Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type means

Transcription

1 Annals of the University of Craiova, Mathematics and Computer Science Series Volume 391, 01, Pages ISSN: Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type means Josip Pečarić and Jurica Perić Abstract. Improvements of the Giaccardi and the Petrović inequality are given. The notion of n onentially convex functions is introduced. An elegant method of producing n onentially convex and onentially convex functions is applied using the Giaccardi and the Petrović differences. Cauchy mean value theorems are proved and shown to be useful in studying Stolarsky type means defined by using the Giaccardi and the Petrović differences. 010 Mathematics Subject Classification. 6D15. Key words and phrases. Giaccardi inequality, Petrović inequality, Cauchy type mean value theorems, n onential convexity, onential convexity, Stolarsky type means. 1. Introduction and preliminaries The Giaccardi inequality states: Theorem 1.1. Let φ be a convex function on an interval I, p a nonnegative n-tuple with n p i = P n 0 and x a real n-tuple. If x I n and x 0 I are such that n p ix i = x I, x x 0 and then where A = x i x 0 x x i 0, i = 1,..., n, n p i φx i Aφ x + B p i 1 φx 0, n p n ix i x 0 n p, B = p ix i n ix i x 0 p. ix i x 0 A simple consequence of the Giaccardi inequality is the Petrović inequality: Corollary 1.1. Let φ be a convex function on [0, a], 0 < a <. Then for every nonnegative n-tuple p and every x [0, a] n such that n p ix i = x 0, a] and p i x i x j, j = 1,..., n, the following inequality holds n p i φx i φ x + p i 1 φ0. For further details on the Giaccardi and the Petrović inequality see [6]. The main goal of this paper is to improve Theorem 1.1 and Corollary 1.1 using the following lemma. Received March 7,

2 66 J. PEČARIĆ AND J. PERIĆ Lemma 1.1. Let φ be a convex function on an interval I, x, y I and p, q [0, 1] such that p + q = 1. Then [ ] x + y min{p, q} φx + φy φ pφx + qφy φpx + qy. 1 Lemma 1.1 is a simple consequence of [5, Theorem 1, p.717]. In Section we also prove Cauchy type mean value theorems, which we use in Section 4 in studying Stolarsky type means defined by the Giaccardi and the Petrović differences. In Section 3 we introduce the notion of n onentially convex functions and deduce an elegant method of producing n onentially convex and onentially convex functions using some known families of functions of the same type.. Improvements of the Giaccardi and the Petrović inequality Next theorem is our main result. Theorem.1. Let φ be a convex function on an interval I, p a nonnegative n-tuple with n p i = P n 0 and x a real n-tuple. If x I n and x 0 I are such that n p ix i = x I, x x 0 and then where A = p i φx i Aφ x x i x 0 x x i 0, i = 1,..., n, n +B p i 1 φx 0 δ φ P n + δ φ x i x 0+ x p i x x 0 n p n ix i x 0 n p, B = p ix i n ix i x 0 p, δ φ = φx 0 + φ x φ ix i x 0, 3 x0 + x Proof. The condition x i x 0 x x i 0, i = 1,..., n, means that either x 0 x i x or x x i x 0, i = 1,..., n. Consider the first case the second is analogous. Let the functions p, q : [x 0, x] [0, 1] be defined by px = x x, qx = x x 0. x x 0 x x 0 For any x [x 0, x] we can write x x φx = φ x 0 + x x 0 x = φpxx 0 + qx x. x x 0 x x 0 By Lemma 1.1 we get for x [x 0, x] [ ] x0 + x min{px, qx} φx 0 + φ x φ pxφx 0 + qxφ x φpxx 0 + qx x and then φx = φpxx 0 + qx x φx 0 + qxφ x min{px, qx} [ φx 0 + φ x φ x0 + x ]..

