Some Mean Inequalities

Transcription

1 Irish Math. Soc. Bulleti 57 (2006), Some Mea Iequalities FINBARR HOLLAND Dedicated to Trevor West o the occasio of his retiremet. Abstract. Let P deote the collectio of positive sequeces defied o the set of atural umbers N. It is proved that if x P, ad s < 0, the ( ) x /s s ( ( ) /s ) s, x N with equality if ad oly if x is a costat sequece. This is a sharp refiemet of a iequality discovered by Kopp i Itroductio Whe I received the ivitatio to participate i the Westfest, I was i the throes of writig up a solutio to the followig problem, due to Joel Zi, which was posed i the America Mathematical Mothly, ad I offered to spea o this topic at the meetig i TCD to mar Trevor s retiremet. I m grateful to the orgaisig committee of the Westfest for givig me the opportuity to do so. Problem (Number 45). Fid the least c such that if, a,..., a > 0, the a c a. I propose to describe a method to hadle a family of similar problems of which this, ad classical oes due to Carlema, ad Kopp, are special cases.

2 70 Fibarr Hollad 2. Bacgroud We deote by P the collectio of positive sequeces x : N (0, ). Clearly, P is a covex set. It is closed uder the usual poitwise operatios of additio ad multiplicatio, ad ordered by the relatio: x y x y, N. I particular, P is a commutative group uder multiplicatio, with the sequece vector e of oes actig as the idetity. We ll write /x for the multiplicative iverse of x P: (/x) x, N. We recall a umber of familiar fuctios that tae P ito itself: A : P P; A(x) x,, 2,... ; G : P P; G(x) x,, 2,... ; H : P P; H(x) x,, 2,... ; mi : P P; mi(x) mi{x :, 2,..., }. These are, respectively, the arithmetic, geometric, harmoic ad miimum meas. (Weighted versios of these exist, but I ll ot have ay eed to refer to them.) It is a well-ow fact [5] that mi(x) x x,, 2,.... x Moreover, the iequalities are strict uless x is a costat sequece. Equivaletly, mi H G A. It s clear from the defiitios that A, G, H ad mi are homogeeous i the sese that, if f {A, G, H, mi}, the f(λx) λf(x), x P, λ > 0.

3 Some Mea Iequalities 7 It s perhaps less obvious, but oetheless true, that they are superadditive: if f {A, G, H, mi}, the f(x) + f(y) f(x + y), x, y P. Hece they are also cocave o P. We also itroduce a oe-parameter family of fuctios {M t : t > 0} that leave P ivariat. If x P, we defie M t (x) by ( M t (x) so that M t (x) (H(x /t )) t, ad x /t ) t,, 2,..., mi(x) M t (x) G(x) A(x), x P, t > 0, lim M t(x) mi(x), t 0 + lim M t (x) G(x), x P. t 3. A Iequality Betwee Compositios of Meas I m iterested i compositioal relatioships betwee these various fuctios. I ll describe the followig result. Theorem. Let t > 0. The A M t M t A. Moreover, A M t (x) M t A(x) if ad oly if x λe for some λ > 0. For istace, whe t, the claim is that A H H A. Equivaletly,,, 2,... x x Eve for small values of this is already fairly challegig, as the reader may discover for him or her self by cosiderig the special case 3. A more geeral weighted iequality of this id was first postulated by Naudiah [3] i 952, but he offered o proof, ad ideed his coecture is ot true geerally. A special case of it was coectured by myself [6] i 992, ad Kedlaya [8] supplied a proof of this i 994, amely that A G G A. I 996, Mod ad Pečarić [2] proved a aalogue of the iequality A H H A, the case t, for Hermitia matrices. To establish the theorem, we begi by provig a lemma.

4 72 Fibarr Hollad Lemma. Let t > 0 ad let p t +. Let x P. The, for each, { } M t (x) t if x a p : 0 < a R, a. Proof. Suppose 0 < a R, ad a. Let q p/(p ) p/t. The, by Hölder s iequality, (a p x ) /p x /p Hece Equality holds here if It follows that ( a x /t ( ) /p ( a p x ) /q, x q/p ( x q/p x q/p ) t The stated result follows. ) p/q ( x q/p a p x. ( ) p/q x q/p ),, 2,...,. { if x a p : 0 < a R, } a. A equivalet formulatio is that, with p t +, { } M t (x) p if x a p : 0 < a R, a. () Thus M t (x) p for all probability vectors a R. x a p (2)

5 x /t Some Mea Iequalities 73 Remar. Already this result reveals that M t is super-additive ad hece cocave. The result we wat to prove is the followig: if x P, ( ) t M t (x) M t (A(x)) ( A(x) /t ) t,, 2,..., with equality if ad oly if x is a costat sequece. Our idea is this: with fixed, suppose a is a probability vector i R. The, by the previous lemma, with p t +, M t (A(x)) p a p A(x) p p a p x x a p, after iterchagig the order of summatio, ad there is equality here for a suitable a. But, also, if u i R i is a probability vector, i M t (x) i i p u p i x, i, 2,...,, whece M t (x) i i i i p i u p i x x i i p u p i. So, we ca accomplish our obective if, give a probability vector a R, we ca costruct similar vectors u i R i so that i p u p i a p p,, 2,...,. (3) i

6 74 Fibarr Hollad To reach our goal, ad to show that these iequalities ca be solved, we costruct a certai lower triagular row-stochastic matrix from a probability vector. To this ed, we use the followig result due to Kedlaya [8]: Lemma 2. The ratioal umbers ( i ) ( i ) α (i, ) ), i,,, ( satisfy the followig coditios () α (i, ) 0, for all i,, ; (2) α (i, ) 0 for all > mi (i, ); (3) α (i, ) α (, i) for all i,, ; (4) α (i, ) for all i, ; { (5) α (i, ), for, 0, for >. i Give a probability vector a R 2, costruct the matrix A [a i ] by a i α (i, )a, i,. The each row of A is a probability vector, because a i α (i, )a a α (i, ) a (by 4.) for all i. Also, a i 0 for all > i. Thus A is a lower triagular row-stochastic matrix. But, for each pair of idices i,, a i is a covex combiatio of a, a 2,..., a, ad so, if p, a p i α (i, )a p, i,, 2,...,.

