New generalized trapezoid type inequalities for complex functions defined on unit circle and applications
UDC 517.5 We establish new generalized trapezoid type inequalities for complex functions defined on unit circle via the function of bounded variation and the functions satisfying H¨older type condition. Using these results, quadrature rule formula is also provided.
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2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512236532924416 |
|---|---|
| author | Budak, H. Budak, H. |
| author_facet | Budak, H. Budak, H. |
| author_sort | Budak, H. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:49:28Z |
| description | UDC 517.5
We establish new generalized trapezoid type inequalities for complex functions defined on unit circle via the function of bounded variation and the functions satisfying H¨older type condition. Using these results, quadrature rule formula is also provided. |
| doi_str_mv | 10.37863/umzh.v72i12.6035 |
| first_indexed | 2026-03-24T03:25:35Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i12.6035
UDC 517.5
H. Budak (Univ. Düzce, Turkey)
NEW GENERALIZED TRAPEZOID TYPE INEQUALITIES
FOR COMPLEX FUNCTIONS DEFINED ON UNIT CIRCLE
AND APPLICATIONS
НОВI УЗАГАЛЬНЕНI НЕРIВНОСТI ТИПУ ТРАПЕЦIЇ
ДЛЯ КОМПЛЕКСНИХ ФУНКЦIЙ НА ОДИНИЧНОМУ КОЛI
ТА ЇХ ЗАСТОСУВАННЯ
We establish new generalized trapezoid type inequalities for complex functions defined on unit circle via the function of
bounded variation and the functions satisfying Hölder type condition. Using these results, quadrature rule formula is also
provided.
За допомогою функцiй обмеженої варiацiї та функцiй, якi задовольняють умову типу Гельдера, отримано новi
узагальненi нерiвностi типу трапецiї для комплексних функцiй на одиничному колi. З цих результатiв також виведено
квадратурну формулу.
1. Introduction. Over the past two decades, the field of inequalities for the function of bounded
variation has undergone explosive growth. The many research paper related to some type inequalities
such as Ostrowski, trapezoid, Gruss for the function of bounded variation have been written. Re-
cently, some works have focused on Ostrowski and trapezoid type inequalities for complex functions
defined on unit circle. Inspired by these inequalities, we will obtain some generalized trapezoid type
inequalities.
The overall structure of the study takes the form of three sections including introduction. The
remainder of this work is organized as follows: first we give the definitions of the function of bounded
variation and total variation and present a trapezoid type inequality for complex functions defined on
unit circle proved by Dragomir. In Section 2, a new generalized versions of this trapezoid inequality
are obtained. We give also some special cases of these inequalities. Utilizing the results established
in Section 2, quadrature rule formula is provided in Section 3.
First of all, we start to give the definitions the function of bounded variation and total variation.
Let P : a = x0 < x1 < . . . < xn = b be any partition of [a, b] and let \Delta f(xi) = f(xi+1) - f(xi).
Then f(x) is said to be of bounded variation if the sum
m\sum
i=1
| \Delta f(xi)|
is bounded for all such partitions.
Let f be of bounded variation on [a, b] and
\sum
(P ) denotes the sum
\sum n
i=1
| \Delta f(xi)| corre-
sponding to the partition P of [a, b] . The number
b\bigvee
a
(f) := \mathrm{s}\mathrm{u}\mathrm{p}
\Bigl\{ \sum
(P ) : P \in \itP ([a, b])
\Bigr\}
is called the total variation of f on [a, b] . Here, \itP ([a, b]) denotes the family of partitions of [a, b] .
c\bigcirc H. BUDAK, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12 1621
1622 H. BUDAK
In [7], Dragomir proved following trapezoid type inequalities for complex functions defined on
unit circle C (0, 1):
Theorem 1.1. Assume that f : C (0, 1) \rightarrow \BbbC satisfies the Hölder’s type condition
| f(z) - f(w)| \leq H | z - w| r (1.1)
for any w, z \in C (0, 1) , where H > 0 and r \in (0, 1] are given.
