Grüss-type and Ostrowski-type inequalities in approximation theory
We discuss the Grass inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain a Grass inequality for the functional $L(f) = H(f; x)$, where $H: C[a,b] \rightarrow C[a,b]$ is a positive linear operator...
Gespeichert in:
| Datum: | 2011 |
|---|---|
| Hauptverfasser: | , , , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2011
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2758 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508724240580608 |
|---|---|
| author | Acu, A.-M. Gonska, H. Ra¸sa, I. Асу, А.-М. Гонська, Х. Раса, І. |
| author_facet | Acu, A.-M. Gonska, H. Ra¸sa, I. Асу, А.-М. Гонська, Х. Раса, І. |
| author_sort | Acu, A.-M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:35:28Z |
| description | We discuss the Grass inequalities on spaces of continuous functions defined on a compact metric space.
Using the least concave majorant of the modulus of continuity, we obtain a Grass inequality for the functional
$L(f) = H(f; x)$, where $H: C[a,b] \rightarrow C[a,b]$ is a positive linear operator and $x \in [a,b]$ is fixed.
We apply this inequality in the case of known operators, for example, the Bernstein, Hermite-Fejer operator the interpolation operator, convolution-type operators.
Moreover, we derive Grass-type inequalities using Cauchy's mean value theorem, thus generalizing results of Cebysev and Ostrowski.
A Grass inequality on a compact metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained.
The latter in turn leads to one further version of Grass' inequality.
In an appendix, we prove a new result concerning the absolute first-order moments of the classical Hermite-Fejer operator. |
| first_indexed | 2026-03-24T02:29:45Z |
| format | Article |
| fulltext |
UDC 517.5
A. M. Acu (Lucian Blaga Univ. Sibiu, Romania),
H. Gonska (Univ. Duisburg-Essen, Germany),
I. Raşa (Techn. Univ., Cluj-Napoca, Romania)
GRÜSS-TYPE AND OSTROWSKI-TYPE INEQUALITIES
IN APPROXIMATION THEORY
НЕРIВНОСТI ТИПУ ГРЮССА ТА ОСТРОВСЬКОГО
В ТЕОРIЇ НАБЛИЖЕНЬ
We discuss the Grüss inequalities on spaces of continuous functions defined on a compact metric space. Using
the least concave majorant of the modulus of continuity, we obtain a Grüss inequality for the functional
L(f) = H(f ;x), where H : C[a, b] → C[a, b] is a positive linear operator and x ∈ [a, b] is fixed. We
apply this inequality in the case of known operators, for example, the Bernstein, Hermite – Fejér operator the
interpolation operator, convolution-type operators. Moreover, we derive Grüss-type inequalities using Cauchy’s
mean value theorem, thus generalizing results of Čebyšev and Ostrowski. A Grüss inequality on a compact
metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained.
The latter in turn leads to one further version of Grüss’ inequality. In an appendix, we prove a new result
concerning the absolute first-order moments of the classical Hermite – Féjer operator.
Розглянуто нерiвностi Грюсса на просторах неперервних функцiй, якi визначено на компактному мет-
ричному просторi. З використанням найменшої опуклої мажоранти модуля неперервностi одержано
нерiвнiсть Грюсса для функцiонала L(f) = H(f ;x), де H : C[a, b] → C[a, b] — додатний лiнiйний
оператор, а x ∈ [a, b] зафiксовано. Цю нерiвнiсть застосовано до випадку вiдомих операторiв, наприклад
оператора Бернштейна, iнтерполяцiйного оператора Ермiта – Фейєра, операторiв типу конволюцiї. Крiм
того, виведено нерiвностi типу Грюсса на основi теореми Кошi про середнє, що узагальнює результати
Чебишова та Островського. Представлено нерiвнiсть Грюсса на компактному метричному просторi
для бiльш нiж двох функцiй та отримано аналогiчну нерiвнiсть типу Островського, яка, в свою чергу,
приводить до ще однiєї версiї нерiвностi Грюсса. У додатку доведено новий результат щодо абсолютних
моментiв першого порядку класичного оператора Ермiта – Фейєра.
1. Introduction. The original form of Grüss’ inequality estimates the difference be-
tween the integral of a product of two functions and the product of integrals of the two
functions and was published by G. Grüss in 1935 [11]:
Theorem A. Let f and g be two functions defined and integrable on [a, b]. If
m ≤ f(x) ≤M and p ≤ g(x) ≤ P for all x ∈ [a, b], then∣∣∣∣∣∣ 1
b−a
b∫
a
f(x)g(x)dx− 1
b−a
b∫
a
f(x)dx
1
b−a
b∫
a
g(x)dx
∣∣∣∣∣∣ ≤ 1
4
(M−m)(P−p).
The constant 1/4 is the best possible.
Grüss’ inequality attracted considerable interest after its publication. Here we men-
tion only papers by E. Landau [14], J. Karamata [12], and a particularly useful one by
A. M. Ostrowski [21]. We also note that a whole chapter in a book by D. S. Mitrinović
et al. [19] is devoted to the inequality we discuss here.
Our present work is to a large extent motivated by a theorem which can be found in
the paper [2] by D. Andrica and C. Badea. Here we cite a special form of it.
c© A. M. ACU, H. GONSKA, I. RAŞA, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6 723
724 A. M. ACU, H. GONSKA, I. RAŞA
Theorem B. Let I = [a, b] be a compact interval of the real axis, B(I) be the
space of real-valued and bounded functions defined on I and L : B(I) → R be a
positive linear functional satisfying L(e0) = 1 where e0 : I 3 x 7→ 1. Assuming that for
f, g ∈ B(I) one has m ≤ f(x) ≤M, p ≤ g(x) ≤ P for all x ∈ I, the following holds:
|L(fg)− L(f)L(g)| ≤ 1
4
(M −m)(P − p).
Another celebrated classical inequality was proved by A. M. Ostrowski [20] in 1938
which we cite below in the form given by Anastassiou in 1995 (see [3]).
Theorem C. Let f be in C1[a, b], x ∈ [a, b]. Then
|f(x)− µ(f)| ≤ ϕ(x)‖f ′‖∞,
where µ(f) :=
1
b− a
∫ b
a
f(t)dt, ϕ(x) :=
(x− a)2 + (b− x)2
2(b− a)
.
There is a relationship between the classical inequalities of Grüss and Ostrowski ob-
served by S. S. Dragomir and S. Wang [7] in 1997 and further studied by
X.-L. Cheng [5] in 2001. The two first-named authors proved that Grüss’ classical
inequality basically implies the following result (which we cite in its improved form
given by Cheng in his Theorem 1.5).
Theorem D. Let f ∈ C1[a, b] satisfy m ≤ f ′(x) ≤M for x ∈ [a, b]. Then∣∣∣∣∣∣f(x)− 1
b− a
b∫
a
f(t)dt− f(b)− f(a)
b− a
(
x− a+ b
2
)∣∣∣∣∣∣ ≤ 1
8
(b− a)(M −m).
Corollary E. Under the assumptions of Theorem D we also have
(i)
∣∣∣∣∣∣f(x)− 1
b− a
b∫
a
f(t)dt
∣∣∣∣∣∣ ≤
∣∣∣∣f(b)− f(a)b− a
∣∣∣∣ ∣∣∣∣x− a+ b
2
∣∣∣∣+ 1
8
(b− a)(M −m);
(ii) if f(b) = f(a), then∣∣∣∣∣∣f(x)− 1
b− a
b∫
a
f(t)dt
∣∣∣∣∣∣ ≤ 1
8
(b− a)(M −m);
(iii) if we choose m = inf
x∈[a,b]
f ′(x), M = sup
x∈[a,b]
f ′(x), then
∣∣∣∣∣∣f(x)− 1
b− a
b∫
a
f(t)dt− f(b)− f(a)
b− a
(
x− a+ b
2
)∣∣∣∣∣∣ ≤ 1
4
(b− a)‖f ′‖.
