Algebraic Phenomenon - Daniel Sitaru - Libris.ro Phenomenon - Daniel Sitaru.pdfDaniel Sitaru Keywords: Algebraic Phenomenon - Daniel Sitaru Created Date: 11/30/2018 11:06:25 AM ...

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+T*+...+t*+l,z(uo + ux) + ... + ,(zn + zx) + ,(rn + rx) lt

zc+ zq+ ro + r(o+x)t1_+ rr|+ zq+ ztl\+ zp+ zxl\; r(t+q+o)+ r(z+K+x)t

{llL + rt + "x ft 5 z(q + t) + r(e + x) I'

rg;pnbaul-s.IXS111OXNIhtr

*N 3u 1(*'0) )r'q'D:T*u =Qu*o +nu+r +tu+q EJqD(*'o) )r'q'D(A):!=q+o *'*t +t+q EJqD ux+...+zx+rx ux

u'l)1:0<lxlg33?r1276 126

(*'0) 3 z't'x,J,q,n tzc-+ ['q + xn

<'- +{ *! zQ*Q*D) r q n

(*'o) 3 z'[.'xiu I r'qn:-?! J x . < " +!*L zQ*Q*n) zr zQ z0

(*'o) ),('x1u r q'e1 J J t-=! * Lz(Q * D) zQ zD

u,r )1 n :,r,,o,(;*Tx,, $) =,(,,,"3)

N = z'K'x')'q'n i(rz * rt + zx)(zJ + zq + zn) j r(zc + te I xn)

U f f'x'q'o ;(r[. * zx)(.zQ + zD) =

,(r(.q a xn)

I1;pnbau 1 - s.Zyvl1,tHf, S -IHf, nVf,

suaroarr snouer ll tl il t_-JilJoro eq)

HOLDER's-Inequality

q3 b3 c3 (a+ b + c\3

T * ; + ; > ffi; a,b, c,x,!, z e (0, oo)

q4 b4 c4 (q+ b + c)4

T *, + ; > ffi; Q,b,c,x,!,z e (0,o)

an bn cn (a* b + c)n

T * ; + ; > ffi; a,b,c,x,Y,z E (o,m); n E N; n > 2

a.n bn bl b)

7 * ; > ffit; Q'b'x'Y e (o'a); n > 2; n € N

x3 v3 23 (x*v*z\3i * F + 7 > ffi i x,!, Z, a,b, c € (0, -)x4 v4 24 (x*v*z\ai * F+ F > ffiix,!,z,a,b,c E (0,*)

xn+I vn+I zn+r k*v+z)n+L**" b" *--'-ffi-ix,Y,z,a,b,c € (0,*); n c N

(;'t)(;")(I't) = (I o'u"')' ;ai'bi'c; € r0'");n € N.

t7

io,nu= (ir',r,)' €,,,,) , p,q e(1,*)i'=1

L L

p* q= ttxbli € IR; i e I,un € N*

HIIYGENS's- Inequality

(1 + ar)(1 * ar) > (t + .! ara)2 ; a1, e2 e [0, o1

(1 + ar)(1 + a)(L* or) > (t +'"lararar)t; or,a2,d3 € [0, *)(1 + ar)(1 + ar)(L+ a3)(1 * o*) > (t + -rfarprara-)n;or,e.2,d3,aa e [0,oo)

t+ .

-n

I l(t * xi) > (t + :,[xrxr-i)" iat E[0,*); n € N; n > 2

i=t

-,

z(a, * b)(az * b) > (,[ara, +,[nrnr)-

(a1+ b)(az * b)(az* bs) > C,talafls +'JUrnrUr)3(a1*b)(aztb). ....(c,n*b) > (\,larar. -.a-+!UrUr. ...61"

8

fi(I

( (xt<xz<J cxr) xz,'l

("rtr* xzlz*...*xn.

c

1@t=x)n| *tY' +

f(xr<xzll.t)n(yt=I rrr, * x2y2*.

(a1* b1* c1)(o,(a, * b1* c1)

> (Wat :-.

HOLDI

)ctlt *

(x1<.x2<(xr ) xz >-

tlt*xzlz+...+x.,,)xl

\ 1xz) n

I[(n

< xz 1 xs) ^(h2

I rrnl I x2y2 *:

6

u'I)l:*Nfu1g3?d'!r I("r( + ... + zt + r0)("x+ ... + zx + rx)!; u4u* + ... + ztzx + tp*l

("t>...>zt>tt)v ("x<...<zx<rx) Ito (ut

(t,t +zf+"Q(*+zx+*)i= EtEx+ztzcaIWhg ] (Ett+z'(.+ rO v (tx<zx7-rx) n (Et<z,t <"[)v (Ex> zx>tq)

(zt + r,t)(zx + rx)!; z6z7s + rtrx ]

(',( > z,t) v (zx q tx) to (z,C < r,() v (zx > rx))

("t +... + z[. + rt)("x +... + zx + rx)! q"6"*+... + ztzx + tntr]

("t<...<zt<t[)v("x<...<zx<rx) f to (uK.

