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I am stuck in the following problem:

Consider the following set $$A=\{f:f \text{ is an entire function and} |f(z)|\leq 10+ |z|^\frac{3}{2},f(\Bbb N )\subset \Bbb N\}.$$ Then cardinality of $A=$?

I just know that using extended Liouville Theorem ,$f(z)$ is linear. But how to proceed after this.

Any help would be appreciated.

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    Try using Cauchy's Integral formula for a certain derivative of $f$. Can you bound it? – 2017-01-29
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    The Count:i have still not got it...could you explain a bit more – 2017-01-29
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    having just reread the problem, I was mistaken. I apologize. – 2017-01-29

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You know that $f(z)=a+b\,z$ for some constants $a,b$. Set $z=0$. The condition $f(\Bbb N )\subset \Bbb N\}$ implies that $a\in\Bbb N$ and $a\le10$. Set now $z=1$. This gives $a+b\in\Bbb N$ and $a+b\le11$. Evaluating at the integer values of $z$ gives further restrictions on $b$. For instance, $z=4$ gives $a+4\,b\le 18$.

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    :thanks a lot for your answer...still,i am not able to get the answer..could you give the answer – 2017-01-30
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    Yes I could, but I will not. You have enough information now to find it yourself. – 2017-01-30