On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment
We study the following modification of the Landau-Kolmogorov problem: Let $k, r \in \mathbb{N}, \quad 1 \leq k \leq r -1$ and $p, q, s \in [1, \infty]$. Also let $MM^m,\; m \in \mathbb{N}$, be the class of nonnegative functions defined on the segment $[0,1]$ whose derivatives of orders $1, 2,......
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| author | Skorokhodov, D. S. Скороходов, Д. С. |
| author_facet | Skorokhodov, D. S. Скороходов, Д. С. |
| author_sort | Skorokhodov, D. S. |
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| description | We study the following modification of the Landau-Kolmogorov problem:
Let $k, r \in \mathbb{N}, \quad 1 \leq k \leq r -1$ and $p, q, s \in [1, \infty]$.
Also let $MM^m,\; m \in \mathbb{N}$, be the class of nonnegative functions defined on the segment $[0,1]$
whose derivatives of orders $1, 2,... , m$ are nonnegative almost everywhere on $[0,1]$. For every $\delta > 0$, find the exact value of the quantity
$$w^{k, r}_{p, q, s}(\delta; MM^m) := \sup \left\{ ||x^{(k)}||_q : \; x \in MM^m,\; ||x||_p \leq \delta, \;\; ||x^{(r)}||_s \leq 1\right\}$$
We determine the quantity $w^{k, r}_{p, q, s}(\delta; MM^m)$ in the case where $s = \infty$ and $m \in \{r,\; r — 1,\; r — 2\}$. In addition, we consider certain generalizations of the above-stated modification of the Landau-Kolmogorov problem. |
| first_indexed | 2026-03-24T02:26:26Z |
| format | Article |
| fulltext |
UDC 517.5
D. S. Skorokhodov (Dnepropetrovsk Nat. Univ.)
ON INEQUALITIES FOR THE NORMS OF INTERMEDIATE DERIVATIVES
OF MULTIPLY MONOTONE FUNCTIONS DEFINED ON A FINITE SEGMENT
ПРО НЕРIВНОСТI ДЛЯ НОРМ ПРОМIЖНИХ ПОХIДНИХ
КРАТНО-МОНОТОННИХ ФУНКЦIЙ, ЩО ЗАДАНI НА СКIНЧЕННОМУ
ВIДРIЗКУ
We study the following modification of the Landau – Kolmogorov problem: Let k, r ∈ N, 1 ≤ k ≤ r − 1, and p, q, s ∈
∈ [1,∞]. Also let MMm, m ∈ N, be the class of nonnegative functions defined on the segment [0, 1] whose derivatives
of orders 1, 2, . . . ,m are nonnegative almost everywhere on [0, 1]. For every δ > 0, find the exact value of the quantity
ωk,r
p,q,s (δ;MMm) := sup
{∥∥∥x(k)∥∥∥
q
: x ∈MMm, ‖x‖p ≤ δ,
∥∥∥x(r)∥∥∥
s
≤ 1
}
.
We determine the quantity ωk,r
p,q,s (δ;MMm) in the case where s =∞ and m ∈ {r, r− 1, r− 2}. In addition, we consider
certain generalizations of the above-stated modification of the Landau – Kolmogorov problem.
Дослiджується наступна модифiкацiя задачi Ландау – Колмогорова. Нехай k, r ∈ N, 1 ≤ k ≤ r − 1, p, q, s ∈ [1,∞] i
MMm, m ∈ N, — клас невiд’ємних функцiй, що заданi на вiдрiзку [0, 1] та мають майже скрiзь на [0, 1] невiд’ємнi
похiднi порядкiв 0, 1, . . . ,m. Для кожного δ > 0 необхiдно знайти величину
ωk,r
p,q,s (δ;MMm) := sup
{∥∥∥x(k)∥∥∥
q
: x ∈MMm, ‖x‖p ≤ δ,
∥∥∥x(r)∥∥∥
s
≤ 1
}
.
У данiй роботi величину ωk,r
p,q,s (δ;MMm) знайдено у випадку s = ∞ та m ∈ {r, r − 1, r − 2}. Також розглянуто
деякi узагальнення вказаної модифiкацiї задачi Ландау – Колмогорова.
1. Introduction and statement of the problem. Estimates for the norm of intermediate derivative
of function with prescribed bounds on the norm of function itself and the norm of its higher order
derivative have various applications in different areas of Mathematics. Sharp estimates of such type
are of the most interest. A plenty of remarkable results were obtained in this direction. However, a
large number of important questions are still waiting for their solution. For example, sharp estimates
for the norm of intermediate derivative of functions given on a finite interval are know only in
few exceptional situations. In this paper we find sharp estimates of such type for nonnegative and
nondecreasing functions which have several nondecreasing derivatives.
By Lp, p ∈ [0,∞], we denote the space of functions x : [0, 1]→ R for which the quantity
‖x‖p :=
exp
1∫
0
ln |x(t)| dt
, if p = 0,
1∫
0
|x(t)|p dt
1/p
, if 0 < p <∞,
ess sup{|x(t)| : t ∈ [0, 1]}, if p =∞,
is finite. Obviously, the quantity ‖ · ‖p is the norm in the space Lp for every p ∈ [1,∞]. For r ∈ N,
let Lrp be the space of functions x : [0, 1] → R such that there exists derivative x(r−1) (x(0) := x)
that is absolutely continuous on [0, 1], and x(r) ∈ Lp.
c© D. S. SKOROKHODOV, 2012
508 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
ON INEQUALITIES FOR THE NORMS OF INTERMEDIATE DERIVATIVES OF MULTIPLY MONOTONE . . . 509
Let numbers k, r ∈ N, 1 ≤ k ≤ r − 1, and p, q, s ∈ [1,∞] be fixed. The Landau – Kolmogorov
problem on the interval [0, 1] can be stated as follows.
Problem 1. For every δ > 0, find
ωk,rp,q,s (δ;Lrs) := sup
{∥∥∥x(k)
∥∥∥
q
: x ∈ Lrs, ‖x‖p ≤ δ, ‖x
(r)‖s ≤ 1
}
. (1)
Following Steckin [27, 28] we shall call the quantity ωk,rp,q,s (δ;Lrs) the modulus of continuity of
differential operator of order k on the unit ball W r
s :=
{
x ∈ Lrs : ‖x(r)‖s ≤ 1
}
.
