RAD HAZU. MATEMATIČKE ZNANOSTI Vol. 18 = 519 014: 107-13 ON POTENTIAL INEQUALITY FOR THE ASOLUTE VALUE OF FUNCTIONS Neven Eleović, Josip Pečarić and Marjan Praljak Abstract. Potential inequality was introduced in [4] and later extended to the case of general convex and concave functions in [1]. The main goal of this paper is to derive the potential inequality for the case where the function at which the potential is evaluated is replaced by its absolute value. The results obtained, together with methods from [], are used to construct new families of exponentially convex functions. 1. introduction Rao and Šikić [4] introduced the potential inequality for kernels and functions that satisfy the maximum principle see below, but they proved it only for a special class of convex and concave functions. Eleović, Pečarić and Praljak [1] generalied the potential inequality to the class of naturally defined convex functions on 0, +, which enabled them to generate certain exponentially convex functions that were used to refine some of the known inequalities and to derive new ones. In this article, we will look at the same class of kernels and functions and extend the results to the case where the function is replaced by its absolute value. Furthermore, we will generate more families of exponentially convex functions by applying methods from [] that make use of the divided differences. We will start off by introducing notation and the setup. We say that Nx, dy is a positive kernel on if N : [0, + ] is a mapping such that, for every x, A Nx, A is a σ-finite measure, and, for every A, x Nx, A is a measurable function. For a measurable function f, the potential of f with respect to N at a point x is Nfx = fynx, dy, 010 Mathematics Subject Classification. 6D15. Key words and phrases. Potential inequality, exponential convexity. 107
108 N. ELEZOVIĆ, J. PEČARIĆ AND M. PRALJAK whenever the integral exists. The class of functions that have the potential at every point is denoted by POT N. For a measure µ on, and a measurable set C we will denote by ˆN C µ the measure defined by ˆN C µdy = Nx, dyµdx. C If C = we will omit the subscript, i. e. ˆNµ will denote the measure ˆN µ. Definition 1.1. Let N be a positive kernel on and R POT N. We say that N satisfies the strong maximum principle on R with constant M 1 if 1.1 Nfx Mu + N[f + 1 {Nf u} ]x holds for every x, f R and u 0. The main result from [1] is the following theorem Theorem 1. The potential inequality for convex functions. Let Φ : 0, + R be a convex function and Nx, dy a positive kernel on which satisfies the strong maximum principle on R with constant M. Let f R, x and > 0 be such that Nfx/M and denote by the set Then = { y : Nfy }. Φ 1 M Nfx Φ 1 M N[f + ϕnf1 ]x + 1 M ϕn[f f + 1 ]x ϕ.. Potential Inequality for Absolute Values Let N be a kernel that satisfies the strong maximum principle. If both f and f belong to R, then a property similar to 1.1 holds for f see Lemma.1 bellow. First, notice that condition 1.1 is equivalent to.1 Nf + x Mu + N[f + 1 {Nf + u}]x. Indeed, the two conditions are equivalent for u > 0 since the right hand sides of 1.1 and.1 are equal because the sets {Nf > u} and {Nf + > u} are equal. Furthermore, the left hand sides are also equal on the set {Nf > 0} = {Nf + > 0}, while the inequalities 1.1 and.1 are trivially satisfied on the set {Nf 0} = {Nf + = 0} since the right hand sides are nonnegative. Finally, for u = 0 the condition.1 holds trivially since {Nf + 0} = and Nf + Nf +.
ON POTENTIAL INEQUALITY 109 get On the other hand, when.1 holds for every u 0, letting u ց 0 we Nfx Nf + x M 0 + N[f + 1 {Nf + >0}] N[f + 1 {Nf 0} ]. The following lemma was proven in [4]. Lemma.1. Let N be a positive kernel that satisfies the strong maximum principle on R. If f and f belong to R, then. Nf Mu + N [ f 1 { Nf u} ] holds for every u 0. get Proof. Since Nf = [N f] +, applying.1 for both f and f we Nfx = Nf + x + Nf x Mu + N[f + 1 {Nf + u}] + N[f 1 {Nf u}]. Finally, since f + f, f f and {Nf + u} {Nf u} = { Nf u}, the claim of the lemma follows from the last inequality. Theorem. The Potential Inequality for the Absolute Value of Functions. Let Φ : 0, + R be a convex function and let ϕ = Φ + be the right-continuous version of its derivative. Let Nx, dy be a positive kernel on which satisfies the strong maximum principle on R with constant M, let f, f R, x and > 0 be such that M Nfx and denote by the set = { y : Nfy }. Then the following inequality holds Φ 1 M Nfx Φ 1 M N[ f ϕ Nf 1 ]x 1 M ϕ N[ f 1 ]x Nfx Proof. Let τx = 1 M Nfx. Integration by parts gives Φτx Φ = τx ϕudu = uϕu = τϕτx ϕ = τx τx τx τx ϕ. udϕu u ± τxdϕu τx udϕu + ϕτx.
