GENERALIZATIONS AND IMPROVEMENTS OF AN INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA. Sadia Khalid and Josip Pečarić

RAD HAZU. MATEMATIČKE ZNANOSTI Vol. 18 = 519 014: 73-89 GENERALIZATIONS AND IMPROVEMENTS OF AN INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA Sadia Khalid and Josip Pečarić Abstract. Some generalizations of an inequality of Hardy- Littlewood-Pólya are presented. We discuss the n-exponential convexity and log-convexity of the functions associated with the linear functional defined by the generalized inequality and also prove the monotonicity property of the generalized Cauchy means obtained via this functional. Finally, we give several examples of the families of functions for which the results can be applied. 1. Introduction and preliminaries The following theorem is given in the famous Hardy-Littlewood-Pólya book [, Theorem 134]. Theorem 1.1. If f is a convex and continuous function defined on [0, and a k, k N are non-negative and non-increasing, then n 1.1 f a k f 0 + [fka k fk 1a k ]. If f is concave, then the inequality in 1.1 reverses. If f is a strictly increasing function, there is equality only when a k are equal up to a certain point and then zero. An example of the above theorem is given below see [, p.100]. Corollary 1.. Let a k [0, and the sequence a k, k N is nonincreasing. If s > 1, then we have n s 1. a k a s k [k s k 1 s ]. If 0 < s < 1, then the inequality in 1. reverses. 010 Mathematics Subject Classification. 6A48, 6A51, 6D15. Key words and phrases. Non-increasing sequence in mean, Wright-convex function, n-exponential and logarithmic convexity, divided difference. 73

74 S. KHALID AND J. PEČARIĆ Inequality 1.1 is of great interest and has been generalized in many different ways by various mathematicians. In order to obtain our main results we need some definitions. if 1.3 Definition 1.3. A sequence a k, k N R is non-increasing in mean 1 n a k 1 n+1 a k, n N. n + 1 A sequence a k, k N R is non-decreasing in mean, if opposite inequality holds in 1.3. In a similar way we can define when a finite sequence a k, k = 1,..., n R is non-increasing in mean or non-decreasing in mean see [4]. Remark 1.4. If we denote S k = k i=1 a i, k N, then it is easy to see that the sequence a k, k N is non-increasing in mean non-decreasing in mean if and only if S k 1 k 1a k S k 1 k 1a k for k =, 3,.... In 1995, inequality 1. was improved by J. Pečarić and L. E. Persson in [5] and this improvement is given below: Theorem 1.5. If the sequence a k > 0, k N is non-increasing in mean and if s is a real number such that s > 1, then s 1.4 a k a s k [k s k 1 s ] holds. If 0 < s < 1, then the inequality in 1.4 reverses. It is well known and easy to see that if a sequence a k, k N is non-increasing non-decreasing, then it is also non-increasing in mean nondecreasing in mean but the reverse implications do not hold in general. This means that Theorem 1.5 is a genuine generalization of Theorem 1.. The following property of a convex function will be useful further see [7, p.]. Definition 1.6. A function f : I R is convex on I if 1.5 x 3 x f x 1 + x 1 x 3 f x + x x 1 f x 3 0 holds for all x 1, x, x 3 I such that x 1 < x < x 3. Another characterization of a convex function will be needed later see [7, p.]. Proposition 1.7. If f is a convex function defined on an interval I and if x 1 y 1, x y, x 1 x, y 1 y, then the following inequality is valid f x f x 1 1.6 f y f y 1. x x 1 y y 1 If the function f is concave, then the inequality reverses.

INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA 75 By letting x 1 = x, x = x + h, y 1 = y and y = y + h, x y, h 0 in 1.6, we have 1.7 fx + h fx fy + h fy. The following definition of Wright-convex function is given in [7, p.7]. Definition 1.8. A function f : [a, b] R is said to be Wright-convex if for all x, y + h [a, b] such that x y, h 0, 1.7 holds. The function f is said to be Wright-concave if opposite inequality holds in 1.7. Remark 1.9. If K [a, b] and W [a, b] denote the set of all convex functions and the set of all Wright-convex functions defined on [a, b], then K [a, b] W [a, b]. That is, a convex function is also a Wright-convex function but not conversely see[7, p.7]. Wright-convex functions have interesting and important generalization for functions of several variables see [1]. Let R m denote the m-dimensional vector lattice of points x = x 1,..., x m, x i R for i = 1,..., m with the partial ordering x = x 1,..., x m y = y 1,..., y m if and only if x i y i for i = 1,..., m see [7, p.13]. Definition 1.10. A sequence a k, k N R m is non-increasing in mean if 1 1.8 a k 1 n+1 a k, n N, n n + 1 where a k = a 1 k,..., am k. A sequence a k, k N R m is non-decreasing in mean, if opposite inequality holds in 1.8. In [1], H. D. Brunk presented an interesting class of multivariate realvalued functions defined as follows: Definition 1.11. A real-valued function f on an m-dimensional rectangle I R m is said to have non-decreasing increments if 1.9 fx + h fx fy + h fy, whenever x, y + h I, 0 h R m, x y. The function f is said to have non-increasing increments if opposite inequality holds in 1.9. Remark 1.1. It is easy to see that if a function f is defined on [a, b] R and has non-decreasing increment, then it is a Wright-convex function. In this paper, we present some generalizations of the Hardy-Littlewood- Pólya inequality 1.1. First generalization is obtained by using a Wrightconvex function and a sequence a k, k = 1,..., n R which is non-increasing in mean. Since the class of all Wright-convex functions contains the class of all

76 S. KHALID AND J. PEČARIĆ convex functions, our first main result is the generalization of the inequality 1.1 in the sense that we can obtain 1.1 as a special case of our first main result. Second generalization is obtained by using a real-valued function defined on an m-dimensional rectangle I R m, having non-decreasing increments and a sequence a k, k = 1,..., n R m, which is non-increasing in mean. We also discuss the case when the sequence is non-decreasing in mean. Further, we give the n-exponential convexity and log-convexity of the functions associated with the linear functional defined by the generalized inequality and we also deduce Lyapunov-type inequalities for this functional. We prove monotonicity property of the generalized Cauchy means obtained via this functional. Finally, we give several examples of the families of functions for which the results can be applied.. Main results Inequality 1.1 was already proved in [], but we give a new proof in a more general setting. Our first main result states: Theorem.1. Let a k k = 1,..., n be real numbers such that a k 0, S k = k i=1 a i, ka k, k 1a k [a, b] for all k =,..., n and f : [a, b] R be a Wright-convex function. i If the sequence a k, k = 1,..., n is non-increasing in mean, then we have n.1 f a k f a 1 + [f ka k f k 1 a k ]. k= ii If the sequence a k, k = 1,..., n is non-decreasing in mean, then we have n. f a k f a 1 + [f ka k f k 1 a k ]. k= If the function f is Wright-concave, then opposite inequalities hold in.1 and.. Proof. i Since the sequence a k, k = 1,..., n R is non-increasing in mean, by definition we have, S k 1 k 1 a k for k =,..., n. As f is a Wright-convex function, by setting x = k 1 a k, y = S k 1 and h = a k k =,..., n in 1.7, and also using the fact that S k 1 + a k = S k k =,..., n, we have fs k fs k 1 fka k fk 1a k. Summing over k from to n, we have [f S k fs k 1 ] [fka k fk 1a k ], k= k=

INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA 77 which is equivalent to fs n fs 1 [fka k fk 1a k ], k= and so.1 holds. ii Since the sequence a k, k = 1,..., n R is non-decreasing in mean, by definition we have, S k 1 k 1a k for k =,..., n. As f is a Wright-convex function, by setting x = S k 1, y = k 1a k and h = a k k =,..., n in 1.7, we have fs k fs k 1 fka k fk 1a k. Now summing over k from to n and after simplification, we have.. If f is a Wright-concave function, then opposite inequality holds in 1.7 and so opposite inequalities hold in.1 and.. Since the class of Wright-convex Wright-concave functions properly contains the class of convex concave functions see for example [7, p.7], the following result is valid. Corollary.. Let a k k = 1,..., n be real numbers such that a k 0, S k = k i=1 a i, ka k, k 1a k [a, b] for all k =,..., n and f : [a, b] R be a convex function. i If the sequence a k, k = 1,..., n is non-increasing in mean, then.1 holds. ii If the sequence a k, k = 1,..., n is non-decreasing in mean, then. holds. If the function f is concave, then opposite inequalities hold in.1 and.. The following corollary is an application of Corollary.. Corollary.3. Let fx = x s, where x 0, and s R. i If the sequence a k > 0, k = 1,..., n is non-increasing in mean and s R such that s < 0 or s > 1, then n s n.3 a k a s 1 + a s k [k s k 1 s ] k= holds. If 0 < s < 1, then the inequality in.3 reverses. ii If the sequence a k > 0, k = 1,..., n is non-decreasing in mean and s R such that s < 0 or s > 1, then n s n.4 a k a s 1 + a s k [k s k 1 s ] k= holds. If 0 < s < 1, then the inequality in.4 reverses.

