sol108c7

# sol108c7 - (1) Show that there do not exist non-constant...

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(1) Show that there do not exist non-constant harmonic functions u : C R with u ( z ) 0 for all z C . Do there exist non-constant subharmonic func- tions with this property? What about non-constant subharmonic functions u satisfying u ( z ) 0 for all z C ? Solution: Suppose u is a harmonic function with u ( z ) 0 for all z C . On the disc D (0 ,r ) it is the real part of some holomorphic function f r , which we can choose such that = ( f r (0)) = 0. Considering g r,r 0 = f r - f r 0 for r 0 < r on the disc D (0 ,r 0 ) we see that < ( g r,r 0 ( z )) = 0 for all z D (0 ,r 0 ). This shows that g has to be constant (the derivatives of < ( g r,r 0 ) are 0, and by the Cauchy-Riemann equations, so are the derivatives of = ( g r,r 0 ). As g r,r 0 (0) = 0 we ﬁnd that f r and f r 0 agree on D (0 ,r 0 ). This allows us to deﬁne f : C C with < ( f ) = u (indeed if | z | < r we just deﬁne f ( z ) = f r ( z )). Now exp( f ) is an entire function, which is bounded by 1, so by Liouville’s theorem, it is constant. Thus u ( z ) = log | exp( f ) | is constant as well. Another proof is as follows: By Harnack’s inequality (on D (0 ,r ) for - u ) we ﬁnd that for any r > 0 and | z | < r we have the inequalities r + | z | r - | z | u (0) u ( z ) 0 . Taking r → ∞ we see that u (0) u ( z ) for all z C . Thus u has an internal minimum at 0; this can only happen if u is constant. Now suppose u ( z ) is a non-constant subharmonic function. Deﬁne m ( r ) = max t ∂D (0 ,r ) u ( t ). Then m is a non-constant, increasing, continuous func- tion. We will show m ( e x ) is convex (as a function of

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sol108c7 - (1) Show that there do not exist non-constant...

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