Giaccardi s Inequality for Convex-Concave Antisymmetric Functions and Applications

Transcription

1 Southeast Asian Bulletin of Mathematics 01) 36: Southeast Asian Bulletin of Mathematics c SEAMS 01 Giaccardi s Inequality for Convex-Concave Antisymmetric Functions and Applications J Pečarić Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore 54600, Pakistan Faculty of Textile Technology, Univ of Zagreb, Pierottijeva 6, Zagreb, Croatia pecaric@mahazuhazuhr Atiq ur Rehman Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan atiq@mathcityorg Received 31 July 007 Accepted 15 June 009 Communicated by WS Cheung AMS Mathematics Subject Classification000): 6D15, 6D0, 6D99 Abstract We give a proof of Giaccardi s inequality for convex-concave antisymmetric functionalso we give the exponential convexity of improvement of Giaccardi s inequality by using positive semidefinite matrix and related mean value theorems of Cauchy type Keywords: Convex-concave; Giaccardi s inequality; Antisymmetric function; Exponentially convex function; Log-convex functions; Positive semidefinite matrix; Mean value theorems 1 Introduction and Preliminaries A well known Giaccardi s inequality [] is given in the following result see also [8, p 155]) 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

2 864 J Pečarić and A Rehman Theorem 11 Let f : I R, where I is an interval Let p be a positive n-tuple, x I n and x 0 I such that x n := n k=1 p kx k I and x i x 0 ) x n x i ) 0 for i = 1,,n, x n x 0 1) If f is a convex function, then p k f x k ) Af x n )+Bfx 0 ) ) holds, where A = k=1 n p ix i x 0 ), B = n p i 1) x n 3) x n x 0 x n x 0 We will keep the notation that x n = n k=1 p kx k through out the paper In [4], the following version of Jensen s inequality for convex-concave antisymmetric functions was proved Theorem 1 Let f : a,b) R be a convex function on a,a + b)/] and fx) = fa+b x) for every x a,b) If x k a,b), p k > 0, x k +x n k+1 )/ a,a + b)/] and p k x k + p n k+1 x n k+1 /p k + p n k+1 ) a,a + b)/] for k = 1,,n, then ) 1 f p k x k 1 p k fx k ), 4) P n P n where P n = n k=1 p k k=1 We shall denote a square matrix of order m by [a ij ], with elements a ij, i,j = 1,, m Recall that a real symmetric matrix is said to be positive semidefinite provided the quadratic form m Qx) = a ij x i x j i,j=1 is nonnegativefor all non-trivial sets of the real variable x i, ie for x 1,,x m ) 0,, 0) Positive semidefinite matrices are very important in theory of inequalities, so in [1] one of the five chapters second chapter) is devoted to them Definition 13 A function f : a,b) R is exponentially convex if it is continuous and ξ i ξ j fx i +x j ) 0 i,j=1 for all n N and all choices ξ i R and x i +x j a,b), 1 i,j n Proposition 14 Let f : a, b) R The following propositions are equivalent: k=1

3 Giaccardi s Inequality for Convex-Concave 865 a) f is exponentially convex b) f is continuous and ) xi +x j ξ i ξ j f 0 i,j=1 for every ξ i R and every x i,x j a,b), 1 i,j n Remark 15 Ifapositivefunctionf : a,b) Risexponentiallyconvexfunction then it is log-convex function [5, page 373] In this paper first we give the proof of Giaccardi s inequality for convexconcave antisymmetric function by using Theorem 1 Further we introduce a positive semidefinite real symmetric matrix to give an exponential convexity of improvement of Giaccardi s inequality for different classes of 3-convex functions and give related results Also we give related mean value theorems of Cauchy type Main Results Theorem 1 Let f : a,b) R be a convex function on a,a + b)/] and fx) = fa+b x) for every x a,b) If x k,x 0 + x n )/ a,a+b)/] for k = 1,,n and 1) is valid, then ) holds Proof Taking n = in an inequality 4), we have ) p1 x 1 +p x f p 1 +p p 1fx 1 )+p fx ) p 1 +p 5) Let p 1 = x n x i, p = x i x 0, x 1 = x 0, x = x n, where x n x i > x 0, then ) ) p1 x 1 +p x xn x i )x 0 +x i x 0 ) x n f = f, p 1 +p x n x i x i x 0 ) xn x i +x i x 0 = f, x n x 0 = fx i ) Also p 1 fx 1 )+p fx ) p 1 +p = x n x 0 )fx 0 )+x i x 0 )f x n ) x n x 0, = x i x 0 ) x n x 0 f x n )+ x n x 0 ) x n x 0 fx 0 )