78 S. KHALID AND J. PEČARIĆ The multidimensional generalization is stated as follows: Theorem.4. Let a k R m be such that a k 0 for all k = 1,..., n, S k = k i=1 a i, ka k, k 1a k I R m for all k =,..., n and f : I R m R be a real valued function having non-decreasing increments. i If the sequence a k, k = 1,..., n is non-increasing in mean, then we have n.5 f a k fa 1 + [f ka k f k 1 a k ]. k= ii If the sequence a k, k = 1,..., n is non-decreasing in mean, then we have n.6 f a k f a 1 + [f ka k fk 1a k ]. k= If the function has non-increasing increments, then opposite inequalities hold in.5 and.6. Proof. The idea of the proof is the same as in Theorem.1. i Since the sequence a k, k = 1,..., n R m is non-increasing in mean, by definition we have, S k 1 k 1a k for k =,..., n. By setting x = k 1a k, y = S k 1 and h = a k k =,..., n in 1.9, where f has non-decreasing increments and also using the fact that S k 1 + a k = S k k =,..., n, we have f S k f S k 1 f ka k f k 1 a k. Summing over k from to n, we have [f S k f S k 1 ] [fka k fk 1a k ], k= which is equivalent to fs n fs 1 k= [fka k fk 1a k ], k= and so.5 holds. ii Since the sequence a k, k = 1,..., n R m is non-decreasing in mean, by definition we have, S k 1 k 1a k for k =,..., n. By setting x = S k 1, y = k 1a k and h = a k k =,..., n in 1.9, where f has non-decreasing increments, we have f S k f S k 1 f ka k f k 1 a k. Now summing over k from to n and after simplification, we have.6.

INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA 79 If f has non-increasing increments, then opposite inequality holds in 1.9 and so opposite inequalities hold in.5 and.6. Consider the inequality.1 and define a functional n.7 Φ f = f a k f a 1 [f ka k f k 1 a k ], where a k k = 1,..., n are real numbers such that a k 0, S k = k i=1 a i, ka k and k 1 a k [a, b] for all k =,..., n. If the function f is convex on [a, b] and the sequence a k, k = 1,..., n R is non-increasing in mean, then Corollary. i implies that Φf 0. Now, we give mean value theorems for the functional Φ. These theorems enable us to define various classes of means that can be expressed in terms of a linear functional. Theorem.5. Let a k k = 1,..., n be real numbers such that a k 0, S k = k i=1 a i, ka k, k 1a k [a, b] for all k =,..., n and the sequence a k, k = 1,..., n is non-increasing in mean. Suppose that Φ is a linear functional defined as in.7 and f C [a, b]. Then there exists ξ [a, b] such that Φ f = f ξ Φ f 0, where f 0 x = x. k= Proof. Analogous to the proof of Theorem. in [6]. We can prove the following Cauchy type mean value theorem by following the proof of Theorem.4 in [6]. Theorem.6. Let a k k = 1,..., n be real numbers such that a k 0, S k = k i=1 a i, ka k, k 1 a k [a, b] for all k =,..., n and the sequence a k, k = 1,..., n is non-increasing in mean. Suppose that Φ is a linear functional defined as in.7 and f, g C [a, b]. Then there exists ξ [a, b] such that Φ f.8 Φ g = f ξ g ξ, provided that the denominators are non-zero. Remark.7. i By taking fx = x s and gx = x q in.8, where s, q R \ {0, 1} are such that s q, we have ξ s q = q q 1 Φ xs s s 1 Φ x q.

