DOI: 10.68381/jca15021 ISSN: 0944-6532
An Application of the Krein-Milman Theorem to Bernstein and Markov Inequalities
Gustavo A. Muñoz-Fernández, Yannis Sarantopoulos, Juan B. Seoane-Sepúlveda
Given a trinomial of the form
p(x)=ax^m+bx^n+c
p
(
x
)
=
a
x
m
+
b
x
n
+
c
with
a,b,c\in{\mathbb R}
a
,
b
,
c
∈
R
, we obtain, explicitly, the best possible constant
\mathcal{M}_{m,n}(x)
M
m
,
n
(
x
)
in the inequality
|p'(x)| \le \mathcal{M}_{m,n}(x) \cdot \|p\|,
∣
p
′
(
x
)
∣
≤
M
m
,
n
(
x
)
⋅
∥
p
∥
,
where
x\in[-1,1]
x
∈
[
−
1
,
1
]
is fixed and
\|p\|
∥
p
∥
is the sup norm of
p
p
over
[-1,1]
[
−
1
,
1
]
. This answers a question to an old problem, first studied by Markov, for a large family of trinomials. We obtain the mappings
\mathcal{M}_{m,n}(x)
M
m
,
n
(
x
)
by means of classical convex analysis techniques, in particular, using the Krein-Milman approach.