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So I think the proof is straight forward, but looking for verifications:

$$g(z)=v-iu=-i[iv-i^2u]=-i[iv-(-1)u]=-i[u+iv]=-if(z)$$

So $g$ is a multiple of $f$ and thus $g$ would satisfy the Cauchy-Riemann equations as $f$ did, and so $g$ is also analytic.

Is this enough?

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    OK. Simple proof. – 2017-01-29
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    I suspect the point of the problem was practice in applying the CR conditions. But noting that $g$ is a multiple of $f$ and therefore satisfies CR by a known result indicates a good eye on your part. – 2017-01-29
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    Thank you! Saw how the terms were rotated and kinda intuited from there. – 2017-01-29

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