and monotonicity property of these means is proved. The related mean value theorems of Cauchy type are also given.

Transcription

1 Rad HAZU Volume , 1-10 ON LOGARITHMIC CONVEXITY FOR GIACCARDI S DIFFERENCE J PEČARIĆ AND ATIQ UR REHMAN Abstract In this paper, the Giaccardi s difference is considered in some special cases By utilizing two classes of the convex functions, the logarithmic convexity of the Giaccardi s difference is proved The positive semi-definiteness of the matrix generated by Giaccardi s difference is shown Related means of Cauchy type are defined and monotonicity property of these means is proved The related mean value theorems of Cauchy type are also given 1 Introduction and preliminaries The well known Giaccardi s inequality[1] is given in the following result see also [5, page 153, 155] Theorem 11 Let f : I R, where I R is an interval, p 1,, p n be a non-negative n-tuple, x 1,, x n be n-tuple in I n and x 0 I such that x n := n p kx k I and 1 x i x 0 x n x i 0 for i =1,, n, x n x 0 If f is a convex function, then p k f x k Af x n +Bfx 0 holds, where n i=1 3 A = p ix i x 0, B = n i=1 p i 1 x n x n x 0 x n x 0 Remark 1 Condition that f is convex function can be replaced with fx fx 0 /x x 0 is an increasing function, then inequality is also valid [5, pages ] 000 Mathematics Subject Classification Primary 6D15, 6D0, 6D99; Key words and phrases Convex function, log-convex function, Giaccardi s inequality, mean value theorems This research was partially funded by Higher Education Commission, Pakistan The research of the first author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant

2 Rad Hrvat akad znan umjet 515 Matematičke znanosti , str 1-10 Moreover, we proved the following result in [3] Theorem 13 Let f : I R, where I R is an interval, p 1,, p n be a non-negative n-tuple, x 1,, x n be n-tuple in I n and x 0 I such that x n := n p kx k I and 1 is satisfied If fx/x x 0 is an increasing function on I\{x 0 }, then 4 p k f x k Af x n holds, where A is the same as in Theorem 11 In the same paper [3], we considered the following Giaccardi s type difference, 5 Υ t = { 1 t 1 A x n x 0 t n } p k x k x 0 t, t 1, A x n x 0 log x n x 0 n p k x k x 0 logx k x 0, t =1, where t R and 6 x n x i >x 0 for i =1,, n If t Υ t is a positive valued function, then for r, s, t R such that r s t, we proved 7 Υ s t r Υ r t s Υ t s r An interesting question is to remove the condition 6 ie consider the case when for some i we have x i x 0 and for some i, x i >x 0 To give an answer to this question, we can consider the case of convex function instead of that in Theorem 13 Namely, we can use the following theorem [] Theorem 14 Let f : I R, where I R is an interval, p 1,, p n be a real n-tuple, x 1,, x n be n-tuple in I n and x 0 I such that x 1 x m x 0 x m+1 x n m {0, 1,,, n}, x n := n p kx k I and k p i x n x i 0 1 k m, 8 i=1 p i x n x i 0 m +1 k n i=k Then, for every convex function f on I, the inequality holds If the reverse inequalities hold in 8, then the reverse inequality holds in

3 Rad Hrvat akad znan umjet 515 Matematičke znanosti , str 1-10 Let us consider the following Giaccardi s difference: 9 Φx; p; f =Af x n p k f x k +Bfx 0 In this paper, we consider two classes of parameterized convex functions and give logarithmic convexity of the difference defined in 9, as a function of parameter We introduce a symmetric matrix generated by the difference and prove that this matrix is positive semi-definite We construct certain Cauchy type means and show that these means are monotone in each variable Also we prove related mean value theorems of Cauchy type Main results Lemma 1 For all t R, let φ t : 0, R be the function defined as x t tt 1, t 0, 1; φ t x = log x, t =0; x log x, t =1 Then φ t x is convex on 0, Proof Since φ t x =x t > 0 for all x 0,,t R, therefore φ t x is convex on 0, for every t R Lemma [4] A positive function f is log-convex in the Jensen sense on an interval I, that is, for each s, t I s + t fsft f, if and only if the relation s + t u fs+uwf + w ft 0 holds for each real u, w and s, t I The following lemma is equivalent to the definition of convex function [5, page ] Lemma 3 If x 1,x,x 3 I are such that x 1 x x 3, then the function f : I R is convex if and only if inequality x 3 x fx 1 +x 1 x 3 fx +x x 1 fx 3 0 holds 3

