GENERALIZATIONS OF CONVERSE JENSEN S INEQUALITY AND RELATED RESULTS

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1 Joural of Mathematical Iequalities Volume 5, Number 20, GENERALIZATIONS OF CONVERSE JENSEN S INEQUALITY AND RELATED RESULTS S IVELIĆ AND J PEČARIĆ Commuicated by A Guessab Abstract I this paper we prove geeralizatios of Coverse Jese s iequality for covex fuctios defied o covex hulls As cosequeces we get geeralizatios of the Hermite- Hadamard iequality for covex fuctios defied o k -simplices i R k We also preset some related results which geeralize results i [8] Itroductio Let U be a covex subset of R k ad N If f : U R is a covex fuctio, x,, x U ad p,, p oegative real umbers with P = p i, the the well kow Jese s iequality f p i x i P i f x i p P holds If the followig coditios are satisfied the Reversed Jese s iequality f p > 0, p i 0 i = 2,, P > 0, P P p i x i U, p i x i P i f x i 2 p holds see [4] The covex hull of vectors x,,x R k is represeted by K = cov{x,, x } Barycetric coordiates over K are cotiuous fuctios λ,λ 2,,λ o K with followig properties: Mathematics subject classificatio 200: 26D5 Keywords ad phrases: Jese s iequality, Coverse Jese s iequality, covex hull, covex fuctios, Hermite-Hadamard iequality, k -simplex, barycetric coordiates c D l,zagreb Paper JMI

2 44 S IVELIĆ AND J PEČARIĆ λ i x 0, i =,,, 2 3 x = λ i x=, λ i xx i If x 2 x,, x x are liearly idepedet vectors, the each x K ca be writte i uique way as covex combiatio of x,, x i the form 3 We also cosider k-simplex S =[v,, v k+ ] i R k which is covex hull of its vertices v, v 2,, v k+ R k Barycetric coordiates λ,λ 2,,λ k+ over S are oegative liear polyomials o S ad have special form see the third sectio The ext variat of Jese s iequality was proved by A MatkovićadJPečarić [8] THEOREM A Let U be a covex subset i R k, x,, x U ad y,, y m cov{x,, x } If f is a covex fuctio o U, the the iequality f p i x i m j= P W m w j y j p i f x i m w j f y j j= 3 P W m holds for all positive real umbers p,, p ad w,,w m satisfyig the coditio p i W m for all i =,,, where P = p i ad W m = m j= w j I the followig, let E be a oempty set ad L be a liear class of fuctios f : E R havig the properties: L if f,g L the af + bg L for all a,b R L2 L where t= forall t E We cosider positive liear fuctioals A : L R That is, we assume: A Aaf + bg=aa f +bag for all f,g L, a,b R liearity A2 if f L, f t 0 forall t E the A f 0 positivity From A we obtai k A A a i g i = k a i Ag i for g,,g k L, a,,a k R liearity

3 CONVERSE JENSEN S INEQUALITY 45 If i additio A= is satisfied, we say that A is a positive ormalized liear fuctioal With L k we deote a liear class of fuctios g : E R k defied by gt=g t,,g k t, g i L i =,,k We also cosider liear operators à : L k R k defied by Ãg=Ag,,Ag k If A= is satisfied, the usig A we also have A3 A f g = f Ãg for every liear fuctio f o Rk Next we itroduce the fuctioal versios of Jese s iequality ad some related results which we geeralize i sequel B Jesse [4, p 47] gave the followig geeralizatio of Jese s iequality for positive liear fuctioals THEOREM B Jesse s iequality Let L satisfy properties L, L2 o oempty set E ad A be a positive ormalized liear fuctioal o L Let f be a cotiuous covex fuctio o a iterval I R The for all g L such that ge I ad f g L, we have Ag I ad f Ag A f g 4 The ext theorem, proved by J Pečarić ad P R Beesack, presets geeralizatio of Theorem Lah-Ribarić see [0, p 98], [4, p 98] THEOREM C Coverse Jesse s iequality Let L satisfy properties L, L2 ad A be a positive ormalized liear fuctioal o L Let f be a covex fuctio o a iterval I =[m,m] R < m < M < The for all g L such that ge I ad f g L, we have A f g M Ag M m f m+ag m f M 5 M m Usig Theorem C, Beesack ad Pečarić also proved the ext result [4, p 0] THEOREM D Let L, A ad f be as i Theorem C Let J be a iterval i R such that f I J If F : J J R is a icreasig fuctio i the first variable, the for all g L such that ge I ad f g L, we have M x FA f g, f Ag max F x [m,m] M m f m+ x m f M, f x 6 M m = max F θ f m+ θ f M, f θm + θm θ [0,]

