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$\text {let}\ f\ \text {be entire such that }|f(z)|\leq e^{Re(z)} \text{for all $z\in\mathbb{C}$ .}$

what can you say about $f$.

i know an entire function which is bounded is cosnstant.is their any to link this to this theorem ? or any other suggestiion

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    The result for bounded entire functions is known as Liouville's Theorem, see here: https://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis) – 2017-02-14
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    Note that $e^{Re(z)}=\left|e^z\right|$. – 2017-02-14

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Hint: The condition implies $$ |f(z)e^{-z}| \leq e^{{\rm Re\,} z} e^{-{\rm Re\,} z}=1$$