ln x dx.2 N b sup { f f b } where N b : t b t ln t b t dt. Theorem.7. Let f : I R + R be differentible funtion on I nd f L b] for b I with < b. If f is geometrilly usionvex on b] for > then ln bfb ln f b fx ln b ln ln b ln x dx ] / M b] N / b / sup { f f b } ].3 where M b : ln t b t dt.

4 5 Brni Mlmir Theorem.8. Let f : I R + R be differentible funtion on I nd f L b] for b I with < b. If f is geometrilly usionvex on b] for > nd > r > then ln bfb ln f b fx ln b ln ln b ln x dx / / N r b r ].4 r r N r/ b r/ ] / sup { f f b } ]. Theorem.9. Let f : I R + R be differentible funtion on I nd f L b] for b I with < b. If f is geometrilly usionvex on b] then fb /2 ln b ln b fx dx sup {f fb}..5 x In 6] M. E. Özdemir introdued the notion of geometrilly onvex funtions on the o-ordintes s follows: Definition.. Let + : b] d] be retngle in R 2 + with < b nd < d. A funtion f : + R is sid to be geometrilly onvex on the o-ordintes if for every y d] nd x b] the prtil mppings f y : b] R f y u fu y nd f x : d] R f x v fx v re geometrilly onvex funtion. This mens tht for every x y z w + nd t s ] fx t z t y s w s tsfx y + s tfz y + t sfx w + t sfz w. The min purpose of this pper is to estblish some new results onneted to the Hermite- Hdmrd type ineulity for geometrilly usionvex funtions on the o-ordintes. 2. The min results In this setion we introdue the notion of geometrilly usionvex funtions on the o-ordintes for funtions defined on retngles in R 2 + whih is generliztion of the notion geometrilly onvex funtions on the o-ordintes given in definition.. Then we estblish some Hermite- Hdmrd type ineulities for this lss of funtions. Definition 2.. Let + : b] d] be subset of R 2 + with < b nd < d. A funtion f : + R is sid to be geometrilly usionvex on the o-ordintes on + R 2 + if for every y d] nd x b] the prtil mppings nd f y : b] R f x : d] R f y u fu y f x v fx v re geometrilly usionvex. This mens tht for every x y z w + nd s t ] fx t z t y s w s mx{fx y fx w fz y fz w}.

5 Hermite-Hdmrd ineulity for geometrilly usionvex No Note tht every geometrilly onvex funtion on o-ordintes is geometrilly usionvex on oordintes but the onverse is not holds. In the following exmple we give geometrilly usionvex funtion on o-ordintes whih is not geometrilly onvex funtion on the o-ordintes. Exmple 2.2. Let + : 4] 4 9] nd onsider the funtion f : + R defined by It is esy to see tht the funtions nd fx y : x 2 y 2. f y x x 2 y 2 x 4] f x y x 2 y 2 y 4 9] re geometrilly usionvex. Hene f is geometrilly usionvex on o-ordintes on +. This funtion is not geometrilly onvex on o-ordintes on +. Indeed if we hoose two points x y 4 z w 4 9 nd s t 2 then fx t z t y s w s f nd tsfx y + s tfz y + t sfx w + t sfz w {fx y fx w fz w fz y} 4 4 < fx t z t y s w s. To reh our gol we introdue the following lemm whih plys ruil role in this pper. Lemm 2.3. Let + : b] d] be subset of R + 2 with < b nd < d. Suppose tht f : + R is prtil differentible funtion on int +. If 2 f L + then where nd ln b ln ln d ln b + + d ln f y y fx ln ln d x ln b fb y ] dy + y fx d x ] dx b d fx y yx t b t s d s ln t b t ln s d s 2 f t b t s d s dtds C : ln dln bfb d ln f d] D : ln ln f ln bfb ]. 2.

