AREFIN巳MENT oF THE PoLYA−szEGo lNEQuALITY
AND A REFINBD UPPER BOUND OF CV
MOtosaburo MASUYAMA 〔Received June 25,.1985)’ Let(1) °<m≦Xi≦M・.:f°r.㌍1・2・…㌔些2∼・
and set〔・) 詔r・・ヱ・.x、・……。ノ/n
and
〔・) ・・2ピ、2…22・……。2一言・ 、 ,
Th・n, setti・g謬物r・ヱノ,・・h・v・p・・v・d[・]9。。m。t。ica・・y,h。 fo!lowing inequality for the coefficient of variation:(4) ・・4。m≦..r・・/2−R−・/・〃、1 『 ’.
Whi・h is equiva1・nt t・』 狽?EP61ya−S・egb ih・qu・1ity.In thi・pap…we t・y t・r・fin・thir b・und: .
Let us set as before(・) ・4・・、.・・2.…叫4・Xlx・ヱ/2f’,
and . (・〕.ヒ?S(・.・..∴・/n1/2.’.. 』... .
Then we have to maximize ’ 『 〔・) ・2/Eli2 一 r・’x一品/r・52) 一 xi・/r品一・一・/r・・f/2−・:ゴ ・rt・皿・輌e c・・2θ一relfノ㌔deゆ・.・・n・tra・h・ ’Cb. lh・mi・i・um・f。。.2θi、 att・i。。d by〔8〕 Qr。μr。瑠ノ2/[n(。 ・ bR2/L .㌧
if we set(9) ・』/(k…1ヱノ,・nd.』・.4カ.一・一・/tR・力r・αノ∴ ’
201
202
M.MASUYAMA
provided thatα is an integer [1]. Ifαis not an fnteger, we raise the decimals to a mit, or o皿it the figure below the unit. Let us co頂pare(?(α≠e) and Q(αチε 一 2ノ,where we set [α≠ε]=α≠εby raising. Hence we have O < ε< ヱ. Setting .(・・〕 。42紐.(R2.力。, a。dβ4頑.rR2.、ノ、,
we obtain
(11) er… ノー・2/[賦・ヱノ2].
Anatural conditionαチε≦ni皿plies that .
〔12〕 ε ≦ n/θR ≠ ヱノ,and consequently
〔13〕 。≦nk/rR2.力. ・
Hence we have an inequality 〔14〕 0、< β < α . On th・・ther h・nd, setti。g。4R2.ヱr・の,w。。bt。i。 (15〕 ar・≠ε一カーrα≠。ノ2/[rβ・。)。rR ・ 02]. Now,1et A∼Bmean.that、4 is to be compared with B in.that order. Then without interchanging two sides, we have(16〕 Qr・≠・ノ∼Qr。≠。.力÷。2/β∼rα.。ノ2/rβ +。ノ→
αε ∼ β ÷ 0 ∼ oε2 _ ro ≠ 2アzR)ε ≠ π石∼≦妾 hrε). Since th・di・cri皿i・ant・f・h r・ノ,・㎝・1y r・≠2・R/2.4。nk・一一。2・4。2R2, Is・P°sltlve・tw°「°°ts°f h「・)=O・・ay・1・nd・2…di・tin・t・tS・um・th・t ε1〈ε2・ Since we have(17〕 rR・+ Ohrn/rR≠ヱ〃一(R.力。2.rR2.1≠2nR)n≠nRrR + 1)
一 rR・・〃。.。2ノくOf。r。・2, it follows that (18〕 ・ヱ<・/rR・+ヱノ<ε2・ H・nce th・ni・im㎝・f…2θi・att・i・・d・by・Qr。≠。ノ, if・hr。)≧0,。r ε≦ε1,and otherwise by Q rαチε 一 ヱノ, respectively. The refined inequality(19〕 CV・rヱ/。。。2θ.が/2
,is obtained with such a eos2θ. Exa珂ple 1. 1、et n=4 and R=2. Then, by definition ofα, we have α= 8/3 and b= 4/3, which are not integers. Thus we set ε=1/3 to make [8/3≠1/3]= 3. Since we have hrヱ/3ノニ=2>0,
ll li211°灘1・;,i・二;鷺蒜1。1「α≠ε”=Q「3/as−2θ・ ely・・
0γ・r28/25.が/2と0.3464. The bound by the P61ya−SzegU inequality is approximately O.3536. Example 2. Let n=5 and R=2. Then, we haveα=10/3, Z)=5/3, and ε=2/3. This ti皿e we have 》2r2/3) = − 4 < 0, ,。th。t w。 t。k。 Qr。≠,.ヱノー eC3ノ。、 nin。。。2θ. m。n w。.。bt。in。。。2θ一 72/r5・1ヱノー49/55. H。nce CV・r55/49.が/2≡。.3499. The bound by the P61ya−SzegU inequality is approximately O.3536 as before, since this bound is determined by R and independent of n. Exa]町)1e 3. Let n= 4 and R= 1.5 r</3). Then we have α・=.ヱ2/5 and hence ε= 3/5. Since h r 3/5ノ = − 3/2 < 0, we take rα ≠ ε 一 1) = Q(2),but not Q r3), ・・W。・・2θ,・v・n・th・。gh。・bh・1dS i。 th。 P61ya−S、egU、。1。ti。n〔9). L・t。・d・t。min。 th・i。t。gral v。1u。・f。,。、 a f。n。ti。。・f R2,砲i。h gives the minimu皿of nQ(α). For the sake of si町)1icity, 1et the vertex with mα times 、md M● times as coordinates be denoted by {α}. Now comPare {α}with{αチ ヱ}. Then ・ r・+bR)2/r。チ励∼[。≠ヱ・の.・川2/[。≠ヱ・rわ.ヱ)R2](20〕 4・∼。r。≠ヱノ.rカ.ヱ)bR2−frR2ノ,
where 伽D coefficients αrα ≠ 1/ and rb 一 力力 are nK)notone increasing anddecreasing functions ofαrespectively.