4 46 S IVELIĆ AND J PEČARIĆ REMARK If we choose Fx,y =x y, as a simple cosequece of Theorem D it follows A f g f Ag max [θ f m+ θ f M f θm + θm] 7 θ [0,] Choosig Fx,y= x, it follows y [ ] A f g θ f m+ θ f M max 8 f Ag θ [0,] f θm + θm It is obviously that the mai results i [5], [6] ad [7] ca be obtaied as direct cosequeces of Theorem D published may years earlier Additioal geeralizatio of Jesse s iequality 4 is proved by E J McShae see [9], [4, p 48] THEOREM E McShae s iequality Let L satisfy properties L, L2, A be a positive ormalized liear fuctioal o L ad à =A,,A : L k R k a liear operator Let f be a cotiuous covex fuctio o a closed covex set U R k The for all g L k such that ge U ad f g L, we have that Ãg U ad f Ãg A f g 9 It is kow that for a covex fuctio f : [a,b] R the Hermite -Hadamard iequality a + b f b f a+ f b f xdx 0 2 b a 2 a holds I this paper, as our mai results we preset geeralizatios of Theorem C ad Theorem D for covex fuctios defied o covex hulls As cosequeces, we obtai geeralizatios of the Hermite-Hadamard iequality 0 for covex fuctios defied o k-simplices i R k Some related results ca be foud i [5], [6], [7] We also preset related results which geeralize results i [8] For N we deote Δ = 2 Mai results { Λ,,Λ : Λ i 0, i {,,}, The ext theorem presets geeralizatio of Theorem C } Λ i =

5 CONVERSE JENSEN S INEQUALITY 47 THEOREM Let L satisfy properties L, L2 o oempty set E ad A be a positive ormalized liear fuctioal o L Let x,, x R k ad K = cov{x,, x } Let f be a covex fuctio o K ad λ,,λ barycetric coordiates over K The for all g L k such that ge K ad f g,λ i g L i =,, we have A f g Aλ i g f x i 2 Proof For each t E we have gt K The there exist barycetric coordiates λ i gt 0 i =,, such that λ igt = ad Sice f is covex o K, the f gt = f gt= λ i gtx i λ i gtx i λ i gt f x i Now, applyig a fuctioal A o the last iequality we get A f g A = Aλ i g f x i λ i g f x i REMARK 2 If all the assumptios of Theorem are satisfied ad i additio f is cotiuous, the f Ãg A f g Aλ i g f x i The first iequality is cosequece of Theorem E ad the secod of Theorem Usig Theorem we prove geeralizatio of Theorem D THEOREM 2 Let L satisfy properties L, L2 o oempty set E, A be a positive ormalized liear fuctioal o L ad à =A,,A : L k R k a liear operator Let x,, x R k ad K = cov{x,, x } Let f be a covex fuctio o K ad λ,,λ barycetric coordiates over K If J is a iterval i R such that f K J ad F : J J R is a icreasig fuctio i the first variable, the for all g L k such that ge K ad f g,λ i g L i =,, we have F A f g, f Ãg F Aλ i g f x i, f Ãg 22 max Λ i f x i, f i x i Λ Λ,,Λ Δ F