7 Some Mea Iequalities 75 Hece i p a p i i i p a p i i p p p i α (i, )a p a p i a p α (i, ) (by 5.). Looig bac at (3) we ow see that we ca solve this by selectig u i a i,, 2,..., i. We re ow ready to provide a proof of the theorem. Fix x P, ad a positive iteger. Let a be a probability vector i R. Choose the correspodig lower triagular row-stochastic matrix A [a i ] as above. By Lemma, if i, Hece M t (x) i i p M t (x) i i i i a p i x. i p i a p i x x i x p p i p a p i a p a p x p a p A(x).

8 76 Fibarr Hollad Whece t i M t (x) i is a lower boud for the set { a p A(x) : 0 < a R, whose ifimum is t M t (A(x)). Hece M t (x) i M t (A(x)), i } a, ad we re doe, apart from dealig with the case of equality, which is easily settled. 4. A Number of Corollaries We deduce a umber of special cases of Theorem. Corollary. A mi mi A. This is obtaied by lettig t 0 +. Corollary 2 (Kedlaya). i.e., x P, A G G A. G(x) i G(A(x)), x P, ; i or, more explicitly, i i x i This is obtaied by lettig t. classical iequality [3, 5, 7]: Corollary 3 (Mod & Pečarić). x i. i G(x) e x, x P. A H H A. This implies Carlema s

9 Some Mea Iequalities 77 This is obtaied by lettig t. It says that, x P,,, 2,... x x Sice the sequece x,, 2,... is strictly icreasig we deduce that the right-had side does ot exceed x 2 x, + whece 2 x < 2 x, + x which gives a solutio to Zi s Mothly problem. Moreover, the costat 2 caot be replaced by a smaller umber, as ca be see by taig x /,, 2,...,. Thus, if x P l, so does H(x), ad H(x) < 2 x. Corollary 4. t > 0 ad x P, ( ) t +/t M t (x) i /t i ad the iequality is strict uless. Proof. M t (A) p ( p A /t x,, 2,... ) t /t ( x ) /t ( ) t p x /t ( ) t +/t /t x. t

10 78 Fibarr Hollad Sice lim +/t /t + t, a simple cosequece of the fact that, with s /t, s s+ ( )s is a Riema sum for the itegral 0 x s dx + s, Corollary 4 implies a result of Kopp [0] to the effect that M t (x) ( + /t) t x, x P. (4) 5. Compaio Results Whe t < 0 The meas M t also mae sese whe t < 0. Similar methods to those employed i the previous sectio lead to the followig statemet. Theorem 2. If < t < 0, the A M t M t A. Moreover, A M t (x) M t A(x) if ad oly if x λe for some λ > 0. Lettig p /t, we ca recast this i terms of p: If p, the, for all x P, ad all, ( ( ) p ) /p ( ) /p x i x p i. (5) i There is equality oly whe x is a costat sequece. This is a substatial improvemet of a very well-ow result due to Hardy [4, 5], which states that, if x l p, the A(x) l p, ad A(x) p p p x p. Iequality (5) was foud by Beett [2], who poited out that the reversed iequality holds whe 0 < p <. A stroger form of (5) was established by B. Mod ad J. E. Pečarić [], ad a weighted versio of their result was outlied by Kedlaya [9]. But results of this id were aouced much earlier by Naudiah [3], though he appears ot to have published a proof. i

11 Some Mea Iequalities 79 Refereces [] G. Beett, Factorizig the Classical Iequalities, Mem. Amer. Math. Soc. 20 (996), o. 576, viii+30pp. [2] G. Beett, Summability matrices ad radom wal, Housto J. Math 28 (2002), o. 4, [3] T. Carlema, Sur les fuctios quasi-aalytiques, Fifth Scad. Math. Cogress (923), [4] G. H. Hardy, Notes o some poits of the itegral calculus (LX), Messeger of Math. 54 (925), [5] G. H. Hardy, J. E. Littlewood, ad G. Pólya, Iequalities, Cambridge Uiversity Press, 934. [6] F. Hollad, O a mixed arithmetic-mea, geometric-mea iequality, Math. Competitios 5 (992), [7] M. Johasso, L. E. Persso, A. Wedestig, Carlema s iequality history, proofs ad some ew geeralizatios, J. Ieq. Pure & Appl. Math. 4 (3), (2003), 9. [8] K. Kedlaya, Proof of a mixed arithmetic-mea, geometric-mea iequality, Amer. Math. Mothly 0 (994), [9] K. Kedlaya, A weighted mixed-mea iequality, Amer. Math. Mothly 06 (999), [0] K. Kopp, Über Reihe mit positive Glieder, J. Lodo Math. Soc., 3 (928), [] B. Mod ad J. E. Pečarić, A mixed meas iequality, Austral.Math. Soc. Gaz., 23(2) (996), [2] B. Mod ad J. E. Pečarić, A mixed arithmetic-mea-harmoic-mea iequality, Li. Algebra Appl. 237/238 (996), [3] T. S. Naudiah, Sharpeig of some classical iequalities, Math. Studet 20 (952), Fibarr Hollad, Mathematics Departmet, Uiversity College, Cor, Irelad f.hollad@ucc.ie Received o 8 Jue 2006.