3 IMPROVEMENTS OF THE GIACCARDI AND THE PETROVIĆ INEQUALITY 67 Multiplying φx i by p i and summing, we get p i φx i [ ]] x0 + x p i [px i φx 0 + qx i φ x min{px i, qx i } φx 0 + φ x φ x i x 0 x x i = φ x p i + φx 0 p i δ φ p i min{px i, qx i } x x 0 x x 0 = Aφ x + B p i 1φx 0 δ φ P n + δ φ x i x 0+ x p i x x 0 Remark.1. Obviously, Theorem.1 is an improvement of Theorem 1.1 since under the required assumptions we have δ φ n p i min{px i, qx i } 0. A simple consequence of the Giaccardi inequality is the Petrović inequality, so we give its refinement too. Corollary.1. Let φ be a convex function on [0, a], 0 < a <. Then for every nonnegative n-tuple p and every x [0, a] n such that n p ix i = x 0, a] and p i x i x j, j = 1,..., n, 4 the following inequality holds n p i φx i φ x + p i 1 φ0 where δ φ = φ0 + φ x φ x. δ φ P n + δ φ. x i p i x 1, 5 Proof. This is a special case of Theorem.1; choose x 0 = 0. Motivated by inequalities 3 and 5, we define two functionals: n Φ 1 x, p, f = Af x + B p i 1 fx 0 δ f P n + + δ f n x i x 0+ x p i x x 0 n p i fx i, 6

4 68 J. PEČARIĆ AND J. PERIĆ where f is a function on interval I, p is a nonnegative n-tuple, x a real n-tuple and x, P n, δ f, A, B as in Theorem.1, and n Φ x, p, f = f x + p i 1 f0 δ φ P n n x i + δ f p i x 1 n p i f x i, 7 where f is a function on interval [0, a], p is a nonnegative n-tuple, x a real n-tuple and x, P n, δ f as in Corollary.1. If f is a convex function, then Theorem.1 and Corollary.1 imply that Φ i x, p, f 0, i = 1,. Now, we give mean value theorems for the functionals Φ i, i = 1,. Theorem.. Let I = [a, b], p be a nonnegative n-tuple with n p i = P n 0 and x a real n-tuple. Let x I n and x 0 I be such that n p ix i = x I, x x 0 and holds. Let f C I. Then there exists ξ I such that where f 0 x = x. Φ 1 x, p, f = f ξ Φ 1 x, p, f 0, 8 Proof. Since f C I therefore there exist real numbers m = min x [a,b] f x and M = max x [a,b] f x. It is easy to show that the functions f 1 x, f x defined by f 1 x = M x fx, f x = fx m x are convex. Therefore and we get Φ 1 x, p, f 1 0 Φ 1 x, p, f 0, Φ 1 x, p, f M Φ 1x, p, f 0 9 Φ 1 x, p, f m Φ 1x, p, f From 9 and 10 we get m Φ 1x, p, f 0 Φ 1 x, p, f M Φ 1x, p, f 0. If Φ 1 x, p, x = 0 there is nothing to prove. Suppose Φ 1 x, p, x > 0. We have Hence, there exists ξ I such that m Φ 1x, p, f Φ 1 x, p, x M. Φ 1 x, p, f = f ξ Φ 1 x, p, f 0.