6 48 S IVELIĆ AND J PEČARIĆ Proof For each t E we have gt K The there exist barycetric coordiates λ i gt 0 i =,, such that λ igt = ad gt= λ i gtx i Sice A is a positive ormalized liear fuctioal o L ad à =A,,A a liear operator o L k, we have Ãg=Ag,,Ag k = Aλ i g x i, where ad Aλ i g 0, i =,, Aλ i g = A λ i g = A= Therefore, Ãg K Sice F : J J R is a icreasig fuctio i the first variable, usig 2 we have F A f g, f Ãg F Aλ i g f x i, f Ãg 23 By substitutios it follows Aλ i g = Λ i i =,,, Ãg= Now we have F Aλ i g f x i, f Ãg = F Λ i x i Λ i f x i, f max Λ,,Λ Δ F Λ i x i Λ i f x i, f i x i 24 Λ By combiig 23 ad 24 we get 22 REMARK 3 If we choose Fx,y=x y, as a simple cosequece of Theorem 2 it follows i x i 25 Λ A f g f Ãg max Λ i f x i f Λ,,Λ Δ

7 CONVERSE JENSEN S INEQUALITY 49 Choosig Fx,y= x, it follows y A f g f Ãg max Λ,,Λ Δ f Λ i f x i i x i Λ 26 The iequalities 25 ad 26 preset geeralizatios of 7 ad 8 Replacig F by F i Theorem 2 we get the ext theorem THEOREM 3 Let L satisfy properties L, L2 o oempty set E, A be a positive ormalized liear fuctioal o L ad à =A,,A : L k R k a liear operator Let x,, x R k ad K = cov{x,, x } Let f be a covex fuctio o K ad λ,,λ barycetric coordiates over K If J is a iterval i R such that f K J ad F : J J R is a decreasig fuctio i the first variable, the for all g L k such that ge K ad f g,λ i g L i =,, we have F A f g, f Ãg F Aλ i g f x i, f Ãg mi Λ,,Λ Δ F Λ i f x i, f i x i Λ 3 Covex fuctios o k-simplices i R k I this sectio we give aalogs to Theorem ad Theorem 2 for covex fuctios defied o k-simplices i R k As a cosequece we obtai geeralizatios of the Hermite-Hadamard iequality 0 Let S =[v, v 2,, v k+ ] be k-simplex i R k with vertices v, v 2,, v k+ R k The barycetric coordiates λ,,λ k+ over S are oegative liear polyomials that satisfy Lagrage s property: {, i = j λ i v j =δ ij = 0, i j Therefore, it is kow that for each x S the barycetric coordiates λ x,,

8 50 S IVELIĆ AND J PEČARIĆ λ k+ x have the form λ x= Vol k [x, v 2,, v k+ ], λ 2 x= Vol k [v,x,v 3,, v k+ ], λ k+ x= Vol k [v,, v k, x], 3 where Vol k deotes k-dimesioal Lebesgue measure o S Here, for example, [v, x,, v k+ ] deotes the subsimplex obtaied by replacig v 2 by x, ie the subsimplex opposite to v 2, whe addig x as a ew vertex I other words, we see that the barycetric coordiates λ,,λ k+ for each x S ca be preseted as the ratios of the volume of subsimplex with oe vertex i x ad the volume of S see Picture Picture 2-simplex S =[v, v 2, v 3 ] i R 2 divided ito 3 subsimplices The siged volume is give by k + k + determiat = v v 2 v k+ v 2 v 22 v k+2, k! v k v 2k v k+k where v =v,v 2,,v k,, v k+ =v k+,v k+2,,v k+k see [8] Sice vectors v 2 v,, v k+ v are liearly idepedet, the each x S ca be writte i uique way as covex combiatio of v,, v k+ i the form x = Vol k [x, v 2,, v k+ ] v + + Vol k [v,, v k, x] v k+ 32

