POTENTIAL INEQUALITY WITH HILBERT TYPE KERNELS

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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N. MATEMATICĂ, Tomul LXII, 06, f. POTENTIAL INEQUALITY WITH HILBERT TYPE KERNELS BY NEVEN ELEZOVIĆ, JOSIP PEČARIĆ and MARJAN PRALJAK Abstract. We generalize to the n-dimensional case the set of sufficient conditions on the kernel under which the maximum principle and the potential inequality hold, given by Rao and Šikić in the -dimensional case. These conditions are satisfied for Hilbert-type kernels and we are able to construct new families of exponentially convex functions. Mathematics Subject Classification 00: 6D5. Key words: potential inequality, exponential convexity.. Introduction π The famous Hilbert inequality states Hf p sin(π/p f p, where f is a nonnegative function and H is the classical Hilbert operator ( (Hf(x = + 0 f(y x+y dy. The inequality was proven by Hilbert for p = and by Riesz for p >, and the constant π/ sin(π/p is the best possible (see []. Boundedness of the Hilbert operator, but without the best constant, can be obtained as a consequence of a more general approach via potential inequality. Potential inequality, introduced by Rao and Šikić in [4], is a very general inequality that holds for kernels that satisfy the maximum principle (see below and which lies behind many classical inequalities. Among other applications, they gave a set of sufficient conditions on a kernel in the

2 90 NEVEN ELEZOVIĆ, JOSIP PEČARIĆ and MARJAN PRALJAK -dimensional case under which the maximum principle, and, hence, the potential inequality, is satisfied. These conditions are satisfied by Hilbert-type kernels N with the densities (for positive α and β { (x α +y α β, if x y, G(x,y = 0, if 0 < y < x and their dual kernels Ñ with the densities G(x,y = G(y,x. Notice that the Hilbert operator is equal to H = N +Ñ, with α = β =. Integral form of the potential inequality gives, for nonnegative f and p > (see below for definition of the measure ˆNµ (Nf p dµ p β(p f(nf p d( ˆNµ. In the special case when µ(dx = dx is the Lebesgue measure and αβ =, it holds ˆNµ = Kdx, where ( K = 0 0 (z α + β dz is a finite constant, and application of Hölder s inequality gives the boundedness of the integral operator N in the L p norm, i.e. Nf p C p f p, where C p = pk β(p. We cannot prove the boundedness of the dual operator Ñ directly as before, since the corresponding integral in ( would diverge. But, since bounded dual operators in L p spaces have the same norm (see [5], pg. 7, we have Ñf p C q f p, where q = p/(p is the dual exponent of p. Therefore, by the triangle inequality, Hf p (C p +C q f p = (p p + q q log( f p, i. e., the general approach via potential inequality gives boundedness of the classical Hilbert operator, but not the best constant. The goal of this paper is to build on the earlier results in two ways. Firstly, we will generalize to the n-dimensional case the set of sufficient conditions on a kernel under which the maximum principle and the potential inequality hold given by Rao and Šikić in the -dimensional case. These conditions are satisfied for the n-dimensional generalization N of the classical Hilbert kernel given by the densities (for positive α and β G(x,y = ( x α + y α β, for x y 0, otherwise,

3 3 POTENTIAL INEQUALITY WITH HILBERT TYPE KERNELS 9 as well as for its dual kernel Ñ given by the densities G(x,y = G(y,x. Secondly, we will use potential inequality proven by Elezović, Pečarić and Praljak in [] that generalized the inequality of Rao and Šikić [4] to the class of naturally defined convex and concave functions on (0, +. This will allow us to construct linear functionals nonnegative on convex functions which, together with the methods from [3] that make use of the divided differences, will enable us to generate many families of exponentially convex functions. As a consequence, we will refine some of the known inequalities and derive new ones. We will start off by introducing notation and the setup. We say that N(x,dy is a(positive kernel onx if N : X B(X [0,+ ] isamapping such that, for every x X, A N(x,A is a σ-finite measure, and, for every A B(X, x N(x,A is a measurable function. For a measurable function f, the potential of f with respect to N at a point x X is (Nf(x = X f(yn(x,dy, 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 (X,B(X and a measurable set C B(X we will denote by ˆN C µ the measure defined by ( ˆN C µ(dy = C N(x,dyµ(dx. If C = X we will omit the subscript, i. e. ˆNµ will denote the measure ˆNX µ. Definition. Let N be a positive kernel on X and R POT (N. We say that N satisfies the strong maximum principle on R (with constant M if (3 (Nf(x Mu+N[f + {(Nf u} ](x holds for every x X, f R and u 0. The main result from [] is the following theorem Theorem (The potential inequality for convex functions. Let Φ : (0,+ R be a convex function and N(x,dy a positive kernel on X which satisfies the strong maximum principle on R with constant M. Let f R, x X and z > 0 be such that z (Nf(x/M and denote by B z