If [a, b] \subseteq [0, 2\pi ] and the function u : [a, b] \rightarrow \BbbC is of bounded variation on [a, b] , then\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f
\bigl(
eib
\bigr)
+ f
\bigl(
eia
\bigr)
2
[u(b) - u(a)] -
b\int
a
f
\bigl(
eit
\bigr)
du(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2r - 1H \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a,b]
Br(a, b; t)
b\bigvee
a
(u) \leq 1
2r
H(b - a)r
b\bigvee
a
(u) (1.2)
for any t \in [a, b] , where the bound Br(a, b; t) is given by
Br(a, b; t) := \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
+ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
\leq 1
2r
[(b - t)r + (t - a)r] .
Ostrowski’s type inequalities for complex functions defined on unit circle C (0, 1) was considered
by Dragomir in [10] and the author give some application for unitary operators in Hilbert spaces.
Recently, Dragomir proved also trapezoid type inequalities for complex functions defined on unit
circle C (0, 1) and give some application in [9, 11]. The purpose of this paper is to obtain new
generalized trapezoid type inequalities for complex functions defined on unit circle C (0, 1) . For
other inequalities for Riemann – Stieltjes integral, see [1 – 8, 12 – 17].
2. Main results. In this section, we present some generalized trapezoid type inequalities for
complex functions defined on unit circle C (0, 1) .
Theorem 2.1. Suppose that f : C (0, 1) \rightarrow \BbbC satisfies Hölder’s type condition (1.1). If [a, b] \subseteq
\subseteq [0, 2\pi ] and the mapping u : [a, b] \rightarrow \BbbC is of bounded variation on [a, b] , then, for all s \in
\in
\biggl[
a,
a+ b
2
\biggr]
, we have the inequalities
| Tc(f, u; a, b; s)| \leq
\leq 2rH
\Biggl\{
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [a,s]
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr) s\bigvee
a
(u)+
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [s,a+b - s]
\left[ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
+ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
2
\right] a+b - s\bigvee
s
(u)+
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a+b - s,b]
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr) b\bigvee
a+b - s
(u)
\Biggr\}
\leq
\leq H
2r
(b - a)r
b\bigvee
a
(u), (2.1)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
NEW GENERALIZED TRAPEZOID TYPE INEQUALITIES FOR COMPLEX FUNCTIONS . . . 1623
where Tc(f, u; a, b; s) defined by
Tc(f, u; a, b; s) := f
\bigl(
eib
\bigr)
u(b) - f
\bigl(
eia
\bigr)
u(a) -
-
f
\bigl(
eib
\bigr)
- f
\bigl(
eia
\bigr)
2
[u(s) + u(a+ b - s)] -
b\int
a
f
\bigl(
eit
\bigr)
du(t).
