I am trying on problems from Bak and Newman's complex analysis. On the part of regarding Liouville's theorem, there were quite a number of problems concerns and an inequality and then ask you to prove that the complex-valued function is a linear one. An example is the following:  Suppose $f$ is entire and $|f(z)|\leq A+B|z|^{3/2}$ for constants $A,B$. Show that $f$ is a linear polynomial.  I hope someone can solve one of them for me. It will be helpful for me to understand problems of similar type once I understand this one!  Thanks in advance for answering.
Suppose $f$ is entire and $|f(z)| \leq A + B |z|^{3/2}$. Show that $f$ is a linear polynomial.   My attempt: Since $f$ is entire, we know that it has a Taylor series expansion on the circle $|z| = R$, that is $f(z) = \sum_{n = 0}^\infty a_n z^n$. By using Cauchy's estimate, we can write   $|f^n(0)| \leq \frac{n! M}{R^n}$. Since this holds for any $R&gt;0$, it follows that for $n\geq 2$, $|f^n(0)| \leq 0$. This implies that $f^n(0) = 0$ for $n\geq 2$. Hence $f$ is a polynomial of degree at most $1$.  Edit 1 : On the circle $|z| = R$, $|f(z)|\leq A + B R^{3/2}$.  Thank you very much.  Kindly suggest me if I am wrong? Is there any other apporoach to solve such kind of problems?
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