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1 Matematiqki ilten ISSN X rint 37LXIII No. ISSN online UDC: :57.5 Skoje, Makedonija GENERALIZED POTENTIAL INEQUALITY AND EXPONENTIAL CONVEXITY NEVEN ELEZOVIĆ, JOSIP PEČARIĆ, AND MARJAN PRALJAK Abstract. In this aer we generalize the otential inequality which was introduced in [6] and extended to the class of naturally defined convex functions in []. The generalization is achieved by relacing the st order Taylor exansion of a convex function in the roof of the otential inequality with the n-th order Taylor exansion of an n + -convex function. Furthermore, by using methods develoed in [4] and [] we construct several families of n-exonentially convex functions by making use of linearity of the generalized otential inequality.. Introduction Potential inequality, introduced by Rao and Šikić in [6], is a very general inequality that holds for kernels that satisfy the maximum rincile see next section, an imortant roerty from the otential theory that is satisfied by many classical kernels. Rao and Šikić roved the inequality only for a secial class of convex and concave functions, which was later extended to the naturally defined class of convex and concave functions by Elezović, Pečarić and Praljak in []. The latter form of the otential inequality is given in the following theorem Theorem 5 in [] Theorem The otential inequality for convex functions. Let g :0, + R be a convex function, g its right-continuous derivative and Nx, dy a ositive kernel on X which satisfies the strong maximum rincile on R with constant M. Letf R, x X and z>0 be such that z Nfx/M 00 Mathematics Subject Classification. 6D5. Key words and hrases. Potential inequality, n-convex function, exonential convexity. Corresonding author 83

2 84 N. ELEZOVIĆ, J. PEČARIĆ, AND M. PRALJAK and denote by the set Then = { y X :Nfy z }. g M Nfx gz M N[f + g Nf z ]x + M g zn[f f + z ]x zg z. In addition to the maximum rincile, another crucial ste in the roof of the otential inequality is an integration by arts which gives the st order Taylor exansion of the convex function with the remainder in its integral form. In this aer we will generalize the otential inequality by relacing the st order Taylor exansion of the convex function with the n-th order Taylor exansion of an n + -convex function. Furthermore, the differences generated by the generalized otential inequality are linear functionals and, by using methods develoed in [3], [] and [4], we construct several families of exonentially convex functions.. Generalized otential inequality We will first introduce the notation and the set u. We say that Nx, dy is a ositive kernel on X if N : X X [0, + ] is a maing such that, for every x X, A Nx, A isaσ-finite measure, and, for every A X, x Nx, A is a measurable function. For a measurable function f, theotential of f with resect to N at a oint x X is Nfx= fynx, dy, X whenever the integral exists. The class of functions that have the otential at every oint is denoted by POT N. For a measure μ on X, X, a measurable set C X andk N 0 we will denote by ˆN C k μ the measure defined by ˆN Cμdy k = Nx, X k Nx, dyμdx. If C = X we will omit the subscrit, i. e. ˆN X k μ. C ˆN k μ will denote the measure Definition. Let N be a ositive kernel on X and R POTN. We say that N satisfies the strong maximum rincile on R with constant

3 GENERALIZED POTENTIAL INEQUALITY 85 M if Nfx Mu+ N[f + {Nf u} ]x holds for every x X, f Rand u 0. We will also make use of the divided differences. Let g be a real-valued function on an interval I R. The divided difference of order n of the function g at distinct oints z 0,z,...,z n I is defined see [5] recursively by and [z i ]g = gz i, i =0,...,n [z 0,z,...,z n ]g = [z,...,z n ]g [z 0,...,z n ]g. z n z 0 The value [z 0,...,z n ]g is indeendent of the order of the oints z 0,...,z n. The definition may be extended to include the case in which some or all of the oints coincide by assuming z 0 z... z n and letting [z,...,z]g = gj z }{{} j! j times rovided that f j z exists. A function g is said to be n-convex, n 0, if the n-th order divided difference satisfies [z 0,...,z n ]g 0, for all choices of distinct oints z 0,...,z n I. If a function g is n + -convex, n, then the derivatives g k exist for k n, while for k = n the right sided derivative g n + exists and it is right-continuous and non-decreasing see [5], Theorem.4. We will denote this right sided derivative simly by g n and dg n will denote the ositive measure generated by g n. Theorem Generalized otential inequality. Let g :0, + R be an n+ convex function and Nx, dy a ositive kernel on X which satisfies the strong maximum rincile on R with constant M. Let f R, x X and z>0 be such that z Nfx/M < + and let the set be defined