80 S. KHALID AND J. PEČARIĆ ii If the inverse of the function f /g exists, then.8 gives f 1 Φf ξ = g. Φg 3. n-exponential convexity and log-convexity of the functions associated with the difference of the generalized inequality We begin this section by recollecting definitions and properties which are going to be explored here and also some useful characterizations of these properties. In the sequel, let I be an open interval in R. Definition 3.1. A function h : I R is n-exponentially convex in the Jensen sense on I if xi + x j α i α j h 0 i,j=1 holds for every α i R and x i I, i = 1,..., n see [6]. Definition 3.. A function h : I R is n-exponentially convex on I if it is n-exponentially convex in the Jensen sense and continuous on I. Remark 3.3. From the above definition, it is clear that 1-exponentially convex functions in the Jensen sense are non-negative functions. Also, n- exponentially convex functions in the Jensen sense are k-exponentially convex functions in the Jensen sense for all k N, k n. By the definition of positive semi-definite matrices and some basic linear algebra, we have the following proposition. Proposition 3.4. If h : I R is n-exponentially convex in the Jensen [ ] k sense, then the matrix h is a positive semi-definite matrix for xi+x j i,j=1 all k N, k n. Particularly, [ ] k xi + x j det h 0 for every k N, k n, x i I, i = 1,..., n. i,j=1 Definition 3.5. A function h : I R is exponentially convex in the Jensen sense if it is n-exponentially convex in the Jensen sense for all n N. Definition 3.6. A function h : I R is exponentially convex if it is exponentially convex in the Jensen sense and continuous. Lemma 3.7. A function h : I 0, is log-convex in the Jensen sense, that is, for every x, y I, x + y h h x h y

INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA 81 holds if and only if the relation x + y α hx + αβ h + β hy 0 holds for every α, β R and x, y I. Remark 3.8. It follows that a function is log-convex in the Jensen sense if and only if it is -exponentially convex in the Jensen sense. Also, by using basic convexity theory, a function is log-convex if and only if it is -exponentially convex. The following definition of divided difference is given in [7, p.14]. Definition 3.9. The second-order divided difference of a function f : [a, b] R at mutually distinct points y 0, y 1, y [a, b] is defined recursively by 3.1 [y i ; f] = f y 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. Remark 3.10. 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 see [7, p.16]. Namely, taking the limit y 1 y 0 in 3.1, 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 that f exists; and furthermore, taking the limits y i y 0, i = 1,, in 3.1, we get lim lim [y 0, y 1, y ; f] = [y 0, y 0, y 0 ; f] = f y 0, y y 0 y 1 y 0 provided that f exists. Remark 3.11. Convex functions can be characterized by second order divided difference see [7, p.16]: a function f : [a, b] R is convex if and only if for all choices of three distinct points y 0, y 1, y, [a, b], [y 0, y 1, y ; f] 0. Next, we study the n-exponential convexity and log-convexity of the functions associated with the linear functional Φ defined in.7. Theorem 3.1. Let Ω = {f s : s I R} be a family of functions defined on [a, b] such that the function s [y 0, y 1, y ; f s ] is n-exponentially convex in the Jensen sense on I for every three mutually distinct points y 0, y 1, y [a, b]. Let Φ be a linear functional defined as in.7. Then the following statements hold:

8 S. KHALID AND J. PEČARIĆ i The function s [ Φf s is ] n-exponentially convex in the Jensen sense on m I and the matrix is a positive semi-definite matrix for all Φf s j +s k j, m N, m n and s 1,..., s m I. Particularly, [ ] m det 0, m N, m n. Φf s j +s k j, ii If the function s Φf s is continuous on I, then it is n-exponentially convex on I. Proof. The idea of the proof is the same as that of Theorem 3.1 in [6]. i Let α j R j = 1,..., n and consider the function ϕy = α j α k f s j +s k y, where s j I and f s j +s k [y 0, y 1, y ; ϕ] = j, Ω. Then j, α j α k [y 0, y 1, y ; f s j +s k [ ] and since y 0, y 1, y ; f s j +s k is n-exponentially convex in the Jensen sense on I by assumption, it follows that ] [y 0, y 1, y ; ϕ] = α j α k [y 0, y 1, y ; f s j +s k 0. j, And so, by using Remark 3.11, we conclude that ϕ is a convex function. Hence, Φ ϕ 0, which is equivalent to j, α j α k Φ f s j +s k 0 and so we conclude that the function s Φf s is n-exponentially convex in the Jensen sense on I. The remaining part follows from Proposition 3.4. ii If the function s Φ f s is continuous on I, then from i and by Definition 3., it follows that it is n-exponentially convex on I. ] The following corollary is an immediate consequence of the above theorem.

INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA 83 Corollary 3.13. Let Ω = {f s : s I R} be a family of functions defined on [a, b] such that the function s [y 0, y 1, y ; f s ] is exponentially convex in the Jensen sense on I for every three mutually distinct points y 0, y 1, y [a, b]. Let Φ be a linear functional defined as in.7. Then the following statements hold: i The function s Φf s is exponentially convex in the Jensen sense on [ I and the matrix Φf s j +s k ] n j, n N and s 1,..., s n I. Particularly, [ ] n det Φf s j +s k is a positive semi-definite matrix for all j, 0, n N. ii If the function s Φf s is continuous on I, then it is exponentially convex on I. Corollary 3.14. Let Ω = {f s : s I R} be a family of functions defined on [a, b] such that the function s [y 0, y 1, y ; f s ] is -exponentially convex in the Jensen sense on I for every three mutually distinct points y 0, y 1, y [a, b]. Let Φ be a linear functional defined as in.7 and also assume that Φf s is strictly positive for f s Ω. Then the following statements hold: i If the function s Φf s is continuous on I, then it is -exponentially convex on I and so it is log-convex on I and for r, s, t I such that r < s < t, we have 3. [Φf s ] t r [Φf r ] t s [Φf t ] s r, known as Lyapunov s inequality. If r < t < s or s < r < t, then opposite inequality holds in 3.. ii If the function s Φf s is differentiable on I, then for every s, q, u, v I such that s u and q v, we have 3.3 µ s,q Φ, Ω µ u,v Φ, Ω, where 3.4 µ s,q Φ, Ω = for f s, f q Ω. 1 Φfs s q, s q, Φf q d ds exp Φf s, s = q, Φf s Proof. The idea of the proof is the same as that of Corollary 3. in [6]. i The claim that the function s Φf s is log-convex on I is an immediate consequence of Theorem 3.1 and Remark 3.8 and 3. can be obtained by replacing the convex function f with the convex function

84 S. KHALID AND J. PEČARIĆ 3.5 fz = log Φf z for z = r, s, t in 1.5, where r, s, t I such that r < s < t. ii Since by i the function s Φf s is log-convex on I, that is, the function s log Φf s is convex on I. Applying Proposition 1.7 with setting fz = log Φf z, we get log Φf s log Φf q s q log Φf u log Φf v u v for s u, q v, s q, u v; and therefore, we conclude that µ s,q Φ, Ω µ u,v Φ, Ω. If s = q, we consider the limit when q s in 3.5 and conclude that µ s,s Φ, Ω µ u,v Φ, Ω. The case u = v can be treated similarly. Remark 3.15. Note that the results from Theorem 3.1, Corollary 3.13 and Corollary 3.14 still hold when two of the points y 0, y 1, y [a, b] 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-exponentially convex in the Jensen sense exponentially convex in the Jensen sense, log-convex in the Jensen sense on I; 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 3.10 and by using suitable characterizations of convexity. 4. Examples In this section, we present several families of functions which fulfil the conditions of Theorem 3.1, Corollary 3.13 and Corollary 3.14 and Remark 3.15. This enables us to construct large families of functions which are exponentially convex. Example 4.1. Consider the 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 dx g s x = e sx > 0, which shows that g s is convex on R for every s R and s d dx g s x is exponentially convex by definition see also [3].

INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA 85 In order to prove that the function s [y 0, y 1, y ; g s ] is exponentially convex, it is enough to show that 4.1 n ] j, α jα k [y 0, y 1, y ; g s j +s k = [ y 0, y 1, y ; ] n j, α jα k g s j +s k 0, n N, α j, s j R, j = 1...., n. By Remark 3.11, 4.1 will hold if Υx := n j, α jα k g s j +s k is convex. Since s g s x is exponentially convex, i.e., n j, α jα k g sj 0, n N, α +sk j, s j R, j = 1,..., n, which shows the convexity of Υx and so 4.1 holds. Now, as the function s [y 0, y 1, y ; g s ] is exponentially convex, s [y 0, y 1, y ; g s ] is exponentially convex in the Jensen sense and by using Corollary 3.13, we have s Φg s is exponentially convex in the Jensen sense. Since this mapping is continuous although the mapping s g s is not continuous for s = 0, s Φg s is exponentially convex. For this family of functions, by taking Ω = Ω 1 in 3.4, µ s,q Φ, Ω 1 become µ s,q Φ, Ω 1 = 1 Φgs s q Φg q exp Φid gs Φg s s exp Φid g0 3Φg 0 By using Theorem.6, it can be seen that, s q, M s,q Φ, Ω 1 = log µ s,q Φ, Ω 1, s = q 0,, s = q = 0. satisfy a M s,q Φ, Ω 1 b, which shows that M s,q Φ, Ω 1 is a mean. Example 4.. Consider the family of functions Ω = {f s : 0, R : s R} defined by f s x = x s ss 1, s 0, 1, ln x, s = 0, x ln x, s = 1. d Here, dx f s x = x s = e s ln x > 0, which shows that f s is convex for x > 0 and s d dx f s x is exponentially convex by definition see also [3]. It is easy to prove that the function s [y 0, y 1, y ; f s ] is exponentially convex. Arguing as in Example 4.1, we have s Φf s is exponentially convex. If r, s, t R are such that r < s < t, then from 3., we have 4. Φ f s [Φ f r ] t s t r [Φ f t ] s r t r.

86 S. KHALID AND J. PEČARIĆ If r < t < s or s < r < t, then opposite inequality holds in 4.. Particularly, for r, s, t R \ {0, 1} such that r < s < t, we have 4.3 n a k s a s 1 n k= as k ks k 1 s ss 1 [ n a k r a r 1 n k= ar k kr k 1 r rr 1 [ n a k t a t 1 n k= at kk t k 1 t tt 1 ] t s t r ] s r t r, where a k > 0 k = 1,..., n are real numbers, S k = k i=1 a i, ka k and k 1a k [a, b] for all k =,..., n. In fact, for s < 0 or s > 1, 4.3 is the improvement of the inequality.3 and for 0 < s < 1, the inequality in 4.3 reverses. By taking Ω = Ω in 3.4, Ξ s,q := µ s,q Φ, Ω are of the form Ξ s,q = qq 1 ss 1. n a s k a s 1 n k= as k k s k 1 s n a q k a q 1 n k= aq k kq k 1 q n 1 Ξ s,0 = ss 1. a k s a s 1 n k= as k ks k 1 s ln na 1 ln n a k Ξ s,1 = s 0, 1, Ξ 0,1 = n a k 1 ss 1 ln n n a k a k s a s n 1 n + k ln ka k k= as k ks k 1 s 1 s q, s q 0, 1, 1 s k= a kk 1 ln k 1 a k, s 0, 1, 1 s 1 n a n k ln a k k ln kak + n a k= kk 1 ln k 1 a k ln na 1 ln n a, k 1 s Ξ s,s = exp ss 1 n s n n exp a k ln a k n s k a a s 1 s 0, 1, n as k ks ln ka k + k= as k k 1s ln k 1 a k n k= as k ks k 1 s,, Ξ 0,0 = exp ln n a k ln a 1 ln e n n a 1 a k ln n a k ln na 1 k= ln k ln k 1 ln e kk 1a k,

Ξ 1 1,1 = exp n exp INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA 87 a k ln a k n a k ln n k= a k a k ln e n a k k ln ka k ln e ka k n k lnka k + k= a kk 1 ln k 1a k a k k 1 ln k 1a k ln e k 1a k ln n a k n k lnka k + a kk 1 ln k 1a k k=. If Φ is positive, then Theorem.6 applied for f = f s Ω and g = f q Ω yields that there exists ξ [a, b] such that ξ s q = Φf s Φf q. Since the function ξ ξ s q is invertible for s q, we have 1 Φfs s q 4.4 a b Φf q which, together with the fact that µ s,q Φ, Ω is continuous, symmetric and monotonous by 3.3 shows that µ s,q Φ, Ω is a mean. If a = 0 and we consider functions defined on [0,, then we can obtain inequalities and means of the same form, but for parameters s and q restricted to 0,. More precisely, we consider the family of functions defined by Ω = { f s : [0, R : s 0, } f s x = { x s ss 1, s 1, x ln x, s = 1, with the convention that 0 ln 0 = 0. If r, s, t 0, \ {1} are such that r < s < t, then from 4., we have n a k s n as k ks k 1 s ss 1 [ n a k r n ar k kr k 1 r rr 1 [ n a k t n at kk t k 1 t tt 1 ] t s t r ] s r t r, which is in fact the improvement of inequality 1. for s > 1. For s > 0 and q > 0, by taking Ω = Ω in 3.4, Ξ s,q =: µ s,q Φ, Ω are of the same form as Ξ s,q. Example 4.3. Consider the family of functions Ω 3 = {h s : 0, 0, : s 0, }