The above stated problem is closely related to the problem of finding sharp additive Kolmogorov
type inequalities for derivatives of functions defined on the interval [0, 1]. Below we give the rigorous
setting of correspondent problem.
Problem 2. Find the set Γk,rp,q,s(Lrs) of all pairs (A,B) of positive real numbers which satisfy
conditions:
1) for every x ∈ Lrs, there holds inequality∥∥∥x(k)
∥∥∥
q
≤ A ‖x‖p +B‖x(r)‖s; (2)
2) for every ε > 0, there exists a function xε ∈ Lrs such that∥∥∥x(k)
ε
∥∥∥
q
> A ‖xε‖p + (B − ε)
∥∥∥x(r)
ε
∥∥∥
s
.
Remark that the set Γk,rp,q,s(Lrs) is nonempty for all admissible values of parameters k, r ∈ N,
1 ≤ k ≤ r − 1, and p, q, s ∈ [1,∞] (see, for instance, [3], Theorem 4.6.2, or [2]).
Let us discuss the connection between Problems 1 and 2. Thus, assume that we were able to
evaluate the quantity ωk,rp,q,s(δ;Lrs). Then the set Γk,rp,q,s(Lrs) can be represented as the union of all
pairs (A,B) where z = Aδ +B is the line of support to the graph of function z = ωk,rp,q,s(δ;Lrs). On
the other hand, if we know the set Γk,rp,q,s (Lrs) then we can provide the following upper estimate:
ωk,rp,q,s(δ;L
r
s) ≤ inf
(A,B)∈Γk,r
p,q,s(Lr
s)
(Aδ +B).
Up to nowardays there was not given any complete solution (in the sense of all possible orders
k, r of intermediate and upper derivatives) to Problems 1 and 2, even in the case p = q = s = ∞.
To the best of our knowledge, partial solutions are known only in the following four situations:
1) p = q = s = ∞, r = 2 – E. Landau [19] (Problem 2) and C. K. Chui, P. W. Smith [12]
(Problem 1);
2) p = q = s =∞, r = 3 – A. I. Zviagintsev and A. J. Lepin [31], and M. Sato [24] (Problem 1);
3) p = q = ∞, s ∈ [1,∞), r = 2 – Yu. V. Babenko [5] (Problem 2), and V. I. Burenkov and
V. A. Gusakov [11] (Problem 1);
4) p = s = ∞, q ∈ [1,∞), r = 2 – B. Bojanov and N. Naidenov [7], and N. Naidenov [21]
(Problem 1).
Other results in this direction can be found in books [23, 3] and papers [22, 15, 9, 10, 25, 2, 13,
4, 29].
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
510 D. S. SKOROKHODOV
Remark that in papers [22, 14, 6, 30, 29, 26] it was shown that Problems 1 and 2 could be solved
completely if they are considered not on the whole space Lrs but on some its subset X. In this paper
we concern with the study of Problems 1 and 2 as well as their generalizations on the classes of
functions that are multiply monotone on the interval [0, 1].
In what follows we shall use notation Z+ for nonnegative integers.
Definition 1. Let m ∈ Z+. A nonnegative and nondecreasing function x : [0, 1]→ R is called
m-multiply monotone on [0, 1] and is written x ∈ MMm, if its derivatives x(1), . . . , x(m−1) are
nondecreasing on [0, 1].
For r,m ∈ N and s ∈ [1,∞], by Lr,ms we denote the subspace of Lrs consisting of m-multiply
monotone functions. Before we state generalizations of Problems 1 and 2 let us introduce some
auxiliary definitions.
Definition 2 [16, p. 25]. A function Φ : [0,+∞) → R is called N -function, if it is continuous,
convex and nonnegative on [0,+∞), and Φ(0) = 0.
Definition 3 [17, p. 95]. Let Φ be an arbitrary N -function. The Luxembourg norm in the space
of continuous functions x : [0, 1]→ R is inroduced as follows:
‖x‖(Φ) := inf
µ > 0:
1∫
0
Φ
(
|x(t)|
µ
)
dt ≤ 1
.
From the definition it follows that the Luxembourg norm generalizes and in the case Φ(t) = tq,
q ∈ [1,∞), coincides with the usual Lq-norm.
For an arbitrary continuous function x : [0, 1] → R, we denote by P (x; ·) its nonincreasing
rearrangement on the interval [0, 1] (see [16, p. 17, 18]). The next proposition is the well-known
criterion for N -functions (see, for instance, [16], Theorem 3.1.11).
Theorem A. Let x and y be continuous on [0, 1] functions such that for every t ∈ [0, 1],
t∫
0
P (|x|;u) du ≤
t∫
0
P (|y|;u) du. (3)
Then for an arbitrary N -function Φ,
‖x‖(Φ) ≤ ‖y‖(Φ). (4)
Conversely, if inequality (4) holds true for every N -function Φ, then inequality (3) holds true as well.
Now let us state the generalizations of Problems 1 and 2. Fix numbers k, r ∈ N, 1 ≤ k ≤ r − 1,
s ∈ [1,∞], p ∈ [0,∞] and N -function Φ. Let also X ⊂ Lrs be a given class of functions.
Problem 3. For every δ > 0, find
ωk,rp,Φ,s(δ;X) := sup
{∥∥∥x(k)
∥∥∥
(Φ)
: x ∈ X, ‖x‖p ≤ δ, ‖x
(r)‖s ≤ 1
}
.
Problem 4. Find the set Γk,rp,Φ,s(X) of all pairs (A,B) of positive real numbers which satisfy
conditions:
1) for every x ∈ X, there holds inequality
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
ON INEQUALITIES FOR THE NORMS OF INTERMEDIATE DERIVATIVES OF MULTIPLY MONOTONE . . . 511∥∥∥x(k)
∥∥∥
(Φ)
≤ A ‖x‖p +B‖x(r)‖s; (5)
2) for every ε > 0, there exists a function xε ∈ X such that∥∥∥x(k)
ε
∥∥∥
(Φ)
> A ‖xε‖p + (B − ε)
∥∥∥x(r)
ε
∥∥∥
s
.
Here we also study one more problem which is connected with Problems 1 – 4. Let n ∈ N and
Pn be the set of all algebraic polynomials of degree at most n. In addition, let X be an arbitrary
subset of Lrs such that Pn ∩X 6= ∅.