There are two cases:204
M.MASUYAMA
1. アz = 2k. . . … ; . ・ ・ :..・ w〔i〕frR2ノ・O fO・.・−1。2、_.k一ヱ
and . . 』 .. f’CR2)・・f・rα一・一・,・…gP・cti・・ly・日2;::‘『. ’‘ ・ i . . 〔ii〕 For Z ・= 03 ヱ3 ... 3 k − 2, we set α= k ≠i, if〔2・) rk.i−、ノrk.iノ/[rk−i+・ノrk 一川≦♂≦
:’”
@
,・・リた・2・.・1/[・・一.・賑仁力.]・.、.
II. η = 2た 一 2.(i)frR2ノ・。 f。rα一、,・2。...パ..、’⊃.・
.頑幽・..』.s..∴.し.∵.”.‘詞.ン.’」.ジ「∵吊
frR2ノ・。 f。。。.。.、, i。。e、pecti。。・y。f R・.. _. (ii〕 For i = 03 13 ... , k − 3, we set α = k +−i,’・if・ ・.:. 「 ・ (22) rk・+i一ヱノrk・+・i)/[rκ.〃rた一i 一 O]・ s R2≦・・… ・. ・’.. ド rk +・幻.(−k≠−i≠力/[rk−i」力.でk−’t−:.2)L.REFERENCES
[1].呼ワ剛・1・(1?82〕:ぷ叩pe「輌4 fb珂・.卿.興g頑rr.ζ.、,18−1.・.25−28・ [2]岨・・in.1 vie・D・C・〔lg1ZP〕・鋤4W孕卿,協』旬輌9・エ・PP・59−6∼・、.、 [3.]・・P5鑑1:。1言ie;P,11 GGb519,71}三、㌘φZc{』Z・・』麟・:・・1・.!’・ DEPARTMENT OF APPLIED MA[[HHMATICS 『 SCIENCE. UNIVERSITY OF [IOKYO ’N°te・ P°ints X「xヱ・x2・・… xnノ・ati・fying the c・nditi・n(1〕c・nstitut・ a h)?ercube. Each h)rli)erplane containing a facet of this h)互)ercube intersects vertically one of coordinate axes on its positive side and parallel to other axes, so that no line on this hypeエplane passes through the origin O. Step 1. Let X be a point of this h)T)ercube, other than two vertices with identical coordinates Cm3 m⊃ ...3m), say A, and(M, M. ....M), say B. If X is one of its vertices, go to Step 5. Step 2. Let a line through a point X, other than vertices, of the hyper∼ cube, but not through the origin O, be Z, and the foot of the peエpendicular from O to Z be H. Th・。 th。 mi。迦㎝・f。。。2θ一re l f/2。1。。g thi、1i。。 i。 att。i。。d。t。 P・i・tT・n・fac・t・f th。 hyp。rcub。,,ince。。。2θi, a m。n。t。。・decrea、i。g function of lHX l. In ca・e・f・1in・Z passing th・・ugh b・th・O ・nd・X,。。。2θi,麺vari、nt. But, sti11, 1et an intersection point between Z and a facet be T. If T is a vertex of the hypercUbe, go to Step 5. Step 3. Draw a line passing through T and a vertex U of this facet. The Inini皿um along this line is atained either by U, or a point on an edge of this facet. If attained by U, go to Step 5. Step 4. Find the Ininimum a↓ong this edge to reach a vertex Y. St・p 5. 脆・h・uld tワt・fi。d・ut th。㎜血㎝。f。。。2e 、m。ng v。rtices。f the hypercube, except two vertices A and B, which give clearly its maxima. 1.et m and〃apPearα・andカ ti皿es respectively among n coordinates of a vertex. Then we have rN.1〕 ・r・・f/2−r。m ・ bM)2/r。m2 ・ bM2)−r。≠励2/r。+ bR2). Hence we select an integerαwhich∬ini皿izes this under the collui ’1三〇王よ5 〔N.2〕 α ≠ゐ=n⊃ and α, Z)> 0. Remark. Since both the Cassels inequality and the Kantrovich inequality are equivalent to the P61ya−SzegU inequality, the former two inequalities can be improved similarly. .