9 CONVERSE JENSEN S INEQUALITY 5 Now we preset a aalog of Theorem for covex fuctios defies o k- simplices i R k THEOREM 4 Let L satisfy properties L, L2 o oempty set E, A be a positive ormalized liear fuctioal o L ad à =A,,A : L k R k a liear operator Let f be a covex fuctio o k-simplex S =[v, v 2,, v k+ ] i R k ad λ,,λ k+ barycetric coordiates over S The for all g L k such that ge S ad f g L we have k+ A f g Aλ i g f v i 33 ] Vol k [Ãg, v2,, v k+ = ] Vol k [v, v 2,,Ãg f v + + f v k+ Proof For each t E we have gt S The there exist the barycetric coordiates g t v 2 v k+ k! λ gt = Vol k [gt, v 2,, v k+ ] g k t v 2k v k+k =, v v 2 v k+ k! v k v 2k v k+k λ k+ gt = Vol k [v,, v k, gt] = v v k g t k! v k v kk g k t v v 2 v k+ k! v k v 2k v k+k such that k+ k+ λ i gt = adgt= Sice f is covex o S, the k+ f gt λ i gtv i λ i gt f v i

10 52 S IVELIĆ AND J PEČARIĆ Usig the Laplace expasio of the determiat we ca easily check that λ i g L for all i =,,k + Now, applyig A o the last iequality we have where A f g A k+ λ i g f v i k+ = Aλ i g f v i, 34 Ag v 2 v k+ k! ] Ag Aλ g = k v 2k v Vol k [Ãg, v2,, v k+ k+k =, v v 2 v k+ k! v k v 2k v k+k 35 v v k Ag k! Aλ k+ g = v k v kk Ag k = v v 2 v k+ k! v k v 2k v k+k By combiig 34 ad 35 we obtai 33 Usig Theorem 4 we prove a aalog of Theorem 2 ] Vol k [v,, v k,ãg THEOREM 5 Let L satisfy properties L, L2 o oempty set E, A be a positive ormalized liear fuctioal o L ad à =A,,A : L k R k a liear operator Let f be a covex fuctio o k-simplex S =[v, v 2,, v k+ ] i R k ad λ,,λ k+ barycetric coordiates over S If J is a iterval i R such that f S J ad F : J J R a icreasig fuctio i the first variable, the for all g L k such that ge S ad f g L we have F A f g, f Ãg max F Volk [x, v 2,, v k+ ] f v + + Vol k [v,, v k, x] f v k+, f x x S = max Λ,,Λ k+ Δ k+ F k+ Λ i f v i, f k+ Λ i v i 36,

11 CONVERSE JENSEN S INEQUALITY 53 Proof Sice for each t E we have gt S, the it follows Ãg S see the first part of proof of Theorem 2 Sice F : J J R is a icreasig fuctio i the first variable, by Theorem 4 we have F A f g, f Ãg F Vol k[ãg,v 2,,v k+] f v + + Vol k[v,,v k,ãg] max F Volk [x,v 2,,v k+ ] x S f v + + Vol k[v,,v k,x] The equality i 36 is simple cosequece of substitutios Λ = Vol k [x, v 2,, v k+ ] f v k+, f Ãg f v k+, f x,,λ k+ = Vol k [v,, v k, x], ad k+ x = Λ i v i REMARK 4 Replacig F by F i Theorem 5 we ca get a aalog of Theorem 3 for covex fuctios defies o k-simplices i R k REMARK 5 If all the assumptios of Theorem 4 are satisfied ad i additio f is cotiuous, the f Ãg A f g k+ Aλ i g f v i 37 ] Vol k [Ãg, v2,, v k+ = ] Vol k [v,, v k,ãg f v + + f v k+ The first iequality is cosequece of Theorem E ad the secod of Theorem 4 EXAMPLE Let p,, p k+ 0 such that k+ p i = We defie the fuctioal A : L R by k+ Ag= p i gt i It is obviously that A is positive ormalized liear fuctioal o L The the liear operator à =A,,A : L k R k is defied by k+ Ãg= p i gt i

