Extremal problems of Turan-type involving the location of all zeros of a polynomial

fIn If P(z) = a(n)Pi(n)(v=1)(z - z(v)) is a polynomial of degree n having all its zeros in |z| <= k, k >= 1 then Aziz [Inequalities for the derivative of a polynomial. Proc Am Math Soc. 1983;89(2):259-266] proved thatmax(|z|=1)|P'(z)| >= 2 /1+ k(n) sigma(n)(v=1) k/k + |z(v)| max(|z|=1) |P(z)|.Recently, Kumar [On the inequalities concerning polynomials. Com-plex Anal Oper Theory. 2020;14(6):1-11 (Article ID 65)] established a generalization of this inequality and proved under the same hypothesis for a polynomial P(z) = a(0) + a(1)z + a(2)z(2) + middot middot middot + a(n)z(n) = a(n)sigma(n)(v=1)(z - z(v)), thatmax(|z|=1)|P'(z)|>= (2/1 + k(n) + (|a(n)|k(n )- |a(0)|)(k - 1)/(1 + k(n))(|a(n)|k(n) + k|a(0)|) sigma(n)(v=1)k/k + |z(v)| max( |z|=1) |P(z)|.In this paper, we sharpen the above inequalities and further extend the obtained results to the polar derivative of a polynomial. As a consequence, our results also sharpens considerably some results of Dewan and Upadhye [Inequalities for the polar derivative of a polynomial. J Ineq Pure Appl Math. 2008;9:1-9 (Article ID 119)].