Problem 5. For k ∈ N, 1 ≤ k ≤ n, p ∈ [0,∞] and N -function Φ find the lowest possible
constant Mk,n
p,Φ(X) in inequality∥∥∥Q(k)
∥∥∥
(Φ)
≤Mk,n
p,Φ(X)‖Q‖p, Q ∈ Pn ∩X. (6)
Inequality (6) is usually called the Markov – Nikolskii type inequality. For a plenty of interesting
and important results concerning the solution of Problem 5 we refer reader to books [16, 20, 8].
In this paper we solve Problems 3 and 4 in the case s = ∞ for classes X = Lr,r∞ , X = Lr,r−1
∞
and partially for the class X = Lr,r−2
∞ . In addition, we find the lowest possible constant in the
Markov – Nikolskii type inequality for (n−1)-multiple monotone algebraic polynomials of degree at
most n, n ∈ N.
The paper is organized as follows. In the next section we state the main results of this paper.
Section 3 is devoted to proofs of several auxilliary statements. In Section 4 we prove main results of
this paper.
2. Main results. For given numbers n ∈ N and c ∈ (0, 1], we set
en(t) :=
tn
n!
and ϕn;c(t) :=
(t− 1 + c)n+
n!
, t ∈ [0, 1].
According to given definition functions en and ϕn;1 are coincide. Let also e0 ≡ 1.
Now we define the following set of indices:
I =
{
(λ, c) ∈ R+ × (0, 1] : λ = 0 for every c < 1
}
, (7)
and by Θn denote the set of functions ψ : [0, 1]→ R represented in the form
ψ = λen−1 + ϕn;c, (λ, c) ∈ I.
Evidently, for every δ > 0 and p ∈ [0,∞], there exists unique function ψ = ψn,δ;p ∈ Θn such that
‖ψn,δ;p‖p = δ. (8)
In some cases the function ψn,δ;p can be found explicitly. For instance, if we take p ∈ (0,∞] and
δ ≤ 1
n!(np+ 1)1/p
then
ψn,δ;p = φn;c, where c =
(
δn!(np+ 1)1/p
)1/(n+1/p)
.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
512 D. S. SKOROKHODOV
In addition, if p ∈ {1,∞} then for every δ >
1
n!(np+ 1)1/p
, we have
ψn,δ;p = en +
n! (n+ 1/p)1/p δ − 1
n+ 1/p
en−1.
The main results of this paper are given by the the following statements.
Theorem 1. Let numbers r ∈ N, r ≥ 2, m ∈ {r − 2, r − 1, r}, p ∈ [0,∞] and N -function Φ
be given. Then for every k ∈ N, 1 ≤ k ≤ m− 1, and δ > 0,
ωk,rp,Φ,∞ (δ;Lr,m∞ ) =
∥∥∥ψ(k)
r,δ;p
∥∥∥
(Φ)
, (9)
where the function ψr,δ;p is determined by (8). Moreover,
ωr−1,r
p,Φ,∞
(
δ;Lr,r−1
∞
)
= ωr−1,r
p,Φ,∞ (δ;Lr,r∞ ) . (10)
Theorem 2. Let r ∈ N, r ≥ 2, and p ∈ [0,∞]. Then for every δ > 0,
ωr−2,r
p,∞,∞
(
δ;Lr,r−2
∞
)
= ωr−2,r
p,∞,∞
(
δ;Lr,r−1
∞
)
,
ωr−1,r
p,∞,∞
(
δ;Lr,r−2
∞
)
= ωr−1,r
p,∞,∞
(
δ;Lr,r−1
∞
)
,
(11)
and in the case r ≥ 3,
ωr−2,r
p,1,∞
(
δ;Lr,r−2
∞
)
= ωr−2,r
p,1,∞
(
δ;Lr,r−1
∞
)
. (12)
Theorems 1 and 2 allow us to solve Problem 4 for classes Lr,r∞ , L
r,r−1
∞ and partially for the class
Lr,r−2
∞ . Before we formulate this solution we firstly solve Problem 5 for multiply monotone algebraic
polynomials. For n,m ∈ N, by Pn,m we denote the set of algebraic polynomials Q of degree at most
n which are nonnegative on the interval [0, 1] along with their derivatives of all orders up to and
including m.
Theorem 3. Let n ∈ N, p ∈ [0,∞] and N -function Φ be given. Then for every k ∈ N,
1 ≤ k ≤ n, and every algebraic polynomial Q ∈ Pn,n−1 there holds exact inequality∥∥∥Q(k)
∥∥∥
(Φ)
≤
(∥∥∥e(k)
n
∥∥∥
(Φ)
‖en‖−1
p
)
‖Q‖p. (13)
Remark that the case when p = ∞ and L∞-norm is taken instead of the Luxembourg norm
inequality (13) for polynomials Q ∈ Pn,n was earlier independently established in papers [18]
and [26]. In addition, in paper [26] the cases of when p ∈ {1,∞} and Lq-norms, q ∈ {1,∞}, are
taken instead of the Luxembourg norms were considered.
Now we introduce an auxiliary function. For k, r ∈ N, 1 ≤ k ≤ r − 1, p ∈ [0,∞], A ≥
≥
∥∥∥e(k)
r−1
∥∥∥
(Φ)
‖er−1‖−1
p and N -function Φ, define
Bk,r
p,Φ(A) := sup
ψ∈Θr
(∥∥∥ψ(k)
∥∥∥
(Φ)
−A‖ψ‖p
)
. (14)
An important property of above-introduced function is given by the following proposition.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
ON INEQUALITIES FOR THE NORMS OF INTERMEDIATE DERIVATIVES OF MULTIPLY MONOTONE . . . 513
Proposition 1. Let numbers k, r ∈ N, 1 ≤ k ≤ r − 1, p ∈ [0,∞] and N -function Φ be given.
Then for every A ≥
∥∥∥e(k)
r−1
∥∥∥
(Φ)
‖er−1‖−1
p , the function Bk,r
p,Φ(A) is finite and nonnegative.
In some certain situations we can provide an explicit formula for the function Bk,r
p,Φ(A). For
instance, the following proposition holds true.