Note that for f(b) = f(a) the left-hand side in (iii) is Ostrowski’s classical expres-
sion. The right-hand side is in terms of ‖f ′‖; however, it is not pointwise. Note that
(x− a)2 + (b− x)2
2(b− a)
=
1
4
(b− a) for x =
b+ a
2
. The right-hand side in the theorem is
of Grüss-type, i.e., it contains M − m, a difference of upper and lower bounds. It is
thus justified to call an inequality, as given in the theorem, an Ostrowski – Grüss-type
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
GRÜSS-TYPE AND OSTROWSKI-TYPE INEQUALITIES IN APPROXIMATION THEORY 725
inequality (although, historically speaking, Grüss – Ostrowski-type inequality might be
the more adequate term).
In [1] the first two authors gave a generalization of Ostrowski’s inequality for arbi-
trary f ∈ C[a, b] and certain linear operators. In order to formulate the result given here
we need the following definition.
Definition 1. Let f ∈ C[a, b]. If for t ∈ [0,∞) the quantity
ω(f ; t) = sup {|f(x)− f(y)| , |x− y| ≤ t}
is the usual modulus of continuity, its least concave majorant is given by
ω̃(f ; t) = sup
{
(t− x)ω(f ; y) + (y − t)ω(f ;x)
y − x
; 0 ≤ x ≤ t ≤ y ≤ b− a
}
.
Let I = [a, b] be a compact interval of the real axis and f ∈ C(I). In [24] the
following result for the least concave majorant is proved:
K
(
t
2
, f ;C[a, b], C1[a, b]
)
:= inf
g∈C1(I)
(
‖f − g‖∞ +
t
2
‖g′‖∞
)
=
1
2
ω̃(f ; t), t ≥ 0.
Theorem F. Let L : C[a, b] → C[a, b] be non-zero, linear and bounded, and such
that L : C1[a, b] → C1[a, b] with ‖(Lg)′‖ ≤ cL‖g′‖ for all g ∈ C1[a, b]. Then for all
f ∈ C[a, b] and x ∈ [a, b] we have
|Lf(x)− µ(Lf)| ≤ ‖L‖ω̃
(
f ;
cL
‖L‖
ϕ(x)
)
.
If L = Id is the identity on C[a, b], then ‖L‖ = cL = 1, and in this case we get
|f(x)− µ(f)| ≤ ω̃(f ;ϕ(x)), f ∈ C[a, b]. (1)
Remark G. If f ∈ C1[a, b], then the inequality (1) can be written as
|f(x)− µ(f)| ≤ ω̃(f ;ϕ(x)) ≤ ϕ(x)‖f ′‖∞.
This is Ostrowski’s classical inequality in Anastassiou’s form (see above). If f ∈
∈ LipMα, 0 < α ≤ 1, then |f(x) − µ(f)| ≤ ω̃(f ;ϕ(x)) ≤ M(ϕ(x))α. For α = 1
we obtain Dragomir’s inequality [6].
It is the aim of this paper to look again at Grüss’ inequality from a somewhat
different point of view, and to eventually relate it again to Ostrowski’s inequality. In
doing so we will be guided by the contribution of Andrica and Badea. That is: how
non-multiplicative is a linear functional in the worst case? This is quite an intriguing
question from the point of view of approximation theory.
2. A pre-Grüss inequality on a compact metric space. In 2004 A. Mc. D. Mercer
and P. R. Mercer [17] gave the following pre-Grüss inequality for a positive linear
functional L : B(I)→ R, with L(1) = 1:
|L(fg)− L(f)L(g)| ≤ 1
2
min {(M −m)L(|g −G|), (P − p)L(|f − F |)} , (2)
where m ≤ f(x) ≤M, p ≤ g(x) ≤ P for all x ∈ I, F := Lf and G := Lg.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
726 A. M. ACU, H. GONSKA, I. RAŞA
In this section we will prove a pre-Grüss inequality on a compact metric space. Let
L : C(X) → R be a linear bounded functional, L(1) = 1, where C(X) is a compact
metric space with metric d. Then there are positive linear functionals L+, L−, |L|
such that L = L+ − L− and |L| = L+ + L−. If L is a positive functional we have
|L| = L+ = L.
Since M − m = ω(f ; d(X)), P − p = ω(g; d(X)), where m = inf f(x), M =
= sup f(x), p = inf g(x), P = sup g(x), we can prove, using the idea of A. Mercer
and P. Mercer’s proof, the following inequality:
Theorem 1. Let L : C(X) → R be a linear, bounded functional, L(1) = 1,
defined on the compact metric space C(X). Then the inequality
|L(fg)−L(f)L(g)| ≤ 1
2
min {ω(f ; d(X))|L|(|g−G|), ω(g; d(X))|L|(|f−F |)} (3)
holds.
Remark 1. The inequality is sharp in the sense that a non-positive functional A
with A(1) = 1 exists such that equality occurs.
Example 1. Let us consider the following non-positive functional
A : C[0, 1]→ R, A(f) = 2f(0)− f(1).
For this functional we have A(1) = 1, A+(f) = 2f(0), A−(f) = f(1), |A|(f) =
= 2f(0) + f(1) and A(fg)−A(f)A(g) = 2(f(1)− f(0))(g(0)− g(1)). If we choose
f(t) = g(t) = t, then F = G = −1 and
|A(fg)−A(f)A(g)| = 2 =
1
2
min {ω(f ; 1)|A|(g + 1), ω(g, 1)|A|(f + 1)} .
Corollary 1. If L : C(X) → R is a positive linear (and thus bounded) functional
with L(1) = 1, then for all f, g ∈ C(X) we have
|L(fg)−L(f)L(g)| ≤ 1
2
min {ω(f ; d(X))L(|g−G|), ω(g; d(X))L(|f−F |)} , (4)
|L(fg)− L(f)L(g)| ≤ 1
4
ω(f ; d(X))ω(g; d(X)). (5)
Proof. Since L is a positive functional it follows |L| = L; so the first inequality is
proved.
In [17] A. Mercer and P. Mercer show that the inequalities
L(|g −G|) ≤ 1
2
(P − p) and L(|f − F |) ≤ 1
2
(M −m) (6)
hold. The inequality (5) can be obtained by using in (4) the inequalities (6).
Corollary 1 is proved.
In [8], B. Gavrea and I. Gavrea raised the following problem.
Problem. Let L be a linear positive functional defined on C[0, 1] with L(1) = 1
and f, g be two continuous functions. Do positive numbers δ1 = δ1(f) < 1 and δ2 =
= δ2(f) < 1 exist such that
|L(fg)− L(f)L(g)| ≤ 1
4
ω̃(f ; δ1)ω̃(f ; δ2)?
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
GRÜSS-TYPE AND OSTROWSKI-TYPE INEQUALITIES IN APPROXIMATION THEORY 727
We will show that the answer to Gavreas’ question is negative. Let us consider
L(f) = B1
(
f ;
1
2
)
=
1
2
(f(0) + f(1)) , f ∈ C[0, 1],
where B1 is the first Bernstein operator on C[0, 1].
