JENSEN S INEQUALITY FOR QUASICONVEX FUNCTIONS S. S. DRAGOMIR 1 and C. E. M. PEARCE 2. Victoria University, Melbourne, Australia

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1 JENSEN S INEQUALITY FOR QUASICONVEX FUNCTIONS S. S. DRAGOMIR ad C. E. M. PEARCE School o Computer Sciece & Mathematics Victoria Uiversity, Melboure, Australia School o Mathematical Scieces The Uiversity o Adelaide, Adelaide, Australia Preseted i hoour o Josip Pečarić o the occasio o his 60th birthday Abstract. Some iequalities o Jese type ad coected results are give or quasicovex uctios o covex sets i real liear spaces. AMS classiicatio: 6D07, 6D5. Key words: quasicovex uctios, Jese s iequality Itroductio Throughout this paper X deotes a real liear space ad C X a covex set, so that x, y C with λ 0, implies that λx + λy C. Deiitio. A mappig : C IR is called quasicovex o the covex set C i λx + λy x, y or all x, y C ad λ 0,. This class o uctios strictly cotais the class o covex uctios deied o a covex set i a real liear space. See 8 ad citatios therei or a overview o this issue. Some recet studies have show that quasicovex uctios have quite close resemblaces to covex uctios see, or example, 4, 6, 7, 0 or quasicovex ad eve more geeral extesios o covex uctios i the cotext o Hadamard s pair o iequalities. Apart rom geeralizatios to theory, weakeig the covexity coditio ca icrease applicability. Thus i 9 use is made o quasicovexity to obtai a global extremum with rather less eort tha via covexity. I this article we pursue the cocept urther ad derive a umber o Jese type iequalities or quasicovex uctios. See also 5 or uctios o Goduova Levi type i the cotext o Jese s iequality. Prelimiaries For a arbitrary mappig : C IR ad x, y two ixed elemets i C, we ca deie the map g x,y : 0, IR by g x,y t = tx + ty. This provides a characterizatio o quasicovexity. Propositio. The ollowig statemets are equivalet: i is quasicovex o C; ii or every x, y C, the mappig g x,y is quasicovex o 0,. Proo. Suppose i holds. Let t, t 0, ad α, α 0 with α + α =. The g x,y α i t i = α i t i x + α i t i y = α i t i x + t i y t ix + t i y = g x,yt i,,, which shows that the mappig g x,y is quasicovex o 0,. For the reverse implicatio, suppose ii holds. The tx + ty = g x,y t = g x,y t.0 + t. g x,y 0, g x,y = x, y, which shows that is quasicovex o C.

2 Propositio. Suppose that φ k is quasicovex o 0, or k =,...,. The k φ k is quasicovex o 0,. Proo. Let t, t 0, ad α, α 0 with α + α =. Put φt = k φ k t. The φα t + α t = φ kα t + α t φ kt i = φ kt i = φt i, k k,, k, establishig the quasicovexity o φ. Lemma.3 I φ is quasicovex o 0, ad φt = φ t or all t 0,, the φt φ/ or all t 0,. Proo. From the give coditios, or each t 0,, φt = φt, φ t φ /t + t = φ/. For a give mappig : C IR we may also deie a map G t : C IR by G t x, y = tx + ty or ixed t 0,. Agai we have a characterizatio o quasicovexity. Propositio.4 We have the ollowig: i i is quasicovex o C, the G t is quasicovex o C or all t 0, ; ii i C is a coe i X ad G t is quasicovex o C t 0,, the is quasicovex o C. Proo. i Fix t 0, ad let x, y, z, u C. The or all λ 0, G t λx, y + λz, u = G t λx + λz, λy + λu which shows that G t is quasicovex o C. = tλx + λz + tλy + λu = λtx + ty + λtz + tu tx + ty, tz + tu = G t x, y, G t z, u, ii Let x, y C ad t 0,. I C is a coe i X, that is, C + C C ad αc C or all α 0, the t x, t y C ad t x, 0, 0, t y C. O the other had, sice G t is quasicovex o C, we have tx + ty = G t x, y = G t t t x, 0 + t 0, t y G t t x, 0, G t 0, t y = x, y or all t 0,. The iequality holds also or t = 0,, so the propositio is proved. 3 Jese s iequality Hereater x i C i =,...,. We assume p i > 0 i ad deie = p i. Theorem 3. I is quasicovex, the p i x i... p i x i, x p i x i, x, x p x + p x p + p, x 3,..., x x i. i