5 IMPROVEMENTS OF THE GIACCARDI AND THE PETROVIĆ INEQUALITY 69 Theorem.3. Let I = [0, a], p be a nonnegative n-tuple and x a real n-tuple. Let x [0, a] n such that n p ix i = x I and 4 holds. Let f C I. Then there exists ξ I such that Φ x, p, f = f ξ Φ x, p, f 0 11 where f 0 x = x. Proof. Analogous to the proof of Theorem. Theorem.4. Let I = [a, b], p be a nonnegative n-tuple with n p i = P n 0 and x real n-tuple. Let x I n and x 0 I be such that n p ix i = x I, x x 0 and holds. Let f, g C I. Then there exists ξ I such that provided that the denominators are non-zero. Proof. Define h C [a, b] by where Φ 1 x, p, f Φ 1 x, p, g = f ξ g ξ, 1 h = c 1 f c g, c 1 = Φ 1 x, p, g, c = Φ 1 x, p, f. Now using Theorem. there exists ξ [a, b] such that f ξ g ξ c 1 c Φ 1 x, p, f 0 = 0. Since Φ 1 x, p, f 0 0 otherwise we have a contradiction with Φ 1 x, p, g 0 by Theorem., we get Φ 1 x, p, f Φ 1 x, p, g = f ξ g ξ. Theorem.5. Let I = [0, a], p be a nonnegative n-tuple and x a real n-tuple. Let x [0, a] n be such that n p ix i = x I and 4 holds. Let f, g C I. Then there exists ξ I such that Φ x, p, f Φ x, p, g = f ξ g 13 ξ provided that the denominators are non zero. Proof. Analogous to the proof of Theorem n onential convexity and onential convexity of the Giaccardi and the Petrović differences We begin this section by notions which are going to be lored here and some characterizations of these properties. Definition 3.1. A function ψ : I R is n onentially convex in the Jensen sense on I if xi + x j ξ i ξ j ψ 0 holds for all choices ξ i R and x i I, i = 1,..., n.

6 70 J. PEČARIĆ AND J. PERIĆ A function ψ : I R is n onentially convex if it is n onentially convex in the Jensen sense and continuous on I. Remark 3.1. It is clear from the definition that 1 onentially convex functions in the Jensen sense are in fact nonnegative functions. Also, n onentially convex functions in the Jensen sense are k onentially convex in the Jensen sense for every k N, k n. By definition of positive semi-definite matrices and some basic linear algebra we have the following proposition. Proposition 3.1. If ψ is an n-onentially convex in the Jensen sense, then the [ ] k xi + x j matrix ψ is a positive semi-definite matrix for all k N, k n. [ ] k xi + x j Particularly, det ψ 0 for all k N, k n. Definition 3.. A function ψ : I R is onentially convex in the Jensen sense on I if it is n onentially convex in the Jensen sense for all n N. A function ψ : I R is onentially convex if it is onentially convex in the Jensen sense and continuous. Remark 3.. It is known and easy to show that ψ : I R is a 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 onentially convex in the Jensen sense. Also, using basic convexity theory it follows that a function is log-convex if and only if it is onentially convex. We will also need the following result see for example [6]. Proposition 3.. If Ψ is a convex function on an interval I and if x 1 y 1, x y, x 1 x, y 1 y, then the following inequality is valid Ψx Ψx 1 x x 1 Ψy Ψy 1 y y If the function Ψ is concave, the inequality reverses. When dealing with functions with different degree of smoothness divided differences are found to be very useful. Definition 3.3. The second order devided difference of a function f : I R, I an interval in R, at mutually different points y 0, y 1, y I is defined recursively by [y i ; f] = fy i, i = 0, 1, [y i, y i+1 ; f] = fy i+1 fy i y i+1 y i, i = 0, 1 [y 0, y 1, y ; f] = [y 1, y ; f] [y 0, y 1 ; f] y y 0. 15

