If p (z) is a polynomial of degree n having all its zeros in |z| k, k 1, then for r 1, Aziz J. Approx. Theory, 55 (1988), 232? 239 proved (₀^2 |1 + kn e^i|ʳ \, d) ^1/r |ₙ|=₁ |p' (z) | n (₀^2 |p (e^i) |ʳ \, d) ^1/r, whereas Devi et al. Note Mat. , 41 (2021), 19? 29 proved that if p (z) is a polynomial of degree n having no zero in |z| 0, kⁿ n (₀^2 |p (e^i) |ʳ \, d) ^1/r (₀^2 |e^i + kⁿ|ʳ \, d) ^1/r (n |ₙ|=₁ |p (z) | - |ₙ|=₁ |p' (z) |), provided |p' (z) | and |q' (z) | attain their maxima at the same point on |z| = 1, where q (z) = zⁿ p (1/z). We do not only obtain improved extensions of the above inequalities into polar derivative by involving the leading coefficient and the constant term of the polynomial, but also give integral analogues of inequalities on polar derivative recently proved by Mir and Dar Filomat, 36 (16) (2022), 5631? 5640.
Singha et al. (Wed,) studied this question.