3 3 Proo. We employ iductio o. The case = provides a trivial basis. Assume that the stated iequality holds or =,..., k k. By quasicovexity ad the iductive assumptio P k+ k+ p i x i P k = P k+ P k P k... k p i x i + p k+ x k+ P k+ k p i x i, x k+ P k k p i x i, x k, x k+ x i, x k+ i k This may be writte as the result o the theorem with = k+, givig the iductive step ad so completig the proo. Corollary 3. For quasicovex pi x i p i x i p i x i mi p i p i where the miimum is over all distict i,..., i,...,. I particular, we have the ollowig or the uweighted case. Corollary 3.3 For quasicovex x x x x x x... x + x ad x x mi, x., x, x, x i,, x 3,..., x x,..., x xi x i where the miimum is over the same domai as i the previous corollary., x i, We ow cosider the mappig η give by ηi, p, x, = x i P I p ix i. Here I P IN, the collectio o iite sets o atural umbers, p = p i with each p i > 0 ad P I := p i, ad x = x i with each x i C. Theorem 3.4 For quasicovex, i the mappig ηi,, x, is quasi-superadditive; ii the mappig η, p, x, is quasi-superadditive as a idex set mappig o P IN. Proo i Let p, q > 0 with P I, Q I > 0 I P IN. The ηi, p + q, x, = x i p i + q i x i P I + Q I = x P I i p i x i + Q I q i x i P I + Q I P I P I + Q I Q I x i p i x i, q i x i. P I Q I

4 4 Sice a, b = /a + b + a b or a, b IR, we have rom the deiitio o η that the last imum i ca be writte as x i ηi, p, x, ηi, q, x, ηi, p, x, ηi, q, x,. Because mia, b = /a + b a b or a, b IR, we thus have ηi, p + q, x, miηi, p, x,, ηi, q, x,, which establishes part i. For ii, let I, J P IN with I J = ad suppose p > 0 with P I, P J > 0. The ηi J, p, x, = x i p i x i J P I J J = x i, x j P I p i x i + P J p j x j j J P I + P J P I P I + P J P J j J x i + x j + x i x J j J j J p i x i, p j x j P I P J j J = x i + x j + x i x j j J j J x i + x j ηi, p, x, ηj, p, x, ηi, p, x, ηj, p, x, j J = ηi, p, x, + ηj, p, x, ηi, p, x, ηj, p, x, + x i x j j I miηi, p, x,, ηj, p, x,, ad we are doe. 4 Two mappigs associated with Jese s iequality Suppose x i, y j C or i =,..., ad j =,..., m ad θ i,j = θ i,j t = tx i + ty j. I what ollows, the mappigs H, F : 0, IR are give by Ht = θ i,j, i j m Ft = Ht, H t. Theorem 4. For quasicovex, i H, F are quasicovex o 0, ; ii Ft = F t or t 0, ; iii F/ Ft F0 = F or t 0,. Proo Part i ollows rom Propositios. ad. ad part ii rom the deiitio o F. The irst iequality i iii derives rom part ii ad Lemma.3. The remaider o iii is a cosequece o ii ad the quasicovexity o F. Put µ := / i=i p ix i. I the special case m = ad y = µ we write H = H 0 ad F = F 0. I the special case m = ad y i = x i i =,...,, we write H = H ad F = F. These mappigs were itroduced by Dragomir i the case o covex but have more geeral applicability. For otatioal coveiece we rebadge the correspodig orms o θ i,j as ψ i t = tx i + tµ ad ψ i,j t = tx i + tx j. Theorem 4. For quasicovex ad t 0,, we have a H 0 0 H 0 t H 0 ;