6 52 Brni Mlmir Proof. If we denote the right hnd side of 2. by I nd integrting by prts on + then we hve ln b ln ln d ln I ln b ln ln d ln t b t s d s ln t b t ln s d s 2 f t b t s d s dtds ln b ln ln d ln s d s ln s d s t b t ln t b t 2 f ] t b t s d s dt ds ln b ln ln d ln ln s d s ln s d s t b t f ln b ln s t b t s d s f ] s t b t s d s dt ds ln b ln ln d ln s d s ln s d s ln b f ln b ln s b s d s ln f ln b ln s s d s f ] s t b t s d s dt ds ln d ln ln b s d s ln s d s f s b s d s ds ln d ln ln s d s ln s d s f s s d s ds ln b ln ln d ln s d s ln s d s f ] s t b t s d s ds dt. Similrly integrting by prts in the right hnd side of 2.2 dedue tht ln b ln ln d ln I ln b ln s d s fb s d s ln d ln ln ln s d s f s d s ln d ln ln b ln ln s d s f t b t s d s dt + ln b ln ln d ln f t b t s d s dtds ln b ln dfb d ln fb ] fb s d s ds f s d s ds 2.2

7 Hermite-Hdmrd ineulity for geometrilly usionvex No ln b ln ln d ln d ln fb s d s ds ln ln df d ln f ] ln d ln f s d s ds + ln b ln ln d ln f t b t ddt ln f t b t s d s dtds. f t b t dt If we using the hnge of vribles x t b t nd y s d s for t s ] we obtin ln b ln ln d ln I d fb y ln b ln dfb d ln fb ] dy y d f y ln ln df d ln f ] y b fx d b fx b d ln d + ln dx + x x dy fx y yx. Dividing both sides of 2.3 by ln b ln ln d ln implies tht the eution 2. holds nd proof is ompleted. Theorem 2.4. Let + : b] d] be subset of R 2 + with < b nd < d. Suppose tht f : + R is prtil differentible funtion on int + nd 2 f L +. If 2 f is geometrilly usionvex funtion on the o-ordintes on + then the following ineulity holds: b d fxy ln b ln ln d ln + yx ln b ln ln d ln B N b N d { mx d b } b d where C D re defined in Lemm 2.3. b B : ln b ln ln d ln d fb y + ln b ln y f y y fx d ln d x ] dy. ln Proof. From Lemm 2.3 it follows tht b d fxy ln b ln ln d ln + yx ln b ln ln d ln B t b t s d s ln t b t ln s d s 2 f t b t s d s dtds. ] fx dx x

8 54 Brni Mlmir Sine 2 f is geometrilly usionvex on the o-ordintes we hve 2 f t b t s d s { mx d b } b d where t s ]. From this ineulity nd Lemm.5 it follows tht t b t s d s ln t b t ln s d s 2 f t b t s d s dtds { mx 2 f d 2 fb 2 fb d } t b t s d s ln t b t ln s d s dtds N b N d { mx d b } b d whih is the reuired ineulity 2.4. Note tht t b t s d s ln t b t ln s d s dtds t b t ln t b t dt s d s ln s d s ds N b N d. The proof of theorem is ompleted. The following orollry is n immedite onseuene of theorem 2.4. Corollry 2.5. Suppose tht the onditions of the theorem 2.4 re stisfied. Additionlly if 2 f is inresing on the o-ordintes on + then 2 2 f b d fxy ln b ln ln d ln + yx ln b ln ln d ln B N b N d 2. fb d is deresing on the o-ordintes on + then b d fxy ln b ln ln d ln + yx ln b ln ln d ln B N b N d 2 f

9 Hermite-Hdmrd ineulity for geometrilly usionvex No where C D nd B re defined respetively in Lemm 2.3 nd Theorem 2.4. Theorem 2.6. Let + : b] d] be subset of R 2 + with < b nd < d. Suppose tht f : + R is prtil differentible funtion on int + nd 2 f L +. If 2 f is geometrilly usionvex funtion on the o-ordintes on + nd p > + then the p following ineulity holds: b d fxy ln b ln ln d ln + yx ln b ln ln d ln B N p b p N p d p ] { p mx d f b } ] / b d where C D nd B re defined respetively in Lemm 2.3 nd Theorem 2.4. Proof. Suppose tht p >. From Lemm 2.3 nd well-known Hölder ineulity for double integrls we obtin b d fxy ln b ln ln d ln + yx ln b ln ln d ln B t b t s d s ln t b t ln s d s 2 f t b t s d s dtds 2.8 p t b pt p s d ps ln t b t ln s d s p p dtds Sine 2 f 2 f t b t s d s dtds. is geometrilly usionvex on the o-ordintes on + we obtin 2 f t b t s d s mx { d b b d Note tht p t b pt p s d ps ln t b t ln s d s p dtds p t b pt ln t b t p dt p s d ps ln s d s p ds N p b p N p d p. } Combintion of nd 2. gives the desired ineulity 2.7. Hene the proof of the theorem is ompleted. The following orollry is n immedite onseuene of theorem 2.6.