If we choose f(t) = g(t) = t we have, with ei(t) := ti,
|L(fg)− L(f)L(g)| = |L(e2)− L(e1)2| =
∣∣∣∣∣B1
(
e2;
1
2
)
−B1
(
e1;
1
2
)2
∣∣∣∣∣ = 1
4
.
Moreover, for 0 ≤ t ≤ 1, ω1(e1; t) = ω̃(e1; t) = t, implying
1
4
ω̃(f ; t)ω̃(g; t) =
1
4
t2 <
1
4
for 0 ≤ t < 1.
Hence the conjecture of the two Gavreas is not true.
One more question is if the upper bound in (5) has a corresponding lower bound,
i.e., if there is a constant c > 0 such that for all f, g ∈ C(X) we also have
cω(f ; d(X))ω(g; d(X)) ≤ |L(f · g)− L(f)L(g)|. (7)
The following example shows that this is not the case.
Example 2. Suppose A : C(X) → R is a positive linear functional satisfying
A(1) = 1. Write D(f, g) := A(f · g)−A(f)A(g).
Case 1: supp A = {x} for x ∈ X. Then A = δx, the point evaluation functional at
x. Hence D(f, g) = 0 for all f, g ∈ C(X), and for appropriate choices of X, f and g
the left-hand side of (7) is non-zero.
Case 2: supp A = {x, y}, meaning that A = α · δx + β · δy, where α, β > 0 and
α+ β = 1. Hence
D(f, g) = α · β(f(y)− f(x))(g(y)− g(x)) = 0
if and only if f or/and g is/are constant on supp A. Again for suitable choices of X, f
and g the left-hand side of (7) is non-zero.
Case 3: |supp A| ≥ 3. Then there is an h ∈ C(X) taking at least 3 distinct values
on supp A. Let a := A(h), b := A(h2), c := A(h3).
For all t ∈ R we have (h − t · 1)2 ≥ 0, implying A(h2) − 2tA(h) + t2 ≥ 0.
Taking t = A(h) shows that A(h2) ≥ A2(h). If A(h2) = A2(h), then there is a t0 ∈ R
such that A(h2)− 2t0A(h) + t20 = 0, i.e., A((h− t01)2) = 0. This implies that h− t01
is constant on supp A, which is a contradiction. Thus A(h2) − A2(h) = b − a2 > 0.
Let f := h− a, g := h2 +
ab− c
b− a2
h. Then A(f) = 0, A(f · g) = 0, and so D(f, g) = 0.
Clearly f is non-constant on supp A. Assuming that g = d is constant on A, means
h2+
ab− c
b− a2
h = d, or h2+
ab− c
b− a2
h−d = 0 on supp A. But this means that h attains at
most two values on supp A, again a contradiction. Thus f and g are both non-constant
on supp A and again the right-hand side of (7) is non-zero.
3. Grüss-type inequalities for positive linear operators. LetHn : C[a, b]→ C[a, b]
be positive linear operators which reproduce constant functions. For x ∈ [a, b] we con-
sider L = εx ◦Hn, so L(f) = Hn(f ;x). Denote by
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
728 A. M. ACU, H. GONSKA, I. RAŞA
D(f, g) := Hn(fg;x)−Hn(f ;x)Hn(g;x).
The following result suggests how non-multiplicative the functional L(f) = Hn(f ;x)
is for a given x ∈ [a, b].
Theorem 2. If f, g ∈ C[a, b] and x ∈ [a, b] is fixed, then the inequality
|D(f, g)| ≤ 1
4
ω̃
(
f ; 2
√
2Hn ((e1 − x)2;x)
)
ω̃
(
g; 2
√
2Hn ((e1 − x)2;x)
)
holds.
Proof. Using the Cauchy – Schwarz inequality for positive linear functionals we can
write
|Hn(f ;x)| ≤ Hn(|f |;x) ≤
√
Hn(f2;x)Hn(1;x) =
√
Hn(f2;x),
so
D(f, f) = Hn(f
2;x)−Hn(f ;x)
2 ≥ 0.
Then D is a positive semidefinite bilinear form on C[a, b]. For f, g ∈ C[a, b], using
Cauchy – Schwarz for D, it follows that
|D(f, g)| ≤
√
D(f, f)D(g, g) ≤ ‖f‖∞‖g‖∞. (8)
Since Hn : C[a, b] → C[a, b] is a positive linear operator which reproduces constant
functions, Hn(f ;x), with x ∈ [a, b] fixed, is a positive linear functional and can be
represented as Hn(f ;x) =
∫ b
a
f(t)dµ(t), where µ is a probability measure on [a, b],
i.e.,
∫ b
a
dµ(t) = 1.
We have
Hn(f
2;x)−Hn(f ;x)
2 =
b∫
a
f2(t)dµ(t)−
b∫
a
f(s)dµ(s)
2
=
=
b∫
a
f(t)− b∫
a
f(s)dµ(s)
2
dµ(t) =
b∫
a
b∫
a
(f(t)− f(s))dµ(s)
2
dµ(t) ≤
≤
b∫
a
b∫
a
(f(t)− f(s))2dµ(s)
dµ(t) ≤ ‖f ′‖2∞
b∫
a
b∫
a
(t− s)2dµ(s)
dµ(t) =
= ‖f ′‖2∞
b∫
a
t2 − 2t
b∫
a
sdµ(s) +
b∫
a
s2dµ(s)
dµ(t) =
= ‖f ′‖2∞
b∫
a
t2dµ(t)− 2
b∫
a
sdµ(s)
b∫
a
tdµ(t) +
b∫
a
s2dµ(s)
=
= 2‖f ′‖2∞
[
Hn(e2;x)−Hn(e1;x)
2
]
≤ 2‖f ′‖2∞Hn
(
(e1 − x)2;x
)
, f ∈ C1[a, b].
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
GRÜSS-TYPE AND OSTROWSKI-TYPE INEQUALITIES IN APPROXIMATION THEORY 729
Therefore
D(f, f) = Hn(f
2;x)−Hn(f ;x)
2 ≤ 2‖f ′‖2∞Hn
(
(e1 − x)2;x
)
. (9)
Using relation (9) for differentiable functions r, s ∈ C1[a, b], we obtain the following
estimate:
|D(r, s)| ≤
√
D(r, r)D(s, s) ≤ 2‖r′‖∞‖s′‖∞Hn
(
(e1 − x)2;x
)
. (10)
Moreover, if f ∈ C[a, b], s ∈ C1[a, b], then
|D(f, s)| ≤
√
D(f, f)D(s, s) ≤ ‖f‖∞
√
2‖s′‖∞
√
Hn ((e1 − x)2;x). (11)
Likewise, for r ∈ C1[a, b], g ∈ C[a, b], we have
|D(r, g)| ≤ ‖g‖∞
√
2‖r′‖∞
√
Hn ((e1 − x)2;x). (12)
Now let f, g ∈ C[a, b] be fixed, r, s ∈ C1[a, b] arbitrary. Then
|D(f, g)| = |D(f − r + r, g − s+ s)| ≤
≤ |D(f − r, g − s)|+ |D(f − r, s)|+ |D(r, g − s)|+ |D(r, s)| ≤
≤ ‖f − r‖‖g − s‖+
√
2‖f − r‖‖s′‖
√
Hn ((e1 − x)2;x)+
+
√
2‖g − s‖‖r′‖
√
Hn ((e1 − x)2;x) + 2‖r′‖‖s′‖Hn
(
(e1 − x)2;x
)
=
=
{
‖f − r‖+ ‖r′‖
√
2Hn ((e1 − x)2;x)
}{
‖g − s‖+ ‖s′‖
√
2Hn ((e1 − x)2;x)
}
.