Proposition 2. Let k, r ∈ N, 1 ≤ k ≤ r − 1, p ∈ [0, 1] ∪ {∞} and q ∈ [1,∞]. Then for every
A ≥
∥∥∥e(k)
r−1
∥∥∥
q
‖er−1‖−1
p ,
Bk,r
p,(·)q(A) = λ(1− λ)1/λ−1
∥∥∥e(k)
r
∥∥∥1/λ
q
‖er‖1−1/λ
p A1−1/λ, λ =
k − 1/q + 1/p
r + 1/p
.
The solution to Problem 4 is given by the following theorem.
Theorem 4. Let numbers r ∈ N, m ∈ {r−2, r−1, r}, p ∈ [0,∞] and N -function Φ be given.
Then for every k ∈ N, 1 ≤ k ≤ r − 1,
Γk,rp,Φ,∞ (Lr,m∞ ) =
{(
A,Bk,r
p,Φ(A)
)
: A ≥
∥∥∥e(k)
r−1
∥∥∥
(Φ)
‖er−1‖−1
p
}
.
Futhermore,
Γr−1,r
p,Φ,∞
(
Lr,r−1
∞
)
= Γr−1,r
p,Φ,∞ (Lr,r∞ ) .
Theorem 5. Let r ∈ N, r ≥ 2, and p ∈ [0,∞]. Then for every δ > 0,
Γr−2,r
p,∞,∞
(
Lr,r−2
∞
)
= Γr−2,r
p,∞,∞
(
Lr,r−1
∞
)
,
Γr−1,r
p,∞,∞
(
Lr,r−2
∞
)
= Γr−1,r
p,∞,∞
(
Lr,r−1
∞
)
,
and in the case r ≥ 3,
Γr−2,r
p,1,∞
(
Lr,r−2
∞
)
= Γr−2,r
p,1,∞
(
Lr,r−1
∞
)
.
3. Auxilliary results. This section is devoted to several auxiliary statements which will be used
to prove main results of this paper. For r,m ∈ N, we set
W r,m
∞ :=
{
x ∈ Lr,m∞ : ‖x(r)‖∞ ≤ 1
}
.
Lemma 1. Let numbers r ∈ N, r ≥ 2, m ∈ {r − 2, r − 1, r} and a function ψ ∈ Θr be given.
Then for every x ∈W r,m
∞ and j = 0, 1, . . . ,m, the difference x(j)−ψ(j) has at most one sign change
on [0, 1].
Proof. To prove the assertion of lemma we use ideas from paper [1]. Let a function ψ ∈ Θr be
fixed. According to its definition there exists the pair (λ, c) ∈ I such that
ψ = λer−1 + φr;c.
Hence, the following equalities hold true:
ψ(j)(t) = 0, j = 0, 1, . . . , r − 2 and t ∈ [0, 1− c], (15)
ψ(r)(t) =
∥∥∥ψ(r)
∥∥∥
∞
= 1, t ∈ [1− c, 1]. (16)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
514 D. S. SKOROKHODOV
Let us show that for every number j = 0, 1, . . . , r − 2 and function x ∈ W r,m
∞ , the difference
x(j) − ψ(j) has at most one sign change on the interval [0, 1]. To this end we consider the function
g(t) := x(t)− ψ(t), t ∈ [0, 1],
and assume to the contrary that g(j) has at least two sign changes on [0, 1]. Since j ≤ r − 2
and m ≥ r − 2 we conclude that the function x(j) is nonnegative on the interval [0, 1]. Thus, by
property (15) we have
g(j)(t) = x(j)(t) ≥ 0 for every t ∈ [0, 1− c]. (17)
Now, in view of our assumption there exist points ξj , ηj , 1− c < ξj < ηj ≤ 1, such that
g(j) (ξj) < 0 and g(j) (ηj) > 0. (18)
By the Lagrange theorem we obtain from (17) and (18) that there exist points ξj+1, ηj+1, 1 − c <
< ξj+1 < ηj+1 < 1, for which
g(j+1)(ξj+1) =
g(j)(ξj)− g(j)(1− c)
ξj − 1 + c
< 0,
and
g(j+1)(ηj+1) =
g(j)(ηj)− g(j)(ξj)
ηj − ξj
> 0.
If j+1 < r−1 then g(j+1)(t) = x(j+1)(t) ≥ 0 for every t ∈ [0, 1− c], and the function g(j+1) has at
least two sign changes on the interval [0, 1]. Therefore, we can apply the above arguments to prove
that each of functions g(j+1), g(j+2), . . . , g(r−2) has at least two sign changes on [0, 1]. Moreover,
we obtain that there exist points ξr−1, ηr−1, 1− c < ξr−1 < ηr−1 < 1, such that
g(r−1)(ξr−1) < 0 and g(r−1)(ηr−1) > 0.
Since x(r−1) is absolutely continuous on [0, 1] we obtain
ηr−1∫
ξr−1
g(r)(t) dt = g(r−1) (ηr−1)− g(r−1) (ξr−1) > 0. (19)
On the other hand, by the choice of function x we have
ηr−1∫
ξr−1
g(r)(t) dt =
ηr−1∫
ξr−1
x(r)(t) dt−
ηr−1∫
ξr−1
ψ(r)(t) dt ≤
(
‖x(r)‖∞ − 1
)
(ηr−1 − ξr−1) ≤ 0,
which contradicts to inequality (19).
The case r − 1 ≤ j ≤ m of this lemma is trivial.
Lemma 1 is proved.
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Lemma 2. Let k, r ∈ Z+, 0 ≤ k ≤ r − 1, r ≥ 2, p ∈ [0,∞] and ψ ∈ Θr. Then for every
function x ∈W r,r−2
∞ such that ‖x‖p ≤ ‖ψ‖p, there holds inequality∥∥∥x(k)
∥∥∥
∞
≤
∥∥∥ψ(k)
∥∥∥
∞
. (20)
Proof. Firstly, we prove the assertion of lemma for k ≤ r − 3. Assume to the contrary that∥∥∥x(k)
∥∥∥
∞
>
∥∥∥ψ(k)
∥∥∥
∞
. (21)
Since both functions x and ψ are (r − 2)-monotone on [0, 1], we can rewrite inequality (21) in the
following form:
x(k)(1)− ψ(k)(1) > 0. (22)
At the same time, for every t ∈ [0, 1− c], we have x(k)(t) ≥ 0 = ψ(k)(t). Now let us show that
there exists a point ξ ∈ (1− c, 1) for which x(k)(ξ) < ψ(k)(ξ). Indeed, let
x(k)(t) ≥ ψ(k)(t) for every t ∈ [0, 1].
