For z ∈ \(\mathbb{C}\), let ℜz denotes its real part. Let f be an entire function satisfying |f(z)| ≤ |z| |ℜz| on \(\mathbb{C}\). Which of the following statements are true?
Let the given property for the entire function \(f: \mathbb{C} \to \mathbb{C}\) be:
|f(z)| ≤ |z| |ℜz| for all \(z \in \mathbb{C}\).
We need to analyze the truthfulness of the given statements based on this property.
Entire Function Evaluation at 0
Let's evaluate the inequality at \(z=0\).
Substitute \(z=0\) into the inequality:
|f(0)| ≤ |0| |ℜ(0)|
|f(0)| ≤ 0 * 0
|f(0)| ≤ 0
Since the modulus of a complex number must be non-negative, the only way for |f(0)| to be less than or equal to 0 is if |f(0)| = 0. This implies \(f(0) = 0\).
Thus, statement 1, f(0) = 0, is true.
Entire Function Derivative at 0
We know that \(f(0) = 0\). Since \(f\) is an entire function, it has a Taylor series expansion around \(z=0\):
\(f(z) = a_0 + a_1 z + a_2 z^2 + \dots\)
where \(a_0 = f(0)\) and \(a_1 = f'(0)\).
Since \(f(0)=0\), we have \(a_0 = 0\). So, \(f(z) = a_1 z + a_2 z^2 + \dots = z(a_1 + a_2 z + \dots)\).
For \(z \neq 0\), we can write \(f(z)/z = a_1 + a_2 z + \dots\). Let \(h(z) = f(z)/z\). Since \(f(0)=0\), the singularity at \(z=0\) is removable, and \(h(z)\) can be defined at \(z=0\) as \(h(0) = a_1 = f'(0)\). The function \(h(z)\) is also an entire function.
The given inequality is |f(z)| ≤ |z| |ℜz|.
For \(z \neq 0\), we can divide by \(|z|\):
|f(z)/z| ≤ |ℜz|
|h(z)| ≤ |ℜz|
Now consider the limit as \(z \to 0\).
\(\lim_{z \to 0} |h(z)| \le \lim_{z \to 0} |ℜz|\)
Since \(h(z)\) is continuous at \(z=0\), \(\lim_{z \to 0} |h(z)| = |h(0)| = |f'(0)|\).
Since \(\lim_{z \to 0} ℜz = ℜ0 = 0\), \(\lim_{z \to 0} |ℜz| = 0\).
So the inequality becomes:
|f'(0)| ≤ 0
Again, since |f'(0)| must be non-negative, this implies |f'(0)| = 0, which means \(f'(0) = 0\).
Thus, statement 2, f'(0) = 0, is true.
Identifying the Entire Function
We have established that \(f(0)=0\) and \(f'(0)=0\). Let's look at the original inequality again: |f(z)| ≤ |z| |ℜz|.
Consider points on the imaginary axis. For such points, \(z = iy\) where \(y\) is a real number. The real part is \(\real(iy) = 0\).
Substitute \(z=iy\) into the inequality:
|f(iy)| ≤ |iy| |ℜ(iy)|
|f(iy)| ≤ |y| |0|
|f(iy)| ≤ 0
This implies |f(iy)| = 0 for all real \(y\), which means \(f(iy) = 0\) for all real \(y\). The function \(f(z)\) is zero for all points on the imaginary axis.
The imaginary axis contains an infinite number of points, including accumulation points (e.g., \(z=0\)). By the Identity Principle for analytic functions, if an entire function is zero on a set with an accumulation point, the function must be identically zero everywhere.
Therefore, the only entire function satisfying the given property is \(f(z) \equiv 0\).
Thus, statement 3, The only entire function satisfying the given property is f(z) ≡ 0, is true.
Consequently, statement 4, There exists a non-constant entire function satisfying the given property, must be false.
Conclusion Summary
Based on our analysis:
- We showed \(f(0)=0\) by evaluating the inequality at \(z=0\).
- We showed \(f'(0)=0\) by analyzing \(f(z)/z\) and the inequality as \(z \to 0\).
- We showed that \(f(iy)=0\) for all real \(y\) using the inequality on the imaginary axis.
- Using the Identity Principle, we concluded that \(f(z)\) must be identically zero.
Statements 1, 2, and 3 are true.
Important Questions from Complex Analysis
If f (z) \( = \begin{cases} \frac {|z|}{Re(z)} &, \quad \text{if } {} \text{Re} (z)\neq 0 \\ 0 &, \quad \text{if } \text{ Re (z) = 0} \end{cases} \), then-
For z ∈ ℂ let f(z) = \(\left\{ \begin{matrix} \rm \frac{\bar{z}^2}{z}\ if \ z \ne 0,\\\ \rm 0 \ \ otherwise. \end{matrix} \right.\)
Then which of the following statements is false?
Let f, g be entire functions such that \(\lim _{z \rightarrow \infty} \frac{f(z)}{z^n}=\lim _{z \rightarrow \infty} \frac{g(z)}{z^n}=1\) for some fixed positive integer n. Which of the following statements is true?
A function f : \(\mathbb{C}\)\(\longmapsto \mathbb{C}\) is said to be analytic at ∞, if the function g defined by \(g(w)=f\left(\frac{1}{w}\right)\) is analytic at 0 with an appropriate value given for g(0). Which of the following statements is true?
For every n ≥ 1, consider the entire function \(p_n(z)=\sum_{k=0}^n \frac{z^k}{k !}\). Which of the following statements are true?