Passing to the infimum over r and s ∈ C1[a, b], respectively, shows
|D(f, g)| ≤ K
(√
2Hn ((e1−x)2;x), f ;C0, C1
)
×
×K
(√
2Hn ((e1−x)2;x), g;C0, C1
)
=
=
1
2
ω̃
(
f ;
√
8Hn ((e1 − x)2;x)
)1
2
ω̃
(
g;
√
8Hn ((e1 − x)2;x)
)
=
=
1
4
ω̃
(
f ; 2
√
2Hn ((e1 − x)2;x)
)
ω̃
(
g; 2
√
2Hn ((e1 − x)2;x)
)
,
which concludes the proof.
Remark 2. If we choose Hn = Bn, the Bernstein operator, then this gives
|Bn(fg;x)−Bn(f ;x)Bn(g;x)| ≤
≤ 1
4
ω̃
(
f ; 2
√
2Bn ((e1−x)2;x)
)
ω̃
(
g; 2
√
2Bn ((e1−x)2;x)
)
=
=
1
4
ω̃
(
f ; 2
√
2x(1−x)
n
)
ω̃
(
g; 2
√
2x(1−x)
n
)
≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
730 A. M. ACU, H. GONSKA, I. RAŞA
≤ ω̃
(
f ;
1√
2n
)
ω̃
(
g;
1√
2n
)
, f, g ∈ C[0, 1].
Remark 3. The above result can be remarkably generalized if we replace ([a, b], |·|)
by a compact metric space (X, d), Hn((e1 − x)2;x) by Hn(d
2(·, x);x), and
K(·, f ;C[a, b], C1[a, b]) by K(·, f ;C(X),Lip1).
4. Grüss-type inequality for the classical Hermite – Fejér interpolation operator.
The classical Hermite – Fejér interpolation operator is a positive linear operator and can
be written as
Ln(f ;x) =
n∑
k=1
f(xk)(1− xxk)
(
Tn(x)
n(x− xk)
)2
, (13)
where f ∈ C[−1, 1] and xk = cos
2k − 1
2n
π, 1 ≤ k ≤ n, are the zeros of Tn(x) =
= cos (n arccos), the n-th Chebyshev polynomial of the first kind.
For this operator we have Ln
(
(e1 − x)2;x
)
=
1
n
T 2
n(x).
Remark 4. If we choose in Theorem 2 Hn = Ln, the classical Hermite – Fejér
interpolation operator, then this gives
|Ln(fg;x)−Ln(f ;x)Ln(g;x)|≤
1
4
ω̃
(
f ;
2
√
2√
n
|Tn(x)|
)
ω̃
(
g;
2
√
2√
n
|Tn(x)|
)
. (14)
This is disappointing in view of the fact that Ln approximates much faster than Bn.
Indeed, in [9] the following pointwise inequality was proved:
|Ln(f ;x)− f(x)| ≤ 5ω1
(
f ;
|Tn(x)|
n
{√
1− x2 lnn+ 1
})
.
In this section we will give a different approach adapted to the Hermite – Fejér case.
Denote by
D(f, g) := Ln(fg;x)− Ln(f ;x)Ln(g;x).
Theorem 3. If f, g ∈ C[−1, 1] and x ∈ [−1, 1] is fixed, then the following in-
equality is verified:
|D(f, g)| ≤ 1
2
min
{
‖f‖∞ω̃
(
g;
40 lnn
n
)
, ‖g‖∞ω̃
(
f ;
40 lnn
n
)}
. (15)
Proof. For f ∈ C[−1, 1], s ∈ C1[−1, 1] proceed as follows:
|D(f, s)| = |Ln(f · s;x)− Ln(f ;x)Ln(s;x)| = |Ln(f(s− Ln(s;x));x)| =
= |Ltn(f(t)(s(t)− s(x) + s(x)− Ln(s;x));x)| ≤
≤ ‖f‖∞Ltn(|s(t)− s(x)|+ |s(x)− Ln(s;x)|;x) ≤
≤ ‖f‖∞Ln(‖s′‖|e1 − x|+ ‖s′‖Ln(|e1 − x|;x);x) =
= 2‖f‖∞‖s′‖∞Ln(|e1 − x|;x).
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
GRÜSS-TYPE AND OSTROWSKI-TYPE INEQUALITIES IN APPROXIMATION THEORY 731
Now, for f, g ∈ C[−1, 1] fixed and s ∈ C1[−1, 1] arbitrary we get
|D(f, g)| = |D(f, g − s+ s)| ≤ |D(f, g − s)|+ |D(f, s)| ≤
≤ ‖f‖∞‖g − s‖∞ + 2‖f‖∞‖s′‖∞Ln(|e1 − x|;x) =
= ‖f‖∞ {‖g − s‖∞ + 2Ln(|e1 − x|;x)‖s′‖∞} .
Passing to the infimum over s ∈ C1 yields
|D(f, g)| ≤ ‖f‖∞K(2Ln(|e1 − x|;x), g;C0, C1) =
= ‖f‖∞
1
2
ω̃(g, 4Ln(|e1 − x|;x)).
By symmetry the same holds with f and g interchanged. Hence
|D(f, g)| ≤ 1
2
min {‖f‖∞ω̃(g, 4Ln(|e1 − x|;x)); ‖g‖∞ω̃(f, 4Ln(|e1 − x|;x))}.
But in [9] it was proved that (see the appendix for a detailed proof)
Ln(|e1 − x|;x) ≤
4
n
|Tn(x)|(
√
1− x2 lnn+ 1) ≤ 10
lnn
n
, n ≥ 2, (16)
and so
|D(f, g)| ≤ 1
2
min
{
‖f‖∞ω̃
(
g;
40 lnn
n
)
, ‖g‖∞ω̃
(
f ;
40 lnn
n
)}
.
Remark 5. If one of the functions f or g is in Lip1, we have |D(f, g)| =
= O
(
lnn
n
)
, n → ∞. The relation (14) implies in this case only |D(f, g)| =
= o
(
1√
n
)
. Also, the relation (14) implies |D(f, g)| = o
(
1
n
)
for f, g ∈ Lip 1.
This cannot be concluded from (15).
5. A Grüss inequality for convolution-type operators.
Definition 2. For every function f ∈ C(I), I = [−1, 1], and any natural number
n, the operator Gm(n) is defined by
Gm(n)(f, t) := π−1
π∫
−π
f (cos(arccos t+ v))Km(n)(v)dv,
where the kernel Km(n) is a trigonometric polynomial of degree m(n) with the following
properties:
(i) Km(n) is positive and even;
(ii)
∫ π
−π
Km(n)(v)dv = π, i.e., Gm(n)(1, t) = 1 for t ∈ I.
For each f ∈ C(I) the integral Gm(n)(f, ·) from Definition 2 is an algebraic poly-
nomial of degree m(n). Moreover, in view of (i) and (ii) one has
Km(n)(v) =
1
2
+
m(n)∑
k=1
ρk,m(n) cos kv, v ∈ [−π, π].