Proof. Obviously, we have the equality
Tc(f, u; a, b; s) =
s\int
a
\bigl[
f
\bigl(
eia
\bigr)
- f
\bigl(
eit
\bigr) \bigr]
du(t)+
+
a+b - s\int
s
\Biggl[
f
\bigl(
eia
\bigr)
+ f
\bigl(
eib
\bigr)
2
- f
\bigl(
eit
\bigr) \Biggr]
du(t)+
+
b\int
a+b - s
\Bigl[
f
\bigl(
eib
\bigr)
- f
\bigl(
eit
\bigr) \Bigr]
du(t). (2.2)
It is known that if P : [c, d] \rightarrow \BbbC is a continuous function and v : [c, d] \rightarrow \BbbC is of bounded variation,
then the Riemann – Stieltjes integral
\int d
c
p(t)dv(t) exists and the inequality holds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
d\int
c
p(t)dv(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
t\in [c,d]
| p(t)|
d\bigvee
c
(v). (2.3)
Taking modulus in (2.2) and using the inequality (2.3), we have
| Tc(f, u; a, b; s)| \leq \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a,s]
\bigm| \bigm| f \bigl(
eia
\bigr)
- f
\bigl(
eit
\bigr) \bigm| \bigm| s\bigvee
a
(u)+
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [s,a+b - s]
\bigm| \bigm| \bigm| \bigm| \bigm| f
\bigl(
eia
\bigr)
+ f
\bigl(
eib
\bigr)
2
- f
\bigl(
eit
\bigr) \bigm| \bigm| \bigm| \bigm| \bigm|
a+b - s\bigvee
s
(u)+
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a+b - s,b]
\bigm| \bigm| f\bigl( eib\bigr) - f
\bigl(
eit
\bigr) \bigm| \bigm| b\bigvee
a+b - s
(u) \leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a,s]
\bigm| \bigm| f \bigl(
eia
\bigr)
- f
\bigl(
eit
\bigr) \bigm| \bigm| s\bigvee
a
(u)+
+
1
2
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [s,a+b - s]
\Bigl[ \bigm| \bigm| f \bigl(
eia
\bigr)
- f
\bigl(
eit
\bigr) \bigm| \bigm| + \bigm| \bigm| f\bigl( eib\bigr) - f
\bigl(
eit
\bigr) \bigm| \bigm| \Bigr] a+b - s\bigvee
s
(u)+
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a+b - s,b]
\bigm| \bigm| f\bigl( eib\bigr) - f
\bigl(
eit
\bigr) \bigm| \bigm| b\bigvee
a+b - s
(u).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1624 H. BUDAK
Since f satisfies the Hölder’s type condition (1.1), we obtain
| Tc(f, u; a, b; s)| \leq \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a+b - s,b]
\bigm| \bigm| eia - eit
\bigm| \bigm| r s\bigvee
a
(u)+
+
1
2
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [s,a+b - s]
\Bigl[ \bigm| \bigm| eia - eit
\bigm| \bigm| r + \bigm| \bigm| eib - eit
\bigm| \bigm| r\Bigr] a+b - s\bigvee
s
(u)+
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a+b - s,b]
\bigm| \bigm| eib - eit
\bigm| \bigm| b\bigvee
a+b - s
(u).
By using the fact that \bigm| \bigm| eix - eiy
\bigm| \bigm| 2 = \bigm| \bigm| eix\bigm| \bigm| 2 - 2\mathrm{R}\mathrm{e}
\bigl(
ei(x - y)
\bigr)
+
\bigm| \bigm| eiy\bigm| \bigm| 2 =
= 2 - 2 \mathrm{c}\mathrm{o}\mathrm{s}(x - y) =
= 4 \mathrm{s}\mathrm{i}\mathrm{n}2
\biggl(
x - y
2
\biggr)
for any x, y \in \BbbR , we have \bigm| \bigm| eix - eiy
\bigm| \bigm| r = 2r
\bigm| \bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n}\biggl( x - y
2
\biggr) \bigm| \bigm| \bigm| \bigm| r
for any x, y \in \BbbR .
Since [a, b] \subseteq [0, 2\pi ] , we get
\bigm| \bigm| eia - eit
\bigm| \bigm| r = 2r \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
s - a
2
\biggr)
and \bigm| \bigm| eib - eit
\bigm| \bigm| r = 2r \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - s
2
\biggr)
for any s \in
\biggl[
a,
a+ b
2
\biggr]
.
This completes the proof of the first inequality in (2.1).
