Practice Questions for CSIR NET Complex Analysis : Unit disk Automorphism, Schwarz Lemma I

1. Let \(D=\{z \in \mathbb{C}:|z|<1\}\). Then there exists a holomorphic function \(f: D \rightarrow \bar{D}\) with \(f(0)=0\) with the property

(a.) \(f^{\prime}(0)=\frac{1}{2}\)
(b.) \(\left|f\left(\frac{1}{3}\right)\right|=\frac{1}{4}\)
(c.) \(f\left(\frac{1}{3}\right)=\frac{1}{2}\)
(d.) \(\left|f^{\prime}(0)\right|=\sec \left(\frac{\pi}{6}\right)\)

2. Let \(f(z)=\frac{1+z}{1-z}\). Which of the following is/are true?

(a.) \(f\) maps \(\{|z|<1\}\) onto \(\{\operatorname{Re}(z)>0\}\)
(b.) \(f\) maps \(\{|z|<1, \operatorname{Im}(z)>0\}\) onto \(\{\operatorname{Re}(z)>0, \operatorname{Im}(z)>0\}\)
(c.) \(f\) maps \(\{|z|<1, \operatorname{Im}(z)<0\}\) onto \(\{\operatorname{Re}(z)<0, \operatorname{Im}(z)<0\}\)
(d.) \(f\) maps \(\{|z|>1\}\) onto \(\{\operatorname{Im}(z)>0\}\)

3. Let \(\mathbb{D}=\{z \in \mathbb{C}:|z|<1\}\). Which of the following are correct?

(a.) There exists a holomorphic function \(f: \mathbb{D} \rightarrow \mathbb{D}\) with \(f(0)=0\) and \(f^{\prime}(0)=2\)
(b.) There exists a holomorphic function \(f: \mathbb{D} \rightarrow \mathbb{D}\) with \(f\left(\frac{3}{4}\right)=\frac{3}{4}\) and \(f^{\prime}\left(\frac{2}{3}\right)=\frac{3}{4}\)
(c.) There exist a holomorphic function \(f: \mathbb{D} \rightarrow \mathbb{D}\) with \(f\left(\frac{3}{4}\right)=\frac{-3}{4}\) and \(f^{\prime}\left(\frac{3}{4}\right)=\frac{-3}{4}\)
(d.) There exists a holomorphic function \(f: \mathbb{D} \rightarrow \mathbb{D}\) with \(f\left(\frac{1}{2}\right)=\frac{-1}{2}\) and \(f^{\prime}\left(\frac{1}{4}\right)=1\)

4. The minimum possible value of \(|z|^{2}+|z-3|^{2}+|z-6 i|^{2}\), where \(z\) is a complex number and \(i=\sqrt{-1}\), is

(a.) 15
(b.) 45
(c.) 30
(d.) 20

5. Let \(f: \mathbb{D} \rightarrow \mathbb{D}\) be a holomorphic function with \(f(0)=0\), where \(\mathbb{D}\) is the open unit disc \(\{z \in \mathbb{C}:|z|<1\}\). Then

(a.) \(\left|f^{\prime}(0)\right|=1 \)
(b.) \(\left|f\left(\frac{1}{2}\right)\right| \leq \frac{1}{2}\)
(c.) \(\left|f\left(\frac{1}{2}\right)\right| \leq \frac{1}{4}\)
(d.) \(\left|f^{\prime}(0)\right| \leq \frac{1}{2}\)

6. Let \(f(z)=z+\frac{1}{z}\) for \(z \in \mathbb{C}\) with \(z \neq 0\). Which of the following are always true?

(a.) \(f\) is an analytic function on \(\mathbb{C} \backslash\{0\}\)
(b.) \(f\) is a conformal map on \(\mathbb{C} \backslash\{0\}\)
(c.) \(f\) maps the unit circle to a subset of the real axis
(d.) The image of any circle in \(\mathbb{C} \backslash\{0\}\) is again a circle.

7. Let \(f: D \rightarrow D\) be holomorphic with \(f(0)=0\) and \(f\left(\frac{1}{2}\right)=0\) where \(D=\{z:|z|<1\}\) which of the following statements are correct?

(a.) \(\left|f^{\prime}\left(\frac{1}{2}\right)\right| \leq \frac{4}{3}\)
(b.) \(\left|f^{\prime}(0)\right| \leq 1\)
(c.) \(\left|f^{\prime}\left(\frac{1}{2}\right)\right| \leq \frac{4}{3}\) and \(\left|f^{\prime}(0)\right| \leq 1\)
(d.) \(f(z)=z, z \in D\)

8. For \(z=x+i y \in \mathbb{C}\), define

\( \begin{aligned} & H^{+}=\{z \in \mathbb{C}: y>0\} \\ & H^{-}=\{z \in \mathbb{C}: y<0\} \\ & L^{+}=\{z \in \mathbb{C}: x>0\} \\ & L^{-}=\{z \in \mathbb{C}: x<0\} \end{aligned} \)

The function \(f(z)=\frac{2 z+1}{5 z+3}\)

(a.) Maps \(H^{+}\) onto \(H^{+}\) and \(H^{-}\) onto \(H^{-}\)
(b.) Maps \(H^{+}\) onto \(H^{-}\) and \(H^{-}\) onto \(H^{+}\)
(c.) Maps \(H^{+}\) onto \(L^{+}\) and \(H^{-}\) onto \(L^{-}\)
(d.) Maps \(H^{+}\) onto \(L^{-}\) and \(H^{-}\) onto \(L^{+}\)

9. Let \(U\) be an open subset of \(\mathbb{C}\) containing \(D=\{z \in \mathbb{C}:|z| \leq 1\}\) and let \(f: U \rightarrow \mathbb{C}\) be the map defined by

\(f(z)=e^{i \psi} \frac{z-a}{1-\bar{a} z}\) for \(a \in D\), and \(\psi \in[0,2 \pi)\).

Which of the following statements are true?

(a.) \(\left|f\left(e^{i \theta}\right)\right|=1\) for \(0 \leq \theta<2 \pi\)
(b.) \(f\) maps \(\{z \in \mathbb{C}:|z|<1 \mid\}\) onto itself
(c.) \(f\) maps \(\{z \in \mathbb{C}:|z|<1 \mid\}\) into itself
(d.) \(f\) is one-one

10. Let \(f: \mathbf{D} \rightarrow \mathbf{D}\) be holomorphic with \(f(0)=\frac{1}{2}\) and \(f\left(\frac{1}{2}\right)=0\), where \(\mathbf{D}=\{z \in \mathbb{C}:|z|<1\}\).

Which of the following is correct?

(a.) \(\left|f^{\prime}(0)\right| \leq \frac{3}{4}\)
(b.) \(\left|f^{\prime}\left(\frac{1}{2}\right)\right| \leq \frac{4}{3}\)
(c.) \(\left|f^{\prime}(0)\right| \leq \frac{3}{4}\) and \(\left|f^{\prime}\left(\frac{1}{2}\right)\right| \leq \frac{4}{3}\)
(d.) \(f(z)=z, z \in \mathbf{D}\)