The Arithmetic-Geometric mean inequality in an external formula. Yuki Seo. October 23, 2012

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1 Sc. Math. Japocae Vol. 00, No , The Arthmetc-Geometrc mea equalty a exteral formula Yuk Seo October 23, 2012 Abstract. The classcal Jese equalty ad ts reverse are dscussed by meas of terally dvdg pots. J.I. Fuj poted out that the cocavty s also expressed by exterally dvdg pots. I ths paper, we shall dscuss a exteral verso of the arthmetc-geometrc mea equalty: For postve real umbers x, y 0 for = 1, 2,, ad r 0 1 x1 + x2 + + x y1 + y2 + + y r x1 + x x 1+r y 1 y 2 y r. 1 Itroducto. The Jese qualty for cocave fuctos s oe of the most mportat equaltes the fuctoal aalyss. Let f be a real valued fucto o a terval J. The classcal Jese equalty s expressed by terally dvdg pots: If f s cocave o J, the 1.1 α fx f α x for all x J ad all α 0 = 1, 2,, such that α = 1, see [3, Theorem 1.1]. Mod-Pečarć [4] showed the followg operator verso of 1.1: If A s a selfadjot operator o a Hlbert space H wth the spectrum J, the 1.2 fax, x f Ax, x for every ut vector x H. I [1], J.I. Fuj poted out that the cocavty s also expressed by exterally dvdg pots: f s cocave o J f ad oly f 1.3 f1 x ry 1 fx rfy for all x, y J ad r > 0 wth 1 x ry J. Thus, a exteral verso of the classcal Jese equalty s as follows: f s cocave o J f ad oly f f α x β j y j 1 α f α α x β j fy j for all x, y j J ad α, β j 0 = 1,, ad j = 1,, k such that α k β j = 1, ad α x k β jy j J, also see [5, p83]. I ths paper, by vrture of a exteral formula, we shall dscuss the arthmetc-geometrc mea equalty. Moreover, we show reverses of the Jese operator equalty by meas of exterally dvdg pots Mathematcs Subject Classfcato. 47A63, 47A64. Key words ad phrases. Cocave fucto, Jese equalty, Reverse equalty, Postve operator, Arthmetc-Geometrc mea equalty.

2 2 Y. SEO 2 Arthmetc-Geometrc mea equalty. The arthmetc-geometrc mea equalty says that for o-egatve real umbers x 1, x 2,, x 2.1 x 1 x 2 x x 1 + x x. By vrture of a exteral formula 1.3, the equalty 2.1 s regarded as oe by meas of terally dvdg pots. I [6], Specht estmated the upper boudary of the arthmetc mea by the geometrc oe for postve umbers: For x 1, x 2,, x [m, M] wth 0 < m M, 2.2 x 1 + x x ad the Specht rato Sh s defed by Sh x 1 x 2 x, 2.3 Sh = h 1h 1 h 1 e log h h 1, h > 0 ad S1 = 1. We call 2.2 the Specht theorem, see [3, Theorem 2.49]. We also have the weghted Specht theorem: For x 0 ad ω 0 for = 1, 2,, wth ω = ω 1 x 1 + ω 2 x ω x Shx ω1 1 xω2 2 xω. Frst of all, we cosder the arthmetc-geometrc mea equalty by vrture of a exteral formula. Theorem 1. For postve real umbers x, y for = 1, 2,, ad r x1 + x x r y1 + y y Proof. We may assume that 1 x1+x2+ +x fucto log t s cocave, t follows that x1 + x x 1 log 1 = 1 log 1 x x log 1 x x = log 1 x x r y y log 1 x x r y y 1+r x1 + x x y 1 y 2 y r. r y1+y2+ +y r y y r y y > 0. Sce the logrthm 1 log y y y y 1 log y y log y log y ad ths mples 1+r x1 + + x log log y 1 y r log 1 x x r y y. By takg the expoet of both sdes, we have the desred equalty 2.5.

