Let
K
n
≔
{
Q
n
:
Q
n
(
z
)
=
∑
k
=
0
n
a
k
z
k
,
a
k
∈
C
,
|
a
k
|
=
1
}
.
\begin{equation*} {\mathcal {K}}_n \coloneq \left \{Q_n: Q_n(z) = \sum _{k=0}^n{a_k z^k}, \quad a_k \in {\mathbb {C}}\,, \quad |a_k| = 1 \right \}\,. \end{equation*}
A sequence
(
P
n
)
(P_n)
of polynomials
P
n
∈
K
n
P_n \!\in \! {\mathcal {K}}_n
is called ultraflat if
(
n
+
1
)
−
1
/
2
|
P
n
(
e
i
t
)
|
(n + 1)^{-1/2}|P_n(e^{it})|
converge to
1
1
uniformly in
t
∈
R
t \!\in \! {\mathbb {R}}
. In this paper we prove that
1
2
π
∫
0
2
π
|
(
P
n
−
P
n
∗
)
(
e
i
t
)
|
q
d
t
∼
2
q
Γ
(
q
+
1
2
)
Γ
(
q
2
+
1
)
π
n
q
/
2
\begin{equation*} \frac {1}{2\pi } \int _0^{2\pi }{\left | (P_n - P_n^*)(e^{it}) \right |^q \, dt} \sim \frac {{2}^q \Gamma \left (\frac {q+1}{2} \right )}{\Gamma \left (\frac q2 + 1 \right ) \sqrt {\pi }} \,\, n^{q/2} \end{equation*}
for every ultraflat sequence
(
P
n
)
(P_n)
of polynomials
P
n
∈
K
n
P_n \in {\mathcal {K}}_n
and for every
q
∈
(
0
,
∞
)
q \in (0,\infty )
, where
P
n
∗
P_n^*
is the conjugate reciprocal polynomial associated with
P
n
P_n
,
Γ
\Gamma
is the usual gamma function, and the
∼
\sim
symbol means that the ratio of the left and right hand sides converges to
1
1
as
n
→
∞
n \rightarrow \infty
. Another highlight of the paper states that
1
2
π
∫
0
2
π
|
(
P
n
′
−
P
n
∗
′
)
(
e
i
t
)
|
2
d
t
∼
2
n
3
3
\begin{equation*} \frac {1}{2\pi }\int _0^{2\pi }{\left | (P_n^\prime - P_n^{*\prime })(e^{it}) \right |^2 \, dt} \sim \frac {2n^3}{3} \end{equation*}
for every ultraflat sequence
(
P
n
)
(P_n)
of polynomials
P
n
∈
K
n
P_n \in {\mathcal {K}}_n
. We prove a few other new results and reprove some interesting old results as well.