4 9 NEVEN ELEZOVIĆ, JOSIP PEČARIĆ and MARJAN PRALJAK 4 the set B z = {y X : (Nf(y z}. Then Φ ( M (Nf(x Φ(z M N[f+ ϕ(nf Bz ](x + M ϕ(zn[f f+ Bz ](x zϕ(z. We shall also need the following result, which is a limiting case of the potential inequality (here B = z>0 B z = {x X : (Nf(x > 0} Theorem ([], Theorem 5. Under the assumptions of Theorem, if f is nonnegative and lim z 0 zϕ(z = 0, then Φ( M (Nf(x Φ(0+ M N[fϕ(Nf B](x. Furthermore, for a finite measure µ on (X, B(X, the following inequality holds BΦ( M (Nf(xµ(dx Φ(0+µ(B f(xϕ((nf(x( M ˆN B µ(dx. B. Hilbert-type kernels and the maximum principle In this section we will give a set of sufficient conditions on the density of a kernel under which the maximum principle holds. Theorem 3. Let X = R n and let N be a kernel with the density G : X X [0,+ that satisfies the following three conditions: (i G(x,y = 0 for x < y ; (ii for every y,x,x X such that y x x, G(x,y MG(x,y; (iii for every bounded, measurable function f : X [0,+, the mapping x (Nf(x is continuous. Then, the kernel N satisfies the maximum principle with constant M on the set of nonnegative functions f : X [0,+. Proof. Inequality (3 from the definition of the maximum principle is trivially satisfied if u = 0, or if (Nf(x u.

5 5 POTENTIAL INEQUALITY WITH HILBERT TYPE KERNELS 93 Let f satisfy the assumptions from (iii and let x and u be such that (Nf(x > u > 0. The set C = {y X : (Nf(y u} is closed since the mapping y (Nf(y is continuous and let a C B n (0, x be a point on the boundary of the set C with the greatest norm within the ball B n (0, x, i. e. a = sup{ y : y C B n (0, x }. The point a is well defined. Indeed, the set C B n (0, x is nonempty (since (Nf(x > u and compact (because it s closed and bounded, so the supremum is attained. Notice that (Nf(a = u, since a is on the boundary of the set C and the mapping y (Nf(y is continuous. Moreover, (Nf(x = f(yg(x,ydy + f(yg(x, ydy, B n(0, x \B n(0, a B n(0, a and, by condition (ii, we have f(yg(x,ydy B n(0, a B n(0, a Due to the definition of a, we also have f(yg(x, ydy B n(0, x \B n(0, a = B n(0, x \B n(0, a B n(0, x f(ymg(a,ydy = M(Nf(a = Mu. f(yg(x,y {(Nf(y u} dy f(yg(x,y {(Nf(y u} dy. The last two inequalities imply that inequality (3 is satisfied for every function f that satisfies the assumptions from condition (iii. Since for every nonnegative, measurable function f there exists a sequence f n of measurable, bounded functions such that f n ր f, it follows that inequality (3 is satisfied for f as well. The following theorem, which is proved in a similar way, gives analogous result for the kernels dual to the ones from Theorem 3. Theorem 4. Let X = R n and let N be a kernel with the density G : X X [0,+ that satisfies the following three conditions: (i G(x,y = 0 for x y ;

6 94 NEVEN ELEZOVIĆ, JOSIP PEČARIĆ and MARJAN PRALJAK 6 (ii for every y,x,x X such that x x < y, G(x,y MG(x,y; (iii for every bounded, measurable function f : X [0,+ with bounded support, the mapping x (Nf(x is continuous. Then, the kernel N satisfies the maximum principle with constant M on the set of nonnegative functions f : X [0,+. Now, for positive numbers α and β, let N(x,dy = G(x,ydy be the kernel on R n given by its density (4 G(x,y = ( x α + y α β, for x y 0, otherwise, and let Ñ be its dual kernel, i. e. the kernel given by the density G(x,y = G(y,x. Notice that the kernel of the classical Hilbert operator ( is equal to H = N +Ñ, with n = α = β =,X = [0,+, so the kernels N and Ñ can be interpreted as the n-dimensional generalizations of the classical Hilbert kernel on the upper-left and lower-right triangles of the plane. For x x y we have G(x,y G(x,y = ( x α + y α β ( y α β x α + y α y α = β, so the kernel N satisfies the assumptions of Theorem 4 with constant M = β. Similarly, for y x x we have G(x,y G(x,y = ( x α + y α β x α + y α, sothekernelñ satisfiestheassumptionsoftheorem3withconstantm =. 3. Exponential convexity The Hilbert-type kernels N and Ñ introduced at the end of the last section satisfy the maximum principle, so by applying the potential inequality we can construct linear functionals nonnegative on the set of convex functions. This will enable us to construct new families of exponentially convex functions by applying the methods from [3].