For the proof of the second inequality in (2.1), by using the basic inequality \mathrm{s}\mathrm{i}\mathrm{n}x \leq x for
x \in [0, \pi ] , we obtain
| Tc(f, u; a, b; s)| \leq H
\Biggl\{
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [a,s]
(t - a)r
s\bigvee
a
(u)+
+ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [s,a+b - s]
\biggl[
(t - a)r + (b - t)r
2
\biggr] a+b - s\bigvee
s
(u) + \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a+b - s,b]
(b - t)r
b\bigvee
a+b - s
(u)
\Biggr\}
=
= H
\Biggl\{
(s - a)r
s\bigvee
a
(f) +
\biggl(
b - a
2
\biggr) r a+b - s\bigvee
s
(u) + (s - a)r
b\bigvee
a+b - s
(u)
\Biggr\}
\leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
NEW GENERALIZED TRAPEZOID TYPE INEQUALITIES FOR COMPLEX FUNCTIONS . . . 1625
\leq H
2r
(b - a)r
b\bigvee
a
(u).
Theorem 2.1 is proved.
Remark 2.1. Under assumption of Theorem 2.1, if we take s = a, then the inequalities (2.1)
reduce the inequalities (1.2).
Corollary 2.1. Under assumption of Theorem 2.1 with s =
a+ b
2
, we have the inequality
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f\bigl( eib\bigr) u(b) - f
\bigl(
eia
\bigr)
u(a) -
\Bigl[
f
\bigl(
eib
\bigr)
- f
\bigl(
eia
\bigr) \Bigr]
u
\biggl(
a+ b
2
\biggr)
-
b\int
a
f
\bigl(
eit
\bigr)
du(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2rH
\left\{ \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a,a+b
2 ]
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr) a+b
2\bigvee
a
(u) + \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a+b
2
,b]
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr) b\bigvee
a+b
2
(u)
\right\} \leq
\leq H
2r
(b - a)r
b\bigvee
a
(u).
Corollary 2.2. Assume that f : C (0, 1) \rightarrow \BbbC is Lipschitzian with the constant L > 0 on the unit
circle C (0, 1) . Then we get the inequality
| Tc(f, u; a, b; s)| \leq 2L
\Biggl\{
\mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
s - a
2
\biggr) s\bigvee
a
(u) + \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
b - a
4
\biggr) a+b - s\bigvee
s
(u) + \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
s - a
2
\biggr) b\bigvee
a+b - s
(u)
\Biggr\}
\leq
\leq L
2
(b - a)
b\bigvee
a
(u).
Proof. The proof is obvious from the choosing r = 1 in Theorem 2.1 and the fact that
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
+ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
= 2 \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
b - a
4
\biggr)
\mathrm{c}\mathrm{o}\mathrm{s}
\left( t - a+ b
2
2
\right) .
Remark 2.2. If we take s = a in Corollary 2.2, then we obtain the inequality\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f
\bigl(
eib
\bigr)
+ f
\bigl(
eia
\bigr)
2
[u(b) - u(a)] -
b\int
a
f
\bigl(
eit
\bigr)
du(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2L \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
b - a
4
\biggr) b\bigvee
a
(u) \leq L
2
(b - a)
b\bigvee
a
(u).
The constant 2 in the first inequality is the best possible in the above sense.
This result is same as that given in [11].
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1626 H. BUDAK
Remark 2.3. If we choose [a, b] = [0, 2\pi ] and s = \pi in Corollary 2.2, then we have the inequality\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f(1) [u(2\pi ) - u(0)] -
2\pi \int
0
f
\bigl(
eit
\bigr)
du(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq 2L
2\pi \bigvee
0
(u),
which is given by Dragomir in [11].
Corollary 2.3. If 0 < b - a \leq \pi , then we get
| Tc(f, u; a, b; s)| \leq 2rH \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - s
2
\biggr) b\bigvee
a
(u) \leq 2rH \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - a
2
\biggr) b\bigvee
a
(u)
for s \in
\biggl[
a,
a+ b
2
\biggr]
.
Proof. For 0 < b - a \leq \pi , we obtain
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [a,s]
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
= \mathrm{m}\mathrm{a}\mathrm{x}
t\in [a+b - s,b]
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
= \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
s - a
2
\biggr)
and
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [s,a+b - s]
\left[ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
+ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
2
\right] = \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - s
2
\biggr)
.