88 S. KHALID AND J. PEČARIĆ defined by h s x = { s x ln s, s 1, x, s = 1. We have d dx h s x = s x > 0, which shows that h s is convex for all s > 0. Since s d dx h s x = s x is the Laplace transform of a non-negative function see [3, 8], it is exponentially convex. It is easy to see that the function s [y 0, y 1, y ; h s ] is also exponentially convex. Arguing as in Example 4.1, we have s Φh s is exponentially convex. In this case, by taking Ω = Ω 3 in 3.4, µ s,q Φ, Ω 3 are of the form Φhs Φh q µ s,q Φ, Ω 3 = exp exp 1 s q By using Theorem.6, it follows that, s q, Φid hs sφh s s ln s Φid h1 3Φh 1, s = q 1,, s = q = 1. M s,q Φ, Ω 3 = Ls, q log µ s,q Φ, Ω 3 satisfy a M s,q Φ, Ω 3 b and so M s,q Φ, Ω 3 is a mean, where Ls, q is a logarithmic mean defined by Ls, q =, s q, Ls, s = s. s q log s log q Example 4.4. Consider the family of functions Ω 4 = {k s : 0, 0, : s 0, } defined by s k s x = e x. s d Here, dx k s x = e x s > 0, which shows that k s is convex for all s > 0. Since s d dx k s x = e x s is the Laplace transform of a non-negative function see [3, 8], it is exponentially convex. It is easy to prove that the function s [y 0, y 1, y ; k s ] is also exponentially convex. Arguing as in Example 4.1, we have s Φk s is exponentially convex. In this case, by taking Ω = Ω 4 in 3.4, µ s,q Φ, Ω 4 are of the form 1 Φks s q Φk µ s,q Φ, Ω 4 = q, s q, exp Φid ks sφk 1 s s, s = q. By using Theorem.6, it is easy to see that M s,q Φ, Ω 4 = s + q log µ s,q Φ, Ω 4 satisfy a M s,q Φ, Ω 4 b, showing that M s,q Φ, Ω 4 is a mean. Remark 4.5. From 3.4, it is clear that µ s,q Φ, Ω for Ω = Ω 1, Ω 3 and Ω 4 are monotonous functions in parameters s and q.

INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA 89 Acknowledgements. This research work was partially supported by Higher Education Commission, Pakistan. The second author s research was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grant 117-1170889-0888. References [1] H. D. Brunk, Integral inequalities for functions with non-decreasing increments, Pacific J. Math 14 1964, 783 793. [] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, nd ed., Cambridge Univ. Press, Cambridge, 195. [3] J. Jakšetić and J. Pečarić, Exponential Convexity Method, J. Convex Anal. 0 013, 181 197. [4] C. Jardas, J. Pečarić, R. Roki and N. Sarapa, On some inequalities for entropies of discrete probability distributions, J. Austral. Math. Soc. Ser. B 40 1999, 535 541. [5] J. Pečarić and L. E. Persson, On an inequality of Hardy-Littlewood-Pólya, Math. Gazette 79 1995, 383 385. [6] 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. [7] J. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press Inc., 199. [8] D. V. Widder, The Laplace Transform. Princeton Univ. Press, New Jersey, 1941. Poopćenja i poboljšanja nejednakosti Hardy-Littlewood-Pólya Sadia Khalid i Josip Pečarić Sažetak. U radu su dana neka poopćenja i poboljšanja nejednakosti Hardy-Littlewood-Pólya. Diskutirana je n- eksponencijalna konveksnost i logaritamska konveksnost funkcionala definiranog poopćenom nejednakošću kao i monotonost odgovarajućih poopćenih Cauchyjevih sredina. Sadia Khalid Abdus Salam School of Mathematical Sciences GC University 68-B, New Muslim Town Lahore 54600, Pakistan E-mail: saadiakhalid176@gmail.com Josip Pečarić Faculty of Textile Technology University of Zagreb Prilaz baruna Filipovića 8a 10000 Zagreb, Croatia E-mail: pecaric@element.hr Received: 4.7.013.

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