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
732 A. M. ACU, H. GONSKA, I. RAŞA
Lemma 1 [15]. For x ∈ I the inequality
Gm(n)
(
(e1 − x)2, x
)
= x2
{
3
2
− 2ρ1,m(n) +
1
2
ρ2,m(n)
}
+ (1− x2)
{
1
2
− 1
2
ρ2,m(n)
}
holds. Here e1 denotes the first monomial given by e1(t) = t for |t| ≤ 1.
If Km(n) is the Fejér – Korovkin kernel with m(n) = n − 1, then it is known that
(see [16])
ρ1,n−1 = cos
π
n+ 1
, ρ2,n−1 =
n
n+ 1
cos
2π
n+ 1
+
1
n+ 1
. (17)
Using the relations (17) we get
Gn−1
(
(e1 − x)2;x)
)
≤
∣∣∣∣32 − 2ρ1,n−1 +
1
2
ρ2,n−1
∣∣∣∣+ 1
2
|1− ρ2,n−1| ≤
≤ 3
(
π
n+ 1
)2
+
(
π
n+ 1
)2
= 4
(
π
n+ 1
)2
.
Remark 6. If we consider in Theorem 2 the convolution-type operators with the
Fejér – Korovkin kernel we have
|D(f ; g)| = |Gn−1(fg;x)−Gn−1(f ;x)Gn−1(g;x)| ≤
≤ 1
4
ω̃
(
f ; 4
√
2
π
n+ 1
)
ω̃
(
g; 4
√
2
π
n+ 1
)
= O
(
ω̃
(
f ;
1
n
)
ω̃
(
g;
1
n
))
.
This is an improvement of what we obtained for the Bernstein and Hermite – Fejér
operators.
6. Estimates via Cauchy’s mean value theorem. Let L : C[a, b] → R be a linear
positive functional. We denote by
T (f, g) = L(fg)− L(f)L(g), f, g ∈ C[a, b].
In this section our aim is to establish a Grüss inequality for the functional L us-
ing Cauchy’s mean value theorem. Our work is motivated by B.G. Pachpatte’s re-
sult obtained in [23] for the functional L(f) =
1∫ b
a
w(x)dx
∫ b
a
w(x)f(x)dx, where
w : [a, b]→ [0,∞) is an integrable function such that
∫ b
a
w(x)dx > 0.
Theorem 4. If L : C[a, b] → R is a linear positive functional, with L(1) = 1,
then
i) there is (η, θ) ∈ [a, b]× [a, b] such that T (f, g) =
f ′(η)
h′(η)
g′(θ)
h′(θ)
T (h, h);
ii) |T (f, g)| ≤
∥∥∥∥f ′h′
∥∥∥∥
∞
∥∥∥∥ g′h′
∥∥∥∥
∞
|T (h, h)|, where f, g, h ∈ C1[a, b] and h′(t) 6= 0 for
each t ∈ [a, b].
Proof. Let x, y ∈ [a, b] with y 6= x. Applying Cauchy’s mean value theorem, there
exist points ξ1 and ξ2 between y and x such that
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
GRÜSS-TYPE AND OSTROWSKI-TYPE INEQUALITIES IN APPROXIMATION THEORY 733
f(x)− f(y) = f ′(ξ1)
h′(ξ1)
(h(x)− h(y)), (18)
g(x)− g(y) = g′(ξ2)
h′(ξ2)
(h(x)− h(y)). (19)
Multiplying the left-hand sides and right-hand sides of (18) and (19), we get
(f(x)− f(y))(g(x)− g(y)) = f ′(ξ1)
h′(ξ1)
g′(ξ2)
h′(ξ2)
(h(x)− h(y))2.
If we apply the functional L with respect to x and y it follows
2T (f, g) = LyLx
(
f ′(ξ1)
h′(ξ1)
g′(ξ2)
h′(ξ2)
(h(x)− h(y))2
)
. (20)
If we denote by
m = min
(x,y)∈[a,b]×[a,b]
f ′(x)
h′(x)
g′(y)
h′(y)
,
M = max
(x,y)∈[a,b]×[a,b]
f ′(x)
h′(x)
g′(y)
h′(y)
,
then we can write m ≤ f ′(ξ1)
h′(ξ1)
g′(ξ2)
h′(ξ2)
≤M, namely
m(h(x)− h(y))2 ≤ f ′(ξ1)
h′(ξ1)
g′(ξ2)
h′(ξ2)
(h(x)− h(y))2 ≤M(h(x)− h(y))2.
If apply the functional L with respect to x and y, we get
2mT (h, h) ≤ LyLx
(
f ′(ξ1)
h′(ξ1)
g′(ξ2)
h′(ξ2)
(h(x)− h(y))2
)
≤ 2MT (h, h).
Since
m ≤
LyLx
(
f ′(ξ1)
h′(ξ1)
g′(ξ2)
h′(ξ2)
(h(x)− h(y))2
)
2T (h, h)
≤M,
it follows that there is (η, θ) ∈ [a, b]× [a, b] such that
LyLx
(
f ′(ξ1)
h′(ξ1)
g′(ξ2)
h′(ξ2)
(h(x)− h(y))2
)
2T (h, h)
=
f ′(η)
h′(η)
g′(θ)
h′(θ)
.
Using the above relation in (20), it follows
T (f, g) =
f ′(η)
h′(η)
g′(θ)
h′(θ)
T (h, h). (21)
From (21) we have
|T (f, g)|≤
∥∥∥∥f ′h′
∥∥∥∥
∞
∥∥∥∥ g′h′
∥∥∥∥
∞
|T (h, h)|.
Theorem 4 is proved.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
734 A. M. ACU, H. GONSKA, I. RAŞA
Remark 7. If in Theorem 4 we take h(x) = x, x ∈ [a, b], and L(f) =
=
1
b− a
∫ b
a
f(x)dx, then
(i) there is (η, θ) ∈ [a, b]× [a, b] such that
1
b− a
b∫
a
f(x)g(x)dx− 1
(b− a)2
b∫
a
f(x)dx
b∫
a
g(x)dx =
(b− a)2
12
f ′(η)g′(θ),
this identity was found by Ostrowski [21] in 1970;
(ii)
∣∣∣∣∣∣ 1
b− a
b∫
a
f(x)g(x)dx− 1
(b− a)2
b∫
a
f(x)dx
b∫
a
g(x)dx
∣∣∣∣∣∣ ≤
≤ (b− a)2
12
sup
x∈[a,b]
|f ′(x)| sup
x∈[a,b]
|g′(x)|;
this inequality was proved by Čebyšev [4] in 1882, the constant
(b− a)2
12
is best possi-
ble, as can be seen for [a, b] = [0, 1], f(x) = g(x) = x.
Theorem 5. If L : C[a, b] → R is a linear positive functional, with L(1) = 1,
then the following inequality holds:
|T (f, h) + T (g, h)| ≤ |T (h, h)|
(∥∥∥∥f ′h′
∥∥∥∥
∞
+
∥∥∥∥ g′h′
∥∥∥∥
∞
)
,
where f, g, h ∈ C1[a, b] and h′(t) 6= 0 for each t ∈ [a, b].
Proof. Multiplying both sides of (18) and (19) by h(x) − h(y) and adding the
resulting identities we get
(f(x)− f(y))(h(x)− h(y)) + (g(x)− g(y))(h(x)− h(y)) =
=
f ′(ξ1)
h′(ξ1)
(h(x)− h(y))2 + g′(ξ2)
h′(ξ2)
(h(x)− h(y))2.