12 54 S IVELIĆ AND J PEČARIĆ We set gt i =v i for all i =,,k + Let S =[v, v 2,, v k+ ] be k-simplex i R k ad f be a cotiuous covex fuctio o S such that f g L The as a simple cosequece of 37 it follows f k+ Settig p = = p k+ = k+ f k + p i v i A f g we get k+ Related results are obtaied i [], [20] k+ v i A f g k + p i f v i k+ f v i EXAMPLE 2 Let S =[v, v 2,, v k+ ] be k-simplex i R k ad f a cotiuous covex fuctio o S Let L =E,A,λ be a measure space with positive measure λ We defie the fuctioal A : L R by Ag= gtdλ t λ E E It is obviously that A is positive ormalized liear fuctioal o L The the liear operator à =A,,A : L k R k is defied by Ãg= gtdλ t λ E E We deote g = λ E gtdλ t If ge S ad f g L, the from 37 it follows E f g A f g Vol k [g, v 2,, v k+ ] f v + + Vol k [v,, v k, g] f v k+, 38 Related results are obtaied as cosequeces of Choquet s theory see [4], [], [2], [3], [9] 4 Related results I this sectio we preset geeralizatios of results i [8] The ext theorem geeralizes Theorem A THEOREM 6 Let L satisfy properties L, L2 o oempty set E, A be a positive liear fuctioal o L ad à =A,,A : L k R k a liear operator Let x,, x R k ad K = cov{x,, x } Let f be a covex fuctio o K ad λ,,λ barycetric coordiates over K The for all g L k such that ge K ad f g,λ i g L

13 CONVERSE JENSEN S INEQUALITY 55 i =,, ad positive real umbers p,, p, with P = p i, satisfyig the coditio p i A for all i =,,, 4 we have f p i x i Ãg P A p i f x i Aλ i g f x i P A p i f x i A f g P A 42 Proof For each t E we have gt K The there exist barycetric coordiates λ i gt 0 i =,, such that Sice f is covex o K, the f gt λ i gt = adgt= λ i gtx i Applyig a positive liear fuctioal A o 43 we get where ad Also we have A f g λ i gt f x i 43 Aλ i g f x i, Aλ i g = A λ i g = A A Aλ i g 0foralli =,, Ãg= Aλ i g x i Now we ca write p i x i Ãg = p i x i Aλ i g x i P A P A = p i Aλ i g x i P A We have P A p i Aλ i g =

14 56 S IVELIĆ AND J PEČARIĆ ad sice P A p i Aλ i g 0foralli =,,, p i A Aλ i g for all i =,, Therefore, expressio p ix i Ãg P A is covex combiatio of vectors x,, x ad belogs to K Sice f is covex o K, we have f p i x i Ãg P A = f P A P A = p i Aλ i g x i p i Aλ i g f x i p i f x i Aλ i g f x i P A p i f x i A f g P A COROLLARY Let L satisfy properties L, L2 o oempty set E ad A be a positive ormalized liear fuctioal o L Let f be a covex fuctio o a iterval I =[m,m] R < m < M < The for all g L such that ge I ad f g L, we have f m + M Ag Ag m M Ag f m+ M m M m f M f m+ f M A f g 44 Proof For each t E we have gt I =[m,m] Sice iterval I =[m,m] is -simplex with vertices m ad M, the the barycetric coordiates have the special form: λ gt = M gt M m ad λ 2gt = gt m M m The applyig a fuctioal A we have Aλ g = M Ag M m ad Aλ 2 g = Ag m M m 45