4 Rad Hrvat akad znan umjet 515 Matematičke znanosti , str 1-10 Theorem 4 Let p =p 1,, p n be a real n-tuple, x =x 1,, x n be a positive n-tuple and x 0 0, such that x 1 x m x 0 x m+1 x n m {0, 1,,, n}, x n := n p kx k 0, and 8 is satisfied Also let Φx; p; φ t, where φ t is defined by Lemma 1, be a positive valued function Then t Φx; p; φ t is log-convex on R Also for r s t, where r,s,t R, we have 10 Φx; p; φ s t r Φx; p; φ r t s Φx; p; φ t s r Proof Let fx =u φ s x+uwφ r x+w φ t x where r,s,t R such that r = s+t and u, w R Then we have f x = u x s +uwx r + w x t, = ux s / + wx t / 0 for x 0, This implies that f is convex on 0, By Theorem 14, we have Af x n p k fx k +Bfx 0 0 and hence we obtain u Aφ s x n p k φ s x k +Bφ s x 0 Since therefore +uw Aφ r x n p k φ r x k +Bφ r x 0 + w Aφ t x n Φx; p; φ t =Aφ t p k x k p k φ t x k +Bφ t x 0 0 p k φ t x k +Bφ t x 0, u Φx; p; φ s +uwφx; p; φ r +w Φx; p; φ t 0 Now by Lemma, we have that Φx; p; φ t is log-convex in the Jensen sense Since lim t 1 Φx; p; φ t =Φx; p; φ 1 and lim t 0 Φx; p; φ t =Φx; p; φ 0 it follows that Φx; p; φ t is continuous for all t R, therefore it is a 4

5 Rad Hrvat akad znan umjet 515 Matematičke znanosti , str 1-10 log-convex function [5, page 6], ie log Φx; p; φ t is convex Now by Lemma 3, we have that, for r s t with f = log Φ, t s log Φx; p; φ r +r t log Φx; p; φ s +s r log Φx; p; φ t 0 This is equivalent to inequality 10 Remark 5 We keep the assumption that the function t Φx; p; φ t, is positive valued in the rest of the paper We shall denote a square matrix of order l by [a ij ], with elements a ij, i, j =1,, l Recall that a real symmetric matrix is said to be positive semi-definite provided the quadratic form l Qx = a ij x i x j i,j=1 is non-negative for all non-trivial sets of the real variable x i, ie x 1,, x l 0,, 0 Theorem 6 Let p =p 1,, p n be a real n-tuple, x =x 1,, x n be a positive n-tuple and x 0 0, such that x 1 x m x 0 x m+1 x n m {0, 1,,, n}, x n := n p kx k 0, and 8 is satisfied For l N, let r 1,, r l be arbitrary real numbers, then the matrix [ ] Φ, where 1 i, j l, x; p; φ r i +r j is a positive semi-definite matrix Particularly ] k det [ Φ x; p; φ r i +r j i,j=1 0 for all k =1,, l [ Proof Define a l l matrix M = φ r i +r j v =v 1,, v l be a nonzero arbitrary vector from R l Consider a function l λx =vmv τ = x Now we have λ x = = l i,j=1 l i=1 i,j=1 v i v j x r i +r j for ], where i, j =1,, l, and let v i v j φ r i +r j v i x r i 0 for x 0, 5

