On a polynomial inequality of Paul Erdős. (Q1976269)

Let \({\mathcal P}_n\) denote the set of all real polynomials of degree \(n\) which have no zeros in the open unit disk. The problem considerded by the authors is to find the exact value of \[ \sup_{f\in {\mathcal P}_n}{\| f'\| _\infty \over \| f\| _p},\quad 0\leq p< \infty, \eqno(1) \] where \[ \| f\| _p=\left\{{1\over 2}\int_{-1}^1| f(x)| ^pdx\right\}^{1\over p}, \quad 0<p<\infty, \] \[ \| f\| _0=\lim_{p\to 0+}\| f\| _p=\exp\left\{{1\over 2}\int_{-1}^1\log | f(x)| dx\right\}, \| f\| _\infty=\max \{| f(x)| :\quad x\in [-1,1]\}. \] The first named author and other authors have obtained the following result: Proposition. Let \(f\in {\mathcal P}_n\) satisfy the following condition \[ f(-1)=f(1)=0.\quad \eqno(2) \] If \(f\) is not a constant multiple by \[ q_{n,1}=(1+x)(1-x)^{n-1}\quad \text{ or}\quad q_{n,n-1}=(1+x)^{n-1}(1-x) \] then \[ \| f'\| _\infty <{\| q_{n,1}'\| _\infty \over \| q_{n,1}\| _p}\| f\| _p,\quad 0\leq p<\infty. \] The question whether the condition (2) in the Proposition is superfluous was proposed by the first named author and Q. I. Rahman. In this paper the authors have completely solved the above problem by finding the exact values of (1) and the corresponding extremal polynomials. Their result shows that for \(p>1\) the condition (2) in the Proposition is realy superfluous, i.e., the extremal polynomials must satisfy (2) even if (2) is not a prerequisite. But in the case \(0\leq p<1\), the extremal polynomials of (1) do not satisfy (2). When \(p=1\), there are two kinds of extremal polynomials of (1) among which one satisfies (2) but the other does not.