110 N. ELEZOVIĆ, J. PEČARIĆ AND M. PRALJAK Since dϕu is a positive measure and τx, using. we get Φτx Φ τx 1 M N[ f 1 { Nf u}]dϕu + ϕτx. Applying Fubini s theorem and the fact that f 1 { Nf u} is a nonnegative function, we further get τx N[ f 1 { Nf u} ] dϕu = τx = = fy 1 { Nfy u} Nx, dy dϕu [ ] τx 1 { Nfy u} dϕu fy Nx, dy [ ] + 1 { Nfy u} dϕu fy Nx, dy fy ϕ Nfy ϕ 1 ynx, dy = N[ f ϕ Nf 1 ]x ϕn[ f 1 ]x. Now the inequality from the theorem follows from the two inequalities above and linearity of potential. Let us further denote the set = lim ց0 = { x : Nfx 0 }. y integrating the potential inequality from Theorem. with respect to the variable x we can get the following, integral version of the inequality. Corollary.3. Let the assumptions of Theorem. hold for a function : 0, +, i. e. Mx Nfx for x. Then, for C and a measure µ on,, the following inequality holds Φ 1 C M Nfx Φx µdx 1 fy ϕ Nfy Nx, dyµdx M C x 1 ϕx N[ f 1 ]x Nfx µdx xϕxµdx. M C C
ON POTENTIAL INEQUALITY 111 In particular, for C = and a constant function x, if the measure µ is σ-finite, we get Φ 1 M Nfx µdx Φµ 1 fx ϕ Nfx M ˆN µdx + 1 M ϕ N[ f 1 ]x Nfx µdx ϕµ. Proof. Integrating the potential inequality with respect to the measure µ we get Φ 1 C M Nfx Φx µdx 1 N[ f ϕ Nf 1 x ]µdx M C 1 ϕx N[ f 1 ]x Nfx µdx xϕxµdx, M C which is the first inequality. The second inequality follows by taking C = and x and by applying Fubini s theorem on the first integral of the right-hand side. Let us denote by Φ p the following class of functions τ p pp 1, p 0, 1.3 Φ p τ = log τ, p = 0 τ log τ, p = 1 Corollary.4. Under the assumptions of Corollary.3, for p R\{0, 1} the following inequality holds 1 Nfx pp 1 p µdx Mp 1 fx Nfx p 1 p 1 ˆN µdx Mp 1 N[ f 1 ]x Nfx µdx Mp µ. p 1 p Furthermore, for q = p/p 1 the following inequality holds 1 pp 1 [ Nf p dµ Mp 1 f p d p 1 ˆN µ Mp 1 N[ f 1 ]x Nfx p 1 C p [ Nf p d ˆN µ µdx Mp µ p Proof. Applying the second inequality from Corollary.3 for convex functions Φ p, p R\{0, 1}, and rearranging we get the first inequality. The q.
