Suggestion: Cauchy integral formula. Suppose that |z| < r. Then , where I want to take the integral round the circle of radius 2r centred at the origin. If we substitute then . Now estimate the size of this integral, using the facts that and . That gives

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I'm pretty sure that the improper integral of (cos θ)^{-1/2} converges. That would be enough to show that |f(z)|→0 as |z|→∞, and then you can apply Liouville's theorem.