4 86 N. ELEZOVIĆ, J. PEČARIĆ, AND M. PRALJAK by. Then { Nx, Xn n!m n N[f + n g n Nf z ]x } g n zn[f + n z ]x + g M Nfx gz i! M Nfx zi g i z Proof. The n-th order Taylor exansion of the function g is gτ gz = i! τ zi g i z+r n z, 3 where the remainder R n in its integral form is given by R n z = n! τ z τ u n dg n u. Inserting τ = τx = M Nfx in 3 and using the strong maximum rincile under the integral of the remainder, since dg n u isaositive measure, we get gτx gz τ x n!m n N[f + {Nf u} ] n dg n u+ z i! τx zi g i z. 4 We can bound the integral on the right hand side of the last inequality by alying, resectively, Jensen s inequality, Fubini s theorem and the fact that f + n {Nf u} is a nonnegative function from the third to the fifth line below τ x z N[f + {Nf u} ] n dg n u [ τ x n = f + y {Nf u} ynx, dy] dg n u z X τ x n Nx, dy f + y n {Nf u} y n Nx, dy dg n u z X X

5 = Nx, X n Nx, X n GENERALIZED POTENTIAL INEQUALITY 87 X X [ ] τ x {Nf u} y dg n u f + y n Nx, dy z [ ] + {Nf u} y dg n u f + y n Nx, dy z = Nx, X n f + y n[ g n Nfy ] g n z z ynx, dy X = Nx, X n { } N[f + n g n Nf z ]x g n zn[f + n z ]x. Finally, combining the last inequality with 4 finishes the roof. Let us further denote the set = z>0 = { x X :Nfx > 0 }. The integral version of the generalized otential inequality is obtained by integrating the inequality from Theorem with resect to the variable x. Corollary 3. Let the assumtions of Theorem hold for a function z : 0, +, i.e. zx Nfx/M for x. Then, for C, C X, and a finite measure μ on X, X, the following inequality holds g C M Nfx gzx μdx n!m n f + y n g n NfyNx, dy Nx, X n μdx C x n!m n f + y n Nx, dy g n zxnx, X n μdx C x + i! C M Nfx zxi g i zxμdx.

6 88 N. ELEZOVIĆ, J. PEČARIĆ, AND M. PRALJAK In articular, for C = and the constant function zx z we get g M Nfx μdx gzμ n!m n f + x n g n Nfx ˆN n z μdx gn z n!m n f + x n ˆN n z μdx+ g i z i! M Nfx zi μdx. Proof. Integrating the generalized otential inequality with resect to the measure μ we get C g M Nfx gzx μdx n!m n Nx, X n N[f + n g n Nf zx ]xμdx n!m n C C Nx, X n g n zxn[f + n zx ]xμdx + i! C M Nfx zxi g i zxμdx. which is the first inequality. The second inequality follows by taking C = and zx z and by alying Fubini s theorem on the first two integrals of the right hand side. Corollary 4. Under the assumtions of Corollary 3, for R\{0,,...,n} the following inequalities hold: Nf xμdx n + i=0 M n n! n f + x n Nf n x f + x n ˆN n zm n n! n z i M i! i i n n ˆN μdx μdx M Nfx zi μdx

7 GENERALIZED POTENTIAL INEQUALITY 89 and Nf xμdx n [ ] M n n [ n! n + i=0 f + d ˆN n μ zm n n! n f + x n ˆN n z i M i! i i n Nf d ˆN n μ μdx ] n M Nfx zi μdx. Proof. The first inequality is obtained by alying the second inequality from Corollary 3 for n + -convex functions g z = n z, 5 for R\{0,,...n}, and rearranging. The second inequality follows from the first by alying Hölder s inequality on the first integral of the right hand side for the air of conjugate exonents /n and / n. If Theorem holds for z>0, then it holds for every z,0<z z. Letting z 0, if the function g satisfies certain roerties, we can get further inequalities. In the following theorem we will assume that either g n is nonnegative, or that for every x there exists a function h x L Nx, such that f + n g n Nf h x. In either case, by the monotone convergence theorem in the former and by the dominated convergence theorem in the latter, we have lim N[f + n g n Nf z ]=N[f + n g n Nf ] z 0 since f + n g n Nf z f + n g n Nf ointwise, when z 0.

