Nuttapong Arunrat and Keaitsuda Maneeruk Nakprasit

DOI: 10.14456/sjst-psu.2019.150

pp. 1192 - 1203

Abstract

In this paper, we investigate an upper bound of the polar derivative of a polynomial of degree n

p(z) = (z - z_m)^t_m(z - z_m-1)^{t_{m-1 …}} (z - z₀)^t₀ ( a₀ + ∑ _ν=µ ^{n–(t_m+…+t₀)} a_νz^ν )

where zeros z₀, ... , z_m are in (z : z <1 ) and the remaining  n – (t_m+…+t₀)

zeros are outside  (z : z < k )

where k ≥ 1
. Furthermore, we give a lower bound of this polynomial where zeros z₀, ... , z_m

are outside (z : z ≤ k )

and the
remaining n – (t_m+…+t₀)

zeros are in (z : z ≤ k )

where k ≤ 1.