11 N. ELEZOVIĆ, J. PEČARIĆ AND M. PRALJAK second inequality follows from the first by applying Hölder s inequality on the first integral of the right-hand side. Corollary.5. Under the assumptions of Theorem., if = and ϕ 0, then for every x the following inequality holds Φ 1 1 Nfx Φ N[ f ϕ Nf ]x ϕ. M M Furthermore, for a σ-finite measure µ on,, the following inequality holds Φ 1 Nfx µdx Φµ M 1 fx ϕ Nfx M ˆNµdx ϕµ. Proof. Under the assumptions of the corollary, the second term on the right hand side of the potential inequality from Theorem. is nonpositive, so the first inequality follows. The second inequality follows by integrating the first with respect to the measure µ over the set and applying Fubini s theorem on the right hand side integral. Corollary.6. Under the assumptions of Corollary.5, the following inequality holds for p > 1 Nfx p µdx pm p 1 fx Nfx p 1 ˆNµdx p 1M p µ. Furthermore, for q = p/p 1 the following inequality holds Nfx p µdx [ pm p 1 f p d ˆNµ p [ Nf p d ˆNµ q p 1M p µ. Proof. Applying Corollary.5 for convex functions Φ p, p > 1, and rearranging we get the first inequality. The second inequality follows from the first by applying Hölder s inequality on the right-hand side integral. If Theorem. or Corollary.5 hold for > 0, then they hold for every, 0 <. Letting 0 we can get further inequalities. In the following corollaries we will assume that either ϕ is nonnegative, or that for every x there exists a function g x L 1 Nx, such that
ON POTENTIAL INEQUALITY 113 fϕ Nf g x. In either case, by the monotone convergence theorem in the former and by the dominated convergence theorem in the latter, we have lim ց0 N[ f ϕ Nf 1 ] = N[ f ϕ Nf 1 ] since f ϕ Nf 1 0 f ϕ Nf 1 pointwise. Corollary.7. Under the assumptions of Theorem., if ϕ0+ is finite, then for every x we have.4 Φ 1 M Nfx Φ0+ 1 M N[ f ϕ Nf 1 ]x 1 M ϕ0+ N[ f 1 ]x Nfx Furthermore, if µ is a σ-finite measure on,, then the following inequality holds Φ 1 Nfx µdx Φ0+µ M 1 M fx ϕ Nfx ˆN µdx 1 M ϕ0+. N[ f 1 ]x Nfx µdx. Proof. Since lim 0 ϕ = 0 ϕ0+ = 0, letting 0 in the potential inequality from Theorem., the last term on the right hand side disappears and inequality.4 follows. The second inequality of the corollary follows by integrating the first with respect to the measure µ and applying Fubini s theorem on the first integral of the right hand side. Corollary.8. Under the assumptions of Corollary.7, for p > 1 the following inequality holds Nfx p µdx pm p 1 fx Nfx p 1 ˆNµdx. Furthermore, for q = p/p 1 the following inequality holds [ Nf p dµ pm p 1 f p d ˆNµ p [ Nf p d ˆNµ Proof. The first inequality holds since convex functions Φ p, p > 1, satisfy the assumptions of Corollary.7 with Φ p 0+ = ϕ p 0+ = 0. The second inequality follows from the first by applying Hölder s inequality on the righthand side integral. q.
114 N. ELEZOVIĆ, J. PEČARIĆ AND M. PRALJAK When one or both of the measures µ and ˆN C µ is bounded by the other up to a multiplicative constant, then we can state further inequalities. In that regard, let K 1 and K be positive constants, if they exist, such that.5 K 1 µ ˆN C µ and.6 ˆNC µ K µ. Corollary.9. Let the assumptions of Corollary.4 hold. If N and µ satisfy.6 with C =, then for p > 1 [ [ p q Nf p dµ pk M p 1 f p dµ Nf p dµ pm p 1 N[ f 1 ]x Nfx µdx p 1M p µ. When N and µ satisfy both.5 and.6 with C =, then for 0 < p < 1 [ [ p Nf p dµ pk 1/p 1 K 1/q M p 1 f p dµ pm p 1 Nf p dµ N[ f 1 ]x Nfx µdx p 1M p µ, while for p < 0 [ [ p Nf p dµ pk 1/q 1 K 1/p M p 1 f p dµ pm p 1 Nf p dµ N[ f 1 ]x Nfx µdx p 1M p µ. Proof. The inequalities follow directly from.5,.6 and Corollary.4. Corollary.10. Let the assumptions of Corollary.8 hold. If N and µ satisfy.6 with C =, then for p > 1 [ [ p p Nf p dµ pk M p 1 f p dµ and [ Nf p dµ p [ pk 1/q M p 1 f p d ˆNµ Proof. The inequalities follow directly from.6 and Corollary.8. p. q q