8 90 N. ELEZOVIĆ, J. PEČARIĆ, AND M. PRALJAK Corollary 5. Under the assumtions of Theorem, if g i 0+ is finite for i =0,,...,n, then for every x we have g { M Nfx Nx, Xn n!m n N[f + n g n Nf ]x } g n 0+N[f + n ]x + i!m i Nfxi g i 0+. i=0 Furthermore, if μ is a finite measure on X, X, then the following inequality holds g M Nfx μdx n!m n f + x n g n n Nfx ˆN μdx gn 0+ n!m n f + x n ˆN n μdx+ g i 0+ i!m i Nfx i μdx. i=0 Proof. The first inequality follows from Theorem by letting z 0and rearranging. The second inequality follows by integrating the first with resect to the measure μ over the set and alying Fubini s theorem on the first two integrals of the righ hand side. Corollary 6. Under the assumtions of Corollary 5, for >nthe following inequalities hold: Nf xμdx n +M n n! and Nf xμdx [ n +M n n! f + x n Nf n x f + d ˆN n μ ] n [ n ˆN μdx Nf d ˆN n μ Proof. The first inequality holds since functions g given by 5 for >n satisfy the assumtions of Corollary 5 with g i 0+ = 0 for i =0,,...,n. The second inequality follows from the first by alying Hölder s inequality on the right hand side integral with the air of conjugate exonents /n and / n. ] n.

9 GENERALIZED POTENTIAL INEQUALITY 9 Examle 7 Alication to a Hardy-tye kernel. Let X =0, + and let the kernel Nx, dy=gx, y dy be given by its density {, if y x, Gx, y= 0, otherwise. The kernel N satisfies the maximum rincile with constant M = see [6], and Nx, X= F x =Nfx= + 0 x 0 Gx, y dy = fydy x For a nonnegative function f the set is equal to 0 dy = x. = {x X :Nfx > 0} = {x 0, + :F x > 0} =b, +, where b =essinf{y : fy > 0}. Further, let ν i dx =λ i xdx, i =,, be two σ-finite measures with densities λ i that satisfy [ + ] λ x=λ x n y n n λ ydy. 6 The measure = + b n ˆN ν satisfies ˆN n ν dx= b y n [ y ] dx λ ydy = 0 x + Ny, X n Ny, dxλ ydy = [ + ] y n λ ydy dx, + 0 maxb,x i. e. + n d ˆN ν x= y n λ ydy. maxb,x Dueto6,weseethatforx d ˆN ν x=λ n xλ n x. 7 Alying the first inequality from Corollary 6 with μ = ν, equality 7 and Hölder s inequality with the air of conjugate exonents /n and / n

10 9 N. ELEZOVIĆ, J. PEČARIĆ, AND M. PRALJAK we get F x λ x dx n + fx n F x n n ˆN n! ν dx n + = fx n λ x n F x n λ x n dx n! [ ] n [ n + fx λ x dx F x λ x dx n! ] n, i. e. [ F x λ x dx ] n [ ] n n + fx λ x dx. n! The densities λ x =x k x k and λ x = k n /n, with k>n, satisfy condition 6 and, in this case, the last inequality is equivalent with where [ x k F x dx ] K [ x k fx dx ], 8 n + K = n. n!k n When = k>, the otimal constant in 8 is given by Hardy s inequality and equals /. This otimal constant is attained for n =, while for n> the constant K is strictly greater than the otimal. Remark 8. When both f and f satisfy the maximum rincile, then the absolute value f satisfies a similar condition, roven by Rao and Šikić see [6], Nf Mu+ N [ f { Nf u} ], for every u 0. 9