15 CONVERSE JENSEN S INEQUALITY 57 Choosig = 2, p = p 2 =, x = m, x 2 = M from 42 it follows [ M Ag Ag m f m + M Ag f m+ f M f m+ M m M m = Ag m M Ag f m+ M m M m f M f m+ f M A f g ] f M REMARK 6 The iequalities i 44 are also obtaied i [3] Some related results are obtaied i [2] THEOREM 7 Let L satisfy properties L, L2 o oempty set E, A be a positive liear fuctioal o L ad à =A,,A : Lk R k a liear operator Let x,, x R k ad K = cov{x,, x } Let f be a covex fuctio o K ad λ,,λ barycetric coordiates over K The for all g L k such that ge K ad f g,λ i g L i =,, ad positive real umbers p,, p satisfyig the coditios P A > 0, where P = p i, ad p i x i Ãg K, 46 P A we have f p i x i Ãg P f P A P f P P p i x i A f AÃg P A p i x i P A Aλ i g f x i 47 Proof For each t E we have gt K The there exist barycetric coordiates λ i gt 0 i =,, such that λ igt = ad gt= λ i gtx i Also we have We ca easily see that Ãg= Aλ i g x i AÃg= A Aλ i g x i K,

16 58 S IVELIĆ AND J PEČARIĆ sice A Aλ i g = ad A Aλ ig 0, i =,, Sice f is covex o K,the f A Aλ i g f x i 48 Usig first 2 ad the 48 we have P P p i x i A f P f P A P p i x i A f P A P f P p i x i A A Aλ i g f x i P A REMARK 7 If positive real umbers p,, p satisfy the coditio 4, the the coditio 46 is also satisfied sice K is covexset The 42 ca be exteded as follows P f P p i x i Aλ i g f x i P f P P A f p i x i A f AÃg P A p i x i Ãg P A p i f x i Aλ i g f x i P A p i f x i A f g P A 49 COROLLARY 2 Let L satisfy properties L, L2 o oempty set E ad A be a positive ormalized liear fuctioal o L Let f be a covex fuctio o a iterval I =[m,m] R < m < M < The for all g L such that ge I ad f g L, we have m + M f m + M Ag 2 f f Ag 2 m + M 2 f 2 [ M Ag M m f m+ Ag m M m ] f M 40

17 CONVERSE JENSEN S INEQUALITY 59 Proof Choosig = 2, x = m, x 2 = M, p = p 2 = ad usig 45, the iequalities i 40 easily follows from 47 Next we give geeralizatios of Corollary ad Corollary 2 for covex fuctios defied o k-simplices i R k COROLLARY 3 Let L satisfy properties L, L2 o oempty set E, A be a positive ormalized liear fuctioal o L ad à =A,,A : L k R k a liear operator Let f be a covex fuctio o k-simplex S =[v, v 2,, v k+ ] i R k ad λ,,λ k+ barycetric coordiates over S The for all g L k such that ge S ad f g L we have k + f k+ k+ v i λ i Ãg f v i k+ k k+ k + f k+ v i f Ãg k k+ v i Ãg f k k+ k+ f v i k+ λ i Ãg f v i k f v i A f g k 4 Proof Sice barycetric coordiates λ,,λ k+ over k-simplex S i R k are oegative liear polyomials, the Aλ i g = λ i Ãg for all i =,,k + Choosig x i = v i for all i =,,k + adp = p 2 = = p k+ =, the iequalities i 4 easily follow from 42 ad 47 REFERENCES [] M BESSENYEI, The Hermite-Hadamard iequality o Simplices, Amer Math Mothly, 5, , [2] B GAVREA, J JAKŠETIĆ AND J PEČARIĆ, O a global upper boud for Jesse s iequality, The Australia & New Zelad Idustrial ad Applied Mathematics Joural, , [3] W S CHEUNG, A MATKOVIĆ AND J PEČARIĆ, A variat of Jesse s iequality ad geeralized meas, J Ieq Pure ad Appl Math, 7, 2006 [4] A FLOREA AND C P NICULESCU, A Hermite-Hadamard iequality for covex-cocave symmetric fuctios, Bull Math Soc Sci Math Roumaie Tome, 50, , [5] A GUESSAB, G SCHMEISSER, A Defiiteess Theory for Cubature Formulae of Order Two, Costructive Approximatio, 24, ,

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