7 7 POTENTIAL INEQUALITY WITH HILBERT TYPE KERNELS 95 Before writing down the functionals, notice that for the kernel N given by its density (4 we have B = {x R n : (Nf(x > 0} = B n (0,b c, where b = inf{b : f(y = 0 for almost every y B n (0,b c } and that f = 0 a.e. on the set B c. Therefore, we can replace the integral over the set B on the right hand side of the second inequality from Theorem with the integral over R n. Furthermore, with the convention Φ( β (Nf(x = Φ(0+ for x B c, by adding and subtracting Φ(0+µ(B c to the left hand side of the second inequality from Theorem we get R n Φ( β(nf(xµ(dx Φ(0+µ(Rn β R n f(xϕ((nf(x( ˆN B µ(dx. Let us now define linear functionals A = A ;f,n,x and A = A ;f,n,µ with A (Φ = ( f(yϕ (Nf(y β B n(0, x c ( x α + y α β dy Φ( β(nf(x +Φ(0+ A (Φ = Rn β f(xϕ ( (Nf(x ( µ(dy B n(0, x ( y α + x α β dx Φ ( R n β(nf(x µ(dx+φ(0+µ(r n. Linear functionals A k, k =,, depend on the choices of function f, measure µ, point x and numbers α,β and n, but if these choices are clear from context, we will omit them from notation. Similarly, by taking the kernel Ñ instead of N and constant M = instead of / β, we define linear functionals Ãk, k =,, with à (Φ = B n(0, x Rn à (Φ = f(xϕ ( (Ñf(x( f(yϕ ( (Ñf(y ( x α + y α β dy Φ( (Ñf(x +Φ(0+ B n(0, x c Φ ( (Ñf(x µ(dx+φ(0+µ(r n. R n µ(dy ( y α + x α β dx The Lagrange and Cauchy type mean value theorems for the linear functionals introduced in this section were given in [] (Theorems 33 and 34. Since we will need the results later, we will restate them here, but with the notation from this section. In what follows, Φ denotes the function Φ (τ = τ /.

8 96 NEVEN ELEZOVIĆ, JOSIP PEČARIĆ and MARJAN PRALJAK 8 Theorem 5. Let k {,}, let N be the kernel given by its density (4 and let f be such that the potential Nf is uniformly bounded, i. e. let there exist a constant K R such that (Nf(x K for every x R n. If Ψ C (0,K] with A k (Ψ finite, lim z 0 zϕ(z = 0 and A k (Φ 0, then there exists ξ k [0,K] (assuming that Ψ (0 = lim z 0 Ψ (z exists when ξ k = 0 such that A k (Ψ = Ψ (ξ k A k (Φ. The claim still holds when the kernel N is replaced with Ñ and the linear functional A k with Ãk respectively. Theorem 6. Let k {,}, let N be the kernel given by its density (4 and let there exist K R such that (Nf(x K for every x X. If Ψ and Ψ satisfy the assumptions of Theorem 5 and if A k (Φ 0 and A k ( Ψ 0, then there exists ξ k [0,K] such that (5 Ψ (ξ k Ψ (ξ k = A k(ψ A k ( Ψ. The claim still holds when the kernel N is replaced with functional A k with Ãk respectively. Ñ and the linear We will continue this section with few basic notions and results on exponential convexity that will be used here. Definition. A function ψ : I R is n-exponentially convex in the Jensen sense on I if n i,j= ξ iξ j ψ( x i+x j 0 holds for all choices ξ i R and x i I, i =,...,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. 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 give us the following proposition Proposition 7. If ψ is an n-exponentially convex in the Jensen sense on I, then for every choice of x i I, i =,...,n, the matrix [ψ( x i+x j ] k i,j= is a positive semi-definite matrix for all k N, k n. In particular, for all k N, det[ψ( x i+x j ] k i,j= 0, for all k n.