11 GENERALIZED POTENTIAL INEQUALITY 93 In that case, by using 9 instead of the maximum rincile, one can rove the generalized otential inequality for absolute values: if x X and z>0aresuchthatz Nfx /M < +, then g M Nfx gz Nx, Xn { n!m n N[ f n g n Nf z ]x } g n zn[ f n z ]x + i! M Nfx zi g i z, where = { y X : Nfy z }. Relacing the number z with a function that satisfies zx Nfx /M < + for every x = {x X : Nfx > 0} and integrating with resect to the variable x one gets the integral version of the last inequality, for C, C X, g C M Nfx gzx μdx n!m n fy n g n Nfy Nx, dy Nx, X n μdx x n!m n C C fy n Nx, dy g n zxnx, X n μdx x + i! C M Nfx zxi g i zxμdx. In articular, for C = and the constant function zx z we get g M Nfx μdx gzμ n!m n gn z n!m n fx n ˆN n fx n g n Nfx μdx+ g i z i! 3. Exonential convexity n ˆN μdx M Nfx zi μdx. We will first construct linear functionals nonnegative on the set of n+- convex functions by alying the generalized otential inequality, and its various forms, derived in the revious section. We then use the methods develoed in [3] to generate new families of n-exonentially convex and

12 94 N. ELEZOVIĆ, J. PEČARIĆ, AND M. PRALJAK exonentially convex functions by evaluating these functionals on families of functions satisfying similar roerties. Let us define linear functionals A k, k =,...,6, by { } Nx, Xn A g= n!m n N[f + n g n Nf z ]x g n zn[f + n z ]x + i! M Nfx zi g i z g M Nfx + gz, A g= n!m n gn z n!m n A 3 g= Nx, Xn n!m n f + x n g n Nfx ˆN n μdx f + x n ˆN n g i z μdx+ i! M Nfx zi μdx g M Nfx μdx+gzμ, { } N[f + n g n Nf ]x g n 0+N[f + n ]x + i!m i Nfxi g i 0+ g M Nfx, i=0 A 4 g= n!m n f + x n g n n Nfx ˆN μdx g M Nfx μdx gn 0+ n!m n f + x n n ˆN μdx+ g i 0+ i!m i Nfx i μdx, i=0 { } Nx, Xn A 5 g= n!m n N[ f n g n Nf z ]x g n zn[ f n z ]x + i! M Nfx zi g i z g M Nfx + gz, A 6 g= n!m n fx n g n Nfx ˆN n μdx gn z n!m n fx n ˆN n g i z μdx+ i! g M Nfx μdx+gzμ. M Nfx zi μdx

13 GENERALIZED POTENTIAL INEQUALITY 95 The linear functionals A k deend on the choice of a function f, kernel N, measureμ and oints x and z, but we consider them given and omit from the notation. Also, we assume that a articular choice of f, N, μ, x and z satisfies the assumtions of Theorem for k =, Corollary 3 for k =, Corollary 5 for k = 3 or 4 and Remark 8 for k = 5 or 6. We continue this section with few basic notions and results on exonential convexity that we will use. Definition. A function ψ : I R is n-exonentially convex in the Jensen sense on I if xi + x j ξ i ξ j ψ 0 i,j= holds for all choices ξ i R and x i I, i =,..., n. A function ψ : I R is n-exonentially convex if it is n-exonentially convex in the Jensen sense and continuous on I. Definition 3. A function ψ : I R is exonentially convex in the Jensen sense on I if it is n-exonentially convex in the Jensen sense for every n N. A function ψ : I R is exonentially convex if it is exonentially convex in the Jensen sense and continuous on I. Definition of ositive semi-definite matrices and some basic algebra gives us the following roosition Proosition 9. If ψ is an n-exonentially convex in the Jensen sense on [ ] k I, then for every choice of x i I, i =,..., n, thematrix ψ xi +x j i,j= is a ositive semi-definite matrix for all k N, k n. In articular, for [ ] k all k N, det ψ 0 for all k n. xi +x j i,j= Remark 0. It is known that ψ : I R is log-convex in the Jensen sense if and only if x + y α ψx+αβψ + β ψy 0 holds for every α, β R and x, y I. It follows that a function is log-convex in the Jensen sense if and only if it is -exonentially convex in the Jensen sense. Moreover, a function is log-convex if and only if it is -exonentially convex. The next theorem will enable us to construct n-exonentially convex and exonentially convex functions.

