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1 f() = + b + c n. f n < λ < 1 λ u, v R n f((1 λ)u + λv) (1 λ)f(u)+λf(v) (.1) f u = v f ( 1) f f : 1. (1 λ)u + λv u, v (1 λ)f(u)+λf(v) f(u),f(v) (.1)
2 14 f (u, f(u)) (v, f(v)) f f(a) 1 f(a)+1 f(b) f f(b) 1 a + 1 b O a 1 a + 1 b b.: 1 λ = 1 f (u, f(u)) (v, f(v)) f f(, ) = f u =(1, ) v =(, ).3: z = λ (1 λ)u + λv =(1 λ) + λ =
3 f((1 λ)u + λv) =f(1 + λ, ) = 1 + λ (1 λ)f(u)+λf(u) =(1 λ) 1+λ =1+λ u v f.3 f u =(u 1, ),v =(v 1, ) (u, f(u)) (v, f(v)).1..1 f (a, b, f(a, b)) f z = f(a, b)+f (a, b)( a)+f (a, b)( b). f, (a, b), (, ) R, f(, ) f(a, b)+f (a, b)( a)+f (a, b)( b) (.). n. f (, ) f (, ) f(, ) = f (, )
4 16 f (, ) n f u =( 1,..., n ) f 1 (u) f (u) f(u) =. f n (u) 3. f(, ) = +3 (5, 3) f =, f =6 f (5, 3) 5 1 f(5, 3) = = = f (5, 3) (.1.) 1. n f u, v R n f(v) f(u)+ f(u) (v u) f(u) (v u) f(u) (v u).. u, v R n,< λ < 1 f f((1 λ)u + λv) =f(u + λ(v u)) = f(u)+ f(u) {λ(u v)} + o(λ u v ) (1 λ)f(u)+λf(v) f ((1 λ)u + λv) = f (u + λ(v u)) = f(u)+λ f(u) (v u)+o(λ v u ), λf(v) λf(u)+λ f(u) (v u)+o(λ v u ), f(v) f(u)+ f(u) (v u)+ o(λ v u ) λ λ o(λ u v ) λ
5 f() = f() = f () = 1+, f () = 1 (1 + ) 3 f () > f() 1 h h (t) (t 1 <t h (t 1 ) h (t )) t R h (t) h() = 4 h () =4 3 h () =1 g(, ) = +.1.1, 1 1,. f (, ), [ ] f (, ) f (, ) f(, ) = f (, ) f (, ). f = f
6 18,,, f(, ) = f(, ) = [ f(, ) = [ ] ] 6 6. n f u =( 1,..., n ),. f 1 1 (u) f 1 (u)... f 1 n (u) f. f(u) = 1 (u).... f n 1 (u) f n n (u). n f (1). f a R n, t u f(a)u u R n (). f = a R n, t u f(a)u > u u R n.. b, c R n h(t) =f ((1 t)b + tc) f b, c R n h(t)
7 .. 19 h 1 h(t) h (t) { a =(1 t)b + tc = b + t(c b) u = c b h (t) = t u f(a)u 4. f(, ) = + f(, ) = +4 f(, ) = 4 f. u = t (u 1,u ) (, ) t u 1 u f(, )u =[u 1 u ] =u 1 4u 1 u +4u 4 u u 1 4u 1 u +4u =(u 1 u 1 u +u )= { (u 1 u ) + u } f u.. A t uau
8 . A u =(u 1,u,...,u n ) u 1,...,u n p(u) = t uau 5. A =, 1 u =(, ) t uau =[ ] =[ ] = + 1 B = t ubu =[ ] =[ ] = + + = +. A u R n t uau A A u u R t uau > A A
9 .. 1 u R t uau A 6. 5 A = 1 u =(, ) t uav = + + > (u ) A 1 1 B = 1 1 t ubu = + u =(1, ) t ubu =1> u =(, 1) t ubu = 1 < B 3 ( ). n f f a R n f(a) f = a R n f(a),,.,? 5
10 4. A (1). A A (). A A (3). A A. A n n A P, P 1 AP = Λ, Λ A P 1 = t P v = t Pu u = Pv t uau = t (Pv)A(Pv)= t v( t P AP )v = t vλv = λ 1 v λ n v n v =(v 1,...,v n ) λ 1,...,λ n A t uau u (1) 7. f(, ) = f(, ) =, 6 f(, ) = 6 6 4, 8 f, [. ] a b A =, a. (, ) R, b d [ ]A ( = a +b + d = a + b ) + a ( = a + b a A A ad b a (ad b ) ) + 1 a A (.3)
11 (1). A > a> A (). A > a< A (3). A < A. (1) A > a> (, ) (, ) [ ]A > ( a + b ) + 1 a a A ( a + b ) + 1 a a A = ( + b ) =, = a (, ) =(, ) (.3) (, ) (, ) [ ]A > (, ) =(1, ) a > 4 A > () (3) f(, ) = 6 f(, ) =3> f (, ) =6>
12 4 3 f(v) f(u)+ f(u) (u v) t u f(a)u f(a) u n f(a) = t u f(a)u > u = u n f(a) f(a) f(a) = f (a) f (a) f(a) = f (a) f (a) f (a) f (a) t uau =[ ] a c b d = a +(b + c) + d
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