3 The Arthmetc-Geometrc mea equalty a exteral formula 3 Remark 2. Theorem 1 mples the arthmetc-geometrc mea equalty 2.1. I fact, f we put r = 1 ad x = y for = 1,, 2.5 of Theorem 1, the we have 2.1. Smlarly we have a exteral verso of the weghted arthmetc-geometrc mea equalty. Theorem 3. For postve real umbers x, y for = 1, 2,, ad ω 1 0 for = 1,, such that ω = 1, ω x r 1+r ω y ω x y ω r. By vrture of a exteral formula 1.3, oe mght expect that x1 + x x r y1 + y y x 1 x 2 x 1+r y 1 y 2 y r. However, the equalty 2.7 does ot hold geeral. By the Specht theorem, we have the followg complemetary equalty to the arthmetc-geometrc mea equalty a exteral formula. Theorem 4. Let x ad y be postve real umbers such that x, y [m, M] wth 0 < m M for = 1, 2,,, ad r 0. If 1 x1+x2+ +x r y1+y2+ +y > 0, the Sh 2r 1 x 1 x 2 x 1+r y 1 y 2 y r 1 x1 + x x r y1 + y y Sh 1+r x 1 x 2 x 1+r y 1 y 2 y r, ad the Specht rato Sh s defed as 2.3. Proof. By the Specht theorem 2.2 ad Theorem 1, t follows that 1 x1 + x x r y1 + y y 1+r x1 + x x y 1 y 2 y r Sh 1+r x 1 x 2 x 1+r y 1 y 2 y r ad ths mples the secod part of Theorem 4.

4 4 Y. SEO Sce the weghted verso of the Specht theorem 2.4 holds, we have 1 log x 1 x 2 x 1 log x 1 + x x 1 = 1 log 1 x x r y y 1 1 log Sh log 1 x x 1 log y y = log Sh 1+r + log 1 x x log Sh 1+r + log 1 x x log Sh + log y 1 + log y ad ths mples the frst part of Theorem 4. r y y r y y 1 r y y y y log y y Remark 5. If r = 0 ad x = y for = 1,, Theorem 4, the we have the Specht theorem 2.2. Smlarly we have the followg exteral verso of Theorem 4. Theorem 6. For postve real umbers x, y [m, M] for = 1, 2,, wth 0 < m M ad ω 1 0 for = 1,, such that ω = 1, 1+r r Sh 2r 1 1 ω x r ω y x ω Sh 1+r x ω y ω 1+r y ω r, where ad the Specht rato Sh s defed as Jese operator equalty. By vrture of a exteral formula, the Jese operator equalty 1.2 fax, x f Ax, x s regarded as a equalty a teral formula. I [2], we showed the followg exteral verso of the Jese operator equalty. Theorem A. Let f be a real valued fucto o a terval J. The f s cocave o J f ad oly f f Ax, x By, y x 2 f A x fby, y x for all x, y H such that x 2 y 2 = 1 ad for all selfadjot operators A ad B wth the spectra J such that Ax, x By, y J.

5 The Arthmetc-Geometrc mea equalty a exteral formula 5 To show the fluctuato of cocavty a exteral formula, we eed the followg wellkow lemma whch s regarded as a reverse of the Jese operator equalty a teral formula, also see [3, Remark 2.7]. Lemma 7. Let A be a selfadjot operator o H wth mi A MI for some scalars m M. If f s cocave o [m, M], the 3.1 x 2 f A x x fax, x + µm, M, f x 2 for all ozero vectors x H where the boud µm, M, f of cocavty s defed by { } fm fm µm, M, f = max ft t m + fm : t [m, M]. Proof. For readers covece, we gve a proof of Lemma 7. Put y = x/ x. Sce f s cocave, t follows that fm fm M m A + Mfm mfm M m I fa. Therefore we have fm fm Mfm mfm f Ay, y Ay, y + + µm, M, f fm fm Mfm mfm = A + y, y + µm, M, f fay, y + µm, M, f. If we replace y by x/ x, the we have the desred equalty 3.1. Though the equalty f Ax, x By, y fax, x fby, y does ot hold geeral for x 2 y 2 = 1, we show the fluctuato of cocavty a exteral formula by usg the boud µm, M, f of cocavty. Theorem 8. If f s cocave o [m, M], the µm, M, f x 2 + y 2 f Ax, x By, y fax, x fby, y µm, M, f x 2 for all x, y H such that x 2 y 2 = 1 ad for all selfadjot operators A ad B wth the spectra J such that Ax, x By, y J. Proof. By Theorem A ad Lemma 7 t follows that f Ax, x By, y x 2 f A x fby, y x fax, x + µm, M, f x 2 fby, y ad ths mples the secod part of Theorem 8. For the frst part of Theorem 8, t follows from Lemma 7 that fax, x x 2 f A x x x 2 1 y 2 f Ax, x By, y + 1+ y 2 1+ y 2 B y y, x 2 1 y 2 f Ax, x By, y + 1+ y 2 1+ y 2 f B y y, = f Ax, x By, y + y 2 f B y y, y y y + µm, M, f x 2 y y + µm, M, f y f Ax, x By, y + fby, y + µm, M, f y 2 +µm, M, f x 2