6 Rad Hrvat akad znan umjet 515 Matematičke znanosti , str 1-10 This implies λ is convex on 0, Now applying Theorem 14 for function λ, we have l 0 Φx; p; λ = v i v j Φ x; p; φ r i +r j i,j=1 so the given matrix is positive semi-definite matrix Using well-known Sylvester criterion, we get Φx; p; φ r1 Φ x; p; φ r 1 +r k 11 0 Φ x; p; φ r k +r 1 Φx; p; φ rk for all k =1,, l Remark 7 Note that, we can again deduce that t Φx; p; φ t is log-convex function by taking k =in 11 Definition 1 Let p =p 1,, p n be a real n-tuple, x =x 1,, x n be a positive n-tuple and x 0 0, such that x 1 x m x 0 x m+1 x n m {0, 1,,, n}, x n := n p kx k 0, and 8 is satisfied Then for t, r R, we define Φx; p; φt 1/t r M t,r x; p =, t r, M r,r x; p = exp M 0,0 x; p = exp M 1,1 x; p = exp Φx; p; φ r r 1 rr 1 Φx; p; φ rφ 0 Φx; p; φ r 1 Φx; p; φ 0, Φx; p; φ 0 1 Φx; p; φ 0φ 1 Φx; p; φ 1, r 0, 1, Remark 8 Note that lim t r M t,r x; p =M r,r x; p, lim r 1 M r,r x; p = M 1,1 x; p and lim r 0 M r,r x; p =M 0,0 x; p To prove the monotonicity of M t,r x; p, we shall use the following Lemma [4] Lemma 9 Let f be a log-convex function and assume that if x 1 y 1, x y, x 1 x, y 1 y Then the following inequality is valid: 1 1 fx x x 1 fy y y 1 1 fx 1 fy 1 6

7 Rad Hrvat akad znan umjet 515 Matematičke znanosti , str 1-10 Theorem 10 Let p =p 1,, p n be a real n-tuple, x =x 1,, x n be a positive n-tuple and x 0 0, such that x 1 x m x 0 x m+1 x n m {0, 1,,, n}, x n 0, and 8 is satisfied Also let t,r,v,u R such that r u, t v Then 13 M t,r x; p M v,u x; p Proof As it is proved in Theorem 4 that Φx; p; φ t is log-convex, so taking x 1 = r, x = t, y 1 = u, y = v, where r t, v u and ft =Φx; p; φ t in Lemma 9, we have Φx; p; φt 1/t r Φx; p; φv 1/v u 14 Φx; p; φ r Φx; p; φ u This is equivalent to 13 for r t and v u From Remark 8, we get 14 is also valid for r = t or v = u Similarly, we can consider another class of functions: Lemma 11 For all t R, let φ t : R R be the function defined as Then φ t x is convex on R φ t x = { e tx t, t 0; 1 x, t =0 Proof Since φ t x =e tx > 0 for all x, t R, therefore φ t x is convex on R for each t R Theorem 1 Let p =p 1,, p n, x =x 1,, x n be two real n- tuples, x 0 R such that x 1 x m x 0 x m+1 x n m {0, 1,,, n}, x n := n p kx k R and 8 is satisfied Also let Φx; p; φ t, where φ t is defined by Lemma 11, be a positive valued function Then t Φx; p; φ t is log-convex Also for r s t, where r, s, t R, we have 15 Φx; p; φ t r s Φx; p; φ t s r Φx; p; φ s r t Proof The proof is similar to the proof of Theorem 4 Theorem 13 Let p =p 1,, p n, x =x 1,, x n be two real n- tuples, x 0 R such that x 1 x m x 0 x m+1 x n m {0, 1,,, n}, x n := n p kx k R and 8 is satisfied Also let r 1,, r l be arbitrary real numbers Then the matrix [ ] Φ, where 1 i, j l, x; p; φ r i +r j 7

