On an inequality of L. Fejér and F. Riesz (Q1354691)

Let \(P_n\) be the class of all polynomials of degree at most \(n\). For a given function \(f\) holomorphic in \(|z|<1\) we define \(F(f)= L(f)|l(f)\), where \(L(f)\) is a length of a curve \(f(|z|=1)\) and \(l(f)\) is a length of a curve \(f((-1,1))\). In this paper, the following results are proved: Theorem. If \(f\in P_n\) then \[ \int^\pi_{-\pi}|f(e^{i\theta})|d\theta\leq K_n\int^1_{-1} |f(x)|dx, \] where \(K_n\) is a constant depending only on \(n\). If \(f\in P_n\) is conformal in \(|z|<1\) then \(F(f)\leq C_n\), where \(C_n\) is a constant depending only on \(n\). If \(f\in P_n\) is real and conformal in \(|z|<1\) then \(F(f)\) is bounded by \[ {F(n+1)\over\Gamma({n+1\over 2})} \Biggl({\Gamma({n+1\over 4})\over \Gamma({n+1\over 2})}\Biggr)^2\quad\text{or} \quad {\Gamma(n+1)\over\Gamma({n+1\over 2})}\cdot {\Gamma({n\over 4})\cdot\Gamma({n+1\over 4})\over \Gamma({n\over 2})\cdot\Gamma ({n+2\over 2})} \] according as \(n\) is odd or even respectively. The result is the best possible.