Question

Use the Cauchy inequalities or the maximum modulus principle to solve the following problems:

Prove that if f is an entire function that satisfies
sup |f(z)|≤AR ᵏ + B
|z|=R
for all R0, and for some integer k≥ 0 and some constants A, B > 0, then f is a polynomial of degree≤ k.
Show that if the real part of an entire function f is bounded, then f is constant._____

Answer 1

To prove that if f is an entire function that satisfies the given condition, then f is a polynomial of degree ≤ k, we can use the Cauchy inequalities. To show that if the real part of an entire function f is bounded, then f is constant, we can use the maximum modulus principle.

Step-by-step explanation:

To prove that if f is an entire function that satisfies ∥f(z)∥ ≤ ARk + B ∥z∥ = R for all R>0, and for some integer k≥0 and some constants A, B>0, then f is a polynomial of degree ≤ k, we can use the Cauchy inequalities. The Cauchy inequalities state that for an entire function f(z) and a circle |z-a|=R, the k-th derivative of f at the point a is bounded by k! multiplied by the maximum value of f(z) on the circle.

Using this inequality, we can show that all the higher-order derivatives of f at any point are zero, which implies that f is a polynomial of degree ≤ k.

To show that if the real part of an entire function f is bounded, then f is constant, we can use the maximum modulus principle. The maximum modulus principle states that if f is an entire function and |f(z)| has a maximum value on a closed and bounded domain, then f must be a constant function.

Since the real part of f is bounded, we can consider the function |f(z)|, which is also bounded. Applying the maximum modulus principle, we conclude that f must be constant.