Estimates on the Derivative of a Polynomial with a Curved Majorant Using Convex Techniques

A mapping

is a curved majorant for a polynomial

p p

in one real variable if

|p(x)|\leq \phi(x) ∣ p ( x ) ∣ ≤ ϕ ( x )

for all

x\in[-1,1] x ∈ [ − 1 , 1 ]

. If

{\mathcal P}_n^\phi({\mathbb R}) P n ϕ ( R )

is the set of all one real variable polynomials of degree at most

n n

having the curved majorant

\phi ϕ

, then we study the problem of determining, explicitly, the best possible constant

\mathcal{M}^\phi_{n}(x) M n ϕ ( x )

in the inequality

|p'(x)| \le \mathcal{M}^\phi_n(x)\|p\|, ∣ p ′ ( x ) ∣ ≤ M n ϕ ( x ) ∥ p ∥ ,

for each fixed

x\in[-1,1] x ∈ [ − 1 , 1 ]

, where

p\in {\mathcal P}_n^\phi ({\mathbb R}) p ∈ P n ϕ ( R )

and

\|p\| ∥ p ∥

is the sup norm of

p p

over the interval

[-1,1] [ − 1 , 1 ]

. These types of estimates are known as Bernstein type inequalities for polynomials with a curved majorant. The cases treated in this manuscript, namely

\phi(x) = \sqrt{1-x^2} ϕ ( x ) = 1 − x 2

or

\phi(x) = |x| ϕ ( x ) = ∣ x ∣

for all

x\in[-1,1] x ∈ [ − 1 , 1 ]

(circular and linear majorant respectively), were first studied by Q. I. Rahman [“On a problem of Turán about polynomials with curved majorants”, Trans. Amer. Math. Soc. 163 (1972) 447–455]. In that reference the author provided, for each

n\in{\mathbb N} n ∈ N

, the maximum of

\mathcal{M}^\phi_n(x) M n ϕ ( x )

over

[-1,1] [ − 1 , 1 ]

as well as an upper bound for

\mathcal{M}^\phi_n(x) M n ϕ ( x )

for each

x\in[-1,1] x ∈ [ − 1 , 1 ]

, where

\phi ϕ

is either a circular or a linear majorant. Here we provide sharp Bernstein inequalities for some specific families of polynomials having a linear or circular majorant by means of classical convex analysis techniques (in particular we use the Krein-Milman approach)