Holder Means, Lehmer Means, and x 1 log cosh x

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1 Ž. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 0, ARTICLE NO Holder Meas, Lehmer Meas, ad x 1 log cosh x Keeth B. Stolarsky Departmet of Mathematics, Ui ersity of Illiois, 1409 West Gree Street, Urbaa, Illiois Submitted by Bria S. Thomso Received February 8, 1994 The mootoicity of the two variable Holder mea is a cosequece of a certai iequality for log cosh x. This i tur is show to be a limitig case of a iequality ot ivolvig ay trascedetal fuctios. The proof of this elemetary iequality is based o the prelimiary result that H 1 L where H ad L are the Holder ad Lehmer meas of order. Some further iequalities for log cosh x are also proved Academic Press, Ic. 1. INTRODUCTION The subject of classical iequalities may be thought of as stemmig from the simple observatio that 1 0 Ž xy. Ž x y., x, y 0. Ž 1.1. Amog its cosequeces ad geeralizatios are x y x y, x, y 0,, 0, 1, Ž ž Ł i/ Ý i i i 1 i 1 x t y t 1 t ž / 00-47X 96 $18.00 Copyright 1996 by Academic Press, Ic. All rights of reproductio i ay form reserved. x x, x 0, Ž 1.3. is odecreasig i t for t real, Ž 1.4. u 0 log cosh u u tah, Ž

2 HOLDER AND LEHMER MEANS 811 ad sih u sih u x y x y, sih u sih u x, y 0,, 0, 1, Ž u logž x y.. Ž. Ž. 1 u The left iequality of 1.5 is just 1.1 with y x e, but the right oe is rather more: it ca be show to imply Ž 1.4., of which Ž 1.1. is a special limitig case Ž let t 0.. Our mai result, Theorem of Sectio 3, is a reductioistic versio of Ž It is deduced from Theorem 1 of Sectio, a iequality betwee the meas of Holder ad Lehmer, that is much stroger tha that give i B, p.. Theorem 3 of Sectio 4 is a sequece of iequalities betwee six hyperbolic fuctios that icorporates the Lazarevic iequality, ad cosiderably refies Ž The term reductioism has bee used to describe the process of makig results i mathematical aalysis more elemetary by replacig x each statemet ivolvig e by oe ivolvig Ž 1 x. ; see D-N-S, S1, S, S3 Ž of course, this results i statemets with more variables.. Theorem is yet aother example of reductioism. For Ž 1.6. see S4. ad. AN INEQUALITY BETWEEN HOLDER AND LEHMER MEANS Our termiology here is take from B-B, p. 3 wherei t t 1 t a b / H Ž a, b. Ž.1. t ž a t 1 b t 1 LtŽ a, b. Ž. t t. a b deote the Holder ad Lehmer meas of a, b 0. We ote that B-B, p.69 gives the 1950 paper B as the stadard referece o Lehmer meas, although the ame stems from the 1971 paper L. From B, p. we fid that Ž i our otatio. H Ž a, b. L Ž a, b.. Ž.3. t 1 t

3 81 KENNETH B. STOLARSKY We ow establish THEOREM 1. Let 0 be a real umber. The H Ž a, b. L Ž a, b. 1 ad this is best possible i the sese that 1 caot be replaced by ay larger fuctio of. Proof. To see that this is best possible observe that while M 1 H H 1,1 1 M MŽ. 8 N L L 1,1 1 N NŽ.. 4 Thus HM LN for small implies M 1 N. By homogeeity it suffices to prove the iequality of Theorem 1 for a 1 ad b 1. First observe that Ž. a 1 Ž 1.Ž 1. a Ž 1. 0 Ž.4. sice this is true for a 1 ad the left side is icreasig Ž just differetiate.. This implies that a Ž 1. a 1 Ž 1. a 0 Ž.5. by the same argumet. Now Ž.5. says that a 1 Ž 1. a. Ž a 1 a 1 a 1 From Ž.6. ad logarithmic differetiatio it follows that a 1 Ž a 1 ž 1 a 1 / ž / is icreasig for a 1, ad the result follows. Ž THE MAIN RESULT AND ITS CONSEQUENCES I Theorem 1 let b 1 ad make the chage of variable a Ž 1 s. Ž1 s. so that the case a 1 becomes 0 s 1. Sice 1 s Ž 1 s.,it

