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1 Abstract: Kuniyeda, Montel and Toya had shown that the polyno- mial p(z) = P

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1 Abstract: Kuniyeda, Montel and Toya had shown that the polyno- mial p(z) =Pn

k=0akzk; a06= 0, of degree n, does not vanish in

|z| ≤ {1 + ( Xn j=1

|aj/a0|p)q/p}1/q,

where p > 1, q > 1, (1/p) + (1/q) = 1 and we had proved that p(z) does not vanish in |z| ≤ α1/q, where

α = unique root in (0, 1) of Dnx3− DnSx2+ (1 + DnS)x − 1 = 0, Dn= (

Xn j=1

|aj/a0|p)q/p,

S = (|a1| + |a2|)q(|a1|p+ |a2|p)(q−1),

a refinement of Kuniyeda et al.’s result under the assumption Dn< (2 − S)/(S − 1).

Now we have obtained a generalization of our old result and proved that the function

f (z) = X k=0

akzk, (6≡ aconstant); a06= 0,

analytic in |z| ≤ 1, does not vanish in |z| < α1/qm , where

αm= unique root in (0, 1) of Dxm+1− DMmx2+ (1 + DMm)x − 1 = 0, D = (

X k=1

|ak/a0|p)q/p,

Mm= ( Xm k=1

|ak|)q( Xm k=1

|ak|p)q/p,

m = any positive integer with the characteristic that there exists a positive integer k(≤ m) with ak 6= 0.

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