LetP(z)beapolynomialofdegreenhavingallitszerosin|z|≤k,k≤1,thenforeveryrealorcomplexnumberβ,with|β|≤1andR≤1,itwasshownbyA.Zirehetal.[7]thatfor|z|=1,min|z|=1|P(Rz)+β(R+k/1+k)^nP(z)|≥k^-n|Rn+b(R+k/1+k)^n|min|z|=k|P(z)|.Inthispaper,weshallpresentarefinementoftheaboveinequality.Besides,weshallalsogeneralizesomewell-knownresults.
