Let P(z) be a polynomial of degree n having all its zeros in |z| ≤ 1 then for all (αi)t i=1 ∈ C with |αi| ≥ 1,1 ≤ i ≤t < n , it was proved by Jain[V. K. Jain, Generalization of an inequality involving maximum moduli of a polynomial and its polar derivative, Bull Math Soc Sci Math Roum Tome. 98, 6774 (2007)] that
nt 2t"Aαt max |z|=1|P(z)|+ 2t t Y i=1
|αi|−Aαt!min |z|=1|P(z)|#. where nt = n(n − 1)...(n − t + 1) and Aαt = (|α1|− 1)(|α2|− 1)...(|αt|−1).