10 56 Brni Mlmir Corollry 2.7. Suppose tht the onditions of the Theorem 2.6 re stisfied. Additionlly if 2 f is inresing on the o-ordintes on + then 2 2 f b d fxy ln b ln ln d ln + yx ln b ln ln d ln B N p b p N p d p ] p 2. fb d is deresing on the o-ordintes on + then b d fxy ln b ln ln d ln + yx ln b ln ln d ln B N p b p N p d p ] p 2. f Theorem 2.8. Let + : b] d] be subset of R 2 + with < b nd < d. Suppose tht f : + R is prtil differentible funtion on int + nd 2 f L +. If 2 f is geometrilly usionvex funtion on the o-ordintes on + for > then the following ineulity holds: b d fxy ln b ln ln d ln + yx ln b ln ln d ln B M b M d] / 2 N / b / N / / d / ] { mx d b } ] b d where C D nd B re defined respetively in Lemm 2.3 nd Theorem 2.4. Proof. By Lemm 2.3 Hölder s ineulity nd the geometrilly usionvexity of we hve b d fxy ln b ln ln d ln + yx ln b ln ln d ln B t b t s d s ln t b t ln s d s 2 f t b t s d s dtds t/ b t/ s/ d s/ ln t b t ln s d s dtds ] / ln t b t ln s d s 2 f t b t s d s dtds ] / 2 f 2.3 on +

11 Hermite-Hdmrd ineulity for geometrilly usionvex No t/ b t/ s/ d s/ ] / ln t b t ln s d s dtds mx Note tht by Lemm.5 it follows tht ] / ln t b t ln s d s dtds { d b } ] b d. t/ b t/ s/ d s/ ln t b t ln s d s dtds t/ b t/ ln t b t dt s/ d s/ ln s d s ds 2 2 N / b / N / d /. It is esy to see tht nd proof is ompleted. ln t b t ln s d s dtds M b M d Theorem 2.9. Let + : b] d] be subset of R 2 + with < b nd < d. Suppose tht f : + R is prtil differentible funtion on int + nd 2 f L +. If 2 f is geometrilly usionvex funtion on the o-ordintes on + nd > l > then b d fxy ln b ln ln d ln + yx ln b ln ln d ln B 2 / 2/ N l b l N l d l] / l l N l/ b l/ N l/ d l/ ] / { mx d b } ] b d where C D nd B re defined respetively in Lemm 2.3 nd Theorem

12 58 Brni Mlmir Proof. From Lemm 2.3 Hölder s ineulity nd the geometrilly usionvexity of the o-ordintes on + we get b d fxy ln b ln ln d ln + yx ln b ln ln d ln B t b t s d s ln t b t ln s d s 2 f t b t s d s dtds l t/ b lt/ l s/ d ls/ ln t b t ln s d s dtds ] / 2 f on ln l t b lt ln l s d ls ] / 2 f t b t s d s dtds l t/ b lt/ l s/ d ls/ ln t b t ln s d s dtds ] / ] / l t b lt l s d ls ln t b t ln s d s dtds { mx d b } ] b d 2 / N l/ b l/ l N l/ d l/ ] / l 2/ N l b l N l d l] / mx The proof of theorem is ompleted. { d b } ] b d. Theorem 2.. Let + : b] d] be subset of R 2 + with < b nd < d. Suppose tht f : + R is geometrilly usionvex funtion on the o-ordintes on +. If f L + then f b /2 d /2 ln b ln ln d ln b d mx{f f d fb fb d}. fx y yx 2.5