14 96 N. ELEZOVIĆ, J. PEČARIĆ, AND M. PRALJAK Theorem. Let Ω={g : J}, wherej is an interval in R, beafamily of functions g :0, + R such that the function [z 0,...,z n+ ]g is m-exonentially convex in the Jensen sense on J for every n +mutually different oints z 0,...,z n+ 0, + and A k g, for k 6, is well defined for every J when k =3or 4, additional assumtion is that g l 0+ is finite for every l n and J. Then, the maing A k g is an m-exonentially convex function in the Jensen sense on J. If the function A k g is continuous on J, thenitism-exonentially convex on J. Proof. For ξ i R and i J, i =,...,m, we define the function m gy= ξ i ξ j g i + j y. i,j= Since [z 0,...,z n+ ]g is m-exonentially convex in the Jensen sense, we have m [z 0,...,z n+ ]g = ξ i ξ j [z 0,...,z n+ ]g 0, i,j= which, in turn, imlies that g is an n + -convex function on J for k =3 and 4 it also holds that g0+ is finite. Therefore, by the generalized otential inequality, m A k g = ξ i ξ j A k g i + j 0. i,j= We conclude that the function A k g ism-exonentially convex in the Jensen sense on J. If the function A k g is also continuous on J, then it is m-exonentially convex by definition. Immediate consequence of the revious theorem and roerties of exonentially convex functions is the following corollary. Corollary. Let Ω = {g : J}, where J is an interval in R, be a family of functions g :0, + R such that the function [z 0,...,z n+ ]g is -exonentially convex in the Jensen sense on J for every n+ mutually different oints z 0,...,z n+ 0, + and let the linear functionals A k, k 6, satisfy the same assumtions as in Theorem. Then, the following statements hold: i If the function A k g is continuous on J, thenitis-exonentially convex and, thus, log-convex.

15 GENERALIZED POTENTIAL INEQUALITY 97 ii If the function A k g is strictly ositive and differentiable on J, then for every, q, r, s J, such that r and q s, we have μ k,q Ω μk r,s Ω, where μ k,qω = ex for g,g q Ω. Ak g A k g q d q d A kg A k g, q, = q. 0 Proof. i This is an immediate consequence of Theorem and Remark 0. ii yi, the function A k g is log-convex on J, thatis,the function log A k g is convex. Therefore log A k g log A k g q q log A kg r log A k g s r s for r, q s, r, q s, which imlies that μ k,qω μ k r,sω, k =,..., 6. The cases = r and q = s follow from by taking limits r or q s. Next, we resent several families of functions that satisfy the assumtions of Theorem and Corollary and, in this way, we construct large families of exonentially convex functions. Examle 3. Consider a family of functions Ω = {g : R} given by g y = { e y, n+ 0 y n+ n+!, =0 Similarly as in the roof of Theorem, let us, for ξ i R and i R, i =,..., m, define the function m gy= ξ i ξ j g i + j y. i,j= Since the function dn+ g dy n+ y =e y is exonentially convex follows from the definition, we have that m m g n+ y= i,j= ξ i ξ j g n+ y = i + j ξ i e iy/ 0