7 IMPROVEMENTS OF THE GIACCARDI AND THE PETROVIĆ INEQUALITY 71 Remark 3.3. The value [y 0, y 1, y ; f] is independent of the order of the points y 0, y 1 and y. This definition may be extended to include the case in which some or all the points coincide. Namely, taking the limit y 1 y 0 in 15, we get lim [y 0, y 1, y ; f] = [y 0, y 0, y ; f] = fy fy 0 f y 0 y y 0 y 1 y 0 y y 0, y y 0 provided f exists, and furthermore, taking the limits y i y 0, i = 1, in 15, we get lim y y 0 provided that f exists. lim [y 0, y 1, y ; f] = [y 0, y 0, y 0 ; f] = f y 0 y 1 y 0 We use an idea from [3] to give an elegant method of producing an n onentially convex functions and onentially convex functions applying the functionals Φ 1 and Φ on a given family with the same property. Theorem 3.1. Let Υ = {f s : s J}, where J an interval in R, be a family of functions defined on an interval I in R, such that the function s [y 0, y 1, y ; f s ] is n onentially convex in the Jensen sense on J for every three mutually different points y 0, y 1, y I. Let Φ i i = 1, be linear functionals defined as in 6 and 7. Then s Φ i x, p, f s is an n onentially convex function in the Jensen sense on J. If the function s Φ i x, p, f s is continuous on J, then it is n onentially convex on J. Proof. For ξ i R, i = 1,..., n and s i J, i = 1,..., n, we define the function gy = ξ i ξ j f s i +s j y. Using the assumption that the function s [y 0, y 1, y ; f s ] is n onentially convex in the Jensen sense, we have [y 0, y 1, y ; g] = ξ i ξ j [y 0, y 1, y ; f s i +s j ] 0, which in turn implies that g is a convex function on I and therefore we have Φ i x, p, g 0, i = 1,. Hence ξ i ξ j Φ i x, p, f s i +s j 0. We conclude that the function s Φ i x, p, f s is n onentially convex on J in the Jensen sense. If the function s Φ i x, p, f s is also continuous on J, then s Φ i x, p, f s is n onentially convex by definition. The following corollary is an immediate consequence of the above theorem. Corollary 3.1. Let Υ = {f s : s J}, where J an interval in R, be a family of functions defined on an interval I in R, such that the function s [y 0, y 1, y ; f s ] is onentially convex in the Jensen sense on J for every three mutually different points y 0, y 1, y I. Let Φ i i = 1, be linear functionals defined as in 6 and 7. Then s Φ i x, p, f s is an onentially convex function in the Jensen sense on J. If the function s Φ i x, p, f s is continuous on J, then it is onentially convex on J.

8 7 J. PEČARIĆ AND J. PERIĆ Corollary 3.. Let Ω = {f s : s J}, where J an interval in R, be a family of functions defined on an interval I in R, such that the function s [y 0, y 1, y ; f s ] is onentially convex in the Jensen sense on J for every three mutually different points y 0, y 1, y I. Let Φ i, i = 1,, be linear functionals defined as in 6 and 7. Then the following statements hold: i If the function s Φ i x, p, f s is continuous on J, then it is onentially convex function on J, and thus log-convex function. ii If the function s Φ i x, p, f s is strictly positive and differentiable on J, then for every s, q, u, v J, such that s u and q v, we have where for f s, f q Ω. µ s,q x, Φ i, Ω µ u,v x, Φ i, Ω, i = 1,, 16 µ s,q x, Φ i, Ω = Φix,p,f s Φ i x,p,f q 1 d ds Φix,p,fs Φ i x,p,f s, s q,, s = q. Proof. i This is an immediate consequence of Theorem 3.1 and Remark 3.. ii Since by i the function s Φ i x, p, f s is log-convex on J, that is, the function s log Φ i x, p, f s is convex on J. Applying Proposition 3. we get log Φ i x, p, f s log Φ i x, p, f q s q for s u, q v, s q, u v, and therefrom conclude that log Φ ix, p, f u log Φ i x, p, f v u v µ s,q x, Φ i, Ω µ u,v x, Φ i, Ω, i = 1,. Cases s = q and u = v follows from 18 as limit cases. Remark 3.4. Note that the results from Theorem 3.1, Corollary 3.1, Corollary 3. still hold when two of the points y 0, y 1, y I coincide, say y 1 = y 0, for a family of differentiable functions f s such that the function s [y 0, y 1, y ; f s ] is n onentially convex in the Jensen sense onentially convex in the Jensen sense, log-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. The proofs are obtained by recalling Remark 3.3 and suitable characterization of convexity. 4. Applications to Stolarsky type means In this section, we present several families of functions which fulfil the conditions of Theorem 3.1, Corollary 3.1 and Corollary 3. and Remark 3.4. This enable us to construct a large families of functions which are onentially convex. For a discussion related to this problem see []. Example 4.1. Consider a family of functions defined by Ω 1 = {g s : R [0, : s R} { 1 g s x = s e sx, s 0, 1 x, s = 0. We have d g s dx x = e sx > 0 which shows that g s is convex on R for every s R and s d g s dx x is onentially convex by definition. Using analogous arguing as in 17 18