If we apply the functional L with respect to x and y, we get
2T (f, h) + 2T (g, h) = LyLx
(
f ′(ξ1)
h′(ξ1)
(h(x)− h(y))2
)
+
+LyLx
(
g′(ξ2)
h′(ξ2)
(h(x)− h(y))2
)
. (22)
In a similar way with the proof of Theorem 4 it can be shown that there are η, θ ∈ [a, b]
such that
LyLx
(
f ′(ξ1)
h′(ξ1)
(h(x)− h(y))2
)
= 2T (h, h)
f ′(η)
h′(η)
,
LyLx
(
g′(ξ2)
h′(ξ2)
(h(x)− h(y))2
)
= 2T (h, h)
g′(θ)
h′(θ)
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
GRÜSS-TYPE AND OSTROWSKI-TYPE INEQUALITIES IN APPROXIMATION THEORY 735
Using the above identities in (22), we get
T (f, h) + T (g, h) =
(
f ′(η)
h′(η)
+
g′(θ)
h′(θ)
)
T (h, h).
Therefore
|T (f, h) + T (g, h)| ≤
(∥∥∥∥f ′h′
∥∥∥∥
∞
+
∥∥∥∥ g′h′
∥∥∥∥
∞
)
|T (h, h)|.
Theorem 5 is proved.
In the paper [21] Ostrowski defined the concept of synchronous functions. The func-
tions f, g : [a, b] → R are called synchronous, if we have, for any couple of points x, y
from [a, b], f(x) ≥ f(y) if and only if g(x) ≥ g(y).
In the case that f, g are synchronous, we get T (f, g) ≥ 0.
Theorem 6. If L : C[a, b] → R is a linear positive functional, with L(1) = 1,
then the following inequality is verified:
|T (f, g)| ≤ 1
2
[∥∥∥∥f ′h′
∥∥∥∥
∞
|T (g, h)|+
∥∥∥∥ g′h′
∥∥∥∥
∞
|T (f, h)|
]
, (23)
where f, g, h ∈ C1[a, b], h′(t) 6= 0 for each t ∈ [a, b] and the functions f, g, respectively
g, h are synchronous.
Proof. Multiplying both sides of (18) and (19) by g(x) − g(y) and f(x) − f(y),
respectively, adding the resulting identities, and applying the functionals L with respect
to x and y, we get
4T (f, g) = LyLx
(
f ′(ξ1)
h′(ξ1)
(h(x)− h(y))(g(x)− g(y))
)
+
+LyLx
(
g′(ξ2)
h′(ξ2)
(h(x)− h(y))(f(x)− f(y))
)
.
Using this identity and the reasoning from the proof of the above theorems inequality
(23) follows.
Theorem 6 is proved.
7. Grüss inequality for more than two functions. In this section we will prove a
Grüss inequality on a compact metric space for more than two functions.
Lemma 2. Let C(X) be a compact metric space and fk ∈ C(X), 1 ≤ k ≤ n,
n ≥ 1. Then the following inequality holds:
θ(f1f2 . . . fn) ≤
n∑
i=1
θ(fi)
n∏
k=1,k 6=i
‖fk‖∞, (24)
where θ(f) := max
X
f −min
X
f, f ∈ C(X).
Proof. Inequality (24) can be proved using induction.
Theorem 7. Let A : C(X)→ R be a positive linear functional, A(1) = 1, defined
on the metric space C(X). The inequality
|A(f1f2 . . . fn)−A(f1)A(f2) . . . A(fn)| ≤
1
4
n∑
i,j=1,i<j
θ(fi)θ(fj)
n∏
k=1,k 6=i,j
‖fk‖∞
(25)
holds.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
736 A. M. ACU, H. GONSKA, I. RAŞA
Proof. The inequality (25) can be proved using induction and relation (24).
Remark 8. If f3, . . . , fn are constant, relation (25) reduces to
|A(f1f2)−A(f1)A(f2)| ≤
1
4
(M1 −m1)(M2 −m2),
where Mi = max
X
fi, mi = min
X
fi, i ∈ {1, 2}.
The following result is an extension of Ostrowski’s inequality (1):
Theorem 8. If fi ∈ C[a, b], 1 ≤ i ≤ n, then the following inequality is true:
|f1(x) . . . fn(x)− µ(f1) . . . µ(fn)| ≤
n∑
i=1
ω̃(fi;ϕ(x))
n∏
k=1,k 6=i
‖fk‖∞,
where µ(f) :=
1
b− a
∫ b
a
f(t)dt, ϕ(x) :=
(x− a)2 + (b− x)2
2(b− a)
.
Proof. The inequality can be proved using induction.
Remark 9. Since ϕ(x) ≤ b− a
2
, x ∈ [a, b], we get
|f1(x) . . . fn(x)− µ(f1) . . . µ(fn)| ≤
n∑
i=1
ω̃
(
fi;
b− a
2
) n∏
k=1,k 6=i
‖fk‖∞.
This relation yields the Grüss-type inequality
|µ (f1 . . . fn)− µ(f1) . . . µ(fn)| ≤
n∑
i=1
ω̃
(
fi;
b− a
2
) n∏
k=1,k 6=i
‖fk‖∞.
Appendix. Here we give the proof of inequality (16) for the classical Hermite –
Fejér interpolation operator based on the roots of Čebyšev polynomials of the first kind,
defined in (13).
Lemma 3. For the continuous function |e1−x|, x ∈ [−1, 1] fixed, and n ≥ 1 one
has
Ln(|e1 − x|, x) ≤
c
n
|Tn(x)|
{
(1− x2)1/2 lnn+ 1
}
.
Here e1(t) = t for |t| ≤ 1, and c is a constant independent of n and x and satisfying
1 ≤ c ≤ 4.
Proof. The existence of a constant c independent of n and x in the above estimate
can be derived from papers of R. N. Misra [18] or S. J. Goodenough and T. M. Mills
[10]. In order to show that c is bounded from above by 4 we give a proof similar to that
of Goodenough and Mills using the technique of O. Kiš [13].
Let x ∈ [−1, 1] be fixed. We may assume that x 6= xk, 1 ≤ k ≤ n, since otherwise
the estimate is apparently true.
For n = 1 we have x1 = 0 such that L1(|e1 − x|;x) = |x| = |T1(x)|. Hence the
estimate holds for n = 1. We assume in the sequel that n ≥ 2. Because of x = cos θ,
0 ≤ θ ≤ π, and xk = cos θk with θk = (2k − 1)(2n)−1π, we may proceed as follows.
Let j be chosen in a way such that θj is closest to θ among all θk’s (if θ has the same
distance from θk and θk+1, say, we may choose either of them). Thus the following
situations may occur:
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
GRÜSS-TYPE AND OSTROWSKI-TYPE INEQUALITIES IN APPROXIMATION THEORY 737
“Left” case:
“Right” case:
Note that in the “left” case θj−1 need not exist (θ close to 0); a corresponding remark
applies in the “right” case to θj+1 (θ close to π).
After fixing j as described above, we write
Ln(|e1 − x|, x)=
n∑
k=1
|xk − x|
(1− xxk)T 2
n(x)
n2(x− xk)2
=:
n∑
k=1
|xk − x|hk(x) =:
n∑
k=1
Wk(x) =
=
j−1∑
k=1
Wk(x) +Wj(x) +
n∑
k=j+1
Wk(x) =: I1 + I2 + I3.
Clearly, if j = 1 or j = n, then one of the two sums will not be present. First observe
that for 1 ≤ k ≤ n we have
hk(x) =
(1− xxk)T 2
n(x)
n2(x− xk)2
=
(1− x2)T 2
n(x)
n2(x− xk)2
+
xT 2
n(x)
n2(x− xk)
,
which implies
Wk(x) = |xk − x|hk(x) ≤
(1− x2)T 2
n(x)
n2|x− xk|
+
|x|T 2
n(x)
n2
.
