Question

Suppose f is an entire function. Show that any of the two criteria below imply that f is a constant function. (a) Imf(z) #0 for all z EC and Ref(z) is an entire function. [10] (b) -1 < Ref (2) <1 for all z E C. [8]

Complex Analysis

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Answer 1

Given that f (2) is an entire function with Im f (2) to for all Z€¢ Let g(2) = Re f(2) Im f(2) Also it is given that g(2) is an entire function so that Now sinee Ref (2) ell, Im f (2) ER Re f(z) EIR Im f (2) Thay g(2) is ar real valued entri entire fometion. Then f(2) omit / does not assume infinitely many compler & numer as its value. Bi Pieand little theorem, a constant femetion. that is Re f (Z) g(7) is a constant formation, Im f(7) Hence by a gcz) is Given that f(2) is an entire formation and -1 < Ref(2)=1 Let my suppose f(3) = utin CS Scanned with CamScanner

ther - 1sual sinee fCZ) is an entire formation so, is also an entire fornetion. f(Z) e f(2) Mtive = e Let g(2) Fe انا NG 1 962)) = lemting Fleut teins [:2£1] 2 е = lea - 1 9(2) Ize 5) gez) is bounded g(2) must be By Lionville is Theorem constant formation g(2) = k fC21 -K = ) f(2)= k, = Cory tant proved CS Scanned with CamScanner