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Problem 2

Part of 1997 VJIMC

Problems(2)

convex-type sequence is bounded, hence convergent

Source: VJIMC 1997 1.2

9/26/2021
Let α∈(0,1]\alpha\in(0,1] be a given real number and let a real sequence {an}n=1∞\{a_n\}^\infty_{n=1} satisfy the inequality an+1≤αan+(1−α)an−1for n=2,3,…a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldotsProve that if {an}\{a_n\} is bounded, then it must be convergent.
Sequenceslimitsreal analysis
|f(z)|=1 if |z|=1, find f if holomorphic

Source: VJIMC 1997 2.2

10/7/2021
Let f:C→Cf:\mathbb C\to\mathbb C be a holomorphic function with the property that ∣f(z)∣=1|f(z)|=1 for all z∈Cz\in\mathbb C such that ∣z∣=1|z|=1. Prove that there exists a θ∈R\theta\in\mathbb R and a k∈{0,1,2,…}k\in\{0,1,2,\ldots\} such that f(z)=eiθzkf(z)=e^{i\theta}z^kfor all z∈Cz\in\mathbb C.
complex analysiscalculus