ON POTENTIAL INEQUALITY 115 Remark.11. Concave case: When Φ is concave, dϕu is a negative measure and the inequalities in Theorem., Corollaries.3 and.7 are reversed. The reversed inequality in Corollary.5 holds if ϕ 0. 3. Exponential Convexity Using the potential inequality from the previous section we will construct linear functionals that are nonnegative for convex functions. This will enable us to construct new families of exponentially convex functions by applying a method from []. Let us define linear functionals A 1 = A 1;f,N,,x and A = A ;f,n,,µ with A 1 Φ = 1 M N[ f ϕ Nf 1 ]x 1 M ϕ N[ f 1 ]x Nfx Φ 1 Nfx + Φ ϕ M A Φ = 1 fx ϕ Nfx M ˆN µdx 1 M ϕ N[ f 1 ]x Nfx µdx Φ 1 M Nfx µdx + Φµ ϕµ. Linear functionals A k, k = 1,, depend on function f, kernel N, measure µ and points x and, but if these choices are clear from context, we will omit them from the notation. Similarly, we define linear functionals A 3 = A 3;f,N,x and A 4 = A 4;f,N,µ with A 3 Φ = 1 M N[ f ϕ Nf 1 ]x 1 M ϕ0+ N[ f 1 ]x Nfx Φ 1 Nfx + Φ0+, M fx ϕ Nfx ˆN µdx M ϕ0+ N[ f ]x Nfx µdx Φ 1 Nfx µdx + Φ0+µ. M A 4 Φ = 1 M 1
116 N. ELEZOVIĆ, J. PEČARIĆ AND M. PRALJAK We will first give mean value theorems for the linear functionals A k, k = 1,..., 4. Theorem 3.1. Let k {1,..., 4} and let there exist a constant b such that Nfx b for every x. Furthermore, define the constant a as: i a = for k = 1 and k =, where > 0 is the number from the definition of the linear functionals A 1 and A, respectively ii a = 0 for k = 3 and k = 4. If Ψ C [a, b], then there exists ξ k [a, b] such that where Φ is given by.3. Proof. Let A k Ψ = Ψ ξ k A k Φ, m = min τ [a,b] Ψ τ and M = max τ [a,b] Ψ τ. The function MΦ Ψ is convex since d dτ M τ Ψτ = M Ψ τ 0. From the statement and proofs of the inequalities from Theorem. and Corollaries.3 and.7 it is clear that they are meaningful for convex functions defined on the interval [a, b]. Since, by the assumptions of the theorem, the convex function MΦ Ψ satisfies the assumptions of Theorem. for k = 1, Corollaries.3 for k = and.7 for k = 3 or 4 we have 0 A k MΦ Ψ, k = 1,..., 4, i. e. 3.1 A k Ψ MA k Φ, k = 1,..., 4. Similarly, the inequality 3. ma k Φ A k Ψ, k = 1,..., 4 holds since Ψ mφ is convex. Since Φ is a convex function we have A k Φ 0. If A k Φ = 0, then A k Ψ = 0 and for ξ k we can take any point in [a, b]. If A k Φ > 0, then from from 3.1 and 3. we conclude m A kψ A k Φ M, so the existence of ξ k, k = 1,..., 4, follows from continuity of Ψ.
ON POTENTIAL INEQUALITY 117 Theorem 3.. Let k {1,...4} and let N, f, a and b be as in Theorem 3.1. If Ψ, Ψ C [a, b] and A k Φ 0, then there exists ξ k [a, b] such that 3.3 Ψ ξ k Ψ ξ k = A kψ A k Ψ. Proof. Let us define the function φ by φτ = ΨτA k Ψ ΨτA k Ψ. The function φ satisfies the assumptions of Theorem 3.1 and, hence, there exists ξ k [a, b] such that A k φ = φ ξ k A k Φ. Since A k φ = 0 and A k Φ 0, it follows that 0 = φ ξ k = Ψ ξ k A k Ψ Ψ ξ k A k Ψ, so equality 3.3 follows. We will continue this section with few basic notions and results on exponential convexity that will be used here. Definition 3.3. A function ψ : I R is n-exponentially convex in the Jensen sense on I if n xi + x j ξ i ξ j ψ 0 i,j=1 holds for all choices ξ i R and x i I, i = 1,..., n. A function ψ : I R is n-exponentially convex if it is n-exponentially convex in the Jensen sense and continuous on I. Definition 3.4. A function ψ : I R is exponentially convex in the Jensen sense on I if it is n-exponentially convex in the Jensen sense for every n N. A function ψ : I R is exponentially convex if it is exponentially convex in the Jensen sense and continuous on I. Definition of positive semi-definite matrices and some basic algebra gives us the following proposition Proposition 3.5. If ψ is an n-exponentially convex in the Jensen sense [ ] k xi+x on I, then for every choice of x i I, i = 1,..., n, the matrix ψ j i,j=1 is a positive semi-definite matrix for all k N, k n. In particular, for all [ ] k xi+x k N, det ψ j 0 for all k n. i,j=1 Remark 3.6. 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 -exponentially convex in the Jensen