8 Rad Hrvat akad znan umjet 515 Matematičke znanosti , str 1-10 is a positive semi-definite matrix Particularly [ ] k det Φ x; p; φ r i +r j 0 for all k =1,, l i,j=1 Proof The proof is similar to the proof of Theorem 6 Definition Let p =p 1,, p n, x =x 1,, x n be two real n-tuples, x 0 R such that x 1 x m x 0 x m+1 x n m {0, 1,,, n}, x n := n p kx k R and 8 is satisfied Then for t, r R, Φx; p; M t,r x; p = φ 1 t r t Φx; p; φ, t r, r M r,r x; p = exp r + Φx; p; x φ r Φx; p; φ, r 0, r Φx; p; x M 0,0 x; p = exp φ 0 3Φx; p; φ 0 Remark 14 Note that lim t r Mt,r x; p = M r,r x; p and lim r 0 Mr,r x; p = M 0,0 x; p Theorem 15 Let p =p 1,, p n, x =x 1,, x n be two real n- tuples, x 0 R such that x 1 x m x 0 x m+1 x n m {0, 1,,, n}, x n := n p kx k R and 8 is satisfied Also let r,t,v,u R such that r u, t v Then we have 16 Mt,r x; p M v,u x; p Proof The proof is similar to the proof of the Theorem 10 3 Mean value theorems Lemma 31 Let f C I, where I R is an interval, such that 17 m f x M for all x I Consider the functions ρ 1, ρ defined as, ρ 1 x = Mx fx, ρ x =fx mx Then ρ i x for i =1, are convex on I 8

9 Rad Hrvat akad znan umjet 515 Matematičke znanosti , str 1-10 Proof We have that ρ 1x =M f x 0, ρ x =f x m 0, that is, ρ i x for i =1, are convex on I Theorem 3 Let f : I R, where I R is an interval, p = p 1,, p n be a real n-tuple and x =x 1,, x n be n-tuple in I n and x 0 I such that x 1 x m x 0 x m+1 x n m {0, 1,,, n}, x n := n p kx k I and 8 is satisfied If f C I, then there exists ξ I such that 18 Φx; p; f =f ξφx; p; φ, where φ x =x / Proof Suppose that f is bounded and that m = inf f and M = sup f <m<m< In Theorem 14, setting f = ρ 1 and f = ρ respectively as defined in Lemma 31, we get the following inequalities 19 Φx; p; f MΦx; p; φ, 0 Φx; p; f mφx; p; φ Since Φx; p; φ is positive by assumption, therefore combining 19 and 0, we get, Φx; p; f 1 m Φx; p; φ M Now by condition 17, there exists ξ I such that Φx; p; f Φx; p; φ = f ξ This implies 18 Moreover 19 is valid if for example f is bounded from above and hence 18 is valid Of course 18 is obvious if f is not bounded Theorem 33 Let f,g : I R, where I R is an interval, p = p 1,, p n be a real n-tuple and x =x 1,, x n be n-tuple in I n and x 0 I such that x 1 x m x 0 x m+1 x n m {0, 1,,, n}, x n := n p kx k I and 8 is satisfied If f,g C I, then there exists ξ I such that Φx; p; f Φx; p; g = f ξ g ξ provided that the denominators are non-zero 9

10 Rad Hrvat akad znan umjet 515 Matematičke znanosti , str 1-10 Proof Let a function k C I be defined as where c 1 and c are defined as k = c 1 f c g, c 1 = Φx; p; g, c = Φx; p; f Then, using Theorem 3 with f = k, we have 3 0 = c 1 f ξ c g ξ Φx; p; φ As Φx; p; φ is positive, so we have After putting values, we get c = f ξ c 1 g ξ References [1] F Giaccardi, Su alcune disuguaglilianze, Giorn Mat Finanz 14, [] J Pečarić, On the Petrović inequality for convex functions, Glas Mat 1838, no 1, [3] J Pečarić, and Atiq ur Rehman, On exponential convexity for Giaccardi s type difference, submitted [4] J Pečarić, and Atiq ur Rehman, On logarithmic convexity for power sums and related results, J Inequal Appl 008, Article ID , 008, 9 pp [5] J Pečarić, F Proschan and Y L Tong, Convex functions, partial orderings and statistical applications, Academic Press, New York, Abdus Salam School of Mathematical Sciences, GC University,68-B, New Muslim Town, Lahore 54600, Pakistan Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, Zagreb, Croatia address: pecaric@mahazuhazuhr Abdus Salam School of Mathematical Sciences, GC University,68-B, New Muslim Town, Lahore 54600, Pakistan address: atiq@mathcityorg 10