Then by the Taylor formula we obtain that for every t ∈ [0, 1],
x(t) = x(0) + . . .+
x(k−1)(0)tk−1
(k − 1)!
+
t∫
0
(t− u)k−1
(k − 1)!
x(k)(u) du ≥
≥
t∫
0
(t− u)k−1
(k − 1)!
ψ(k)(u) du = ψ(t).
Hence, in view of inequality (22) and continuity of functions x(k) and ψ(k), we have x(1) > ψ(1).
This yields that ‖x‖p > ‖ψ‖p which contradicts to the choice of function x.
Therefore, we have proved that
x(k)(1− c) ≥ ψ(k)(1− c), x(k)(1) > ψ(k)(1)
and there exists a point ξ ∈ (1− c, 1) such that
x(k)(ξ) < ψ(k)(ξ).
This shows that the difference x(k) − ψ(k) has at least two sign changes on the interval [0, 1], which
is impossible due to Lemma 1. Therefore, inequality (20) holds true for every k ≤ r − 3.
Let us now prove inequality (20) for k = r − 2. Assume to the contrary that∥∥∥x(r−2)
∥∥∥
∞
>
∥∥∥ψ(r−2)
∥∥∥
∞
. (23)
Let ξ ∈ [0, 1] be the point of global maximum of function x(r−2). Let also (λ, c) ∈ I be the pair for
which
ψ = λer−1 + φr;c.
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516 D. S. SKOROKHODOV
From inequality (23) we conclude that
x(r−2)(ξ) >
c2
2
+ λ. (24)
Let us consider three cases: 1) ξ = 0, 2) ξ = 1, 3) ξ ∈ (0, 1).
1) Let ξ = 0. In this case for every t ∈ [0, c],
x(r−2)(t) > τ1(t) :=
1
2
(c− t)2 + λ(1− t). (25)
Indeed, by inequality (24) and nonnegativity of x(r−2) we obtain
x(r−2)(0) >
c2
2
+ λ = τ1(0) and x(r−2)(c) ≥ 0 = τ1(c).
Assume that there exists a point η ∈ (0, c) such that x(r−2)(η) ≤ τ1(η). Then by the Lagrange
theorem there exist points ξ1, η1, 0 < ξ1 < η < η1 < c, for which
x(r−1)(ξ1) < τ ′1(ξ1) and x(r−1)(η1) ≥ τ ′1(η1).
Therefore,
η1∫
ξ1
x(r)(t) dt = x(r−1) (η1)− x(r−1) (ξ1) > τ ′1 (η1)− τ ′1 (ξ1) =
= η1 − ξ1 =
η1∫
ξ1
∥∥∥ψ(r)
∥∥∥
∞
dt ≥
η1∫
ξ1
x(r)(t) dt,
which is impossible. Consequently, inequality (25) holds true.
Since the function x(r−2) is nonnegative on [0, 1], we obtain
∥∥∥x(r−3)
∥∥∥
∞
≥
∥∥∥x(r−2)
∥∥∥
1
>
c∫
0
τ1(u) du =
c3
6
+ λ
c2
2
=
∥∥∥ψ(r−3)
∥∥∥
∞
.
However the latter inequality contradicts to inequality (20) with k = r − 3, which we have already
proved.
The second case when ξ = 1 can be done similarly.
3. Let ξ ∈ (0, 1). Since x(r−1)(ξ) = 0, for every t ∈ [0, 1], we have
x(r−2)(t) ≥ x(r−2)(ξ)− 1
2
(t− ξ)2 >
c2
2
+ λ− 1
2
(t− ξ)2.
Hence,
∥∥∥x(r−3)
∥∥∥
∞
≥
∥∥∥x(r−2)
∥∥∥
1
≥ inf
η∈[0,1−c]
η+c∫
η
x(r−2)(t) dt =
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ON INEQUALITIES FOR THE NORMS OF INTERMEDIATE DERIVATIVES OF MULTIPLY MONOTONE . . . 517
=
c3
2
+ λc− 1
2
c∫
0
t2 dt >
c3
6
+ λc =
∥∥∥ψ(r−3)
∥∥∥
∞
,
which is impossible.
To finish the proof of lemma, we need to verify that inequality (20) holds true for k = r − 1.
Assume to the contrary that there exists a point ξ ∈ [0, 1] such that
|x(r−1)(ξ)| > λ+ c =
∥∥∥ψ(r−1)
∥∥∥
∞
,
where (λ, c) ∈ I is the pair of numbers for which
ψ = λer−1 + φr;c.
Here we have to consider two cases: 1) x(r−1)(ξ) > 0 and 2) x(r−1)(ξ) < 0.
1. If x(r−1)(ξ) > 0 then for every t ∈ [0, 1], we have
x(r−1)(t) ≥ x(r−1)(ξ)− |ξ − t|.
Note that [ξ−c, ξ]∩ [0, 1−c] 6= ∅. The latter inequality yields that for every α ∈ [ξ−c, ξ]∩ [0, 1−c],
x(r−2)(α+ c) ≥ x(r−2)(α+ c)− x(r−2)(α) =
α+c∫
α
x(r−1)(t) dt ≥
≥ x(r−1)(ξ)c− inf
η∈[0,1−c]
η+c∫
η
|ξ − t| dt > λc+
c2
2
=
∥∥∥ψ(r−2)
∥∥∥
∞
.
Hence, we obtain ∥∥∥x(r−2)
∥∥∥
∞
>
∥∥∥ψ(r−2)
∥∥∥
∞
,
which is impossible. The case when x(r−1)(ξ) < 0 can be studied similarly.
Lemma 2 is proved.
The following statement is a corollary of Lemma 2.
Lemma 3. Let numbers k, r ∈ N, 1 ≤ k ≤ r − 2, p ∈ [0,∞] and a function ψ ∈ Θr be given.
If a function x ∈W r,r−2
∞ is such that ‖x‖p ≤ ‖ψ‖p then∥∥∥x(k)
∥∥∥
1
≤
∥∥∥ψ(k)
∥∥∥
1
.