16 98 N. ELEZOVIĆ, J. PEČARIĆ, AND M. PRALJAK is an n + -exonentially convex function. Therefore 0 [z 0,...,z n ]g = m i,j= ξ i ξ j [z 0,...,z n ]g i + j so [z 0,...,z n ]g is m-exonentially convex in the Jensen sense for every m. Using Theorem we conclude that the maing A k g is exonentially convex in the Jensen sense. It is easy to verify that this maing is continuous although the maing g is not continuous at = 0, so it is exonentially convex. For this family of functions, μ k,q Ω from 0 becomes μ k,qω = Ak g q A k g q Ak id g ex A k g n+ ex A k id g 0 n+ A k g 0 where idy = y is the identity function., q, = q 0, = q =0, Examle 4. Let Ω = {g : I k } be a family of functions defined by, g y = { y n, y j ln y n j j!n j!, / {0,,...,n} = j {0,,...,n}, where I k =0, + fork =,, 5 and 6, and I k =n, + fork =3 and 4. The maing dn+ g y =e n ln y is exonentially convex dy n+ follows from the definition and, arguing as in Examle 3, we get that the maings A k g, k 6, are exonentially convex. In this case, the functions 0 are equal to Ak g q A k g q, q μ k,qω = ex n n! A kg 0 g A k g + n k=0 k, = q/ {0,,...,n}, ex n n! A kg 0 g A k g + n k=0 k, = q {0,,...,n}. k Examle 5. Let Ω 3 = {g : 0, + } be a family of functions given by { g y = y, ln n+ y n+ n+!, =.

17 GENERALIZED POTENTIAL INEQUALITY 99 Since dn+ g y = dy y is the Lalace transform of a non-negative n+ function see [7] it is exonentially convex. Arguing as in Examle 3 we get that the maings A k g, k 6, are exonentially convex. For this family of functions, μ k,q Ω 3 from 0 becomes μ k,q Ω 3= ex Ak g q A k g q ex A kid g A k g n+ ln A k id g n+ A k g, q, = q,, = q =. Examle 6. Let Ω 4 = {Ψ : 0, + } be a family of functions defined by g y = e y n+. Since dn+ g y =e dy y is the Lalace transform of a nonnegative n+ function see [7], it is exonentially convex. Arguing as before, we get that A k g, k 6, are exonentially convex functions. For this family of functions, μ k,q Ω 4 from 0 becomes Ak g q μ k,q Ω A 4= k g q, q ex A kid g A k g n+, = q. References [] N. Elezović, J. Pečarić, M. Praljak, Potential inequality revisited, I: General case, Math. Inequal. Al. 5 0, [] R. Ghulam, J. Pečarić, A. Vukelić, n-exonential convexity of divided differences and related Stolarsky tye means, Math. Inequal. Al. 6 03, [3] J. Jakšetić, J. Pečarić, Exonential convexity method, J. Convex Anal. 0 03, [4] J. Pečarić, J. Perić, Imrovements of the Giaccardi and the Petrović inequality and related Stolarsky tye means, An. Univ. Craiova Ser. Mat. Inform. 39 0, [5] J. Pečarić, F. Proschan, Y. L. Tong, Convex Functions, Partial Orderings and Statistical Alication, Academic Press, San Diego, 99. [6] M. Rao, H. Šikić, Potential Inequality, Israel J. Math , [7] J. L. Schiff, The Lalace transform. Theory and alications, Undergraduate Texts in Mathematics, Sringer-Verlag, New York, 999.

18 00 N. ELEZOVIĆ, J. PEČARIĆ, AND M. PRALJAK OOPXTENO POTENCIJALNO NERAVENSTVO I EKSPONENCIJALNA KONVEKSNOST Neven Elezoviḱ, Josi Peqariḱ, Marjan Praljak Rezime Vo ovoj trud go generalizirame otencijalnoto neravenstvo koe e vovedeno vo [6] i roxireno na klasata rirodno definirani konveksni funkcii vo []. Generalizacijata e dostignata so zamena vo dokazot na otencijalnoto neravenstvo, na Tejlorovata eksanzija od rv red na konveksna funkcija so Tajlorovata eksanzija od red n na n +- konveksna funkcija. Osven toa, koristejḱi metodi razvieni vo [4] i [] konstruirame nekolku familii od n-eksonencijalni konveksni funkcii, so koristenje na linearnosta na obxtenoto otencijalno neravenstvo. Faculty of Electrical Engineering and Comuting, University of Zagreb, Unska 3, 0000 Zagreb, Croatia address: neven.elez@fer.hr Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filiovića 8a, 0000 Zagreb, Croatia address: ecaric@element.hr Faculty of Food Technology and iotechnology, University of Zagreb, Pierottijeva 6, 0000 Zagreb, Croatia address: mraljak@bf.hr