ANA LIZA IR 2018/2019 C' WICZENIA15.10.2018 04
WAZNE NIER 'OWNO5c1
lbdowodnic ( na prsyktadimdukcyjnie )
(r) NIEROW NOSE BERNOULLI EGO : th > - t the IN ( Atx ) " > It n x
(2) NIERJWNOS.ci CAUCHY ' EGO ( migobysreolnimi )
Dld Qsl ... , an nieujemnych defining.envy A- (as , . .. .am/=Q2ta2mt...tQh- ,
Ghazi ... , Qu ) = ( as . .. an )Yh
Wytuazacize Alas , ... , an ) > Glas ,. .. , an )
Jes.li ponaolto H (as , ... , an ) = M I Ollie ai to
q t .. . t
G Casi ... , Qu ) 3 HCQs , ... , Qu )
(3) NIENOW NOSE JENSEN Aodcinek -7112CIR I f.ma I I jes.limiowimy , Ze f jest wypukte
Fact EI t Osyth
f ( qat CA - g) b) f Off (a) t G- g) ffb ) wykazoic Ze owe funky's. wypukig. 20, cliooki
Xz,... ,XnEI , Ofzt . .. tqn=1 , Oli 30 ⇒ ffolnxzt.. . tqnxu ) fqsflx , )t . . . tquffxu )
.
(a) NERO'wNOSE SCHWARZ A
III. xiyik ( II.'llxi iyi )
.
NIENOW NOSE BERNOULLI EGO :
Dere m =L mierowuosc ' stage sigrowuoscig
Lakiaolamy, Ze ( It x) " > It Kx Dowoobimy , ze ( ttx ) "
t
> ttfkt1) x
( Atx ) ktt = ( ttx ) ( it x ) " 3 ( Atx ) ( At Kx ) = A txt Kx t k x' = It ( Ktn )xt Kx '
> It ( Ktn ) x On
NIENOW NOSE CAUCHY ' EGO
Dee n = I nierowuosc ' stage rig no'wmosiig
Takeda my A Case ... . Ok ) I G ( as , ... , Ok )
Dowoobimy A ( as ,... , Ock, 01kt, ) 7 G (Qsl . .. I Ok, Qkte )
Alas , ... , ok , a K+a) = Q2tQ2t-i.TO/etQktI=k(Qzt...tQk#Htt k ( Ktn ) =
= kfast.i.tQKJTKQK.tn#=(ktn)(azt...tQk)-(Qst...tQk)tkOlkfK ( Ktn ) =
( KH ) k
( ) ( Qzt . . . tak ) Ktn+
[KQkte-(Qzt..tQkD/=0= k¥1 I =
= Ak ( as , . .. , Qk) t ¥1
Uoojoluieuie mieriwuosci Bernoulli
ego :
( at b)h = ah (A -1 tf )" > a " (A th BE) -- ah t nah " b
atb > O
a > - b I
( A ( as , . .. , a.) t # ) " 's, A C. .. 1kt 't Af .. lkckthn)k¥1 =
=
' II.It. Akc. .. ) [ a * a - AC. .. D=
= ANY... ) + A " (... jaw ,
- A Kt ' (... ) = AKC... )Qk+, 3 Gk( as
, . .. , Oik ) akt ,
=
= as .az . .. . . Qk. Qktn
Alas , ... , akin )kH= ( A ( as , . .. . Ok ) -1 ¥ ) Kt it 3 as . . . aw ,
Alas , . .. ,ak)¥zi...,a = Glass . .. , 0kt , )
•
Haa . .. iak ) - A¥¥T ⇐ = Gla . ' . .. ' " )
A ( as , .. . , an ) > Glas , .. . , an ) 3 H (as , ... , an )
NIENOW NOSE JENSEN A
① he a- 2 mierowuosc ' jest tozsauue a defining fuukyi wypuktg
. .
Laktaolamy ffqzxzt . . . tqkxk ) f q ,f( %) t ... tohaffxk)
Do www.myffqzxzt . .. to
, , )fqf(9k¥ohkti* )t . .. t # )
02ham
my ELI = cfzxzt . . . t OLKXK
T 2h OI = Ofzt ... to ,a= (t - Of K+, )
I = 9zXzt...t0Lk of
f-( Olsxzt . . tqkxkt qk+
, Xkt, )=f( OJI t out , Xkti ) f If ( I ) +
Olktiffxkt )
=.to#exk)tqk+,fkk+i)fdIEo*fCxn)t...t9qIffxkDIf (0¥ xzt . .
+ 9kt , f-Akt, ) =
= qnffxdtqatfxdt.it ohkflxkltcfktiflxkti )
@q
NIER -0WNOSE SCHWARZ A
Uzywamy nierowuosci Jen send ollie fuukyi )artbiorpcof
Oli ai=? axis
Hayat . . . t xnyn)! Ies 2x it ... ¥7 Ex i ) ' =
= ( q za z t . . . ton an )? f Of , Q ft §. . . t qua f = - ¥÷⇐- xp t . . .
. . . 7¥( zxiy !yi Exit . . . t y ! Ext xizgi
I xzyst . . . txuyu I f ( Z x i)% ( Z yi ) %
•