(t,( + zK.+ tf,Xsr + zx +ttlj = EK.Ey + ztzx + rgrx

.|

(tf < zf < rq v (sr <'x < tx) n (Et > zr(. > rt) v (ex =

r* ; rD )

(zt + tt)(zx + rx)! 1 z3zv + rtrx ]

(',t < tQ v ('* 1 tx) n ('t > tt) v (zx > rx))

dlqenbaul-s.Ail HSLBtf,

I=u/A+"'+ztvt+tm

('*u)T=, ("'3)ijpezllerauag flllenbeul-s.Ug('I9H

u("c . "' . zcb k * uq . "' . zqrn

k * uo . "' . znro k) q

< ("t *uq +uo). .(zt +zq +zo)(rt +rq +tn),(?l

+ T3l + Wh 1 (zt + zq + zn)(rt + rq + ro)

'33h+'u'- - r

,(??3ft+hhq

(tqrJt+l z-

Z <u:Il3 u {ofl a

(-'O] ) iD'ED''o''o: r(E

(*'01 a aP'72'\:.[!(- Ol tht:"

r*Irrlf

(-'r) e r*r(rI

*N 3 ulu a lp 'u'\'b: (#

*N f u1(-'01 e !lt'er:!

N3u1(*'0)3"'qT21

(*'0) ) J'q?',

(*'0) 3 t'qbZ

N3u:71ut(a'n

Z?uiN = u:(@'0))z'1

(-'0) 3 z't('x'J

(*'0) 2 z't'x'x

,{r

[i, r,r, (b)p,= (;

^,)p,)(f , * r,') = [ r @) g (bn-i*,) p i

I for a13 azS"'I anibt< bz< "'3bnt pl 2 0; t etn;n € N*; pt*pz+...+ pn= !

(a1*b1*r*!r,

(t"'+b,)) EH,f , g nan-increasing functions

SCHUR's-Inequality

fa'(a - b)(a- c) + b' (b - a)(b - c) + c' (c - a)(c - b) > 0t a,b,c € [0, *); r ) 0

a3 + b3 + c3 + 3abc ) ab(a * b) + bc(b + c) + ca(c * a)abc ) (-a + b + c)(a - b + c)(a + b - c)

(a + b + c)3 + gabc 2 4(a + b + c)(ab + bc + ca)(a - b)2 (a + b - c) + (b - c)' (b * c * a) + (c - a)z (c + a - b) > 0

a2 +b2 + c2 + noru', > z(ab +bc* ca)a*b*c

REAI(z) i=1I at<a

[" = (1 :

( i,,ru,< I-/I i=t(.u (4*r1t;

s

a,b,c € (0, *)

[on + b4 + c4 * abc(a + b + c) > ab(az + b2) + bc(bz + c2) + ca(cz + a2)l. s,b,c G [0,*)

( J- :-/:- \I E ^?=Z(E,1J ;.4€M,,(R)

l^r, ^r,...

, An eigenvaluesol A; n € N*

(ukI Io,,=I ti;!<k1n;n€N*{ i=1 i=1

l^, = lz 2 ... ) 7n eigenvalues of A = (ai)

t o t2 ozz, "' ) dnn

MILNE's-Inequality

(a1* b1r a2 * b,) (ffi;. #h) < fo, + az)(bz + b2)

10

"(:)" ='tffi(\"

J@)V@n D@r1

Wflc'i=7

xi'tt

II

Z < u1N : u :yzp 1;g I .1t,!x

I=! T=!

(zq+zq)(zr+tr)>(

A

(r'o) =, iJ'-N 3 u:u

*N3

(u)"ru : v:

(rn+rt)nti(zr*rq)tq+

__(o+r)(e+q ..---G(D)+rq+qD.

0<(q-D+t)r(o-t)+(ncaq+qo)(r

(t-q+lrxr(nqt)ot+(r+q)rt

0

0<(q-r)(D-r)p.E

suon:

r =ud + "'+"q > ...> zq

l=.'t /td(r+t-uq)ogqlK=

('o

{U*+'lJ <Y(t*'')lJvnt'&tr0h 1 vxtxzxr'xh 1 (t + bx)(tf. + tx)(zt(, + zx)(rt + tx)

1,,

t,C@ q t'xzxrx t < (C ta3[ + ra(t + "U'tttf + yl 1 (zr( + zx)(tt + rx)

lr

d111enbeu1-s,U iI'I HVhtr

,(i)"" = ryu ,()%t> iu > t+vzra

,(?,)wr= ,(?,)'

dtl1enbaul-s.ONI'IUIIS

u > t > t ; 0utsn.,anapuou (r:!rU _ (r) r+.7) nr1

(r+t-uq)t! ( = 1,,,"n)US< (.,U! J

.*';::lJrj ,n,:"=t".]= ; =;}= 1 T.='! I=1 I=? t ru-uqtDK=rrK=rnrrK l

,{41enbeuf-r.r*rrftr"**o#r" )

o < tq, 0 < to tz < u,u,r ), . (, T) (, T) = (WTX,,, .,,

T)

lEq +En (tg + zq +rQ) '(trl + zo lro) )

\ rqr, .W.W)('q +'n *zq + zn +rq +ro)