Find all the entire functions f such that

$\displaystyle f(0) = i $ and $\displaystyle \left |f(z)- \cos z \right |\geq{\sqrt{2}}$

for all $\displaystyle {z\in{\mathbb{C}}}$.

My attempt at this question...

Let $\displaystyle g(z) = \frac{1}{f(z)-\cos z}$.

Since $\displaystyle f(z)$ and $\displaystyle \cos z$ are entre and $\displaystyle f(z) - \cos z \neq 0, \forall {z\in{\mathbb{C}}}$ (since $\displaystyle \left |f(z)- cos z \right |\geq{\sqrt{2}}$).

$\displaystyle \Rightarrow g(z)$ is entire & $\displaystyle \left |g(z) \right |= \frac{1}{\left |f(z)-\cos z\right |} \leq \frac{1}{\sqrt{2}}, \forall {z\in{\mathbb{C}}}$.

By Liouville's Theorem, $\displaystyle g(z) \equiv k$ for some $\displaystyle {k\in{\mathbb{C}}$.

$\displaystyle \Rightarrow \frac{1}{f(z)-\cos z} \equiv k$ for some $\displaystyle {k\in{\mathbb{C}}, k \neq 0$, since $\displaystyle f(z) - \cos z \neq 0, \forall {z\in{\mathbb{C}}}$.

Hence $\displaystyle f(z) = \frac{1}{k} + \cos z$

Now it is given that $\displaystyle f(0) = i$, and $\displaystyle cos(0) = 1$, hence $\displaystyle \frac{1}{k} = i-1$.

$\displaystyle \Rightarrow f(z) = i - 1 + \cos z$

Is this correct? Are there any other solutions? Thanks in advance!.