4 866 J Pečarić and A Rehman Thus 5) implies fx i ) x i x 0 ) f x n )+ x n x 0 ) fx 0 ) x n x 0 x n x 0 n p i f x i ) p ix i x 0 ) f x n )+ n p i 1) x n fx 0 ) x n x 0 x n x 0 This is exactly the ) If x 0 > x i x n, then we can put p 1 = x 0 x i, p = x i x n, x 1 = x n, x = x 0, in 5) to get ) PM Vasić and Lj Stanković [11] proved the following theorem: Theorem Let f : I R, where I = [0,a] Let p be a positive n-tuple, x I n and x 0 I such that x n I and 1) is valid Let us denote Gx;p;f) = p i fx i ) i=0 p i fa x i ) Af x n ) f a x n )) i=0 Bfx 0 ) fa x 0 )) If f is 3-convex function on [0,a], then Gx;p;f) 0 6) J Pečarić and RR Janić [9] gave improvement of the above result and we shall show that the result can be obtained from Theorem 1 Theorem 3 Let p be a positive n-tuple, x [0,a] n and x 0 be nonnegative number such that x 0 + x n )/ [0,a] and 1) is valid Let f : [0,a] R be 3-convex function, then 6) holds Proof Since f : [0,a] R is 3-convex function therefore gx) := fa x) fx) is convex on [0,a] Also note that gx) = ga x), thus applying Theorem 1 for a function g follows 6) To give results related to improvement of Giaccardi s inequality, we consider a following class of 3-convex functions Lemma 4 Let x > x 0, t R + λ t x) = x x 0) t tt 1)t ) if t 1,; x x 0 )lnx x 0 ) if t = 1; 1 x x 0) lnx x 0 ) if t =

5 Giaccardi s Inequality for Convex-Concave 867 Then λ t x) is 3-convex function Here we use a notation that 0ln0 = 0 Proof Since λ t x) = x x 0) t 3 > 0 for each t R +, therefore λ t x) is 3- convex Theorem 5 Let p be a positive n-tuple, x [0,a] n and x 0 be nonnegative number such that x 0 + x n )/ [0,a] and x n x i > x 0 for i = 1,,n Also let r 1,,r m be arbitrary positive real numbers a) The matrix A = [ G )], where 1 i,j m, is a positive- x;p;λr i +r j semidefinite matrix Particularly [ )] k det G x;p;λr i +r j 0 k = 1,,m i,j=1 b) A function t Gx;p;λ t ), where t R +, is a exponentially convex function c) Let Gx;p;λ t ) > 0 Then Gx;p;λ t ) is a log-convex function [ Proof a) Define a m m matrix M = λr i +r j ], where i,j = 1,,m, and let v = v 1,,v m ) be an nonzero arbitrary vector from R m Consider a function Since λ t x) = x x 0 ) t 3, therefore φ t) = φt) = vmv τ m v i t x 0 ) r i 3 ) 0 This implies φt) is 3-convex function Now applying Theorem 3 for function φt), we have ) 0 Gx;p;φ) = v i v j G x;p;λr i +r j Therefore matrix A is positive-semidefinite matrix Specially, we get ) Gx;p;λ r1 ) G x;p;λr 1 +r k 0 7) ) G x;p;λr k +r 1 Gx;p;λ rk ) for all k = 1,,m b) Since lim t i Gx;p;λ t ) = Gx;p;λ i )fori = 1,, it followsthatgx;p;λ t ) is continuous for all t R + Now using Proposition 14 we have exponential convexity of the function t Gx;p;λ t )

