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$f(z)$ and $g(z)$ are entire functions and $g$ doesnt vanish anywhere in thr complex plane.and $|f(z)|\le|g(z)|$ for all $z$. Then we can conclude that

1) $f$ doesnt vanish anywhere

2) $f(0)=0$

3) $f$ is a constant function.

4) $f(z)=C.g(z)$

This was a question in which only one option is correct. but i think that all the answers can be correct for this question.Since the option that is definitely true is 4th but if that is true it would also make one of the above options 1,2,3 correct for different values of C.If it $0$ then 2nd and 3rd option will be correct and if $\ne0$ then 1st option is correct.Can somebody please comment on my views.

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    I think you're making a mistake in quantifiers. This sort of question means that it is true for EVERY such $f$ and $g$. So yes, for some it will be true that $f(0)=0$, or that $f$ doesn't vanish anywhere, or etc; but that doesn't mean it is ALWAYS true. – 2017-01-17

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The only choice which is correct is (4).

Let $h(z)=\dfrac{f(z)}{g(z)}$. Then $h$ is enitre and $|h(z)| \le 1$. Then by Liouvulle's Theorem, $h(z)=h(0) \implies f(z)=\dfrac{f(0)}{g(0)}g(z) $