9 9 POTENTIAL INEQUALITY WITH HILBERT TYPE KERNELS 97 Remark 8. Itis known that ψ : I R is log-convex in thejensen sense if and only if α ψ(x +αβψ( x+y +β ψ(y 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 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 4. 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,τ I is defined recursively by (6 [τ i,;ψ] = Ψ(τ i, i = 0,,, [τ i,τ i+ ;Ψ] = Ψ(τ i+ Ψ(τ i τ i+ τ i, i = 0,, [τ 0,τ,τ ;Ψ] = [τ,τ ;Ψ] [τ 0,τ ;Ψ] τ τ 0. Remark 9. The value [τ 0,τ,τ ;Ψ] is independent of the order of the points τ 0,τ,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 τ 0 in (7 we get lim [τ 0,τ,τ ;Ψ]=[τ 0,τ 0,τ ;Ψ]= Ψ(τ Ψ(τ 0 Ψ (τ 0 (τ τ 0 τ τ 0 (τ τ 0,τ τ 0. Furthermore, if Ψ exists, then by taking the limits τ i τ 0, i =, in (7 we get lim τ τ 0 lim τ τ 0 [τ 0,τ,τ ;Ψ] = [τ 0,τ 0,τ 0 ;Ψ] = Ψ (τ 0. Notice that Ψ [τ 0,τ,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 and Ãk on a family of functions with the same property. Theorem 0. Let Ω = {Ψ p : p J}, where J is an interval in R, be a family of differentiable functions Ψ p : (0,+ R such that lim τ 0 τψ p(τ = 0 and that the function p [τ 0,τ,τ ;Ψ p ] is n-exponentially convex in the Jensen sense on J for every three mutually different points τ 0,τ,τ (0,+. Then, for k = and, the mapping p A k (Ψ p

10 98 NEVEN ELEZOVIĆ, JOSIP PEČARIĆ and MARJAN PRALJAK 0 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. The same claims hold for the mappings p Ãk(Ψ p, k =,. Proof. For ξ i Randp i J, i =,...,n, wedefinethefunctionψ(τ = n i,j= ξ iξ j Ψp i +p j (τ. Due to the linearity of the divided differences and the assumptionthatthefunctionp [τ 0,τ,τ ;Ψ p ]isn-exponentially convex in the Jensen sense we have [τ 0,τ,τ ;Ψ] = n i,j= ξ iξ j [τ 0,τ,τ ;Ψp i +p j ] 0. This implies that Ψ is a convex function and, due to the assumptions, it satisfies the assumptions of Theorem. Hence, by the potential inequality, 0 A k (Ψ = n i,j= ξ ( iξ j A k Ψpi +p j, k =,...,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. By applying the potential inequality to the linear functionals Ãk, we get the same claim for the mappings p Ãk(Ψ p. Corollary. Let Ω = {Ψ p : p J}, where J is an interval in R, be a family of differentiable functions Ψ p : (0,+ R such that lim τ 0 τψ p (τ = 0 and that the function p [τ 0,τ,τ ;Ψ p ] is -exponentially convex in the Jensen sense on J for every three mutually different points τ 0,τ,τ (0,+. Then, for k = and, 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 µ k p,q (Ω µk r,s (Ω, where ( Ak (Ψ p (7 µ k A k (Ψ q p,q(ω = ( d exp p q, p q dp A k(ψ p A k (Ψ p, p = q, for Ψ p,ψ q Ω. The same claims hold for the mappings p Ãk(Ψ p, k =,. Proof. (i This is an immediate consequence of Theorem 0 and Remark 8.

