229 - Fonctions monotones. Fonctions convexes. Exemples et

I⊂Rf:I→R
f∀a, b ∈I, a ≤b=⇒f(a)≤
f(b)f(a)≥f(b)
f
f
E C E
f:C→R
f∀a6=b∈C, ∀t∈[0,1], f(ta + (1 −t)b)≤tf(a) +
(1 −t)f(b)
f
f−f
x7→ xαα≥1 0 ≤α≤1
x7→ ln x
x7→ exp x
f:I→Rg:J→R
f g f +g
f g f.g
f g f ◦g
f◦g
f:I→J f−1
f, g :C→Rf g f +g

f:I→Rg:C→I f g
f◦g
f:I→J f−1
(fi)fi:C→
Rsupifi
f f f0
f:C→R∀x1, . . . , xn∈C,∀λ1, . . . , λn∈
R∗
+, f Pλixi
Pλi≤Pλif(xi) Pλi
f:I→R
f
∀y∈I, x 7→ f(x)−f(y)
x−y
epi(f)f
f
R
f:I→Rf f0
f
f:I→Rf f00
f:I→RA
IRf f
∀x∈I, f(x) = supl∈Al(x)
f:I→R
˚
I
R
[a, b[
b b

f:I→Rf
f
f:I→R
f0
gf0
d˚
I
f0
g≤f0
d
f:x7→ |x|f0
g(0) = −1f0
d(0) = 1
f:I→R
I
f:I→R∃g:I→R
∀x∈˚ I, f(x)−f(a) = Rx
ag(t)dt
ln x=Rx
11/tdt
f:R∗
+→R
f(x)/x l ∈R∪ {+∞} l f(x)−lx
R+∞
f:I→Rf f
f f
(fn) [a, b]
Rf
C2C1

Gln
(Ω,F, P )X∈L1(Ω)
f:R→Rf(X)∈L1(Ω) f(E(X)) ≤E(f(X))
α1, . . . , αn∈Rn
+Pαi= 1
x1, . . . , xnPαi1
xi−1≤Qxαi
i≤Pαixi
p q p, q ∈R
1/p + 1/q = 1 f, g (X, µ)
kfgk1≤ kfkpkgkq
Lp⊂Lq1≤q≤p≤+∞
p≥1f, g
(X, µ)kf+gkp≤ kfkp+kgkp
k.kp
(pn)n
1 +
N
X
n=1
pn≤
N
Y
n=1
(1 + pn)≤exp N
X
n=1
pn!
f∈C2([a, b],R)f0>0 [a, b]f(a)<0< f(b)
f(c)=0 c
F:x7→ x−f(x)
f0(x)
F

J⊂[a, b]c F
J∀x0∈J F n(x0)→c∃K > 0|Fn+1(x0)−c| ≤
K|xn−c|2f[a, b] [c, b]f
x0∈[c, b] (xn)n
c
f:R+→R+, x 7→ x2−y
a=√y
x0≥a0≤xn−a≤
2ax0−a
2a2n
X(Ω,F, P )
X FX:x7→ P(X≤x)
FX(x)→x→−∞ 0FX(x)→x→+∞1
R