5 5 b µ F / F t F = i x i ; c F t F 0 t. Proo The outermost iequality o Theorem 3. may be writte µ i x i, so by the deiitio o H 0 ad the quasicovexity o H 0 t i x i, µ = whece we deduce the secod iequality i a. x i, µ i = i x i, The outermost iequality o Theorem 3. gives also that H 0 t = ψ i i p i ψ i. Sice p iψ i = p ix i, this provides H 0 t µ, whece the irst iequality i a. From Theorem 3., we have successively F / = xi + x j j i j = j j= p j p i xi + x j p i x i + x j p i x i + x j = p i x i givig the irst iequality i b. Further, F t i,j x i, x j = i x i or all t 0,, rom which we have the rest o b. Agai by Theorem 3., H 0 t = ψ i = p j ψ i,j i i P ψ i,j = F t i j j= or all t 0,. Sice F t = F t, we have also H 0 t F t, so c holds. 5 Further related maps Some urther maps o 0, itimately related to H 0, F are Kt := / p iψ i, Lt := P i,j= p i p j ψ i,j, T t := p i ψ i,j, j W t := j p i ψ i,j. The mappigs K ad L were itroduced with dieret otatio i ad their properties studied i the case where is covex. See also, 3. These mappigs provided useul iterpolatios o Jese s discrete iequality. Their behaviour i the covex cotext is similar to that o H 0 ad F respectively o the previous sectio. The preset cotext is more subtle i that a sum o quasicovex uctios eed ot be quasicovex. Remark 5. We have rom the deiitios that or all t 0, W t p i ψ i,j = T t. j

6 6 Propositio 5. For quasicovex Kt Kt miht, T t or all t 0, ad p i x i + µ + p i x i µ 3 p i x i + x j + xi p i j P x j x j. j j Proo. From Theorem 3. we have or t 0, that j= p jψ i,j j ψ i,j, so that Kt = p i p j ψ i,j p i P ψ i,j = T t. j j= O the other had, rom its deiitio, Kt i y i = Ht or all t 0,. Take together, these two results yield. Also, by the deiitios o quasicovexity ad Kt, Kt p i x i, µ = p i x i + µ + x i µ = p i x i + µ + p i x i µ, which provides the irst iequality i 3. For the remaider o 3, Theorem 3. provides p i x i, µ p i x i, P x j j = p i x i + P x j + p i x i j P x j j p i P x j = x j. j j Propositio 5.3 For all t 0,, we have Lt W t ad Lt Lt = P i,j= p i x i + P p i p j ψ i,j = j= i<j Proo. The irst iequality is provided by p j p i p j x i x j i x i. 4 p i ψ i,j j p i ψ i,j. Quasicovexity yields ψ i,j x i, x j or all i, j,..., ad t 0,. Multiplyig by p i p j ad summatio over i, j yields Lt P = = p i p j x i, x j i,j= P i,j= P p i p j x i + x j + x i x j i,j= p i p j x i + x j + P p i p j x i x j, i,j=

