On some variants of Jensen s inequality

Transcription

1 On some varants of Jensen s nequalty S S DRAGOMIR School of Communcatons & Informatcs, Vctora Unversty, Vc 800, Australa EMMA HUNT Department of Mathematcs, Unversty of Adelade, SA 5005, Adelade, Australa and Survellance Systems Dvson, DSTO, PO Box 500, Ednburgh 5, Australa Abstract Some varants of Jensen s dscrete nequalty are derved These nclude nterpolatons of the basc relaton for subaddtve maps and of the generalsed trangle nequalty AMS Subject Classfcaton: 26D5 Key words and Phrases: Jensen s nequalty, generalsed trangle nequalty, subaddtve maps Introducton Let X be a real lnear space and C X a convex set n X, that s, a set such that x, y C and λ [0, ] mply λx + λy C

2 If f : C R s convex, f satsfes fλx + λy λfx + λfy for all x, y C and λ [0, ] If p 0 =,, n wth P n := n p > 0 and y C =,, n, we have the Jensen nequalty f p y n p fy P n P n see [2] or [8, p 6] For some recent generalzatons, refnements and applcatons the reader s referred to [] [7], [9] and [8, p 20] In ths paper we show that several new results flow from smple but judcous applcatons of Abel s dentty, whch gves the followng Suppose X s a lnear space, x X =,, n and s n := n x If a s real =,, n, then a x = a s + a s s = n a a + s + a n s n Consequences nclude an nterpolaton of the basc nequalty for subaddtve maps and of the generalsed trangle nequalty 2 Results We wll start wth the followng theorem Theorem 2 Let X be a lnear space and f : X R a convex mappng, x,, x n X and 0 a a 2 a n 0 Then f a a x a a fx + a f x j f x j 2

3 Proof Choose p := a a + < n, p n := a n and y = s =,, n n Jensen s theorem We derve [ n ] a a + s n a a + fs f n a a + n, a a + where for notatonal smplcty we have ntroduced a n+ := 0 The desred result now follows by Abel s dentty Corollary 22 Let g : X 0, be logarthmcally concave, that s, let ln g be concave Under the assumptons of the theorem g a n a x [gx ] a g a /a x g x j The result follows from the theorem for the convex mappng f = ln g Suppose that the mappng ϕ : X R s subaddtve, that s, for α, β nonnegatve we have ϕαx + βy αϕx + βϕy By mathematcal nducton we have for all α 0 and y X =, n that n ϕ α y α ϕy Ths nequalty may be nterpolated as follows Corollary 23 Let ϕ : X R be subaddtve, y,, y n X and α α 2 α n 0 Then n ϕ α y α ϕy + α ϕ y j ϕ y j α ϕy Proof As ϕ s subaddtve, t s convex The frst desred nequalty follows from Theorem 2 3

4 For the second, we observe that for 2 n, ϕ ϕ ϕy y j y j Multplyng the th nequalty by α and summng over provdes the desred result Our second man result s the followng Theorem 24 Let f : X R be convex and x X =,, n Suppose that m =,, n satsfy m j 0 n and Then f n + m > 0 n m x n n + m n nj= f x j x j+ n n + m, where agan we put x n+ := 0 for notatonal convenence Proof Let s = m j n Then by Abel s dentty m x = s x + s s x = s x x + Applyng Jensen s nequalty provdes [ n ] s x x + n s fx x + f n s n x The numerator on the rght hand sde may be wrtten as n m fx j x j+ j= 4

5 and we have the desred result Corollary 25 Let g : X 0, be logarthmcally concave Wth the above assumptons n m m x / n n n n+ m g n gx n + m j x j+ j= The result follows from the theorem wth the choce of convex mappng f = ln g 3 Applcatons We now derve some partcular applcatons relatng to homely choces of convex functon Let x > 0 =,, n wth 0 a a 2 a n 0 Then a x a [x a n a ] /a x j x j The result follows from Corollary 22 wth the mappng g : 0, 0, gven by gx = x Suppose x x 2 x n 0 and n R wth m 0 In the same way we have from Corollary 25 that m n m x / n n n n+ m n x n + m j x j+ j= 2 Let x > 0 =,, n and 0 a a 2 a n 0 Then n a 2 a x a a x x k= x j x k Ths follows from Theorem 2 appled to the convex mappng fx = /x on the nterval 0, 5

6 3 Let x R and a a 2 a n 0 Then n 2 a x a a x 2 + a x x + 2 x j Ths follows from Theorem 2 appled for the convex mappng fx = x 2 x R 4 Consder the mappng f : R R gven by fx = ln + e x We have f x = e x / + e x and f x = e x / + e x 2, whch shows that f s convex on R Let 0 a a 2 a n 0 and x,, x n R Then by Theorem 2 [ ] ln + exp a a x a ln[ + e x ] + a ln + exp x j ln + exp x j n = ln + ex a + exp a x j + exp x, j whence + exp a n a x [ + e x ] a + exp x j + exp a x j 5 Let X be a real normed space and α α 2 α n 0 Then for x X =,, n we have the refnement α x α x + α x x α x 6

7 of the generalsed trangle nequalty The result follows from Corollary 23 References [] S S Dragomr, Some refnements of Ky Fan s nequalty, J Math Anal Appl , [2] S S Dragomr, Some refnements of Jensen s nequalty, J Math Anal Appl , [3] S S Dragomr, On some refnements of Jensen s nequalty and applcatons, Utltas Math , [4] S S Dragomr, Two mappngs assocated wth Jensen s nequalty, Extracta Math 8 993, [5] S S Dragomr, A further mprovement of Jensen s nequalty, Tamkang J Math 5 994, [6] S S Dragomr and N M Ionescu, Some remarks on convex functons, Anal Num Theor Approx 2 992, 3 36 [7] S S Dragomr and D M Mloševć, A sequence of mappngs connected wth Jensen s nequalty and applcatons, Mat Vesn , 3 2 [8] D S Mtrnovć, J E Pečarć and A M Fnk, Classcal and New Inequaltes n Analyss Kluwer Academc Publshers, Dordrecht/Boston/London 993 [9] J E Pečarć and S S Dragomr, A refnement of Jensen nequalty and applcatons, Studa Unv Babes Bolya , 5 9 7