Erdös and Lorentz showed that by considering the special kind of the polynomials better bounds for the derivative are possible. Let us denote by
H
n
{H_n}
the set of all polynomials whose degree is n and whose zeros are real and lie inside
[
−
1
,
1
)
[ - 1,1)
. Let
P
n
∈
H
n
{P_n} \in {H_n}
and
P
n
(
1
)
=
1
{P_n}(1) = 1
; then the object of Theorem 1 is to obtain the best lower bound of the expression
∫
−
1
1
|
P
n
′
(
x
)
|
p
d
x
\smallint _{ - 1}^1|P_n’(x){|^p}\,dx
for
p
≥
1
p \geq 1
and characterize the polynomial which achieves this lower bound. Next, we say that
P
n
∈
S
n
[
0
,
∞
)
{P_n} \in {S_n}[0,\infty )
if
P
n
{P_n}
is a polynomial whose degree is n and whose roots are all real and do not lie inside
[
0
,
∞
)
[0,\infty )
. In Theorem 2, we shall prove Markov-type inequality for such a class of polynomials belonging to
S
n
[
0
,
∞
)
{S_n}[0,\infty )
in the weighted
L
p
{L_p}
norm (p integer). Here
‖
P
n
‖
L
p
=
(
∫
0
∞
|
P
n
(
x
)
|
p
e
−
x
d
x
)
1
/
p
{\left \| {{P_n}} \right \|_{{L_p}}} = {(\smallint _0^\infty |{P_n}(x){|^p}{e^{ - x}}\,dx)^{1/p}}
. In Theorem 3 we shall consider another analogous problem as in Theorem 2.