13 Hermite-Hdmrd ineulity for geometrilly usionvex No Proof. By geometrilly usionvexity of f on o-ordintes on + for every t ] we hve f b /2 d /2 mx{f t b t s d s f t b t s d s } mx{f f d fb fb d}. 2.6 Sine f t b t s d s dtds by integrting in 2.6 we get nd proof is ompleted. ln b ln ln d ln f b /2 d /2 { mx f t b t s d s dtds ln b ln ln d ln b d mx{f f d fb fb d}. f t b t s d s dtds b d fx y yx fx y yx } f t b t s d s dtds Theorem 2.. Let + : b] d] be subset of R 2 + with < b nd < d. Suppose tht f g : + R re geometrilly usionvex funtions on the o-ordintes on +. If fg L +. Then ln b ln ln d ln b d fx y gx y yx mx { fu v gw z u w { b} v z { d} }. Proof. Let x t b t y s b s s t ]. By using the geometrilly usionvexity of f g on + we hve b d fx y gx y ln b ln ln d ln yx nd proof is ompleted. f t b t s d s g t b t s d s dtds mx{f f d fb fb d} mx{g g d gb gb d} Referenes ] A. Brni nd S. Brni Hermite-Hdmrd ineulities for funtions when power of the bsolute vlue of the first derivtive is P-onvex Bull. Aust. Mth. So ] A. Brni A.G. Ghznfri nd S.S. Drgomir Hermite-Hdmrd ineulity for funtions whose derivtives bsolute vlues re preinvex J. Ine. ppl

14 6 Brni Mlmir 3] A.G. Ghznfri nd A.Brni Some Hermite-Hdmrd type ineulities for the produt of two opertor preinvex funtions Bnh J. Mth. Anl ] S.S. Drgomir On the Hdmrd s ineulity for onvex funtions on the o-ordintes in retngle from the pln Tiwn. J. Mth ] S.S. Drgomir nd C.E.M. Pere Seleted Topis on Hermit-Hdmrd type ineulites nd pplitions RGMIA Monogrphs Vitori University 2. 6] K.C. Hsu Some Hermite-Hdmrd type ineulities for differentible o-ordinted onvex funtions nd pplitions Adv. Pure Mth ] İ. İsn New generl integrl ineulities for usi geometrilly onvex funtions vi frtionl integrls J. Ine. Appl ] İ. İsn Hermite-Hdmrd type ineulities for hrmonilly onvex funtions Hettepe J. Mth. Stt ] U.S. Kirmi Ineulites for differentible mppings nd pplitions to speil mens of rel numbers to midpoint formul Appl. Mth. Comput ] M.A. Ltif nd S.S. Drgomir Some Hermite-Hdmrd type ineulities for funtions whose prtil derivtives in bsolute vlue re preinvex on the o-ordintes Ft Universittis Series Mthemtis nd Informtis ] M.A. Ltif S. Hussin nd S.S. Drgomir On Some new ineulity for o-ordinted usi-onvex funtions ] M.S. Moslehin Mtrix Hermite-Hdmrd type ineulities Houston J. Mth ] M.E. Özdemir A.O. Akdemir nd Ç. Yıldız On the o-ordinted onvex funtions Applied Mthemtis nd Informtion Sienes ] M.E. Özdemir Ç. Yıldız nd A.O. Akdemir On some new Hdmrd-type ineulities for o-ordinted usionvex funtions Hettepe J. Mth. Sttis ] M.E. Özdemir A.O. Akdemir nd Ç. Yıldız On o-ordinted usi-onvex funtions Czeh. Mth. J ] M.E. Özdemir On the o-ordinted geometrilly onvex funtions Abstrts of MMA23 nd AMOE23 My Trtu Estoni. 7] J. Pečrić F. Proshn nd Y.L. Tong Convex Funtion Prtil Orderings nd Sttistil Applitions Ademi Press ] F. Qi nd B.Y. Xi Some Hermite-Hdmrd type ineulities for geometrilly usi-onvex funtions Pro. Indin Ad. Si ] M.Z. Srıky S. Erhn M.E. Özdemir nd S.S. Drgomir New some Hermit-Hdmrd s type ineulities for o-ordinted onvex funtions Tmsui Oxford J. Inform. Mth. Si ] D.Y. Wng K.L. Tseng nd G.S. Yng Some Hdmrd s ineulity for o-ordinted onvex funtions in retngle from the plne Tiwnese J. Mth ] B.Y. Xi J. Hu nd F. Qi Hermit-Hdmrd type ineulities for extended s-onvex funtions on the o-ordintes in retngle J. Appl. Anl

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