4 HOLDER AND LEHMER MEANS 813 yields Ž 1 s.ž 1 s s. Ž 3.1. Let x 0 ad choose N so that 0 x N 1. The we have THEOREM. For 0, x 0, ad x N, Ž 1 x N.Ž 1 x N. Ž 1 x N.Ž 1 x N. x Ž 1 x N. Ž 1 x N. 1 N Ž 1 x N. Ž 1 x N. 1. Ž 3.. For a special limitig case take logarithms, set N, ad let N. We obtai x logž cosh x. x tah. Ž 3.3. We ow use Ž 3.3. Ž ideed we oly eed a weakeed form of it. to obtai Ž Note that ad 1 1 t H a, b ' tž. ab Ž cosh ut., u logž a b., Ž 3.4. d tutah tu log cosh ut log H t dt t tu tahž tu. log cosh ut 0. Ž 3.5. t

5 814 KENNETH B. STOLARSKY It is curious that Theorem 1, eve if restricted to positive itegers, implies Ž 1.4., a statemet about all real t. Iequality Ž 3.3. may also be proved by differetiatio. We have tah x x for x 0, so ad hece This leads to ž / ž / x x x x sih x cosh sih x cosh sih x sih x sih x x Ž sih x x. tah. 4 sihž x. coshž x. x x sih cosh x 1 tah Ž x. ad x x tah x tah. cosh Ž x. Hece for x 0 the left side of Ž 3.3. icreases more slowly tha the right side Ž just differetiate.. We ow prove THEOREM A STACK OF CONCAVE FUNCTIONS 1 1 3x coth x log cosh x coth x x 3x x tah tah x Ž 3 x. cothž 3 x., Ž 4.1. sih Ž 3 x. the graphs of all of these fuctios are coca e dowwards, ad each fuctio icreases from 0 to 1 as x icreases from 0 to.

6 HOLDER AND LEHMER MEANS 815 For y tah kx Ž for ay k 0. the cocavity is straightforward, ad for y Ž log cosh x. x it is proved i P-S, pp Next, 3 ž x / ž x/ ž / d 1 sih x coth x cosh x ; Ž 4.. dx sih x the opositivity of this expressio is precisely Lazarevic s iequality M, p. 70. For the rightmost expressio we eed the followig lemma, which is yet aother geeralizatio of the fact that 1 cosh x. LEMMA x sih 3 x x cosh 3 x x 3 x cosh 3 x. Proof. Oly the ceter iequality poses ay difficulty. From 6 sih 3 x 3 sih 3 x 9 x cosh 3 x we obtai cosh 3 x 3 x sih 3 x by differetiatio, ad from 3 cosh 3 x cosh 3 x 3 x sih 3 x we obtai the ceter iequality by differetiatio. Now simply ote that ž / d 3 x 3 x 7Ž x x cosh 3 x sih 3 x. coth. 4 dx sih Ž 3 x. 4Ž sihž 3 x.. Ž 4.3. Ž. That each fuctio i the iequality 4.1 is icreasig is easily see by examiig first derivatives. Thus it remais to verify the iequalities themselves. This requires some further lemmas. LEMMA 4.. For x 0, 3 x x coth tah. 3x Proof. sice For t 1 we have Ž t 3 1.Ž t 1. fž t. 3 log t 0 Ž t 1. t 4 Ž t 1. Ž t t 1. f Ž t.. t Ž 1 t.

7 816 KENNETH B. STOLARSKY Now set t 1 u where 0 u 1. The 1 u 3 1 u Ž u u u 1 u Ž 1 u.ž 1 u. 3 logž 1 u. Now set u e x to obtai the result. LEMMA 4.3. For x 0, 3 x Ž 3 x. tah x coth. sih Ž 3 x. so Ž. Ž. 3 Proof. From 0 t 1 1 t 1 3t t we derive by differetiatio. Thus or 3 cosh x cosh x, 3 x tah x sih x ž / 3 x x 3x cosh x sih x sih x sih cosh sihž 3 x x. sih x 3 x cosh x. This i tur yields Ž cosh 3 x 1. sih x Ž sih 3 x 3 x. cosh x. Now write 3 x 3x Ž., apply both hyperbolic double agle formulas, ad divide by cosh x. The result is 3 x 3 x 3 x sih tah x sih cosh 3 x Ž 4.4. which immediately implies the desired iequality. Sice 3 x log cosh x x coth 3 holds Ž i the limitig sese. at x 0, it follows for x 0 by differetiatio ad Lemma 4.3. Fially, a simple proof of the left-most iequality was give i S5, so Ž 4.1. is established.