6 6 Y. SEO as desred. Remark 9. If y = 0 Theorem 8, the we have Lemma 7. exteso of 3.1. Hece Theorem 8 s a As a applcato of Theorem 8, we show the fluctuato of the logarthm fucto a exteral formula by meas of the Specht rato. Corollary 10. If x, y j [m, M] wth 0 < m M ad a, b j 0 for = 1,, ad j = 1,, k such that a k b j = 1 ad a x k b jy j > 0, the log Sh a + b j log a x b j y j a log x b j log y j log Sh a ad the Specht rato Sh s defed by Arthmetc-Geometrc mea operator equalty. Let A ad B be postve operators o H. For each α [0, 1], the weghted arthmetc meaa A α B s defed as A α B = 1 αa + αb ad the weghted geometrc mea A α B s defed as A α B = A 1 2 A 1 2 BA 1 2 α A 1 2. The the followg arthmetc-geometrc mea operator equalty holds: 4.1 A α B A α B for all postve operators A ad B, ad α [0, 1]. The equalty 4.1 s regarded as oe a teral formula. I the followg theorem we propose the arthmetc-geometrc mea operator equalty the exteral formula. Theorem 11. Let A, B, C ad D be postve operators o a Hlbert space H such that C, D are vertble. The for each α [0, 1] 2A α B C α D A α BC α D 1 A α B. Proof. Sce the arthmetc-geometrc mea operator equalty C α D C α D holds for each α [0, 1], t follows that A α BC α D 1 A α B 2A α B + C α D = C α D 1 2 A α B C α D 1 2 C α D 1 2 A α B C α D as desred. C α D + C α D Remark 12. Theorem 11 s a exteso of the arthmetc-geometrc mea operator equalty. I fact, f we put C = A ad D = B Theorem 11, the we have A α B A α B.

7 The Arthmetc-Geometrc mea equalty a exteral formula 7 The equalty 2A α B C α D A α BC α D 1 A α B does ot hold geeral. However, we have the followg theorem by vrture of the Specht theorem. Theorem 13. Let A, B, Cad D be postve vertble operators o H such that mi A, B MI for some scalars 0 < m M. The for each α [0, 1] 2A α B C α D Sh 2 A α BC α D 1 A α B, ad the Specht rato Sh s defed as 2.3. Proof. Sce t follows from [7] that the Specht theorem A α B Sh A α B holds for each α [0, 1], we have Sh 2 A α BC α D 1 A α B 2A α B + C α D = ShC α D 1 2 A α B C α D 1 2 ShC α D 1 2 A α B C α D as desred. + 2ShA α B 2A α B C α D + C α D Refereces [1] J.I. Fuj, A exteral verso of the Jese operator equalty, Sc. Math. Japo., Ole, e-2011, [2] J.I. Fuj, J. Pečarć ad Y. Seo, The Jese equalty a exteral formula, to appear J. Math. Iequal. [3] T. Furuta, J. Mćć Hot, J. Pečarć ad Y. Seo, Mod-Pečarć Method Operator Iequaltes, Moographs Iequaltes 1, Elemet, Zagreb, [4] B. Mod ad J. Pečarć, Covex equaltes Hlbert space, Housto J. Math., , [5] J. Pečarć, F. Proscha ad Y.L. Tog, Covex fuctos, Partal Ordergs, ad Statstcal Applcatos, Academc Press, Ic [6] W. Specht, Zur Theore der elemetare Mttel, Math. Z , [7] M. Tomaga, Specht s rato the Youg equalty, Sc. Math. Japo., , Departmet of Mathematcs Educato, Osaka Kyoku Uversty, Asahgaoka Kashwara Osaka Japa. E-mal address : yuks@cc.osaka-kyoku.ac.jp