Forapolynomialp(z)ofdegreenwhichhasnozerosin|z|<1,Dewanetal.,(K.K.DewanandSunilHans,Generalizationofcertainwellknownpolynomialinequalities,J.Math.Anal.Appl.,363(2010),38–41)establishedzp′(z)+nβ2p(z)≤n2{(β2+1+β2)max|z|=1|p(z)|-(1+β2-β2)min|z|=1|p(z)|},foranycomplexnumberβwith|β|≤1and|z|=1.InthispaperweconsidertheoperatorB,whichcarriesapolynomialp(z)intoB[p(z)]:=λ0p(z)+λ1(nz2)p′(z)1!+λ2(nz2)2p′′(z)2!,whereλ0,λ1,andλ2aresuchthatallthezerosofu(z)=λ0+c(n,1)λ1z+c(n,2)λ2z2lieinthehalfplane|z|≤|z-n/2|.ByusingtheoperatorB,wepresentageneralizationofresultofDewan.Ourresultgeneralizescertainwell-knownpolynomialinequalities.