8 We ote that S5 also established that HOLDER AND LEHMER MEANS 817 ž / p p p s sih Ž p s. u s ž p/ sih pu ž p / 1 1 cosh pu Ž 4.5. Ž. for 0 s 1, 0 u, ad 1 p. The left-most iequality of 4.1 follows also from the limitig case p of the above. 5. REMARKS A sizable catalogue of iequalities betwee hyperbolic fuctios may be gleaed from P-S, S4, S5. The papers P-S, S4 were iteded to cotai all refereces to the theory of relative errors, but B, which begis by solvig the problem of miimizig the sum of the squares of the relative errors, was ot refereced there. I Ž 4.1. the secod ad last iequalities Ž from the left. are fairly tight: some umerical calculatios idicate that the maximal differece betwee the two sides ca be at most ad Žat x ad x , respectively.. Now oe may go yet further ad attach 1 sih x 1 logž / coth x ; x x x this is equivalet to the iequality betwee the logarithmic ad idetric meas. This raises the followig questio. Call Ž as is sometimes doe. a polyomial i x, expž c x.,..., expž c x. 1 a expoomial. Alteratively, a expoomial is a solutio of a costat coefficiet liear differetial equatio. Is there a sequece of fuctios f Ž x., 1,, 3,..., each a ratio of expoomials ad each icreasig from 0 to 1 as x icreases from 0 to, such that Ž. 1 f Ž x. 0 for x 0, Ž. either f Ž x. f Ž x. for all x 0orf Ž x. f Ž x. m m for all x 0, Ž. 3 assertio Ž. remais valid if f Ž x. is replaced by Ž 1 x. m log cosh x Žor by Ž 1 x. logž sih x x.., ad Ž. 4 i some eighborhood of the graph of y Ž 1 x. log cosh x Žor of Ž 1 x. logž sih x x.. the graphs of the f Ž x. are dese with respect to the uiform Ž supremum. orm? ACKNOWLEDGMENT The author thaks the referee for poitig out a umber of misprits ad obscurities i a earlier versio of this paper.

9 818 KENNETH B. STOLARSKY REFERENCES B E. F. Beckebach, A class of mea value fuctios, Amer. Math. Mothly 57 Ž 1950., 1 6. B-B J. M. Borwei ad P. B. Borwei, Pi ad the AGM, Wiley, New York, D-N-S K. Dilcher, J. D. Nulto, ad K. B. Stolarsky, The zeros of a certai family of triomials, Glasgow Math. J. 34 Ž 199., L D. H. Lehmer, O the compoudig of certai meas, J. Math. Aal. Appl. 36 Ž 1971., M D. S. Mitriovic, Aalytic Iequalities, Spriger-Verlag, Berli, P-S H. Porta ad K. B. Stolarsky, Meas that miimize relative error, ad a associated itegral equatio, J. Math. Aal. Appl. 1 Ž 1987., S1 K. B. Stolarsky, Zeros of expoetial polyomials ad reductioism, i Topics i Classical Number Theory, Coll. Math. Soc. Jaos Bolyai, Vol. 34, Elsevier North-Hollad, Amsterdam, S K. B. Stolarsky, A family of polyomials with cocyclic zeros, Proc. Amer. Math. Soc. 88 Ž 1983., S3 K. B. Stolarsky, A family of polyomials with cocyclic zeros, III, Quart. J. Math. Oxford Ser. () 36 Ž 1985., S4 K. B. Stolarsky, Rademacher, calculus of variatios, iequalities, ad relative error, Cotemp. Math. 166 Ž 1994., S5 K. B. Stolarsky, Searchig for commo geeralizatios: the case of hyperbolic iequalities, Amer. Math. Mothly 10 Ž 1995.,