Intereting Posts

All real numbers in $$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$
Equivalence of Archimedian Fields Properties
$\sup\{g(y):y\in Y\}\leq \inf\{f(x):x\in X\}$
Cayley-Hamilton Theorem – Trace of Exterior Power Form
What can we say about $f$ if $\int_0^1 f(x)p(x)dx=0$ for all polynomials $p$?
What is the exact motivation for the Minkowski metric?
Proving that sum of two measurable functions is measurable.
Limit of algebraic function $\ \lim_{x\to\infty} \sqrt{x^5 – 3x^4 + 17} – x$
In how many ways we can place $N$ mutually non-attacking knights on an $M \times M$ chessboard?
Adriaan van Roomen's 45th degree equation in 1593
Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k$?
A uniform bound on $u_n$ in $L^\infty(0,T;L^\infty(\Omega))$
Demystify integration of $\int \frac{1}{x} \mathrm dx$
Forcing Classes Into Sets
What is an example of a nonmetrizable topological space?

Suppose polynomial $P_n$ of degree $n$ is such that $|P_n(x)|\le 1$ for $|x|\le 1$. What can you say about the $|P_n'(x)|$ for $|x|\le 1$?

This question is just a generalization of this result: $f(x)=ax^2+bx+c$ where $a, b, c \in R $ and $|f(x)|\leq 1$ on the interval $|x|\leq1$. Prove that $|f'(x)\leq4|$ on the same interval..

- Only 12 polynomials exist with given properties
- $a,b,c,d\ne 0$ are roots (of $x$) to the equation $ x^4 + ax^3 + bx^2 + cx + d = 0 $
- proving that this ideal is radical or the generator is irreducible
- $x^6+x^3+1$ is irreducible over $\mathbb{Q}$
- How many irreducible monic quadratic polynomials are there in $\mathbb{F}_p$?
- Show that the set of polynomials with rational coefficients is countable.

- Coefficient of $n$th cyclotomic polynomial equals $-\mu(n)$
- Arc length of general polynomial
- Intuition regarding Chevalley-Warning Theorem
- Ideals of Polynomial Rings and Field Extensions
- Find the degree of the splitting field of $x^4 + 1$ over $\mathbb{Q}$
- $p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real
- Minimal polynomial of $\alpha^2$ given the minimal polynomial of $\alpha$
- Cyclotomic polynomials explicitly solvable??
- A polynomial is zero if it zero on infinite subsets
- Using Gröbner bases for solving polynomial equations

I vaguely recall this: among all polynomials with sup norm less than 1, the Chebyshev polynomials (or some shifted/scaled variant) are the ones with maximum first derivative. So, to get a tight bound, you just have to figure out the bound for derivatives of Chebyshev polynomials.

I’ll try to find a reference, unless someone else finds one first.

**Confirmation**:

My memory was correct (surprisingly). There is a result known as the Markov-Bernstein inequality. Suppose we use $\|p\|$ to denote $\sup\{|p(x)| : x \in [-1,1] \}$. Then the result says that, for any polynomial $p$ of degree $n$, we have

$$

\|p^{(k)}\| \le \|T_n^{(k)}\| \cdot \|p\|

$$

where $T_n$ is the Chebyshev polynomial of degree $n$. The bound is tight, and is achieved if $p$ is the Chebyshev polynomial of degree $n$.

Figuring out explicit values for $\|T_n^{(k)}\|$ is straightforward. Putting $n=1$, we see that

$$

\|p\| \le 1 \Longrightarrow \|p’\| \le n^2

$$

which (happily) agrees with your result for the quadratic case.

Twelve different proofs of the general result are given in this paper.

- Proof that the intersection of any finite number of convex sets is a convex set
- $C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.
- Euler's Approximation of pi.
- Application of Polya's Enumeration Theorem on small cases examples
- How many ways there are?
- Why do we use the Euclidean metric on $\mathbb{R}^2$?
- Detail in Conditional expectation on more than one random variable
- Hölder continuity definition through distributions.
- How to prove following integral equality?
- Deciding whether $2^{\sqrt2}$ is irrational/transcendental
- How does one evaluate $\sqrt{x + iy} + \sqrt{x – iy}$?
- Combinatorial proof of $\sum_{i=0}^n {{i}\choose{ k}} = {{n+1}\choose{ k+1}}$
- Using Fermat's little theorem to find remainders.
- The commutator subgroup of a quotient in terms of the commutator subgroup and the kernel
- Borel set preserved by continuous map