By using these inequalities in inequality (2.1), we have
| Tc(f, u; a, b; s)| \leq
\leq 2rH
\Biggl\{
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
s - a
2
\biggr) s\bigvee
a
(u) + \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - s
2
\biggr) a+b - s\bigvee
s
(u) + \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
s - a
2
\biggr) b\bigvee
a+b - s
(u)
\Biggr\}
\leq
\leq 2rH\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
s - a
2
\biggr)
, \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - s
2
\biggr) \biggr\} b\bigvee
a
(u) =
= 2rH \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - s
2
\biggr) b\bigvee
a
(u) \leq 2rH \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - a
2
\biggr) b\bigvee
a
(u).
Theorem 2.2. Suppose that f : C (0, 1) \rightarrow \BbbC satisfies Hölder’s type condition (1.1). If [a, b] \subseteq
\subseteq [0, 2\pi ] and the mapping u : [a, b] \rightarrow \BbbC is Lipschitzian with the constant K > 0 on [a, b] , then,
for all s \in
\biggl[
a,
a+ b
2
\biggr]
, we have the inequality
| Tc(f, u; a, b; s)| \leq
\leq 2rKH
\left\{
s\int
a
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
dt+
1
2
a+b - s\int
s
\biggl[
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
dt+ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr) \biggr]
dt+
+
b\int
a+b - s
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
dt
\right\} \leq KH
(s - a)r+1 + (b - s)r+1
r + 1
. (2.4)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
NEW GENERALIZED TRAPEZOID TYPE INEQUALITIES FOR COMPLEX FUNCTIONS . . . 1627
Proof. Consider the fact that if w : [a, b] \rightarrow \BbbC is a Riemann integrable function and the
mapping v : [a, b] \rightarrow \BbbC is Lipschitzian with the constant M > 0, then the Riemann – Stieltjes
integral
\int b
a
w(t)dv(t) exists and the inequality holds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
b\int
a
w(t)dv(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq M
b\int
a
| w(t)| dt.
Since u is Lipschitzian with the constant K > 0, taking modulus in (2.2) and using the inequality
(2.3), we have
| Tc(f, u; a, b; s)| =
= K
s\int
a
\bigm| \bigm| f \bigl(
eia
\bigr)
- f
\bigl(
eit
\bigr) \bigm| \bigm| dt+K
a+b - s\int
s
\bigm| \bigm| \bigm| \bigm| \bigm| f
\bigl(
eia
\bigr)
+ f
\bigl(
eib
\bigr)
2
- f
\bigl(
eit
\bigr) \bigm| \bigm| \bigm| \bigm| \bigm| dt+
+K
b\int
a+b - s
\bigm| \bigm| f\bigl( eib\bigr) - f
\bigl(
eit
\bigr) \bigm| \bigm| dt \leq
\leq K
s\int
a
\bigm| \bigm| f \bigl(
eit
\bigr)
- f
\bigl(
eia
\bigr) \bigm| \bigm| dt+ K
2
a+b - s\int
s
\Bigl[ \bigm| \bigm| f \bigl(
eit
\bigr)
- f
\bigl(
eia
\bigr) \bigm| \bigm| + \bigm| \bigm| \bigm| f\bigl( eib\bigr) - f
\bigl(
eit
\bigr) \bigm| \bigm| \bigm| \Bigr] dt+
+K
b\int
a+b - s
\bigm| \bigm| f\bigl( eib\bigr) - f
\bigl(
eit
\bigr) \bigm| \bigm| dt.
