In this blog post we present some basic quantitative results that show the polynomials of a fixed degree are good and predictable functions. It is also illustrated with two Olympiad problems.

Let be a fixed natural number. We consider the set of all polynomials with real coefficients and degree at most Suppose, we know some information about a polynomial Then we can predict some other properties of For example, if the magnitude of is small in then it cannot become too large in a slightly expanded interval, say in . Its slope cannot become abruptly too steep. That is, by knowing some local properties of we can predict its behavior in nearby neighborhood. The main reason why it happens is that any polynomial in has finite number degrees of freedom. It is fully determined by finite and fixed number of parameters, for example by its coefficients. Below are some quantitative properties in this spirit.

**Claim 1. (Markov’s inequality)**. Let be a polynomial of degree Then

That is, knowing that is not large in implies also cannot be too large in the same interval.

The following inequality is a generalization for the derivatives of higher order and is also attributed to Markov brothers.

**Claim 2. (Markov brothers’ inequality).** For any polynomial of degree it holds

for

**Claim 3**. Let be a polynomial of degree and we know the behavior of in the interval Then, we can predict the growth rate of of outside this interval. It holds

where is the Chebyshev polynomial of first kind. This is a famous extremal property of the Chebyshev polynomials. One can see a short proof of this fact in this blog post, as well as some of its applications.

**Claim 4**. For any polynomial we have

where is a constant depending only on

*Proof*. There are many possible approaches. We can use Markov brothers’ inequality. We also can use the Langrange interpolation formula for at some fixed knots, for example equally spaced in and then estimate the coefficients. I prefer a rather abstract approach, but it shows the ultimate reason why it holds.

Consider the vector space of all polynomials of degree at most It can be normed by any of the following two norms

where . Since is a finite, dimensional vector space, any two norms are equivalent. It means there exist constants depending eventually only on so that

which proves Of course, the interval is not important, it can be replaced by any other interval, for example

**Claim 5**. For any polynomial we have

where is a constant depending only on

*Proof*. The same approach, by considering the norm

### Two applications.

**Problem 1 (Putnam 1995, A5)**. Prove that there is a constant such that, if is a polynomial of degree , then *Solution*. Straightforward, by Claim 5.

**Problem 2. (Romanian NMO)**. Let be a times differentiable function so that:

and .

Prove that: for , where is the -th derivative of .

*Solution*. It’s natural to use Taylor’s expansion.

where Therefore

We choose a large enough and vary in the interval So, the RHS of can be made as small as we want as and we want to show all are also small.

Let us choose some Taking into account and the given conditions, there exists large enough such that for all and it holds

Using Claim 4, we get

where is a constant that may depend only on . Therefore

**References**.

[1] Markov brothers’ inequality.

[2] Putnam 1999, A5.

[3] Problem 3 of Romanian NMO, grade 11.

[4] On an Extremal Property of Chebyshev Polynomials.