7 7 which equals the right had side o the irst iequality i 4. Sice x i, x j k x k or all i, j,,...,, we have P i,j= p i p j x i, x j P p i p j x k, k i,j= which equals the right had side o the secod iequality i 4. The propositio is proved. Propositio 5.4 For all t 0,, we have T t F t ad T t p i x i + P x i + p i i x i x j, j W t p i x i + x i + i p i x i x j. j Proo. From the deiitio o T, we have or t 0, that T t ψ i,j = ψ i,j, i j i,j whece the irst iequality ollows. Agai by quasicovexity ψ i,j j j x i + x j + x i x j x i + j x j + j x i x j Multiplyig by p i ad summatio over i yields the secod iequality. Similarly quasicovexity supplies or all j,..., ad t 0, that. p i ψ i,j p i x i + x j + x i x j p i x i + x j + p i x i x j. Takig the imum over j provides the third ad ial iequality. 6 Reiemets o Jese s iequality or quasicovex uctios We begi by extedig Theorem 3. to multisums. The ollowig elemetary lemma is useul. Lemma 6. Let K be a positive iteger ad σ, σ,, σ K real umbers. Real umbers ρ,..., ρ K are deied by ρ i = r σ i + r σ i+ + + r K σ i+k, where we iterpret σ l+k = σ l. I K l= r l =, the K l= ρ l = K l= σ l. Lemma 6. Suppose x i,...,i k C, p i,...,i k > 0 or i,..., i k,,...,. For is quasicovex i,...,i k = p i,...,i k x i,...,i k i,...,i k = p i,...,i k i,...,i k x i,...,i k. Proo. The vectors i C may be relabelled by positive itegers via x = x,,...,,, x = x,,...,,,..., x k = x,,...,,, x k = x,,...,,

8 8 with a similar relabellig or p i,...,i k. The relatio i the euciatio the becomes which holds by virtue o Theorem 3.. k l= p lx l k l= p l x l, l k Theorem 6.3 Suppose is quasicovex. Let y,k = y,k x i, x i,..., x ik = /kx i + x i x ik ad a k = i,i,...,i k y,k. The the sequece a k k is oicreasig ad bouded below by µ. Proo. Take x i,i,...,i k = y,k i Lemma 6.. The covexity o C esures that x i,i,...,i k C. The or each k, Lemma 6. gives i,...,i k = p i... p ik y,k i,...,i k = p a k. 5 i... p ik Easy iductios o k provide p i... p ik = P k ad i,...,i k = i,...,i k = p i... p ik y,k = P k p i x i, so the let had side o 5 reduces to the required lower boud. Put σ l = x il l k+ i Lemma 6. with K = k+ ad r i = /k or i k ad r k+ = 0. We may exted the deiitio o y,k to y l,k or l k + by settig y l,k = ρ l. The coditio K l= r l = holds, so k+ l= y l,k = k+ l= x i l ad by Theorem 3. Takig the imum yields y,k+ = k + y,k y k+,k a k+ i,...,i k+ y l,k l k+ = l k+ l k+ y l,k. i,...,i k+ y l,k By symmetry, each o the ier ima takes the value i,...,i k y,k = a k, so we have a k+ a k, ad we are doe. We may also derive a weighted reiemet o Jese s iequality or quasicovex mappigs. Theorem 6.4 Suppose is quasicovex ad q j 0 j k with Q k = k j= q j > 0. Deie z,k = /Q k q x i q k x ik ad b k = i,i,...,i k z,k. The µ a k b k i x i. Proo. We have just established the irst iequality. For the secod, take K = k i Lemma 6. with σ l = x il ad deie r l = q l /Q k. We exted the deiitio o z,k to z l,k or l k by z l,k = ρ l. The K l= r l = ad so y,k = /k k l= x i l = /k k l= z l,k. Thus k y,k = /k z l,k z l,k. l k Takig ima provides a k l= by symmetry, ad we have the secod iequality. z l,k, = z,k = b k, i,...,i k l k i,...,i k Fially, by quasicovexity z,k x i,..., x ik. Takig ima yields b k i,...,i k l k x il = i k x i ad we are doe..

