Extremal problems for a polynomial and its polar derivative

Abstract

If P(z) = zs(a0 + a1z + ::: + an????szn????s); 0 s n; is a polynomial of degree n having
all its zeros in jzj k; k 1, then for jj k, Govil and Kumar
????
Appl. Anal. Discrete
Math. 13 (2019) 711-720

proved
max
jzj=1
jDP(z)j (jj ???? k)

n + s
1 + kn +
(kn????sjan????sj ???? ja0j)
(1 + kn)(kn????sjan????sj + ja0j)

max
jzj=1
jP(z)j:
Very recently, Mir
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Ramanujan J. 56 (2021) 1061-1071

generalized as well as improved
the above inequality and proved under the same hypothesis that
max
jzj=1
jDP(z)j
n
1 + kn

(jj ???? k) max
jzj=1
jP(z)j +

jj +
1
kn????1

mk

+ (jj ???? k)

s
1 + kn +
(kn????sjan????sj ???? ja0j ???? mk)
(1 + kn)(kn????sjan????sj + ja0j ???? mk)



max
jzj=1
jP(z)j ????
mk
kn

;
where mk = minjzj=k jP(z)j.
In this paper, we generalize as well as improve upon the above inequalities and related
results.