Set of Holomorphic functions s.t. |f1|^2 + + |fn|^2 = K

[deluks917]

Homework Statement

Given holomorphic functions f₁, ...,f_n defined on some open connected set s.t. |f₁|² + ... + |f_n|² = K. K is some constant. Prove all the f_j's are constant.

Homework Equations

Cauchy Riemann:
f_k = u +iv

u_x = v_y
u_y = -v_x

The Attempt at a Solution

For n=1

|f|^2 = k.
we have taking derivatives

u*u_x + v*v_x = 0
u*u_y + v*v_y = 0

and hence by CR:
u*u_x + v*v_x = 0
v*u_x + -u*v_x = 0

if u^2 + v^2 = 0 we are done

if not the determinant = -(u^2 + v^2) =/= 0. so the only solution is u_x = v_x = 0.

(f\bar = complex conjugate of f)

also f*f\bar = K (again assume K =/=0)

f\bar = K/f but since f =/= 0 this implies f\bar is holomorphic which is not possible unless f is constant.

I can neither get the linear algebra nor the solving for f\bar to work with more than one function. If anyone could even give me a hint for n=2 I'd appreciate it. This is not for a course (though I am taking Complex) I got the problem from Narasimhan complex analysis in one variable prob 48.6. Thanks in advance.

 

[mighty2000]

First, let's consider the case where n=2. We have f1 = u1 + iv1 and f2 = u2 + iv2. By the given condition, we have |f1|^2 + |f2|^2 = K. Using the Cauchy-Riemann equations, we have u1*ux1 + v1*vx1 = 0 and u1*uy1 + v1*vy1 = 0 for f1, and u2*ux2 + v2*vx2 = 0 and u2*uy2 + v2*vy2 = 0 for f2.

Now, let's consider the function g = f1*f2. Using the product rule, we have g = (u1u2 - v1v2) + i(u1v2 + u2v1). Taking the partial derivatives, we have:

gux1 = u2 + v1*vx1
guy1 = -v2 + u1*vx1
gux2 = u1 + v2*vx2
guy2 = -v1 + u2*vx2

Substituting these into the Cauchy-Riemann equations for g, we have:

u2 + v1*vx1 + v2 - u1*vx1 = 0
-v2 + u1*vx1 - u2 - v1*vx2 = 0

Adding these equations together, we get:

-2v2 + 2v1*vx1 = 0

Since vx1 is a real function, this implies that v1 = v2 = 0. Similarly, we can show that u1 = u2 = 0. Therefore, both f1 and f2 are constant functions.

Now, let's consider the case where n>2. We can use a similar approach as above to show that all the functions are constant. First, we can define a new function g = f1*f2*...*fn. Using the product rule, we can show that g is also a holomorphic function. Then, using the Cauchy-Riemann equations, we can show that all the partial derivatives of g are equal to 0. This implies that g is a constant function. Since g is a product of n holomorphic functions, all of which are non-zero, this implies that each of