Annular regions containing all the zeros of a polynomial

A result due to Tôya, Montel and Kuniyeda concerning the location of the zeros of a polynomial states that if $$P(z)=a_{n}z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\cdots +a_0$$ P ( z ) = a n z n + a n - 1 z n - 1 + a n - 2 z n - 2 + ⋯ + a 0 is a polynomial of degree n then all its zeros lie in the disk $$ |z|\le \left( 1+A_{p}^{q}\right) ^{1/q} $$ | z | ≤ 1 + A p q 1 / q where $$p>1,$$ p > 1 , $$q>1$$ q > 1 with $$1/p+1/q=1$$ 1 / p + 1 / q = 1 and $$A_{p}=\left( \sum _{j=0}^{n-1}\left| \frac{a_j}{a_n}\right| ^p\right) ^{1/p}.$$ A p = ∑ j = 0 n - 1 a j a n p 1 / p . In this paper, we refine this result and among other things obtain ring shaped regions containing all the zeros of a polynomial with complex coefficients.