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Bernstein's theorem (polynomials)

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(Redirected from Bernstein's inequality (mathematical analysis)) Mathematical inequality

In mathematics, Bernstein's theorem is an inequality relating the maximum modulus of a complex polynomial function on the unit disk with the maximum modulus of its derivative on the unit disk. It was proven by Sergei Bernstein while he was working on approximation theory.

Statement

Let max | z | = 1 | f ( z ) | {\displaystyle \max _{|z|=1}|f(z)|} denote the maximum modulus of an arbitrary function f ( z ) {\displaystyle f(z)} on | z | = 1 {\displaystyle |z|=1} , and let f ′ ( z ) {\displaystyle f'(z)} denote its derivative. Then for every polynomial P ( z ) {\displaystyle P(z)} of degree n {\displaystyle n} we have

max | z | = 1 | P ′ ( z ) | ≤ n max | z | = 1 | P ( z ) | {\displaystyle \max _{|z|=1}|P'(z)|\leq n\max _{|z|=1}|P(z)|} .

The inequality cannot be improved and equality holds if and only if P ( z ) = α z n {\displaystyle P(z)=\alpha z^{n}} .

Bernstein's inequality

In mathematical analysis, Bernstein's inequality states that on the complex plane, within the disk of radius 1, the degree of a polynomial times the maximum value of a polynomial is an upper bound for the similar maximum of its derivative. Applying the theorem k times yields

max | z | ≤ 1 | P ( k ) ( z ) | ≤ n ! ( n − k ) ! ⋅ max | z | ≤ 1 | P ( z ) | . {\displaystyle \max _{|z|\leq 1}|P^{(k)}(z)|\leq {\frac {n!}{(n-k)!}}\cdot \max _{|z|\leq 1}|P(z)|.}

Similar results

Paul Erdős conjectured that if P ( z ) {\displaystyle P(z)} has no zeros in | z | < 1 {\displaystyle |z|<1} , then max | z | = 1 | P ′ ( z ) | ≤ n 2 max | z | = 1 | P ( z ) | {\displaystyle \max _{|z|=1}|P'(z)|\leq {\frac {n}{2}}\max _{|z|=1}|P(z)|} . This was proved by Peter Lax.

M. A. Malik showed that if P ( z ) {\displaystyle P(z)} has no zeros in | z | < k {\displaystyle |z|<k} for a given k ≥ 1 {\displaystyle k\geq 1} , then max | z | = 1 | P ′ ( z ) | ≤ n 1 + k max | z | = 1 | P ( z ) | {\displaystyle \max _{|z|=1}|P'(z)|\leq {\frac {n}{1+k}}\max _{|z|=1}|P(z)|} .

See also

References

  1. R. P. Boas, Jr., Inequalities for the derivatives of polynomials, Math. Mag. 42 (1969), 165–174.
  2. M. A. Malik, M. C. Vong, Inequalities concerning the derivative of polynomials, Rend. Circ. Mat. Palermo (2) 34 (1985), 422–426.
  3. P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509–513.
  4. M. A. Malik, On the derivative of a polynomial J. London Math. Soc (2) 1 (1969), 57–60.

Further reading

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