Markov's inequality for typically real polynomials (Q912233)

Let T denote the set of typically real functions in \(\Delta =\{z:\) \(| z| <1\}\). A function \(f\in T\) provided that f is analytic in \(\Delta\), \(f(0)=0\) and f(z) is real if and only if z is real. For each positive integer n let \(T_ n\) denote the subset of T consisting of polynomials of degree at mot n. Also let \(\| p\| =\sup_{- 1<x<1}| p(x)|\) for any function p defined on (-1,1). The authors prove that if \(p\in T_ n\) then \[ \| p'\| \leq \begin{cases} (n+1)/2\| p\| &\text{ if n is odd;} \\ [(n+2)/(n+1)](n/2)\| p\| &\text{ if n is even.}\end{cases} \] Also, if \(p(z)=\sum^{n}_{k=1}p_ kz^ k\in T_ n\) then \[ | p_ 1| \leq \begin{cases} (1+\cos (2\pi (n+3)^{-1}))\| p\| &\text{ if n is odd;} \\ (1+\cos (2\pi (n+2)^{-1}))\| p\| &\text{ if n is even,}\end{cases} \] \(| p_ 1| +| p_ n| \leq 2\| p\|\) if n is odd, and \(| p_ 1| +| p_{n-1}| \leq 2\| p\|\) if n is even. All inequalities above are sharp. Additional coefficient estimates are obtained for functions in T in terms of \(\| f\|\). For example, if \(f(z)=\sum^{\infty}_{n=1}a_ nz^ n\in T\) then \(| a_ n| \leq 2\| f\|\) for \(n=1,2... \).