Inégalités célèbres
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Transcript of Inégalités célèbres
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() [Abels Summation Formula],1 2, , , 1, 2, ,
= 1,2, , = 1 + 2 + + ,
=
=1
+1
+
1
=1
()
[AM-GM (Arithmetic Mean-Geometric Mean)]
,1 2, , 1
12 1
=1
1 = 2 = = .
.((Power Mean Inequality
()
[AM-HM (Arithmetic Mean-Harmonic Mean)]
,1 2, , 1 11 1=1=1
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1 = 2 = = . .
()
[Bernoullis Inequality] > 1 > 1
1 +
1 +
()
[Cauchy-Schwartzs Inequality], 1 2, , 1, 2, ,
12 + 22 + + 212 + 22 + + 2 11 + 22 + + 2 1,2, , .
()
[Cauchy-Schwartzs Inequality for Integrals] , > , : [, ]
2
2
2
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()
[Chebyshevs Inequality ] 1 2 1, 2, , .
(I) 1 2 1
=1
=1
=1
(II) 1 2
1
=1
=1
=1
()
[
Chebyshevs Inequality for Integrals]
<
, : [, ]
.
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() [Convex Function]
f , , 0 + = 1 + +
()
[Convexity]
,["" ] ""
1,1 1,1 1 < < 1 .
"" ""
.
.
f
[,
]
1,
2,
,
1
11 + 22 + + 11 + 22 + + 1, 2 , , [, ].
.((Jensen .
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() [Cyclic Sum]
n
n f .
,1 2, , 1, 2, , =1, 2, , +2, 3, , , 1
+ + , 1, 2 , 1()
[Holders Inequality]
, =11+1.
1, 2, , 1, 2, , =1
=1
1
=1
1
()
[Huygenss Inequality]
,1 2, , , 1, 2, , , 1, 2, , 1=1 + 2 + +
+
=1
=1+
=1
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() [Maclaurins Inequality]
,1 2 , , 1 2
= 1 2 11
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() [Root Mean Inequality]
,1 2, ,
12 + 22 + + 2 1 + 2 + +
1 =
2 =
=
.
()
[Schurs Inequality ]
, , > 0 + + 0
= 1
1)2
+ 2
+ 2
+ 3 + + + + + 2) + + + 3) + + = 11+9
4 + +
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() [Suranyis Inequality ]
,1 2, ,
1 + =1 1
=1
=1
=1
()
[Turkevicis Inequality], , , .
4 + 4 + 4 + 4 + 2 22 + 22 + 22 + 22 + 22 + 22 ()
[Weighted AM-GM Inequality]
1, 2, , ,1 . 2, ,
1
11 + 22 + + 11 22 1 = 2 = = .