9 IMPROVEMENTS OF THE GIACCARDI AND THE PETROVIĆ INEQUALITY 73 the proof of Theorem 3.1 we also have that s [y 0, y 1, y ; g s ] is onentially convex and so onentially convex in the Jensen sense. Using Theorem 3.1 we conclude that s Φ i x, p, g s, i = 1,, are onentially convex in the Jensen sense. It is easy to verify that these mappings are continuous although mapping s g s is not continuous for s = 0, so they are onentially convex. For this family of functions, µ s,q x, Φ i, Ω 1, i = 1,, from 17 become µ s,q x, Φ i, Ω 1 = 1 Φi x,p,g s Φ i x,p,g q Φi x,p,id g s Φ ix,p,g s s Φi x,p,id g 0 3Φ ix,p,g 0, s q,, s = q 0,, s = q = 0, and using 16 they are monotonous functions in parameters s and q. Using Theorems.4 and.5 it follows that for i = 1, M s,q x, Φ i, Ω 1 = log µ s,q x, Φ i, Ω 1 satisfy min {x 0, x} M s,q x, Φ i, Ω 1 max {x 0, x}, which shows that M s,q x, Φ i, Ω 1 are means of x 0, x 1,... x n, x. Notice that by 16 they are monotonous means. Example 4.. Consider a family of functions defined by Ω = {f s : 0, R : s R} x s ss 1, s 0, 1, f s x = log x, s = 0, x log x, s = 1. Here, d f s dx x = x s = e s log x > 0 which shows that f s is convex for x > 0 and s d f s dx x is onentially convex by definition. Arguing as in Example 4.1 we get that the mapping s Φ 1 x, p, g s is onentially convex. In this case we assume x j > 0, j = 0, 1..., n. Notice that the functional Φ is not defined in this case of course it can be defined for s 0. Functions 17 in this case are equal to: 1 Φ1 x,p,f s Φ 1x,p,f q, s q, 1 s ss 1 µ s,q x, Φ 1, Ω = Φ 1x,p,f s f 0 Φ 1 x,p,f s, s = q 0, 1, 1 Φ1x,p,f 0 Φ 1 x,p,f 0, s = q = 0, 1 Φ1x,p,f0f1 Φ 1 x,p,f 1, s = q = 1. If Φ 1 is positive, then Theorem.4 and Theorem.5 applied for f = f s Ω and g = f q Ω yields that there exists ξ [min {x 0, x}, max {x 0, x}] such that ξ = Φ 1x, p, f s Φ 1 x, p, f q. Since the function ξ ξ is invertible for s q, we then have 1 Φ1 x, p, f s min {x 0, x} max {x0, x}, 19 Φ 1 x, p, f q which together with the fact that µ s,q x, Φ 1, Ω is continuous, symmetric and monotonous by 16, shows that µ s,q x, Φ 1, Ω is a mean. Now, by substitutions x i x t i,