As f satisfies the Hölder’s type condition (1.1), we obtain
| Tc(f, u; a, b; s)| \leq
\leq KH
s\int
a
\bigm| \bigm| eit - eia
\bigm| \bigm| r dt+ KH
2
a+b - s\int
s
\Bigl[ \bigm| \bigm| eit - eia
\bigm| \bigm| r + \bigm| \bigm| eib - eit
\bigm| \bigm| r\Bigr] dt+
+KH
b\int
a+b - s
\bigm| \bigm| eib - eit
\bigm| \bigm| rdt =
= 2rKH
\left\{
s\int
a
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
dt+
1
2
a+b - s\int
s
\biggl[
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
+ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr) \biggr]
dt +
+
b\int
a+b - s
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
dt
\right\} , (2.5)
which completes the proof of first inequality in (2.4).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1628 H. BUDAK
For the proof of the second inequality in (2.4), by using the basic inequality \mathrm{s}\mathrm{i}\mathrm{n}x \leq x for
x \in [0, \pi ] , we have
s\int
a
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
dt \leq 1
2r
s\int
a
(t - a) dt =
(s - a)r+1
2r(r + 1)
, (2.6)
1
2
a+b - s\int
s
\biggl[
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
+ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr) \biggr]
dt \leq 1
2r+1
a+b - s\int
s
[(t - a) + (b - t)] dt =
=
(b - s)r+1 - (s - a)r+1
2r(r + 1)
(2.7)
and
b\int
a+b - s
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
dt \leq 1
2r
b\int
a+b - s
(b - t) dt =
(s - a)r+1
2r(r + 1)
. (2.8)
Substituting the inequalities (2.6) – (2.8) in (2.5), we obtain the required result (2.4).
Theorem 2.2 is proved.
Remark 2.4. If we choose s = a in Theorem 2.2, then we have the inequality\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f
\bigl(
eib
\bigr)
+ f
\bigl(
eia
\bigr)
2
[u(b) - u(a)] -
b\int
a
f
\bigl(
eit
\bigr)
du(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2r - 1KH
b\int
a
\biggl[
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
dt+ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr) \biggr]
dt \leq
\leq KH
(b - a)r+1
r + 1
,
which is proved by Dragomir in [11].
Corollary 2.4. Under assumption of Theorem 2.2 with s =
a+ b
2
, we have the inequality\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f\bigl( eib\bigr) u(b) - f
\bigl(
eia
\bigr)
u(a) -
\Bigl[
f
\bigl(
eib
\bigr)
- f
\bigl(
eia
\bigr) \Bigr]
u
\biggl(
a+ b
2
\biggr)
-
b\int
a
f
\bigl(
eit
\bigr)
du(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2rKH
\left\{
a+b
2\int
a
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
dt++
b\int
a+b
2
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
dt
\right\} \leq KH
(b - a)r+1
2r (r + 1)
.
Theorem 2.3. Suppose that f : C (0, 1) \rightarrow \BbbC satisfies Hölder’s type condition (1.1). If [a, b] \subseteq
\subseteq [0, 2\pi ] and the mapping u : [a, b] \rightarrow \BbbR is monotonic nondecreasing on [a, b] , then, for all
s \in
\biggl[
a,
a+ b
2
\biggr]
, we have the inequality
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
NEW GENERALIZED TRAPEZOID TYPE INEQUALITIES FOR COMPLEX FUNCTIONS . . . 1629
| Tc(f, u; a, b; s)| \leq 2rH
\left\{
s\int
a
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
du(t) +
a+b - s\int
s
\left[ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
+ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
2
\right] du(t)+
+
b\int
a+b - s
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
du(t)
\right\} \leq
\leq H
\left\{
s\int
a
(t - a)r du(t) +
a+b - s\int
s
\biggl[
(b - t)r + (t - a)r
2
\biggr]
du(t)+
+
b\int
a+b - s
(b - t)r du(t)
\right\} . (2.9)
Proof. Consider the fact that if w : [a, b] \rightarrow \BbbC is a continuoud function and the mapping v :