Proof. Indeed, since the function x(k) is nonnegative on the interval [0, 1], inequality (20) shows
us that
∥∥∥x(k)
∥∥∥
1
=
1∫
0
x(k)(t) dt = x(k−1)(1)− x(k−1)(0) =
≤ x(k−1)(1) = ‖x(k−1)‖∞ ≤ ‖ψ(k−1)‖∞ =
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518 D. S. SKOROKHODOV
= ψ(k−1)(1) = ψ(k−1)(1)− ψ(k−1)(0) =
1∫
0
ψ(k)(t) dt =
∥∥∥ψ(k)
∥∥∥
1
.
Lemma 3 is proved.
Lemma 4. Let numbers r ∈ N, r ≥ 2, p ∈ [0,∞], N -function Φ and a function ψ ∈ Θr be
given. If a function x ∈W r,r−1
∞ is such that ‖x‖p ≤ ‖ψ‖p then∥∥∥x(r−1)
∥∥∥
(Φ)
≤
∥∥∥ψ(r−1)
∥∥∥
(Φ)
. (26)
Proof. In Lemma 2 we have established that
∥∥x(k)
∥∥
∞ ≤
∥∥ψ(k)
∥∥
∞ for every k ∈ N, 0 ≤ k ≤
≤ r − 1. Consequently, ∥∥∥x(r−1)
∥∥∥
∞
≤
∥∥∥ψ(r−1)
∥∥∥
∞
,
and, futhermore,∥∥∥x(r−1)
∥∥∥
1
= x(r−2)(1)− x(r−2)(0) ≤
∥∥∥x(r−2)
∥∥∥
∞
≤
∥∥∥ψ(r−2)
∥∥∥
∞
=
∥∥∥ψ(r−1)
∥∥∥
1
.
Now let us prove that the difference P
(
x(r−1); ·
)
− P
(
ψ(r−1); ·
)
has at most one sign change on
the interval [0, 1]. Indeed, by the choice of function x we have that
∣∣x(r)(t)
∣∣ ≤ 1 for almost all
t ∈ [0, 1]. Hence,
∣∣P ′ (x(r−1); t
)∣∣ ≤ 1 for almost all t ∈ [0, 1]. On the other hand, P
(
ψ(r−1); t
)
=
= max{λ+ c− t; 0}. Therefore, the graphs of functions P
(
x(r−1); ·
)
and P
(
ψ(r−1); ·
)
intersect at
most once. This implies that for every t ∈ [0, 1],
t∫
0
P
(
x(r−1);u
)
du ≤
t∫
0
P
(
ψ(r−1);u
)
du.
Now, we can apply Theorem A and verify the validity of inequality (26).
Lemma 4 is proved.
4. Proofs of main results. Proof of Theorem 1. Firstly, we prove equality (9). To this end we
choose an arbitrary function x ∈W r,m
∞ for which ‖x‖p ≤ δ. We need to show that∥∥∥x(k)
∥∥∥
(Φ)
≤
∥∥∥ψ(k)
r,δ;p
∥∥∥
(Φ)
.
It is clear that functions x and ψ = ψr,δ;p satisfy conditions of Lemmas 1, 2 and 3. Hence,∥∥∥x(k)
∥∥∥
∞
≤
∥∥∥ψ(k)
r,δ;p
∥∥∥
∞
,
∥∥∥x(k)
∥∥∥
1
≤
∥∥∥ψ(k)
r,δ;p
∥∥∥
1
,
and the difference x(k)−ψ(k)
r,δ;p has at most one sign change on [0, 1]. Since k ≤ m−1, the functions
x(k) and ψ(k) are nondecreasing on [0, 1], and we conclude that for every t ∈ [0, 1],
P
(
x(k); t
)
= x(k)(1− t) and P
(
ψ
(k)
r,δ;p; t
)
= ψ
(k)
r,δ;p(1− t).
This yields that for every t ∈ [0, 1],
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ON INEQUALITIES FOR THE NORMS OF INTERMEDIATE DERIVATIVES OF MULTIPLY MONOTONE . . . 519
t∫
0
P
(
x(k);u
)
du ≤
t∫
0
P
(
ψ
(k)
r,δ;p;u
)
du.
From the latter inequality and Theorem A we obtain desired inequality (9).
To finish the proof of theorem we note that the validity of equality (10) immediately follows from
Lemma 4.
Theorem 1 is proved.
Remark that Theorem 2 is a corollary of Lemmas 1 and 2.
Proof of Theorem 3. Firstly, we prove that∥∥∥Q(n)
∥∥∥
∞
≤ ‖Q‖p
‖en‖p
. (27)
Indeed, since Q is the polynomial of degree at most n we should consider two cases: 1) Q(n)(0) > 0
and 2) Q(n)(0) < 0.
1. Assume thatQ(n)(0) > 0. Taking into account the fact that each of functionsQ,Q′, . . . , Q(n−1)
is nonnegative on the interval [0, 1] we obtain that
Q(t) =
n−1∑
m=0
Q(m)(0)em(t) +Q(n)(0)en(t) ≥ Q(n)(0)en(t) =
∥∥∥Q(n)
∥∥∥
∞
en(t)
for every t ∈ [0, 1]. Hence,
‖Q‖p ≥
∥∥∥Q(n)
∥∥∥
∞
‖en‖p,
which is inequality (27).
2. Let now Q(n)(0) < 0. Since Q(n−1) is nonnegative on [0, 1], we conclude that Q(n−1)(t) ≥
≥ Q(n)(0)(t − 1) for every t ∈ [0, 1]. If n = 1 then ‖Q‖p ≥
∥∥Q(1)
∥∥
∞ ‖e1‖p which gives desired
inequality (27). If n ≥ 2 then for every t ∈ [0, 1], we obtain
Q(t) =
n−2∑
m=0
Q(m)(0)em(t) +
t∫
0
Q(n−1)(u)en−2(t− u) du ≥
≥
t∫
0
Q(n−1)(u)en−2(t− u) du ≥
∥∥∥Q(n)
∥∥∥
∞
t∫
0
(1− u)en−2(t− u) du =
=
∥∥Q(n)
∥∥
∞
(n− 2)!
t∫
0
(1− u)(t− u)n−2 du ≥
∥∥Q(n)
∥∥
∞
(n− 2)!
t∫
0
u(t− u)n−2 du =
=
∥∥∥Q(n)
∥∥∥
∞
en(t).