11 POTENTIAL INEQUALITY WITH HILBERT TYPE KERNELS 99 (ii By (i, the function p A k (Ψ p is log-convex on J, that is, the function p loga k (Ψ p is convex. Therefore (8 loga k (Ψ p loga k (Ψ q p q loga k(ψ r loga k (Ψ s, r s for p r, q s, p r, q s, which implies that µ k p,q(ω µ k r,s(ω, k =,...,4. The cases p = r and q = s follow from (8 by taking limits p r or q s. Remark. The results from Theorem 0 (Corollary still hold whentwo of thepoints τ 0,τ,τ (0,+ coincide, say τ 0 = τ, forafamily 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 0 and Corollary (and Remark, which, as a consequence, gives us large families of exponentially convex functions. Example. Consider a family of functions Ω = {Φ p : p (0,+ } given by τ p Φ p (τ = p(p, p, τ logτ, p =. We have d Φ dτ p (τ = τ p > 0 which shows that Φ p are convex on (0,+. Similarly as in the proof of Theorem 0, let us, for ξ i R and p i J k, i =,...,n, define the function Φ(τ = n i,j= ξ iξ j Φp i +p j (τ. Since the function p d dτ Φ p (τ = τ p = e (p lnτ is exponentially convex (by definition, it follows that Φ (τ = n i,j= ξ i ξ j Φ p i +p j (τ = ( n ξ i e (pi lnτ 0 isaconvexfunction. Therefore0 [τ 0,τ,τ ;Φ]= n i,j= ξ iξ j [τ 0,τ,τ ;Φp i +p j ], so p [τ 0,τ,τ ;Ψ p ] is n-exponentially convex in the Jensen sense. Now, i=

12 00 NEVEN ELEZOVIĆ, JOSIP PEČARIĆ and MARJAN PRALJAK by Theorem 0, it follows that the mappings p A k (Φ p and p Ãk(Φ p are exponentially convex in the Jensen sense. It is straightforward to check that these mappings are continuous, so they are exponentially convex. For these families of functions, µ k p,q (Ω from (7, for k = and, are equal to ( Ak (Φ p p q, p q, A k (Φ q ( p exp µ k p,q (Ω p(p A k(φ 0 Φ p A k (Φ p, p = q 0,, = ( exp A k(φ 0 Φ, p = q =, A k (Φ ( exp A k(φ 0, p = q = 0. A k (Φ 0 Furthermore, ifalinearfunctionala k andthefunctionsψ = Φ p and Ψ = Φ q are such that the assumptions of Theorem 6 are satisfied, we can define a two-parameter family of means. Indeed, since (Ψ/ Ψ (τ = τ /(p q, the number E k p,q(ω = µ k p,q(ω satisfies 0 E k p,q(ω K. We can construct the means Ẽk p,q (Ω corresponding to the linear functionals Ãk, k =,, in the same way. Remark 3. We have seen in Example that the mapping p ψ(p = A (Φ p is exponentially convex. From this fact we can show that the classical Hilbert operator is bounded. For p > 0, p, ψ(p = (p β p(p pβ ( f(x(nf p (x R n (Nf p (xµ(dx. R n B n(0, x µ(dy ( y α + x α β dx Due to the log-convexity of ψ, for p,s,t (0,+, p, p < s < t or s < t < p, the following inequality holds (9 [ ]t p [ ψ(s t s ψ(t ]p s t s ( (p β f(x(nf p (x R n p(p pβ (Nf p (xµ(dx. R n B n(0, x µ(dy ( y α + x α β dx

13 3 POTENTIAL INEQUALITY WITH HILBERT TYPE KERNELS 0 The potential inequality from Theorem holds for a σ-finite measure µ as long as Φ(0+ = 0 and the integrals are finite (see Remark 7 in [], so we can take the Lebesgue measure µ(dy = dy in the definition of the linear functional A. When α and β are such that αβ = n, a simple calculation shows that the integral ( y α + x α β dy B n(0, x doesn t depend on x R n and is equal to (0 K = ω n 0 r n (r α + β dr, where ω n is the surface of the unit sphere S n in R n. Plugging this into (9 we get the inequality [ ]t p [ ψ(s t s ψ(t ]p s K t s (p β f(x(nf p (xdx R n p(p pβ (Nf p (xµ(dx. R n Applying Hölder s inequality on the first integral of the right hand side and multiplying by pβ [ X (Nfp (xdx] /q we get ( pβ[ ψ(s ]t p[ [ t s ψ(t ]p s t s (Nf p (xdx R n [ ] Kβ(p p f p (xdx p R n ] q [ ] p (Nf p (xdx. p(p R n The left hand side of the last inequality is nonnegative, so multiplying both sides of ( by p(p and rearranging we get that for p > the following inequality holds ( Nf p C p f p, where C p = pk β(p. The last inequality ( was given by Rao and Šikić in [4], while the previous inequality ( is its refinement derived from the exponentially convex functions we constructed in this paper.