6 868 J Pečarić and A Rehman c) It follows from the Remark 15 Definition 6 Let p be a positive n-tuple, x [0,a] n and x 0 be nonnegative number such that x 0 + x n )/ [0,a] and x n x i > x 0 for i = 1,,n Then for t,r R +, ) 1/t r) Gx;p;λt ) A t,r x;p) =, t r Gx;p;λ r ) A r,r x;p) = exp 3r 6r+ rr 1)r ) Gx;p;λ ) r 1λ 1 ), r 1, Gx;p;λ r ) ) Gx;p;λ1 lnx x 0 )) A 1,1 x;p) = exp Gx;p;λ 1 ) A, x;p) = exp 3 + Gx;p;λ ) lnx x 0 )) Gx;p;λ ) Remark 7 Note that lima t,r x;p) = A r,r x;p), lim A r,r x;p) = A 1,1 x;p) t r r 1 and lim A r,r x;p) = A, x;p) r To prove the monotonicity of A t,r x;p), we shall use the following lemma Lemma 8 [6] 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 fx ) fx 1 ) x x 1 fy ) fy 1 ) ) 1 y y 1 8) Theorem 9 Let p be a positive n-tuple, x [0,a] n and x 0 be nonnegative number such that x 0 + x n )/ [0,a] and x n x i > x 0 for i = 1,,n a) Let t,r,u,v R + such that r u, t v Then b) For r < s < t, where r,s,t R + A t,r x;p) A v,u x;p) 9) Gx;p;λ s )) t r Gx;p;λ r )) t s Gx;p;λ t )) s r 10) Proof a) Taking x 1 = r, x = t, y 1 = u, y = v, where r t and ft) = Gx;p;λ t ) in Lemma 8, we have ) 1/t r) Gx;p;λt ) Gx;p;λ r ) ) 1/v u) Gx;p;λv ) 11) Gx;p;λ u )

7 Giaccardi s Inequality for Convex-Concave 869 This is equivalent to 9) for r t and v u From Remark 7, we get 9) is also valid for r = t or v = u b) Put u = t in 9) follows 10) Similarly, we can consider another class of functions: Let x R and e tx x0) λ t x) = t 3 if t 0, 1 6 x x 0) 3 if t = 0 It is clear that function 1) λ t x) = e tx x0) > 0 for each t R, therefore it is 3-convex Theorem 10 Let p be a positive n-tuple, x [0,a] n and x 0 be nonnegative number such that x 0 + x n )/ [0,a] and x n x i > x 0 for i = 1,,n Also let r 1,,r m be arbitrary real numbers a) The matrix A = [ G )], where 1 i,j m, is a positive- x;p; λr i +r j semidefinite matrix Particularly det [ G x;p; λr i +r j )] k i,j=1 0 k = 1,,m ) b) A function t G x;p; λ t, where t R, is an exponentially convex function ) c) Let G x;p; λ t > 0 Then Gx;p;λ t ) is a log-convex function Proof A proof is similar to the proof of Theorem 5 Definition 11 Let p be a positive n-tuple, x [0,a] n and x 0 be nonnegative number such that x 0 + x n )/ [0,a] and x n x i > x 0 for i = 1,,n Then for t,r R Then for t,r R, ) 1/t r) Gx;p; λ t ) B t,r x;p) =, t r Gx;p; λ r ) ) B r,r x;p) = exp 3 r + Gx;p;x x 0) λ r ), r 0 Gx;p; λ r ) ) Gx;p;x x 0 ) λ 0 ) B 0,0 x;p) = exp 4Gx;p; λ 0 ) Theorem 1 Let p be a positive n-tuple, x [0,a] n and x 0 be nonnegative number such that x 0 + x n )/ [0,a] and x n x i > x 0 for i = 1,,n