9 9 7 Associated sequeces o mappigs We itroduce a sequece o mappigs L k+ L k+ t = i,...,i k+ Theorem 7. For quasicovex, L k+ a k+ = L k+ : 0, IR deied by ty,k + tx ik+. is quasicovex o 0, with a k+ L k+ t x i, 6 i k/k + L k+ Proo. Quasicovexity is immediate rom Propositio.. Now put t L k+ 0 = x i. i ω,k = ty,k + tx ik+ ad ω l,k = ty l,k + tx il or l k +. The k+ l= ω l,k = k+ l= x i l = y,k+, while by Theorem 3. k + ω,k + ω,k ω k+,k ω l,k. l k+ Hece y,k+ l k+ ω l,k or all t 0,. Takig ima yields a,k+ i,...,i k+ by symmetry. This gives the irst iequality i 6. j k+ ω j,k = i,...,i k+ ω,k, The quasicovexity o provides ω,k y,k, x ik+. Takig ima gives L k+ t y,k, x ik+ i,...,i k+ = y,k, x i k+ i,...,i k+ i,...,i k+ = y,k, x i. i,...,i k i By Theorem 6.3, y,k i x i, whece we derive the secod iequality i 6. The remaiig iequalities ollow directly. Remark 7.. For k = we recover the mappig L t = i,i ψ i,i = i,j ψ i,j = F t studied i Sectio 4. F k Deie u k = u k x ik+,..., x ik by u k = x ik x ik /k. A urther sequece o mappigs : 0, IR k associated with a quasicovex is give by where agai each x i C i ad k. Theorem 7.3 For quasicovex i F k is quasicovex o 0, ; ii F k t = F k t or all t 0, ; iii we have or all t 0, the bouds F k F k t = i,...,i k ty,k + tu k t F k 0 = F k = a k ad F k t F k / = a k. Proo. The proo ollows amiliar lies. We address oly the pair o iequalities a k F k t a k. By quasicovexity, ty,k + tu k y,k, u k. Takig ima yields ty,k + tu k y,k, u k i,...,i k i,...,i k = y,k, u k i,...,i k i,...,i k = i,...,i k y,k = a k,

10 0 which proves the irst iequality. By the symmetry ad quasicovexity o F k, F k t = F k t, F k t F k /t + t = F k / = a k ad the secod is proved. Fially, we cosider or t 0, the mappigs H k t := i,...,i k ty,k + tu k. Theorem 7.4 The mappig H k is quasicovex o 0, with µ = H k 0 H k t ty,k + tx ik+ = L k+ t, 7 i,...,i k+ a k = H k H k t. 8 Proo. Quasicovexity is immediate. By Lemma 6., H k t P k p i... p ik ty,k + tµ = µ i,...,i k ad the irst two relatios i 7 are established. For the rest, observe that H k t = i,...,i k p ik+ ty,k + tx ik+ i k+ = ty,k + tx ik+ = L k+ t. i,...,i k i k+ By the quasicovexity o, we have ty,k + tµ y,k, µ, so ad 8 is proved. H k t y,k, µ i,...,i k = y,k, µ = y,k = a k i,...,i k i,...,i k Reereces S.S. Dragomir, Two mappigs associated with Jese s iequality, Extracta Math , S.S. Dragomir ad D.M. Milošević, Two mappigs i coectio to Jese s iequality, Zb. Rad. Krajujevac 5 994, S.S. Dragomir ad D.M. Milošević, Two mappigs i coectio to Jese s iequality, Math. Balkaica N.S , S.S. Dragomir ad B. Mod, O Hadamard s iequality or a class o uctios o Goduova ad Levi, Idia J. Math , 9. 5 S.S. Dragomir ad C.E.M. Pearce, O Jese s iequality or a class o uctios o Goduova ad Levi, Periodica Math. Hugar , S.S. Dragomir ad C.E.M. Pearce, Quasi covex uctios ad Hadamard s iequality, Bull. Austral. Math. Soc , S.S. Dragomir, J.E. Pečarić ad L.E. Persso, Some iequalities o Hadamard type, Soochow J. Math. 995, A. Eberhard ad C.E.M. Pearce, Class iclusio properties or covex uctios, Progress i Optimizatio, Eds X. Yag et al., Kluwer Academic Publishers, Dordrecht 000, C.E.M. Pearce, Quasicovexity, ractioal programmig ad extremal traic cogestio, i Frotiers i Global Optimizatio, Kluwer, Dordrecht, Noliear Optimizatio ad its Applicatios , C.E.M. Pearce ad A.M. Rubiov, P uctios, quasi covex uctios ad Hadamard type iequalities, J. Math. Aal. & Applic , 9 04.