Let \(f\) be entire and suppose that for \(\abs{z} \geq M\), \[ \abs{f} \leq A + B\abs{z}^k \] for some constants \(A, B\) and \(k\). Show that \(f\) is a polynomial of degree at most \(k\).
Solution
Apply a Cauchy estimate over a contour of radius \(R> M\) to obtain \[ \abs{f^{(n)}(0)} \leq n!{A+B R^k \over R^n} \asymptotic 1/R^{n-k} ,\] and if \(n>k\) then this goes to zero in \(R\) and \(c_n = 0\) for all \(n>k\).