The Sendov conjecture asserts that if
p
(
z
)
=
∏
k
=
1
n
(
z
−
z
k
)
p(z) = \prod _{k = 1}^n(z - {z_k})
is a polynomial with zeros
|
z
k
|
≤
1
\left | {{z_k}} \right | \leq 1
, then each disk
|
z
−
z
k
|
≤
1
,
(
1
≤
k
≤
n
)
\left | {z - {z_k}} \right | \leq 1,(1 \leq k \leq n)
contains a zero of
p
′
(
z
)
p’(z)
. This conjecture has been verified in general only for polynomials of degree
n
=
2
,
3
,
4
,
5
n = 2,3,4,5
. If
p
(
z
)
p(z)
is an extremal polynomial for this conjecture when
n
=
6
n = 6
, it is known that if a zero
|
z
j
|
≤
λ
6
=
0.626997
…
\left | {{z_j}} \right | \leq {\lambda _6} = 0.626997 \ldots
then
|
z
−
z
j
|
≤
1
\left | {z - {z_j}} \right | \leq 1
contains a zero of
p
′
(
z
)
p’(z)
. (The conjecture for
n
=
6
n = 6
would be proved if
λ
6
=
1
{\lambda _6} = 1
.) It is shown that
λ
6
{\lambda _6}
can be improved to
λ
6
=
63
/
64
=
0.984375
{\lambda _6} = 63/64 = 0.984375
.