Define
S
(
n
,
β
)
S(n,\beta )
to be the set of complex polynomials of degree
n
≥
2
n\ge 2
with all roots in the unit disk and at least one root at
β
\beta
. For a polynomial
P
P
, define
|
P
|
β
|P|_\beta
to be the distance between
β
\beta
and the closest root of the derivative
P
′
P’
. Finally, define
r
n
(
β
)
=
sup
{
|
P
|
β
:
P
∈
S
(
n
,
β
)
}
r_n(\beta )=\sup \{ |P|_\beta : P \in S(n,\beta ) \}
. In this notation, a conjecture of Bl. Sendov claims that
r
n
(
β
)
≤
1
r_n(\beta )\le 1
. In this paper we investigate Sendov’s conjecture near the unit circle, by computing constants
C
1
C_1
and
C
2
C_2
(depending only on
n
n
) such that
r
n
(
β
)
∼
1
+
C
1
(
1
−
|
β
|
)
+
C
2
(
1
−
|
β
|
)
2
r_n(\beta )\sim 1+C_1(1-|\beta |)+C_2(1-|\beta |)^2
for
|
β
|
|\beta |
near
1
1
. We also consider some consequences of this approximation, including a hint of where one might look for a counterexample to Sendov’s conjecture.