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November 2nd, 2011, 08:45 PM   #1
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Derivative Estimates

An entire function f obeys the estimate |f(z)|<= me^(alpha*x) for all points z=x+iy, where m and alpha are positive constants. Verify that f has the form f(z) = Ae^alpha*z for some constant A. Could one draw the same conclusion if one knew only that |f(z)|<=me^alpha*|z| held for every z?
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