The logarithmic mean is a mean

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1 Mthemticl Communictions 2(1997), The logrithmic men is men B. Mond, Chrles E. M. Perce nd J. Pečrić Abstrct. The fct tht the logrithmic men of two positive numbers is men, tht is, tht it lies between those two numbers, is shown to hve number of consequences. Key words: logrithmic men, potentil mens, opertor entropy Sžetk.Logritmsk sredin je sredin. Činjenic d je logritmsk sredin dv pozitivn broj sredin, tj. d leži izmed u t dv broj im niz primjen kko z potencijlne i generlizirne logritmske sredine tko i u teoriji opertor. Ključne riječi: logritmsk sredin, potencijln sredin, opertorsk entropij AMS subject clssifictions: 26D15, 47A63 Received Februry 25, 1997, Accepted My 27, Introduction The logrithmic men of positive numbers x, y is given by L(x, y) = x y ln x ln y if x y nd by L(x, x) = x if x = y. It is men, tht is, we hve min(x, y) L(x, y) mx(x, y). (1) Here we shll show tht this fct lone leds to vriety of interesting results. Tke x 1. Setting y = 1 in (1) provides min(x, 1) x 1 mx(x, 1), ln x School of Mthemtics, L Trobe University, Bundoor, Vic. 3083, Austrli, e-mil: mthbm@lure.ltrobe.edu.u Deprtment of Applied Mthemtics, University of Adelide, Adelide SA 5005, Austrli, e-mil: cperce@mths.delide.edu.u Fculty of Textile Technology, University of Zgreb, Pierottijev 6, HR Zgreb, Croti, e-mil: pecric@cromth.mth.hr

2 36 B. Mond, C. E. M. Perce, nd J. Pečrić so tht ln x x 1 x ln x for x > 1. Similrly, the sme reltion holds for 0 < x < 1. In fct, equlity occurs only for x = 1. So we hve ln x x 1 x ln x for x > 0. (2) Schumberger [7] hs shown tht (2) provides strikingly direct route to severl results involving power mens. We my use similr development to obtin results for integrl mens but with nonuniform rther thn uniform weights. In Section 2 we derive key inequlity for potentil mens in this wy. A discrete version is lso given. In Section 3 we note the existence of nlogous results for other mens which possess integrl representtions. Finlly, in Section 4, we observe pplictions to the reltive opertor entropy of two positive opertors on Hilbert spce. 2. Potentil mens Let f, w : [, b] R be positive, integrble functions. The potentil men of order r of function f with weight function w is given by b 1/r w(t)f(t) r dt M r (f, w) =, r 0, w(t)dt M 0 (f, w) = exp w(t) ln f(t)dt, r = 0. w(t)dt For convenience we write f for f(t), w for w(t), M for M r (f, w) (r 0) nd M 0 for M 0 (f, w). Set x = f r /M r in (2). We get fter multipliction by wm r tht Integrtion gives tht is, for r > 0, M r w ln f dt M r ln M This is equivlent to rwm r ln f M wf r wm r rwf r ln f M. rm r w ln f M dt 0 r wf r ln f M dt, w dt 0 M r ln M 0 M r ln M 0 wf r ln f dt ln M wf r ln f dt M r ln M w dt wf r dt.

3 The logrithmic men is men 37 or Therefore, we hve tht is, in extenso, M r ln M 0 M r ln M wf r ln f dt M r 0 wf r dt M 0 M M 0 (f, wf r ), M 0 (f, w) M r (f, w) M 0 (f, wf r ) for r > 0. (3) If r < 0, the inequlities re reversed. Assuming ll integrls exist, consequence of (3) is tht lim M r(f, w) = M 0 (f, w). r 0 A similr development is vilble in the discrete cse. If, w re positive n tuples, potentil men of order r with weights w is given by { where W n = obtin M [r] n (, w) = M [0] n = 1 W n { n } 1/r n w i r i, r 0, w i i } 1/Wn, n w i. With the nottion w r = (w 1 r 1, w 2 r 2,..., w n r n) we cn M [0] n (, w) M [r] n (, w) M [0] n (, w r ), which is generliztion of result from [6] for the cse of n unweighted men. We hve gin tht lim M n [r] (, w) = M n [0] (, w). r 0 3. Anlogues We now consider the generlized logrithmic men of order r of positive numbers x, y. For x y this is defined by L r (x, y) = { x r y r } 1/(r 1), r 0, 1, r(x y) L 0 (x, y) = L(x, y), L 1 (x, y) = I(x, y) = 1 e (xx /y y ) 1/(x y)

