Affiliation:
1. Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24 , LT-03225 Vilnius , Lithuania
Abstract
Abstract
In this paper, for polynomials with real coefficients
P
,
Q
P,Q
satisfying
∣
P
(
x
)
∣
≤
∣
Q
(
x
)
∣
| P\left(x)| \le | Q\left(x)|
for each
x
x
in a real interval
I
I
, we prove the bound
L
(
P
)
≤
c
L
(
Q
)
L\left(P)\le cL\left(Q)
between the lengths of
P
P
and
Q
Q
with a constant
c
c
, which is exponential in the degree
d
d
of
P
P
. An example showing that the constant
c
c
in this bound should be at least exponential in
d
d
is also given. Similar inequalities are obtained for the heights of
P
P
and
Q
Q
when the interval
I
I
is infinite and
P
,
Q
P,Q
are both of degree
d
d
. In the proofs and in the constructions of examples, we use some translations of Chebyshev polynomials.