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An application of Cauchy's inequality

Let f be an entire function such that |f(z)|<=A|z|. Use Cauchy's inequality to show that f(z)=az for some complex constant a.

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Solution Summary

Under appropriate conditions, Cauchy's inequality bounds the absolute value of the nth derivative of a function at a point in terms of the maximum of the absolute value of that function inside a region. This solution is an application of this result to show that the only entire functions bounded by A|z| are the straight line functions through the origin.

The solution comprises approximately 1/2 page written in Word with equations in Mathtype. Each step is explained.