[nooonuii]

อืมครับ ผมคิดว่าน่าจะเป็น polynomial กับ exp แต่ไม่รู้วิธีคิดเหมือนกันครับ

10. (UMCP Jan 2001) Let f be a conformal mapping of a unit disk D = {|z|<1 } so that
f(0)=0 and f'(0) = 1. Show that there is a point z₀ฮ {|z|=1} so that f(z)น z₀ for all zฮ D.

11. ( UMCP August 1998 ) Let E be a measurable subset of [0,1]. Suppose there is a constant c>0 such that m(Eว[a,b])ณ c(b-a) whenever 0ฃaฃ bฃ1 . Prove that m(E) = 1. ( m is the Lebesgue measure)

12. (UMCP August 1998) Find all f(z) analytic on 0<|z|<ฅ so that f(iz) = if(z).

13. (UMCP Jan 1998) (a) Evaluate \( \large{ k = \lim_{n\to\infty}\int_{0}^{\pi} |\sin{nx}| dx } \)
(b) Prove that \( \large{ \lim_{n\to\infty} \int_{a}^{b} |\sin{nx}| dx = \frac{k}{\pi}(b-a) } \)
(c) Prove that \( \large{ \lim_{n\to\infty} \int_{-\infty}^{\infty} |\sin{nx}|f(x) dx = \frac{k}{\pi} \int_{-\infty}^{\infty} f(x) dx } \) for f ฮ L¹(R)