# For a non-constant entire function which property is possible?

Let $$f$$ be a non-constant entire function.Which of the following properties is possible for each $$z \in \mathbb{C}$$

$$(1) \ \ \mathrm{Re} f(z) =\mathrm{Im} f(z)$$

$$(2) \ \ |f(z)|<1$$

$$(3)\ \ \mathrm{Im} f(z)<0$$

$$(4)\ \ f(z) \neq 0$$

I tried for $$(2)$$ and $$(3)$$ option.For $$(2) f$$ is entire and bounded by Louiville's theorem it has to be constant which is contradiction to hypothesis.for $$(3)$$ if imaginary part or real part is bounded below or above then function has to be constant.How eliminate $$(1)$$ & $$(4)$$? I don't know what are the right option.

Here it is possible that there are more than one answers. Please help me thanks in advance.

• Think of the non-constant entire functions you know. Which of the above properties do they satisfy? Commented Jan 24, 2014 at 16:25
• I suppose first thing you should have tried for first case is cauchy riemann... you tried that?
– user87543
Commented Jan 24, 2014 at 16:29

## 3 Answers

1. Consider the function , $g(z)=e^{f^2(z)}$. Then , $g$ is entire (as $f$ is). Now , $$|g(z)|=\left|e^{(u^2-v^2)+2iuv}\right|=e^{u^2-v^2}=1.$$where , $f=u+iv$ and given $u=v$. So , $g$ is entire and bounded and hence constant. So , $f$ is constant ,which contradicts. So , it is FALSE.

2. Clearly $f$ is bounded and entire and hence constant. So , it is FALSE.

3. Consider , $g(z)=e^{-if(z)}$. Then $g$ is bounded and entire and hence constant and so $f$ is constant. So , it is FALSE.

So, only 4. is TRUE.

$f(z)=u(x,y)+iu(x,y)$

$u_x=u_y$ and $u_y=-u_x$

Can you conclude now?

• In the first option it can't be hold. Commented Jan 24, 2014 at 16:47
• so... what do you want to say?
– user87543
Commented Jan 24, 2014 at 16:47
• So,first is not true. Commented Jan 24, 2014 at 16:50
• How to prove last option by assuming $f(z)=0$ for some $z$ in $\mathbb{C}$ and function is given entire Can i prove that function is zero on complex plane. Commented Jan 24, 2014 at 16:56
• Thanks Cameron Buie and Praphulla Koushik both of you. Commented Jan 24, 2014 at 17:01

Hints: For $1,$ note that $f$ can't be $0$ everywhere, nor can its derivative. Hence, there is some non-empty open set that $f$ maps to an open set. (Why?) Can the line $\operatorname{Re}(w)=\operatorname{Im}(w)$ contain any non-empty open set?

For $4,$ try to think of an example of a non-constant entire function that is never $0.$ (A basic example should do.)

• If I consider $f(z)=e^z$ then $(4)$ holds and there are many entire function. Commented Jan 24, 2014 at 16:49
• Bingo! That tells you that $4$ is a possibility. All that's left is for you to rule out option $1.$ You can proceed in the way I mentioned, or alternately, you can use the Cauchy-Riemann equations. Commented Jan 24, 2014 at 16:51
• Can i prove last one? If I amssuming $f(z)=0$ for some $z$ and function is entire then can i prove fuction is zero everywhere? Commented Jan 24, 2014 at 16:54
• You have already proven the last one. $f(z)=e^z$ is a nonconstant entire function such that $f(z)\ne0$ for all $z.$ Thus, it is possible for a nonconstant entire function to be nonzero everywhere. Proof by example complete. Commented Jan 24, 2014 at 16:56
• You cannot prove that if an entire function has a zero, then it is zero everywhere, since it isn't true. Think of some other basic examples of entire functions to see this. Commented Jan 24, 2014 at 16:57