14 0 NEVEN ELEZOVIĆ, JOSIP PEČARIĆ and MARJAN PRALJAK 4 We cannot derive a similar bound for the dual operator Ñ, since the analogous integralasin(0wouldgofromto+ andwouldbedivergent. But, wecan usethefact that Ñ is thedualof N, so Ñf q C p f q, where q = p/(p is the conjugate exponent of p. Therefore, the integral operator H = N +Ñ is bounded since Hf p Nf p + Ñp p ( C p +C q f p. Example. Let Ω = {Ψ p : p R} be a family of functions defined by p Ψ p (τ = epτ, p 0, τ, p = 0. We have d dτ Ψ p (τ = e pτ > 0 which shows that Ψ p are convex and p d Ψ dτ p (τ is exponentially convex (by definition. Arguing as in Example we get that the mappings p A k (Ψ p and p Ãk(Φ p, k =,, are exponentially convex. In this case, the functions (7 are equal to ( Ak (Ψ p p q, p q A k (Ψ q ( µ k Ak (id Ψ p,q(ω = p exp A k (Ψ p p ( Ak (id Ψ 0 exp 3A k (Ψ 0, p = q 0, p = q = 0, where id(τ = 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 6 are satisfied, then, since (Ψ/ Ψ (τ = ln(τ/(p q, we have 0 E k p,q(ω = lnµ k p,q(ω K, so E k p,q(ω are means. We can construct the means Ẽk p,q (Ω corresponding to the linear functionals Ãk, k =,, in the same way. Example 3. Consider a family of functions Ω 3 = {Ψ p : p (0,+ } given by p τ Ψ p (τ = ln τ, p, τ, p =.

15 5 POTENTIAL INEQUALITY WITH HILBERT TYPE KERNELS 03 Since d dτ Ψ p (τ = p τ is the Laplace transform of a nonnegative function (see [6], it is exponentially convex. Arguing as in Example we get that p A k (Ψ p and p Ãk(Φ p, k =,, are exponentially convex functions. For this family of functions, µ k p,q (Ω 3 from (7 becomes ( Ak (Ψ p A k (Ψ q ( µ k p,q (Ω 3 = exp A k(id Ψ p pa k (Ψ p plnp ( exp A k(id Ψ 3A k (Ψ p q, p q,, p = q,, p = q =. Ifthe assumptionsof Theorem 6are satisfied for thelinear functional A k andthefunctions Ψ = Ψ p and Ψ = Ψ q, thene k p,q (Ω 3 = L(p,qlnµ 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, L(p,q is the logarithmicmeandefinedbyl(p,q = (p q/(lnp lnq, p q, L(p,p = p. We can construct the means Ẽk p,q(ω 3 corresponding to the linear functionals Ãk, k =,, in the same way. Example 4. Consider a family of functions Ω 4 = {Ψ p : p (0,+ } given by Ψ p (τ = e τ p p. Since d Ψ dτ p (τ = e τ p is the Laplace transform of a nonnegative function (see [6], it is exponentially convex. Arguing as before, weget that p A k (Ψ p andp Ãk(Φ p, k =,, areexponentially convex functions. For this family of functions, µ k p,q(ω 4 from (7 becomes ( Ak (Ψ p p q, p q, A µ k k (Ψ q p,q(ω 3 = ( exp A k(id Ψ p pa k (Ψ p, p = q. p If the assumptions of Theorem 6 are satisfied for the linear functional A k and the functions Ψ = Ψ p and Ψ = Ψ q, then Ep,q k (Ω 4 = ( p + q lnµ k p,q (Ω 4 satisfies 0 Ep,q(Ω k 4 K, which shows that Ep,q(Ω k 4 is a mean. We can construct the means Ẽk p,q(ω 4 corresponding to the linear functionals Ãk, k =,, in the same way.

16 04 NEVEN ELEZOVIĆ, JOSIP PEČARIĆ and MARJAN PRALJAK 6 REFERENCES. Elezović, N.; Pečarić, J.; Praljak, M. Potential inequality revisited I: General case, Math. Inequal. Appl., 5 (0, Hardy, G.H.; Littlewood, J.E.; Pólya, G. Inequalities, d ed. Cambridge at the University Press, Pečarić, J.; Perić, J. Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type means, An. Univ. Craiova Ser. Mat. Inform., 39 (0, Rao, M.; Šikić, H. Potential inequality, Israel J. Math., 83 (993, Rudin, W. Real and Complex Analysis, Third edition, McGraw-Hill Book Co., New York, Schiff, J.L. The Laplace Transform. Theory and Applications, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 999. Received: 4.VII.0 Revised: 3.I.03 Accepted:.II.03 Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 0000 Zagreb, CROATIA neven.elez@fer.hr Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 8a, 0000 Zagreb, CROATIA pecaric@element.hr Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 0000 Zagreb, CROATIA mpraljak@pbf.hr