An elementary inequality about the Mahler measure (Q381195)

Let NEWLINE\[NEWLINEP(z)=a_nz^n+\dots+a_0=a_n(z-z_1 )\dots(z-z_n),NEWLINE\]NEWLINE where \(|a_n|=|a_0|=1\), be a polynomial with complex coefficients and Mahler measure NEWLINE\[NEWLINEM(P)=\prod_{j=1}^n \max(1,|z_j|).NEWLINE\]NEWLINE Putting \(\text{td}(P):=\sum_{j=1}^n ||z_j|-1|\), the authors prove that NEWLINE\[NEWLINE\log M(P) \leq \text{td}(P) \leq 2(M(P)-1).NEWLINE\]NEWLINE Furthermore, they show that the stronger inequality \(2 \log M(P) \leq \text{td}(P)\) holds if the polynomial \(P\) is reciprocal. Here, the upper bound for \(\text{td}(P)\) is straightforward, whereas the lower bound follows from the inequality \(\sum_{j=1}^k (1-t_j) \leq \big(\prod_{j=1}^k t_j \big)^{-1}-1\), where \(0 \leq t_1,\dots,t_k \leq 1\), which is checked elementarily by induction on \(k\).