4 38 B. Mond, C. E. M. Perce, nd J. Pečrić nd for x = y by L r (x, x) = x. These mens hve n integrl representtion L r (x, y) = M r 1 (e 1, e 0 ), where e 1 (t) = t, e 0 (t) = 1 for ll t [min(x, y), mx(x, y)]. So, consequence of (3) for r > 1 is tht I(x, y) L r (x, y) I(x r, y r ) 1/r. (4) Reverse inequlities pply for r < 1 (r 0), while for r = 0 we hve We hve immeditely from (4) tht G(x, y) xy L(x, y) I(x, y). lim L r(x, y) = I(x, y). r 1 In plce of the integrl potentil men of the function f : [, b] R we could consider more generl potentil mens of functions g : Ω R, where Ω is n rbitrry set. For such mens, (3) follows in the sme wy. As specil cses we cn gin consider, for exmple, the logrithmic mens on n vribles (see Pittenger [5]), the hypergeometric men (see Brenner nd Crlson [1]) nd other mens which hve representtions in the form of integrl mens (see [1]). 4. Opertor theory A further ppliction lies in the theory of opertors. Fujii nd Kmei [2] introduced the notion of the reltive opertor entropy S(A B) for positive opertors A, B on Hilbert spce. For A nd B invertible, this is given by Fujii nd Kmei [3] proved tht S(A B) = A 1/2 ( ln A 1/2 BA 1/2) A 1/2. A AB 1 A S(A B) B A. (5) We cn derive this (see e.g. [6] pp ) s consequence of (2), which my be re expressed s 1 1/x ln x x 1 for ll x > 0. (6) Since A 1/2 BA 1/2 is positive opertor, we my substitute it for x to obtin I A 1/2 B 1 A 1/2 ln A 1/2 BA 1/2 A 1/2 BA 1/2 I. Pre nd postmultipliction by the positive opertor A 1/2 now provide (5). Recently Furut [4] proved generliztion of this inequlity, nmely, if A nd B re positive invertible opertors, then for ny positive number x 0 we hve (ln x 0 1)A + Bx 1 0 S(A B) (1 log x 0 )A AB 1 Ax 1 0. (7)

5 The logrithmic men is men 39 We show tht this follows from (6). For on substituting x/x 0 for x in (6) we obtin ln x (x 0 /x) ln x ln x 0 + (x/x 0 ) 1 for ll positive x, x 0. (This inequlity is equivlent to (1) for y = x 0 ). Agin we replce x by the positive opertor A 1/2 BA 1/2 to derive (ln x 0 +1)I x 0 A 1/2 B 1 A 1/2 ln A 1/2 BA 1/2 (ln x 0 1)I +A 1/2 BA 1/2 x 1 0. Pre nd postmultipliction by A 1/2 s before yields (ln x 0 + 1)A x 0 AB 1 A S(A B) (ln x 0 1)A + Bx 1 0. (8) We cn replce x 0 by x 1 0 to give (1 ln x 0 )A x 1 0 AB 1 A S(A B) Bx 1 0 (ln x 0 + 1)A. (9) The first inequlity in (9) tken with the second in (8) provides (7). References [1] J. L. Brenner, B. C. Crlson, Homogeneous men vlues: weights nd symptotics, J. Mth. Anl. Appl. 123(1987), [2] J. I. Fujii, E. Kmei, Reltive opertor entropy in noncommuttive informtion theory, Mth. Jpon. 34(1989), [3] J. I. Fujii, E. Kmei, Uhlmnn s interpoltionl method for opertor mens, Mth. Jpon. 34(1989), [4] T. Furut, Furut s inequlity nd its ppliction to the reltive opertor entropy, J. Opertor Theory 30(1993), [5] A. O. Pittenger, The logrithmic men in n vribles, Amer. Mth. Monthly 92(1985), [6] F. Riesz, B. Sz-Ngy, Functionl nlysis, Trnslted from the 2nd French edition by Leo F. Boron, Frederick Ungr Pub. Co., [7] N. Schumberger, Extending fmilir inequlity, Pi Mu Epsilon Journl 10(1992),