- #1
Markov2
- 149
- 0
Denote $D=\{z\in\mathbb C:|z|<1\}$
1) Given $z,w\in\mathbb C,$ prove that the angle between $z$ and $w$ equals $\dfrac\pi2$ iff $\dfrac zw$ is pure imaginary or $\overline zw+z\overline w=0$
2) Let $a,b,c\in\partial D$ so that $a+b+c=0.$ Prove that the triangle $\Delta(a,b,c)$ is equilateral.
3) Let $C>0$ be a fixed constant. Characterize all the $f$ entire functions so that for all $z\in\mathbb C$ with $|z|>1$ is $|f(z)|\le\dfrac{C|z|^3}{\log|z|}.$
4) Consider a function $f\in\mathcal H(D)\cap C(\overline D).$ If exists $a\in\mathbb C$ so that for all $t\in[0,\pi]$ is $f(e^{it})=a,$ prove that for all $z\in D$ is $f(z)=a.$
Attempts:
1) I don't know how to see the stuff of the angles, what's the way to prove it?
2) I don't see a way to work it analytically, how to start?
3) Since $f$ is entire, it has convergent Taylor series then $f(z)=\displaystyle\sum_{k=0}^\infty\frac{f^{(k)(0)}}{k!}z^k,$ now by using Cauchy's integral formula we have $\displaystyle\left| {{f}^{(k)}}(0) \right|\le \frac{k!}{2\pi }\int_{\left| z \right|=R}{\frac{\left| f(z) \right|}{{{R}^{k+1}}}\,dz},$ and $\dfrac{{\left| {{f^{(k)}}(0)} \right|}}{{k!}} \le \dfrac{{C{R^3}}}{{{R^{k + 1}}\log R}} = \dfrac{{C{R^{2 - k}}}}{{\log R}}$ this clearly goes to zero as $R\to\infty,$ but for $k\ge2,$ then $f^{(k)}(0)=0$ for $k\ge2$ so the functions are polynomials of degree 1.
4) I don't see how to do this one.
1) Given $z,w\in\mathbb C,$ prove that the angle between $z$ and $w$ equals $\dfrac\pi2$ iff $\dfrac zw$ is pure imaginary or $\overline zw+z\overline w=0$
2) Let $a,b,c\in\partial D$ so that $a+b+c=0.$ Prove that the triangle $\Delta(a,b,c)$ is equilateral.
3) Let $C>0$ be a fixed constant. Characterize all the $f$ entire functions so that for all $z\in\mathbb C$ with $|z|>1$ is $|f(z)|\le\dfrac{C|z|^3}{\log|z|}.$
4) Consider a function $f\in\mathcal H(D)\cap C(\overline D).$ If exists $a\in\mathbb C$ so that for all $t\in[0,\pi]$ is $f(e^{it})=a,$ prove that for all $z\in D$ is $f(z)=a.$
Attempts:
1) I don't know how to see the stuff of the angles, what's the way to prove it?
2) I don't see a way to work it analytically, how to start?
3) Since $f$ is entire, it has convergent Taylor series then $f(z)=\displaystyle\sum_{k=0}^\infty\frac{f^{(k)(0)}}{k!}z^k,$ now by using Cauchy's integral formula we have $\displaystyle\left| {{f}^{(k)}}(0) \right|\le \frac{k!}{2\pi }\int_{\left| z \right|=R}{\frac{\left| f(z) \right|}{{{R}^{k+1}}}\,dz},$ and $\dfrac{{\left| {{f^{(k)}}(0)} \right|}}{{k!}} \le \dfrac{{C{R^3}}}{{{R^{k + 1}}\log R}} = \dfrac{{C{R^{2 - k}}}}{{\log R}}$ this clearly goes to zero as $R\to\infty,$ but for $k\ge2,$ then $f^{(k)}(0)=0$ for $k\ge2$ so the functions are polynomials of degree 1.
4) I don't see how to do this one.