This yields that ‖Q‖p ≥
∥∥Q(n)
∥∥
∞ ‖en‖p. Therefore, inequality (27) is proved.
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520 D. S. SKOROKHODOV
Let us turn to the proof of Theorem 3. Since
∣∣Q(n)(t)
∣∣ =
∥∥Q(n)
∥∥
∞ for every t ∈ [0, 1], from
inequality (27) we obtain that ∥∥∥Q(n)
∥∥∥
(Φ)
≤
‖e(n)
n ‖(Φ)
‖en‖p
‖Q‖p.
This is desired inequality (13) for k = n.
Let us now prove inequality (13) for 1 ≤ k ≤ n− 1. Consider the polynomial
x(t) :=
‖en‖p
‖Q‖p
Q(t), t ∈ [0, 1].
It is clear that ‖x‖p = ‖en‖p. Moreover, in view of inequality (27) we conclude that x ∈ Wn,n−1
∞ .
Since en ∈ Θn we can apply the assertion of Theorem 1. This yields∥∥∥x(k)
∥∥∥
(Φ)
=
‖en‖p
‖Q‖p
∥∥∥Q(k)
∥∥∥
(Φ)
≤
∥∥∥e(k)
n
∥∥∥
(Φ)
.
Theorem 3 is proved.
Proof of Proposition 1. Note that for every A ≥
∥∥∥e(k)
r−1
∥∥∥
(Φ)
‖er−1‖−1
p ,
Bk,r
p,Φ(A) = max{B1;B2},
where
B1 := sup
c∈(0,1]
(∥∥∥φ(k)
r;c
∥∥∥
(Φ)
−A‖φr;c‖p
)
,
B2 := sup
λ≥0
(∥∥∥e(k)
r + λe
(k)
r−1
∥∥∥
(Φ)
−A‖er + λer−1‖p
)
.
Let us show that both quantities B1 and B2 are finite. Firstly, we prove that B1 < ∞. Indeed, for
every t ∈ [0, 1] and c ∈ (0, 1], we have
φr;c(t) ≤ φr;1(t) = er(t).
Hence,
B1 ≤ sup
c∈(0,1]
(∥∥∥e(k)
r
∥∥∥
(Φ)
−A‖φr;c‖p
)
≤
∥∥∥e(k)
r
∥∥∥
(Φ)
<∞.
Now let us prove that B2 <∞. Indeed,
B2 = sup
λ≥0
(∥∥∥e(k)
r + λe
(k)
r−1
∥∥∥
(Φ)
−A‖er + λer−1‖p
)
≤
≤ sup
λ≥0
∥∥∥e(k)
r
∥∥∥
(Φ)
+ λ
∥∥∥e(k)
r−1
∥∥∥
(Φ)
−
∥∥∥e(k)
r−1
∥∥∥
(Φ)
‖er−1‖p
λ‖er−1‖p
=
∥∥∥e(k)
r
∥∥∥
(Φ)
.
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The nonnegativity of Bk,r
p,Φ(A) follows from inequalities
Bk,r
p,Φ(A) ≥ B1 ≥ lim
c→0
(∥∥∥φ(k)
r;c
∥∥∥
(Φ)
−A ‖φr;c‖p
)
= 0.
Proposition 1 is proved.
Proof of Proposition 2. As in the proof of Proposition 1 we see that for every A ≥
≥
∥∥∥e(k)
r−1
∥∥∥
q
‖er−1‖−1
p ,
Bk,r
p,(·)q(A) = max{B1;B2},
where the quantities B1 and B2 were defined before. In view of the choice of numbers p and q for
every two functions x, y ∈ Lr,r−1
∞ , we have
‖x+ y‖q ≤ ‖x‖q + ‖y‖q and ‖x+ y‖p ≥ ‖x‖p + ‖y‖p.
Hence,
B2 = sup
λ≥0
(∥∥∥e(k)
r + λe
(k)
r−1
∥∥∥
q
−A‖er + λer−1‖p
)
≤
≤ sup
λ≥0
(∥∥∥e(k)
r
∥∥∥
q
+ λ
∥∥∥e(k)
r−1
∥∥∥
q
−A‖er‖p −Aλ‖er−1‖p
)
=
=
∥∥∥e(k)
r
∥∥∥
q
−A‖er‖p ≤ sup
c∈(0,1]
(∥∥∥φ(k)
r;c
∥∥∥
q
−A‖φr;c‖p
)
= B1.
Therefore,
Bk,r
p,(·)q(A) = sup
c∈(0,1]
(∥∥∥φ(k)
r;c
∥∥∥
q
−A‖φr;c‖p
)
=
= sup
c∈(0,1]
(∥∥∥e(k)
r
∥∥∥
q
cr−k+1/q −A‖er‖pcr+1/p
)
.
Simple calculations show us that the function
g(c) :=
∥∥∥e(k)
r
∥∥∥
q
cr−k+1/q −A‖er‖pcr+1/p, c > 0,
achieves its maximum at the point
c0 =
(
(1− λ)
∥∥∥e(k)
r
∥∥∥
q
A−1‖er‖−1
p
) 1
k−1/q+1/p
.
Let us show that c0 ≤ 1. Indeed, since A ≥
∥∥∥e(k)
r−1
∥∥∥
q
‖er−1‖−1
p we have
c
k−1/q+1/p
0 ≤ (1− λ)
∥∥∥e(k)
r
∥∥∥
q
‖er−1‖p∥∥∥e(k)
r−1
∥∥∥
q
‖er‖p
. (28)
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522 D. S. SKOROKHODOV
The generalized Bernoulli inequality states that if x > −1 then inequality (1 + x)α ≥ 1 + αx
holds true for α ∈ (−∞, 0] ∪ [1,+∞), and inequality (1 + x)α ≤ 1 + αx holds true for α ∈ [0, 1].
Applying both inequalities we for obtain
‖e(k)
r ‖q∥∥∥e(k)
r−1
∥∥∥
q
=
(
1− 1
r − k + 1/q
)1/q
≤ 1− 1/q
r − k + 1/q
=
r − k
r − k + 1/q
,
and
‖er‖p
‖er−1‖p
=
(
1− 1
r + 1/p
)1/p
≥ 1− 1/p
r + 1/p
=
r
r + 1/p
.
