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Stolz region
Given two points $z_1, z_2 \in \mathbb{C}$ such that $|z_1| < 1$ and $|z_2|<1$, there exists $K > 0$ such that for any points $z \neq 1$ in the closed triangle with vertices $z_1, z_2,$ and 1,
$$\frac{|1-z|}{1-|z|} \leq K.$$ Find the smallest possible value of $K$ if $z_1 = \frac{1+i}{2}, z_2 = \frac{1-i}{2}.$ I let $z = x + iy$ in the closed triangle and use two dimensional calculus to find the absolute maximum on the closed triangle region. I find that the smallest possible $K$ is 1 in this case(Actually, it is at the point $z= \frac{1}{2} + i \frac{1}{2})$. I am not sure if this is true since the closed triangle region is not closed, so not compact. So I am not sure the method of finding absolute maximum in multivriate calculus can be applied here. Can anyone recommend ?
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เรื่อยๆ เฉื่อยๆ 12 กุมภาพันธ์ 2015 21:09 : ข้อความนี้ถูกแก้ไขแล้ว 4 ครั้ง, ครั้งล่าสุดโดยคุณ B บ .... |
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