118 N. ELEZOVIĆ, J. PEČARIĆ AND M. PRALJAK sense. Moreover, a function is log-convex if and only if it is -exponentially convex. We will also make use of the divided differences. Definition 3.7. The second order divided differences of a function Ψ : I R, where I is an interval in R, at mutually different points τ 0, τ 1, τ I is defined recursively by 3.4 [τ i, ; Ψ] = Ψτ i, i = 0, 1,, [τ i, τ i+1 ; Ψ] = Ψτ i+1 Ψτ i τ i+1 τ i, i = 0, 1, [τ 0, τ 1, τ ; Ψ] = [τ 1, τ ; Ψ] [τ 0, τ 1 ; Ψ] τ τ 0. Remark 3.8. The value [τ 0, τ 1, τ ; Ψ] is independent of the order of the points τ 0, τ 1, τ. This definition may be extended to include the case in which some or all of the points coincide by taking limits. If Ψ exists, then by taking the limit τ 1 τ 0 in 3.4 we get lim [τ 0, τ 1, τ ; Ψ] = [τ 0, τ 0, τ ; Ψ] = Ψτ Ψτ 0 Ψ τ 0 τ τ 0 τ 1 τ 0 τ τ 0, τ τ 0. Furthermore, if Ψ exists, then by taking the limits τ i τ 0, i = 1, in 3.4 we get lim lim [τ 0, τ 1, τ ; Ψ] = [τ 0, τ 0, τ 0 ; Ψ] = Ψ τ 0. τ τ 0 τ 1 τ 0 Notice that Ψ [τ 0, τ 1, τ ; Ψ] is a linear functional that is nonnegative for a convex function Ψ. The following theorem will enable us to construct families of n-exponentially and exponentially convex functions by applying the linear functionals A k on a family of functions with the same property. Theorem 3.9. Let Ω = {Ψ p : p J}, where J is an interval in R, be a family of functions Ψ p : 0, + R such that the function p [τ 0, τ 1, τ ; Ψ p ] is n-exponentially convex in the Jensen sense on J for every three mutually different points τ 0, τ 1, τ 0, +. Then: i for k = 1 or, the mapping p A k Ψ p is an n-exponentially convex function in the Jensen sense on J. If the function p A k Ψ p is continuous on J, then it is n-exponentially convex on J. ii if Ψ p 0+ is finite for every p J, then the same conclusions as in i hold for k = 3 and 4. Proof. For ξ i R and p i J, i = 1,..., n, we define the function n Ψτ = ξ i ξ j Ψ p i +p j τ. i,j=1
ON POTENTIAL INEQUALITY 119 Due to the linearity of the divided differences and the assumption that the function p [τ 0, τ 1, τ ; Ψ p ] is n-exponentially convex in the Jensen sense we have [τ 0, τ 1, τ ; Ψ] = n i,j=1 ξ i ξ j [τ 0, τ 1, τ ; Ψ p i +p j ] 0. This implies that Ψ is a convex functions and, due to the assumptions, it satisfies the assumptions of Theorem. for k = 1 or and Corollary.7 for k = 3 or 4. Hence, by the potential inequality, 0 A k Ψ = n i,j=1 ξ i ξ j A k Ψ pi +p j, k = 1,..., 4 so the function p A k Ψ p is n-exponentially convex in the Jensen sense on J. If it is also continuous on J, then it is n-exponentially convex by definition. Corollary 3.10. Let Ω = {Ψ p : p J}, where J is an interval in R, be a family of functions Ψ p : 0, + R such that the function p [τ 0, τ 1, τ ; Ψ p ] is -exponentially convex in the Jensen sense on J for every three mutually different points τ 0, τ 1, τ 0, +. Then for k = 1 and k = the following statements hold: i If the function p A k Ψ p is continuous on J, then it is - exponentially convex and, thus, log-convex. ii If the function p A k Ψ p is strictly positive and differentiable on J, then for every p, q, r, s J, such that p r and q s, we have where 3.5 µ k p,qω = exp for Ψ p, Ψ q Ω. µ k p,qω µ k r,sω, Ak Ψ p A k Ψ q 1 p q d dp A kψ p A k Ψ p, p q, p = q If, additionally, Ψ p 0+ is finite for every p J, then statements i and ii hold for k = 3 and 4 as well. Proof. i This is an immediate consequence of Theorem 3.9 and Remark 3.6 ii y i, the function p A k Ψ p is log-convex on J, that is, the function p log A k Ψ p is convex. Therefore 3.6 log A k Ψ p log A k Ψ q p q log A kψ r log A k Ψ s r s
