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$f(z)$ be an entire function such that $|zf(z)-1+e^z| \leq 1+|z| \forall z \in \mathbb{C}$ then what can we say about such function value at $z=0$ and its derivative at $z=0$?

I know the basic definition of entire functions and their properties but am still confused about how to approach this problem.

Any hint is appreciated

Maths
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    Do you know that $|g(z)| \le 1+|z|$ for an entire function $g$ implies that $g$ is a polynomial (of degree at most $1$)? See for example https://math.stackexchange.com/q/2418785/42969. – Martin R Oct 12 '23 at 17:38
  • @MartinR Thank you – Maths Oct 12 '23 at 17:53

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