While the second term in this upper bound does not cause difficulties, the first one may
be written in the following way:
(1− x2)T 2
n(x)
n2|x− xk|
=
√
1− x2T 2
n(x)
n2
sin θ
| cos θ − cos θk|
.
Now only the second ratio requires further consideration. First observe that
sin θ
| cos θ − cos θk|
1
| sin 1
2 (θ − θk)|
≤ π
|θ − θk|
;
here the first inequality may be obtained by using Lemma 2 (a) in Goodenough’s and
Mills’ paper [10], while the second one is a consequence of α ≤ 1
2
π sinα, 0 ≤ α ≤ 1
2
π.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
738 A. M. ACU, H. GONSKA, I. RAŞA
Consequently, it remains to investigate the quantities 1/|θ − θk|. It is at this point that
we check the “left” and the “right” case separately. In both cases we first estimate
I1 + I3 =
j−1∑
k=1
Wk(x) +
n∑
k=j+1
Wk(x) and add Wj(x) afterwards;
we have
I1 + I3 =
j−1∑
k=1
Wk(x) +
n∑
k=j+1
Wk(x) ≤
≤
√
1− x2T 2
n(x)
n2
π
j−1∑
k=1
1
|θ − θk|
+
n∑
k=j+1
1
|θ − θk|
+ (n− 1)
|x|T 2
n(x)
n2
.
For the “left” case (i.e., θj−1 < θ < θj) we have for 1 < j < n and k ≤ j − 1
θ − θk ≥ (θj−1 − θk) +
1
2
(θj − θj−1) = (2i− 1)(2n)−1π, if k = j − i,
and for k ≥ j + 1
θk − θ ≥ θk − θj = in−1π, if k = j + i.
In this case
j−1∑
k=1
1
|θ − θk|
+
n∑
k=j+1
1
|θ − θk|
≤ 2nπ−1
(
j−1∑
k=1
1
2k − 1
+
n−j∑
k=1
1
2k
)
≤
≤ 2nπ−1
[
1 +
1
2
ln(2j − 3) +
1
2
(1 + ln(n− j))
]
≤ 2nπ−1
[
3
2
+ ln
(
1√
2
n
)]
.
Note that this estimate is also true if j = 1 or j = n. For the “right” case (i.e., θj < θ <
< θj+1) our estimate for 1 < j < n and k ≤ j − 1 is
θ − θk ≥ θj − θk = in−1π, if k = j − i
and for k ≥ j + 1
θk − θ ≥ θk − θj+1 +
1
2
(θj+1 − θj) = (2i− 1)(2n)−1π, if k = j + i.
Thus for the “right” case we arrive at the symmetric inequality
j−1∑
k=1
1
|θ − θk|
+
n∑
k=j+1
1
|θ − θk
| ≤ 2nπ−1
(
j−1∑
k=1
1
2k
+
n−j∑
k=1
1
2k − 1
)
≤
≤ 2nπ−1
[
1
2
+
1
2
ln(j − 1) + 1 +
1
2
ln(2(n− j)− 1)
]
≤ 2nπ−1
[
3
2
+ ln
(
1√
2
n
)]
.
Note that this inequality is also true for j = 1 or j = n.
The common estimate obtained for both the “left” and the “right” cases is thus
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
GRÜSS-TYPE AND OSTROWSKI-TYPE INEQUALITIES IN APPROXIMATION THEORY 739
I1 + I3 ≤
√
1− x2T 2
n(x)
n2
π
j−1∑
k=1
1
|θ − θk|
+
n∑
k=j+1
1
|θ − θk|
+ (n− 1)
|x|T 2
n(x)
n2
≤
≤
√
1− x2T 2
n(x)
n2
π2nπ−1
(
3
2
+ ln
(
1√
2
n
))
+
|x|T 2
n(x)
n
≤
≤
√
1− x2T 2
n(x)
n
(
3 + 2 ln
(
1√
2
n
))
+
|x|T 2
n(x)
n
.
Using Goodenough’s and Mills’ [10] Lemma 3 we also have that
I2 =Wj(x) = |xj − x|hj(x) ≤ π(2n)−1| cosnθ| = π(2n)−1|Tn(x)|,
so that for n ≥ 2 the following inequality holds:
I1 + I2 + I3 ≤
√
1− x2T 2
n(x)
n
(
3 + 2 ln
(
1√
2
n
))
+
|x|T 2
n(x)
n
+
π
2n
|Tn(x)| =
=
√
1− x2T 2
n(x)
n
(
2 + 2 ln
(
1√
2
n
))
+
√
1− x2T 2
n(x)
n
+
+
|x|T 2
n(x)
n
+
π
2n
|Tn(x)| ≤
≤|Tn(x)|
n
[√
1− x2
(
2+2 ln
(
1√
2
n
))
+
√
1− x2|Tn(x)|+|x||Tn(x)|+
π
2
]
≤
≤ |Tn(x)|
n
[√
1− x2
(
2 + 2 ln
(
1√
2
n
))
+ 2 +
π
2
]
≤
≤ |Tn(x)|
n
[√
1− x24 lnn+ 4
]
= 4
|Tn(x)|
n
[√
1− x2 lnn+ 1
]
.
In order to show that c ≥ 1, it is only necessary to evaluate the left and the right-hand
side of the inequality in the above claim at the point x = 1, say. We have
Ln(|e1 − 1|, 1) = −Ln(e1 − 1, 1) =
1
n
Tn(1)Tn−1(1) =
1
n
.
Using the same point on the right-hand side shows that
c
n
|Tn(1)|
{
(1− 1)1/2 lnn+ 1
}
=
c
n
and thus c ≥ 1.
Lemma 3 is proved.
Remark 10. Numerical evidence suggests that the constant c in Lemma 3 equals
1. However, it does not seem to be possible to use the technique of O. Kiš to obtain
such a good estimate.
1. Acu A. M., Gonska H. Ostrowski-type inequalities and moduli of smoothness // Results Math. – 2009. –
53. – P. 217 – 228.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
740 A. M. ACU, H. GONSKA, I. RAŞA
2. Andrica D., Badea C. Grüss’ inequality for positive linear functionals // Period. math. hung. – 1988. –
19. – P. 155 – 167.
3. Anastassiou G. A. Ostrowski type inequalities // Proc. Amer. Math. Soc. – 1995. – 123. – P. 3775 – 3781.
4. Čebyšev P. L. Sur les expressions approximatives des intégrales définies par les autres prises entre les
méme limites // Proc. Math. Soc. Kharkov. – 1882. – 2. – P. 93 – 98 (in Russian) (Transl.: Oeuvres. –
1907. – 2. – P. 716 – 719).
5. Cheng X. L. Improvement of some Ostrowski – Grüss type inequalities // Comput. Math. Appl. – 2001.
– 42. – P. 109 – 114.
6. Dragomir S. S. On the Ostrowski integral inequality for Lipschitzian mappings and applications //
Comput. Math. Appl. – 1999. – 38. – P. 33 – 37.
7. Dragomir S. S., Wang S. An inequality of Ostrowski – Grüss’ type and its applications to the estimation
of error bounds for some special means and for some numerical quadrature rules // Comput. Math. Appl.
– 1997. – 33, № 11. – P. 15 – 20.