Hence,
c
k−1/q+1/p
0 ≤ (1− λ)
(r − k) (r + 1/p)
r (r − k + 1/q)
=
r − k
r
< 1.
Therefore,
Bk,r
p,(·)q(A) = g(c0) = λ(1− λ)1/λ−1
∥∥∥e(k)
r
∥∥∥1/λ
q
‖er‖1−1/λ
p A1−1/λ.
Proposition 2 is proved.
Proof of Theorem 4. Let us choose an arbitrary pair of numbers (A,B) ∈ Γk,rp,Φ,∞ (Lr,m∞ ) . In
view of inequality (13) we obtain
A ≥
∥∥∥e(k)
r−1
∥∥∥
(Φ)
‖er−1‖−1
p .
Let us show that
B ≤ Bk,r
p,Φ(A). (29)
To this end for every function x ∈ Lr,m∞ \ Pr−1,m we define
y(t) :=
x(t)
‖x(r)‖∞
, t ∈ [0, 1].
Evidently, y ∈ W r,m
∞ . Let ψ ∈ Θr be the function such that ‖ψ‖p = ‖y‖p. Then by Theorem 1 we
have ∥∥∥x(k)
∥∥∥
(Φ)
= ‖x(r)‖∞‖y(k)‖(Φ) ≤ ‖x(r)‖∞
∥∥∥ψ(k)
∥∥∥
(Φ)
≤
≤ ‖x(r)‖∞
[
A‖ψ‖p +Bk,r
p,Φ(A)
]
= A‖x‖p +Bk,r
p,Φ(A)‖x(r)‖∞.
It remains to consider the case when x ∈ Pr−1,m. Since ‖x(r)‖∞ = 0 Lemma 1 shows us that
∥∥∥x(k)
∥∥∥
(Φ)
≤
∥∥∥e(k)
r−1
∥∥∥
(Φ)
‖er−1‖p
‖x‖p ≤ A‖x‖p +Bk,r
p,Φ(A)‖x(r)‖∞.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
ON INEQUALITIES FOR THE NORMS OF INTERMEDIATE DERIVATIVES OF MULTIPLY MONOTONE . . . 523
Hence, inequality (29) holds true according to definition of the set Γk,rp,Φ,∞ (Lr,m∞ ) . Futhermore, we
have just established that for every A ≥
∥∥∥e(k)
r−1
∥∥∥
(Φ)
‖er−1‖−1
p there exists B ≥ 0 such that (A,B) ∈
∈ Γk,rp,Φ,∞ (Lr,m∞ ) .
On the other hand,
B ≥ sup
ψ∈Θr
∥∥ψ(k)
∥∥
(Φ)
−A‖ψ‖p∥∥ψ(r)
∥∥
∞
:= Bk,r
p,Φ(A).
Hence, B = Bk,r
p,Φ(A).
Theorem 4 is proved.
Theorem 5 can be proved similarly to Theorem 4.
5. Acknowledgements. I would like to express my gratitude to Professor Vladislav Babenko for
his interest in this work and helpful advices.
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Received 03.01.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
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| id | umjimathkievua-article-2592 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:26:26Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/21/2fb7b5be2cccb0d5ddb8e1686655b521.pdf |
| spelling | umjimathkievua-article-25922020-03-18T19:30:15Z On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment Про нерiвностi для норм промiжних похiдних кратно-монотонних функцiй, що заданi на скiнченному вiдрiзку Skorokhodov, D. S. Скороходов, Д. С. We study the following modification of the Landau-Kolmogorov problem: Let $k, r \in \mathbb{N}, \quad 1 \leq k \leq r -1$ and $p, q, s \in [1, \infty]$. Also let $MM^m,\; m \in \mathbb{N}$, be the class of nonnegative functions defined on the segment $[0,1]$ whose derivatives of orders $1, 2,... , m$ are nonnegative almost everywhere on $[0,1]$. For every $\delta > 0$, find the exact value of the quantity $$w^{k, r}_{p, q, s}(\delta; MM^m) := \sup \left\{ ||x^{(k)}||_q : \; x \in MM^m,\; ||x||_p \leq \delta, \;\; ||x^{(r)}||_s \leq 1\right\}$$ We determine the quantity $w^{k, r}_{p, q, s}(\delta; MM^m)$ in the case where $s = \infty$ and $m \in \{r,\; r — 1,\; r — 2\}$. In addition, we consider certain generalizations of the above-stated modification of the Landau-Kolmogorov problem. Дослiджується наступна модифiкацiя задачi Ландау – Колмогорова. Нехай $k, r \in \mathbb{N}, \quad 1 \leq k \leq r -1$, $p, q, s \in [1, \infty]$ i $MM^m,\; m \in \mathbb{N}$, — клас невiд’ємних функцiй, що заданi на вiдрiзку $[0, 1]$ та мають майже скрiзь на $[0, 1]$ невiд’ємнi похiднi порядкiв $0, 1, . . . , m$. Для кожного $\delta > 0$ необхiдно знайти величину $$w^{k, r}_{p, q, s}(\delta; MM^m) := \sup \left\{ ||x^{(k)}||_q : \; x \in MM^m,\; ||x||_p \leq \delta, \;\; ||x^{(r)}||_s \leq 1\right\}.$$ У данiй роботi величину $w^{k, r}_{p, q, s}(\delta; MM^m)$ знайдено у випадку $s = \infty$ та$m \in \{r,\; r — 1,\; r — 2\}$. Також розглянуто деякi узагальнення вказаної модифiкацiї задачi Ландау – Колмогорова. Institute of Mathematics, NAS of Ukraine 2012-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2592 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 4 (2012); 508-524 Український математичний журнал; Том 64 № 4 (2012); 508-524 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2592/1943 https://umj.imath.kiev.ua/index.php/umj/article/view/2592/1944 Copyright (c) 2012 Skorokhodov D. S. |
| spellingShingle | Skorokhodov, D. S. Скороходов, Д. С. On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment |
| title | On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment |
| title_alt | Про нерiвностi для норм промiжних похiдних кратно-монотонних функцiй, що заданi на скiнченному вiдрiзку |
| title_full | On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment |
| title_fullStr | On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment |
| title_full_unstemmed | On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment |
| title_short | On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment |
| title_sort | on inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2592 |
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