Intereting Posts

To show that the limit of the sequence $\sum\limits_{k=1}^n \frac{n}{n^2+k^2}$ is $\frac{\pi}{4}$
Probability that a normal distribution is greater than two others
Understanding the symmetries of the Riemann tensor
If $z$ is the unique element such that $uzu=u$, why is $z=u^{-1}$?
Why is the composition of smooth multivariable functions smooth?
Different definitions for submanifolds
A chain ring with Krull dimension greater than one
Determinant of a generalized Pascal matrix
Inequality problems in different cases of $n$
Is there any easy way to understand the definition of Gaussian Curvature?
What is the Arg of $\sqrt{(t-1)(t-2)}$ at the point $t=0$?
Does $\mathbb R^2$ contain more numbers than $\mathbb R^1$?
Fermat's equation with real exponents
Legendre symbol $(-21/p)$
Relationship between subrings and ideals

How can we prove the following?

If $\frac{dP_{n}}{dz}|_{z=z_{0}}=0$ then $|P_{n}(z_{0})|<2$ for all $n>1$, where $P_{n}(z)\equiv P_{n-1}^{2}+z$ and $P_{1}\equiv z$

$z$ is in the complex plane.

- Prove that $\int_0^1|f''(x)|dx\ge4.$
- Does this inequality hold true, in general?
- Gradient Estimate - Question about Inequality vs. Equality sign in one part
- The Functional Inequality $f(x) \ge x+1$, $f(x)f(y)\le f(x+y)$
- Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.
- If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex

It appears that $\lim{}_{n\rightarrow\infty}\max\{|P_{n}(z_{0})|:P_{n}'(z_{0})=0\}=2

$, but all I really need is a proof that it’s always <2 not that it approaches 2.

Both are easy to demonstrate with the Mathematica code below, but I can’t figure out a proof.

```
p[n_, z_] := If[n > 1, p[n - 1, z]^2 + z, z];
Do[Print[Max[Abs[p[n, z] /. Solve[D[p[n, z], z] == 0, z]] // N]], {n, 2, 8}]
```

Output:

0.25

1.15268

1.76235

1.94118

1.98545

1.99638

1.9991

- When is infinite sum bounded by an integral?
- Geometric mean never exceeds arithmetic mean
- How prove this $x^3+y^3+z^3+3\ge 2(x^2+y^2+z^2)$
- Prove $\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$ if $a^2+b^2+c^2=1$
- About the Application of Cauchy-Schwarz (Basic): maximum of $3x+4y$ for $x^2+y^2 \leq 16$
- How prove this inequality $(a^3+b^3+c^3)(ab+bc+ac)\ge 6abc(a^2+b^2+c^2-ab-bc-ac)$
- An inequality $\,\, (1+1/n)^n<3-1/n \,$using mathematical induction
- Show that if $X \succeq Y$, then $\det{(X)}\ge\det{(Y)}$
- Why does Group Theory not come in here?
- Maximum of the sum of cube

- Complexity of verifying proofs
- The kernel and image of $T^n$
- How to prove continuity of $e^x$.
- A finite sum over $\pm 1$ vectors
- solution set for congruence $x^2 \equiv 1 \mod m$
- How do i simplify the following: $\sum_{i=1}^n (3i^2+4) – \sum_{j=2}^{n+1} (3j^2+1)$
- Comparison Statement
- $S^2 \times S^2$ is diffeomophic to $G_2(\mathbb{R}^4)$
- Example where Tietze Extension fails?
- A curious coincidence for Wroblewski's solutions to $1^4+x_2^4+x_3^4+x_4^4+x_5^4 = y_1^4$
- functional equation of type $f(x+f(y)+xf(y)) = y+f(x)+yf(x)$
- The cardinality of a countable union of countable sets, without the axiom of choice
- What happens if an uncountable collection of intervals is used in the definition of the Lebesgue outer measure?
- Power series without analytic continuation
- Laplace transform of integrated geometric Brownian motion