Note on a problem of Q. I. Rahman and P. Turán (Q2266777)

Let \(f(z)=u(z)/v(z)\), \(f(0)=1\), be a rational function with \(v(z)=(z-z_ 1)(z-z_ 2)...(z-z_ n),\) \(0<| z_ 1| \leq | z_ 2| \leq...\leq | z_ n| =1\) and u(z) be a polynomial of degree \(t<n\). Rahman and Turán conjectured, that for all \(p\geq 1\) and \(r<| z_ 1|\) the following inequality \[ I^ p(f,r):=(1/2\pi)\int^{2\pi}_{0}| f(re^{i\theta})|^ pd\theta \geq I^ p(1/(1-z^ n),r) \] holds. In this paper the mentioned conjecture is proved locally. Theorem. There exists a radius \(r_ 0(p,f)\) such that for all \(r<r_ 0(p,f)\) we have \(I^ p(f,r)\geq I^ p(1/(1-z^ n),r).\) There exists a radius \(r_ 2(p,f)\) such that for all \(r<r_ 2(p,f)\) we have \(I^ p(v'/v,r)\geq I^ p(-nz^{n-1}/(1-z^ n),r).\) In the general case the conjecture is undecided at the moment.