Summary: We prove that the inequality \(\| g(\cdot/n)\| _{L_1[-1,1]} \| P_{n+k}\| _{L_1[-1,1]}\leq 2 \| gP_{n+k}\| _{L_1[-1,1]}\), where \(g:[-1,1]\to\mathbb R\) is a monotone odd function and \(P_{ n+k}\) is an algebraic polynomial of degree not higher than \(n+k\), is true for all natural \(n\) for \(k=0\) and all natural \(n\geq 2\) for \(k=1\). We also propose some other new pairs \((n,k)\) for which this inequality holds. Some conditions on the polynomial \(P_{ n+k}\) under which this inequality turns into an equality are established. Some generalizations of this inequality are proposed.