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.
See the attachment for a more complete description of the question and Cauchy's inequality.
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.