8 870 J Pečarić and A Rehman a) Let t,r,u,v R such that r u, t v Then b) For r < s < t, where r,s,t R B t,r x;p) B v,u x;p) 13) ) t r ) t s s r Gx;p; λ s ) Gx;p; λ r ) Gx;p; λ t )) 14) Proof The proof is similar to the proof of the Theorem 9 To give mean value theorems related to improvement of Giaccardi s inequality, we consider the following functions: where f C 3 [0,a] and x > x 0, such that It is clear that ρ 1 x) = Mx x 0) 3 fx), 6 15) ρ x) = fx) mx x 0) 3, 6 16) m f x) M 17) ρ 1 x) = M f x) 0, ρ x) = f x) m 0, that is, ρ 1 x) and ρ x) are 3-convex functions Theorem 13 Let p be a positive n-tuple, x [0,a] n and x 0 be nonnegative number such that x 0 + x n )/ [0,a] and 1) is valid Let f C 3 [0,a] and Gx;p;λ 3 ) > 0, then there exists ξ [0,a] such that Gx;p;f) = f ξ)gx;p;λ 3 ) 18) Proof Suppose that 17) is valid < m < M < ) In Theorem 3, setting f = ρ 1 and f = ρ respectively, we get the following inequalities Combine 19) and 0) to get, Gx;p;f) MGx;p;λ 3 ), 19) Gx;p;f) mgx;p;λ 3 ) 0) m Gx;p;f) M 1) Gx;p;λ 3 )

9 Giaccardi s Inequality for Convex-Concave 871 Now by condition 17), there exists ξ [0,a] such that Gx;p;f) Gx;p;λ 3 ) = 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 14 Let p be a positive n-tuple, x [0,a] n and x 0 be nonnegative number such that x 0 + x n )/ [0,a] and 1) is valid Let f,g C 3 [0,a] and Gx;p;λ 3 ) > 0, then there exists ξ [0,a] such that Gx;p;f) Gx;p;g) = f ξ) g ξ) provided that the denominators are non-zero Proof Let a function k C 3 [0,a] be defined as where c 1 and c are defined as k = c 1 f c g, c 1 = Gx;p;g), c = Gx;p;f) Then, using Theorem 13 with f = k, we have ) 0 = c 1 f ξ) c g ξ))gx;p;λ 3 ) 3) Since Gx;p;λ 3 ) is positive valued by assumption, so we have and our proof is complete c = f ξ) c 1 g ξ) 3 Petrović Inequality for Convex-Concave Antisymmetric Function Taking x 0 = 0 in Theorem 1 gives the well known Petrović inequality for convex-concave antisymmetric function Theorem 31 Let f : [0,a] R be a convex function on [0,a] and fx) = fa x) for every x [0,a] If x k, x n / [0,a] and x n x k for k = 1,,n, then n ) p k fx k ) f x n ) p k 1 f0) 4) k=1 k=1

