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Analytic function in the punctured plane satisfying $|f(z)| \leq \sqrt{|z|} + \frac{1}{\sqrt{z}}$ is constant

Let $f$ be an holomorphic function in $\mathbb C-\{0\} $ so that $|f(z)|\leq\sqrt{|z|}+\frac{1}{\sqrt{|z|}}.$

Prove that $f=\text{const}$.

I'd be happy for a hint.

Thanks.

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