# Which of the following statements are true?[NBHM-2019](2.10)

Which of the following statements are true?

a. There exists an analytic function $$f : \mathbb C → \mathbb C$$ such that its real part is the function $$e^x$$, where $$z = x + iy.$$

b. There exists an analytic function $$f : \mathbb C → \mathbb C$$ such that $$f(z) = z$$ for all $$z$$ such that $$|z| = 1$$ and $$f(z) = z^2$$ for all $$z$$ such that $$|z| = 2.$$

c. There exists an analytic function $$f : \mathbb C → \mathbb C$$ such that $$f(0) = 1, f(4i) = i$$ and for all $$z_j$$ such that $$1 < |z_j | < 3, j = 1, 2,$$ we have $$|f(z_1) − f(z_2)| ≤ |z_1 − z_2|^{\frac{\pi}{3}}$$ .

My Try:-

(a) The real and imaginary part of an analytic functions are harmonic. That is $$\frac{d^2e^x}{d^2 x}+\frac{d^2e^x}{d^2 y}=e^x \neq 0$$. So, $$f$$ is not analytic. This is not the case. So. (a) must be false.

(b) Let $$w=f(z)$$, I think it is possible. $$|z|=1$$, $$f(z)=z$$. That is maps a $$|z|=1$$ to $$|w|=1$$ $$|z|=2 \implies z=2e^i\theta \implies f(z)=4e^{i 2\theta}.$$ which maps to a semicircle of radius $$4$$ and centre $$0.$$ Cant we join suitable sheets and paste with these two curve to form an analytic function.

(c) $$\frac{|f(z_1) − f(z_2)|}{|z_1 − z_2|} ≤ |z_1 − z_2|^{\frac{\pi}{3}-1}$$. So $$z_2 \to z_1$$. we get $$f'(z)=0, \forall z:$$ $$1 < |z | < 3(\text{connected domain})$$. hence $$f(z)=k(k \text{is a constant} )$$. So, there is an analytic function. $$g(z)=f(z)-k$$. Which is not possible. Since zeroes of analytic functions are isolated.

• In b), will $f(z)-z$ have isolated zeroes? – Gerry Myerson Jun 7 at 3:07
• How do I check? – Math geek Jun 7 at 3:13
• It is staring you in the face. If $f(z)=z$ for $|z|=1$, then the zeroes of $f(z)-z$ will incvlude ....? – Gerry Myerson Jun 7 at 3:15
• okay. Thank you :) – Math geek Jun 7 at 4:09
• @Mathgeek But the constant function $f(z) = 0$ is the only entire function which has non-isolated zeroes, by analytic continuation. – Ovi Jun 7 at 18:32