10 87 J Pečarić and A Rehman The following Petrović type inequality can be obtained by taking x 0 = 0 in Theorem 3 also see [9]) Theorem 3 Let f : [0,a] R be 3-convex function and let p be a positive n-tuple, x [0,a] n such that either x i x n a i = 1,,n) 5) or x i a x n a i = 1,,n) 6) is valid, and let us dentoe Px;p;f) = p i fx i ) p i fa x i ) f x n ) +f a x n )+ 1 p i )f0) fa)) Then Px;p;f) 0 The result similar to improvement of Giccardi s inequality can be obtained for Petrovićtype inequalityby takingx 0 = 0 The followingresults, for example, are valid Theorem 33 Let p be a positive n-tuple, x [0,a] n such that either 5) or 6) is valid and x 0 be a non-negative number Also let r 1,,r m be arbitrary positive real numbers [ )] a) The matrix A = P x;p;λr i +r j, where 1 i,j m, is a positivesemidefinite matrix Particularly [ )] k det P x;p;λr i +r j 0 k = 1,,m i,j=1 b) A function t Px;p;λ t ), where t R +, is an exponentially convex function c) Let Px;p;λ t ) > 0, then Px;p;λ t ) is a log-convex function Where λ t is a function defined in Lemma 4 Definition 34 Let p be a positive n-tuple, x [0,a] n such that either 5) or

11 Giaccardi s Inequality for Convex-Concave 873 6) is valid and x 0 be a non-negative number Then for t,r R +, ) 1/t r) Px;p;λt ) C t,r x;p) =, t r Px;p;λ r ) C r,r x;p) = exp 3r 6r+ rr 1)r ) Px;p;λ ) r 1λ 1 ), r 1, Px;p;λ r ) ) Px;p;λ1 lnx x 0 )) C 1,1 x;p) = exp Px;p;λ 1 ) C, x;p) = exp 3 + Px;p;λ ) lnx x 0 )) Px;p;λ ) Remark 35 Note that limc t,r x;p) = C r,r x;p), lim C r,r x;p) = C 1,1 x;p) t r r 1 and lim C r,r x;p) = C, x;p) r Theorem 36 Let p be a positive n-tuple, x [0,a] n such that either 5) or 6) is valid a) Let t,r,u,v R + such that r u, t v Then b) For r < s < t, where r,s,t R + C t,r x;p) C v,u x;p) Px;p;λ s )) t r Px;p;λ r )) t s Px;p;λ t )) s r Theorem 37 Let p be a positive n-tuple, x [0,a] n such that either 5) or 6) is valid Let f C 3 [0,a] and Gx;p;λ 3 ) > 0, then there exists ξ [0,a] such that Px;p;f) = f ξ)px;p;λ 3 ) Theorem 38 Let p be a positive n-tuple, x [0,a] n such that either 5) or 6) is valid Let f,g C 3 [0,a] and Gx;p;λ 3 ) > 0, then there exists ξ [0,a] such that Px;p;f) Px;p;g) = f ξ) g ξ) provided that the denominators are non-zero References [1] E Beckenbach and R Bellman, Inequalities, Springer-Verlag, Berlin, 1961 [] F Giaccardi, Su alcune disuguaglilianze, Giorn Mat Finanz 1 4) 1953)

12 874 J Pečarić and A Rehman [3] KZ Guan, On some inequalities of Ky Fan type related to a new mean, Southeast Asian Bull Math ) [4] S Hussain, J Pečarić, I Perić, Jensen s inequality for convex-concave antisymmetric functions and applications, J Inequal Appl 008, Article ID , 6p [5] DS Mitrinović, J Pečarić, AM Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, The Netherlands, 1993 [6] J Pečarić and A Rehman, On logarithmic convexity for power sums and related results, J Inequal Appl 008, Article ID , 9p [7] J Pečarić, and MR Lipanović, Cauchy type mean-value theorems for mid-point and trapezoidal quadrature formulas, Southeast Asian Bull Math ) [8] J Pečarić, F Proschan, YC Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York, 199 [9] J Pečarić and RR Janić, Remark on an inequality for 3-convex functions, Facta Univ Ser Math Inform ) 3 5 [10] J Pečarić, MR Lipanović, Cauchy type mean-value theorems for mid-point and trapezoidal quadrature formulas, Southeast Asian Bull Math ) [11] PM Vasić and L Stanković, On some inequalities for convex and g-convex functions, Univ Beograd Publ Elektrotehn Fak Ser Mat Fiz ) 11 16