[‰ß‹ŽÛ¸Þ] •s“™Ž®‚ւ̵‘Ò ‘æ‚QÍ (989Ú½)
㉺‘OŽŸ1-V
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‚±‚̽گÄނ͉ߋŽÛ¸Þ‘qŒÉ‚ÉŠi”[‚³‚ê‚Ä‚¢‚Ü‚·¡
ŽŸ½ÚŒŸõ —ðí¨ŽŸ½Ú žxí¨ŽŸ½Ú ‰ß‹ŽÛ¸ÞÒÆ­°
219
(3): 2005/03/24(–Ø)23:05 AAS
AAÈ
222: 2005/03/25(‹à)20:55 AAS
>220,221
@N=2n+1 ‚Æ‚¨‚­B
@w ß cot(kƒÎ/N)^2 = {1+cos(2kƒÎ/N)}/{1-cos(2kƒÎ/N)} ß (1+z)/(1-z).
@z=cos(2kƒÎ/N) ‚Í f(z)={T_N(z)-1}/(z-1)=0 ‚̪‚¾‚©‚çAf((w-1)/(w+1))=0.
@—¼•Ó‚É (w+1)^(N-1) ‚ðŠ|‚¯‚ÄAw^(N-2) ‚ÌŒW”‚ð‹‚ß‚½‚悤‚È‹HƒKƒX‚éB
@‚±‚±‚É T_N(z) ‚Í NŽŸ‚̃`ƒFƒrƒVƒFƒt‘½€Ž®

@ŠO•”ØÝ¸:functions.wolfram.com
@ŠO•”ØÝ¸:functions.wolfram.com

>219 ‚ÌŽQl‚ÉiHlawka‚Ì•s“™Ž®j
@|x| + |y| + |z| + |x+y+z| † |x+y| + |y+z| + |z+x|.
È1
243
(2): 2005/03/30(…)00:57 AAS
>>219
(1)@x(x+y+z) < x(x+y+z) + yz = (x+y)(x+z)@‚ðxæ‚·‚éB‚»‚µ‚ÄzŠÂ“I‚ÉŠ|‚¯‚éB

(2)@—á‚É‚æ‚Á‚Ä x+y+z=s ‚Æ‚¨‚­B
@i•â‘èj‚ÌŽ®‚É s^(xy) > (x+y)^(xy)@‚ðŠ|‚¯‚ÄA
@@z^(z^2)¥s^(2xy+z^2) > [(x+z)(y+z)]^(z^2)¥(x+y)^(2xy).
@‚±‚ê‚ðzŠÂ“I‚ÉŠ|‚¯‚éB
‚Ê‚é‚Û

y•â‘èzz^(z^2)¥s^(xy+z^2) > [(x+z)(y+z)]^(z^2)¥(x+y)^(xy).
(—ªØ)
@‚Q€’è—‚æ‚èAa>1 d>0 ‚̂Ƃ« (1+d)^a > 1+ad.
È4
249: 243-245 2005/04/04(ŒŽ)02:07 AAS
>>219 (1)‚ÌŒn
yŒnz³‚ÌŽÀ” x,y,z ‚ɑ΂µ‚Ä x+y+z=s ‚Æ‚¨‚­‚Æ‚«A
@C … (s/ã3)^(2s) < (x^x)(y^y)(z^z)(s^s) < {(x+y)^(x+y)}{(y+z)^(y+z)}{(z+x)^(z+x)} < (A_x)^x (A_y)^y (A_z)^z < s^(2s).
@‚±‚±‚É A_x=(s^2 +x^2)/2, A_y=(s^2 +y^2)/2, A_z=(s^2 +z^2)/2, C=exp{-2(ã3)/e}.
(—ªØ)@¶‚©‚ç
@v¥Log(v) † -1/e ‚æ‚è. (“™†‚Í s/ã3 = v = 1/e ‚̂Ƃ«)
@Log ‚Íã‚ɓʂ¾‚©‚çJensen‚æ‚è (x^x)(y^y)(z^z) = -x¥Log(1/x) -y¥Log(1/y) -z¥Log(1/z) > -s¥Log(3/s) = s¥Log(s/3).
@xs = x(x+y+z) < (x+y)(x+z) < (s^2 +x^2)/2 = A_x < s^2 ‚ðxæ‚·‚éB‚»‚µ‚ÄzŠÂ“I‚ÉŠ|‚¯‚éB(I)

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