10 N. ELEZOVIĆ, J. PEČARIĆ AND M. PRALJAK for p r, q s, p r, q s, which implies that µ k p,q Ω µk r,s Ω, k = 1,..., 4. The cases p = r and q = s follow from 3.6 by taking limits p r or q s. Remark 3.11. The results from Theorem 3.9 Corollary 3.10 still hold when two of the points τ 0, τ 1, τ 0, + coincide, say τ 0 = τ 1, for a family of differentiable functions Ψ p such that the function p [τ 0, τ 0, τ ; Ψ p ] is n-exponentially convex in the Jensen sense -exponentially convex in the Jensen sense and, furthermore, they still hold when all three points coincide for a family of twice differentiable functions with the same property. We will end this sections with several examples of families of functions that satisfy the assumptions of Theorem 3.9 and Corollary 3.10 and Remark 3.11, which, as a consequence, gives us large families of exponentially convex functions. Example 3.1. Consider a family of functions Ω k 1 = {Φ p : p J k }, where Φ p are the power functions defined by.3 and J 1 = J = R and J 3 = J 4 = 1, +. We have d dτ Φ p τ = τ p > 0 which shows that Φ p are convex on 0, +. Similarly as in the proof of Theorem 3.9, let us, for ξ i R and p i J k, i = 1,..., n, define the function Φτ = n i,j=1 ξ i ξ j Φ p i +p j τ. Since the function p d dτ Φ p τ = τ p = e p ln τ is exponentially convex by definition, it follows that Φ τ = n i,j=1 is a convex function. Therefore n ξ i ξ j Φ p i +p j τ = ξ i e pi ln τ 0 i=1 0 [τ 0, τ 1, τ ; Φ] = n i,j=1 ξ i ξ j [τ 0, τ 1, τ ; Φ p i +p j ], so p [τ 0, τ 1, τ ; Ψ p ] is n-exponentially convex in the Jensen sense. Now, by Theorem 3.9, it follows that the mappings p A k Φ p are exponentially convex in the Jensen sense. It is straightforward to check that these mappings are continuous, so they are exponentially convex.
ON POTENTIAL INEQUALITY 11 For these families of functions, µ k p,q Ωk 1 from 3.5, for k = 1 and k =, are equal to 1 Ak Φ p p q A k Φ q, p q µ k p,q Ωk 1 = exp 1 p pp 1 A kφ 0Φ p A k Φ p, p = q 0, 1 exp 1 A kφ 0Φ 1 A k Φ 1, p = q = 1 exp 1 A kφ 0 A k Φ 0, p = q = 0 while for k = 3 and k = 4 they have the same form, but are only defined for p, q > 1. Furthermore, if a linear functional A k and the functions Ψ = Φ p and Ψ = Φ q are such that the assumptions of Theorem 3. are satisfied, we can define a two-parameter family of means. Indeed, since Ψ/ Ψ 1 τ = τ 1/p q, the number E k p,qω k 1 = µ k p,qω k 1 satisfies 0 E k p,q Ωk 1 K. Example 3.13. Let Ω = {Ψ p : p R} be a family of functions defined by { 1 Ψ p τ = p e px, p 0, 1 τ, p = 0. We have d dτ Ψ p τ = e pτ > 0 which shows that Ψ p are convex and p d dτ Ψ p τ is exponentially convex by definition. Arguing as in Example 3.1 we get that the mappings p A k Ψ p, k = 1,..., 4, are exponentially convex. In this case, the functions 3.5 are equal to 1 Ak Ψ p p q A k Ψ q, p q µ k p,q Ω = Ak id Ψ exp p A k Ψ p p, p = q 0 exp Ak id Ψ 0, p = q = 0, 3A k Ψ 0 where idτ = τ is the identity function. Again, if a linear functional A k and the functions Ψ = Ψ p and Ψ = Ψ q are such that the assumptions of Theorem 3. are satisfied, then, since Ψ/ Ψ 1 τ = lnτ/p q, we have so E k p,q Ω are means. 0 E k p,qω = ln µ k p,qω K, Example 3.14. Consider a family of functions Ω 3 = {Ψ p : p 0, + } given by { p τ Ψ p τ = ln τ, p 1, 1 τ, p = 1.