10 74 J. PEČARIĆ AND J. PERIĆ s s t, q q t t 0, s q from 19 we get min { x t 0, x t} Φ1 x t, p, f s/t Φ 1 x t, p, f q/t t max { x t 0, x t}, where x t = x t 1,..., x t n. We define a new mean as follows { 1/t µ s,q;t x, Φ 1, Ω = µ s t, q xt, Φ t 1, Ω, t 0 µ s,q log x, Φ 1, Ω 1, t = 0. These new means are also monotonous. More precisely, for s, q, u, v R such that s u, q v, s u, q v, we have We know that µ s t, q t x, Φ 1, Ω = 0 µ s,q;t x, Φ 1, Ω µ u,v;t x, Φ 1, Ω. 1 t Φ1 x, p, f s/t Φ 1 x, p, f q/t µ u t, v t x, Φ 1, Ω = t Φ1 x, p, f s/t, Φ 1 x, p, f q/t for s, q, u, v I such that s/t u/t, q/t v/t and t 0, since µ s,q x, Φ 1, Ω are monotonous in both parameters, the claim follows. For t = 0, we obtain the required result by taking the limit t 0. Example 4.3. Consider a family of functions defined by Ω 3 = {h s : 0, 0, : s 0, } h s x = { s x log s, s 1, x, s = 1. Since s d h s dx x = s x is the Laplace transform of a non-negative function see [7] it is onentially convex. Obviously h s are convex functions for every s > 0. For this family of functions, µ s,q x, Φ 1, Ω 3, in this case for x j > 0, j = 0, 1,..., n, from 17 becomes Φ1 x,p,h s Φ 1x,p,h q µ s,q x, Φ 1, Ω 3 = 1, s q, Φ 1x,p,id h s sφ 1x,p,h s s log s Φ1x,p,id h1 3Φ 1 x,p,h 1 and it is monotonous in parameters s and q by 16. Using Theorem.4, it follows that, s = q 1,, s = q = 1, M s,q x, Φ 1, Ω 3 = Ls, q log µ s,q x, Φ 1, Ω 3, satisfies min {x 0, x} M s,q x, Φ 1, Ω 3 max {x 0, x}, which shows that M s,q x, Φ 1, Ω 3 is a mean of x 0, x 1,... x n, x. Ls, q is the logarithmic mean defined by Ls, q =, s q, Ls, s = s. log s log q Example 4.4. Consider a family of functions defined by Ω 4 = {k s : 0, 0, : s 0, } k s x = e x s s

11 IMPROVEMENTS OF THE GIACCARDI AND THE PETROVIĆ INEQUALITY 75 Since s d k s dx x = e x s is the Laplace transform of a non-negative function see [7] it is onentially convex. Obviously k s are convex functions for every s > 0. For this family of functions, µ s,q x, Φ 1, Ω 4, in this case for x j > 0, j = 0, 1,..., n, from 17 becomes µ s,q x, Φ 1, Ω 4 = Φ1x,p,k s Φ 1 x,p,k q 1 Φ1x,p,id ks sφx,p,k s 1 s, s q,, s = q, and it is monotonous function in parameters s and q by 16. Using Theorem.4, it follows that M s,q x, Φ 1, Ω 4 = s + q log µ s,q x, Φ 1, Ω 4 satisfies min {x 0, x} M s,q x, Φ 1, Ω 4 max {x 0, x}, which shows that M s,q x, Φ 1, Ω 4 is a mean of x 0, x 1,... x n, x. References [1] M. Anwar, J. Jakšetić, J. Pečarić and Atiq ur Rehman, Exponential convexity, positive semidefinite matrices and fundamental inequalities, J. Math. Inequal , [] W. Ehm, M. G. Genton, T. Gneiting, Stationary covariances associated with onentially convex functions, Bernoulli , [3] J. Jakšetić and J. Pečarić, Exponential Convexity Method, submitted. [4] Khuram A. Khan, J. Pečarić, I. Perić, Differences of weighted mixed symmetric means and related results, J. Inequal. Appl , Article ID 89730, 16 pages. [5] D. S. Mitrinović, J. E. Pečarić, A. M. Fink, Classical and new inequalities in analysis. Mathematics and its Applications East European Series, 61, Kluwer Academic Publishers Group, Dordrecht, [6] J. E. Pečarić, F. Proschan and Y. L. Tong, Convex functions, partial orderings, and statistical applications, Academic Press Inc, 199. [7] D. V. Widder, The Laplace transform, Princeton Univ. Press, New Jersey, Josip Pečarić Faculty of Textile Technology, University of Zagreb, Zagreb, Croatia address: pecaric@element.hr Jurica Perić Faculty of Science, Departments of Mathematics, University of Split, Teslina 1, 1000 Split, Croatia address: jperic@pmfst.hr