8. Gavrea B., Gavrea I. Ostrowski type inequalities from a linear functional point of view // J. Inequal.
Pure and Appl. Math. – 2000. – 1, Article 11.
9. Gonska H. Quantitative Approximation in C(X) // Habilitationsschrift. – Univ. Duisburg, 1986.
10. Goodenough S. J., Mills T. M. A new estimate for the approximation of functions by Hermite – Fejér
interpolation polynomials // J. Approxim. Theory. – 1981. – 31. – P. 253 – 260.
11. Grüss G. Über das Maximum des absoluten Betrages von
1
b− a
∫ b
a
f(x)g(x)dx −
−
1
(b− a)2
∫ b
a
f(x)dx
∫ b
a
g(x)dx // Math. Z. – 1935. – 39. – S. 215 – 226.
12. Karamata J. Inégalités relatives aux quotients et à la différence de
∫
fg et
∫
f
∫
g // Bull. Acad. Serbe.
Sci. Math. Natur. A. – 1948. – P. 131 – 145.
13. Kiš O. Remarks on the rapidity of convergence of Lagrange interpolation (in Russian) // Ann. Univ. sci.
Budapest. Sect. math. – 1968. – 11. – P. 27 – 40.
14. Landau E. Über einige Ungleichungen von Herrn G. Grüss // Math. Z. – 1935. – 39. – S. 742 – 744.
15. Lehnhoff H. G. A simple proof of A.F. Timan’s theorem // J. Approxim. Theory. – 1983. – 38. –
P. 172 – 176.
16. Matsuoka Y. On the degree of approximation of functions by some positive linear operators // Sci. Rep.
Kagoshima Univ. – 1960. – 9. – P. 11 – 16.
17. Mercer A. Mc. D., Mercer P. R. New proofs of the Grüss inequality // Austral. J. Math. Anal. and Appl.
– 2004. – 1, Issue 2. – P. 1 – 6.
18. Misra R. N. On the rate of convergence of Hermite – Fejér interpolation polynomials // Period. math.
hung. – 1982. – 13. – P. 15 – 20.
19. Mitrinović D. S., Pečarić J. E., Fink A. M. Classical and new inequalities in analysis. – Dordrecht:
Kluwer, 1993.
20. Ostrowski A. Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert
// Comment. math. helv. – 1938. – 10. – P. 226 – 227.
21. Ostrowski A. On an integral inequality // Aequat. math. – 1970. – 4. – P. 358 – 373.
22. Pachpatte B. G. A note on Ostrowski like inequalities // J. Inequal. Pure and Appl. Math. – 2005. – 6,
Article 114.
23. Pachpatte B. G. A note on Grüss type inequalities via Cauchy’s mean value theorem // Math. Inequal.
Appl. – 2007. – 11, № 1. – P. 75 – 80.
24. Semenov E. M., Mitjagin B. S. Lack of interpolation of linear operators in spaces of smooth functions //
Mat. USSR. Izv. – 1977. – 11. – S. 1229 – 1266.
Received 11.02.10,
after revision — 30.03.11
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
|
| id | umjimathkievua-article-2758 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:29:45Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ff/1a83131a0f3b3b069f142b58a9330aff.pdf |
| spelling | umjimathkievua-article-27582020-03-18T19:35:28Z Grüss-type and Ostrowski-type inequalities in approximation theory Нерiвностi типу грюсса та островського в теорiї наближень Acu, A.-M. Gonska, H. Ra¸sa, I. Асу, А.-М. Гонська, Х. Раса, І. We discuss the Grass inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain a Grass inequality for the functional $L(f) = H(f; x)$, where $H: C[a,b] \rightarrow C[a,b]$ is a positive linear operator and $x \in [a,b]$ is fixed. We apply this inequality in the case of known operators, for example, the Bernstein, Hermite-Fejer operator the interpolation operator, convolution-type operators. Moreover, we derive Grass-type inequalities using Cauchy's mean value theorem, thus generalizing results of Cebysev and Ostrowski. A Grass inequality on a compact metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained. The latter in turn leads to one further version of Grass' inequality. In an appendix, we prove a new result concerning the absolute first-order moments of the classical Hermite-Fejer operator. Розглянуто нерiвностi Грюсса на просторах неперервних функцiй, якi визначено на компактному метричному просторi. З використанням найменшої опуклої мажоранти модуля неперервностi одержано нерiвнiсть Грюсса для функцiонала $L(f) = H(f; x)$, де $H\;:\; C[a, b] \rightarrow C[a, b]$ — додатний лiнiйний оператор, а $x ∈ [a, b]$ зафiксовано. Цю нерiвнiсть застосовано до випадку вiдомих операторiв, наприклад оператора Бернштейна, iнтерполяцiйного оператора Ермiта – Фейєра, операторiв типу конволюцiї. Крiм того, виведено нерiвностi типу Грюсса на основi теореми Кошi про середнє, що узагальнює результати Чебишова та Островського. Представлено нерiвнiсть Грюсса на компактному метричному просторi для бiльш нiж двох функцiй та отримано аналогiчну нерiвнiсть типу Островського, яка, в свою чергу, приводить до ще однiєї версiї нерiвностi Грюсса. У додатку доведено новий результат щодо абсолютних моментiв першого порядку класичного оператора Ермiта – Фейєра. Institute of Mathematics, NAS of Ukraine 2011-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2758 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 6 (2011); 723-740 Український математичний журнал; Том 63 № 6 (2011); 723-740 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2758/2272 https://umj.imath.kiev.ua/index.php/umj/article/view/2758/2273 Copyright (c) 2011 Acu A.-M.; Gonska H.; Ra¸sa I. |
| spellingShingle | Acu, A.-M. Gonska, H. Ra¸sa, I. Асу, А.-М. Гонська, Х. Раса, І. Grüss-type and Ostrowski-type inequalities in approximation theory |
| title | Grüss-type and Ostrowski-type inequalities in approximation theory |
| title_alt | Нерiвностi типу грюсса та островського в теорiї наближень |
| title_full | Grüss-type and Ostrowski-type inequalities in approximation theory |
| title_fullStr | Grüss-type and Ostrowski-type inequalities in approximation theory |
| title_full_unstemmed | Grüss-type and Ostrowski-type inequalities in approximation theory |
| title_short | Grüss-type and Ostrowski-type inequalities in approximation theory |
| title_sort | grüss-type and ostrowski-type inequalities in approximation theory |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2758 |
| work_keys_str_mv | AT acuam grusstypeandostrowskitypeinequalitiesinapproximationtheory AT gonskah grusstypeandostrowskitypeinequalitiesinapproximationtheory AT rasai grusstypeandostrowskitypeinequalitiesinapproximationtheory AT asuam grusstypeandostrowskitypeinequalitiesinapproximationtheory AT gonsʹkah grusstypeandostrowskitypeinequalitiesinapproximationtheory AT rasaí grusstypeandostrowskitypeinequalitiesinapproximationtheory AT acuam nerivnostitipugrûssataostrovsʹkogovteoriínabliženʹ AT gonskah nerivnostitipugrûssataostrovsʹkogovteoriínabliženʹ AT rasai nerivnostitipugrûssataostrovsʹkogovteoriínabliženʹ AT asuam nerivnostitipugrûssataostrovsʹkogovteoriínabliženʹ AT gonsʹkah nerivnostitipugrûssataostrovsʹkogovteoriínabliženʹ AT rasaí nerivnostitipugrûssataostrovsʹkogovteoriínabliženʹ |