1 N. ELEZOVIĆ, J. PEČARIĆ AND M. PRALJAK Since d dτ Ψ p τ = p τ is the Laplace transform of a nonnegative function see [5], it is exponentially convex. Arguing as in Example 3.1 we get that p A k Ψ p, k = 1,..., 4, are exponentially convex functions. For this family of functions, µ k p,qω 3 from 3.5 becomes Ak Ψ p A k Ψ q µ k p,qω 3 = exp exp 1 p q, p q A kid Ψ p pa k Ψ p p ln p A kid Ψ 1 3A k Ψ 1, p = q 1, p = q = 1. If the assumptions of Theorem 3. are satisfied for the linear functional A k and the functions Ψ = Ψ p and Ψ = Ψ q, then E k p,q Ω 3 = Lp, q ln µ k p,q Ω 3 satisfies 0 E k p,q Ω 3 K, i. e. E k p,q Ω 3 is a mean. Here, Lp, q is the logarithmic mean defined by Lp, q = p q/ln p ln q, p q, Lp, p = p. Example 3.15. Consider a family of functions Ω 4 = {Ψ p : p 0, + } given by p Ψ p τ = e τ. p Since d dτ Ψ p τ = e τ p is the Laplace transform of a nonnegative function see [5], it is exponentially convex. Arguing as before, we get that p A k Ψ p, k = 1,..., 4, are exponentially convex functions. For this family of functions, µ k p,qω 4 from 3.5 becomes 1 Ak Ψ p p q µ k p,q Ω A 3 = k Ψ q, p q exp A kid Ψ p pa k Ψ 1 p p, p = q. If the assumptions of Theorem 3. are satisfied for the linear functional A k and the functions Ψ = Ψ p and Ψ = Ψ q, then E k p,q Ω 4 = p + q ln µ k p,q Ω 4 satisfies 0 E k p,qω 4 K, which shows that E k p,qω 4 is a mean. References [1] N. Eleović, J. Pečarić and M. Praljak, Potential inequality revisited, I: General case, Math. Inequal. Appl. 1 01, 787 810. [] J. Pečarić and J. Perić, Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type means, An. Univ. Craiova Ser. Mat. Inform. 39 01, 65 75. [3] M. Rao, rownian Motion and Classical Potential Theory, Lecture Notes Series No. 47, Aarhus University, 1977. [4] M. Rao and H. Šikić, Potential inequality, Israel J. Math. 83 1993, 97 17. [5] J. L. Schiff, The Laplace transform. Theory and applications, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1999.
ON POTENTIAL INEQUALITY 13 O potencijalnoj nejednakosti a apsolutne vrijednosti funkcija Neven Eleović, Josip Pečarić i Marjan Praljak Sažetak. Potencijalna nejednakost ivedena je u [4] i kasnije proširena na opću klasu konveksnih i konkavnih funkcija u [1]. Glavni cilj ovog članka je ivesti potencijalnu nejednakost u slučaju kada funkciju u kojoj se računa potencijal amijenimo s apsolutnom vrijednošću te funkcije. Dobiveni reultati, ajedno s metodama i [], koriste se pri konstrukciji novih klasa eksponencijalno konveksnih funkcija. Neven Eleović Faculty of Electrical Engineering and Computing University of Zagreb Unska 3, 10000 Zagreb, Croatia E-mail: neven.ele@fer.hr Josip Pečarić Faculty of Textile Technology University of Zagreb Prila baruna Filipovića 8a, 10000 Zagreb, Croatia E-mail: pecaric@element.hr Marjan Praljak Faculty of Food Technology and iotechnology University of Zagreb Pierottijeva 6, 10000 Zagreb, Croatia E-mail: mpraljak@pbf.hr Received: 15.7.01.
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