[a, b] \rightarrow \BbbC is monotonic nondecreasing on [a, b] , then the Riemann – Stieltjes integral
\int b
a
w(t)dv(t)
exists and the inequality holds \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
b\int
a
w(t)dv(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
b\int
a
| w(t)| dv(t). (2.10)
By using the inequality (2.10), we have, from (2.2),
| Tc(f, u; a, b; s)| \leq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
s\int
a
\bigl[
f
\bigl(
eia
\bigr)
- f
\bigl(
eit
\bigr) \bigr]
du(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
+
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
a+b - s\int
s
\Biggl[
f
\bigl(
eia
\bigr)
+ f
\bigl(
eib
\bigr)
2
- f
\bigl(
eit
\bigr) \Biggr]
du(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
+
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
b\int
a+b - s
\Bigl[
f
\bigl(
eib
\bigr)
- f
\bigl(
eit
\bigr) \Bigr]
du(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
s\int
a
\bigm| \bigm| f \bigl(
eia
\bigr)
- f
\bigl(
eit
\bigr) \bigm| \bigm| du(t) + a+b - s\int
s
\bigm| \bigm| \bigm| \bigm| \bigm| f
\bigl(
eia
\bigr)
+ f
\bigl(
eib
\bigr)
2
- f
\bigl(
eit
\bigr) \bigm| \bigm| \bigm| \bigm| \bigm| du(t)+
+
b\int
a+b - s
\bigm| \bigm| f\bigl( eib\bigr) - f
\bigl(
eit
\bigr) \bigm| \bigm| du(t).
As f satisfies the Hölder’s type condition (1.1), we get
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1630 H. BUDAK
| Tc(f, u; a, b; s)| \leq H
s\int
a
\bigm| \bigm| eia - eit
\bigm| \bigm| r du(t) + H
2
a+b - s\int
s
\Bigl[ \bigm| \bigm| eia - eit
\bigm| \bigm| r + \bigm| \bigm| eib - eit
\bigm| \bigm| r\Bigr] du(t)+
+H
b\int
a+b - s
\bigm| \bigm| eib - eit
\bigm| \bigm| rdu(t) \leq
\leq 2rH
s\int
a
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
du(t) + 2r - 1H
a+b - s\int
s
\biggl[
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
+ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr) \biggr]
du(t)+
+2rH
b\int
a+b - s
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
du(t),
which completes the proof of first inequality in (2.9). The proof of second inequality in (2.9) is
obvious from the fact that \mathrm{s}\mathrm{i}\mathrm{n}x \leq x for x \in [0, \pi ] .
Theorem 2.3 is proved.
Remark 2.5. If we choose s = a in Theorem 2.3, then we have the inequality
| Tc(f, u; a, b)| \leq 2r - 1H
b\int
a
\biggl[
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
+ \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr) \biggr]
du(t) \leq
\leq H
2
a+b - s\int
s
[(b - t)r + (t - a)r] du(t),
which is proved by Dragomir in [11].
Corollary 2.5. Under assumption of Theorem 2.3 with s =
a+ b
2
, we have the inequality
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f\bigl( eib\bigr) u(b) - f
\bigl(
eia
\bigr)
u(a) -
\Bigl[
f
\bigl(
eib
\bigr)
- f
\bigl(
eia
\bigr) \Bigr]
u
\biggl(
a+ b
2
\biggr)
-
b\int
a
f
\bigl(
eit
\bigr)
du(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2rH
\left\{
a+b
2\int
a
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
t - a
2
\biggr)
du(t) +
a+ b
2
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
b - t
2
\biggr)
du(t)
\right\} \leq
\leq H
\left\{
a+b
2\int
a
(t - a)r du(t) +
b\int
a+b
2
(b - t)r du(t)
\right\} .
3. Application to quadrature rule. We now introduce the intermediate points
\xi k \in
\biggl[
xk,
xk + xk+1
2
\biggr]
, k = 0, 1, . . . , n - 1,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
NEW GENERALIZED TRAPEZOID TYPE INEQUALITIES FOR COMPLEX FUNCTIONS . . . 1631
in the partition \Delta n : a = x0 < x1 < . . . < xn = b. Let hk : = xk+1 - xk and v(h) =
= \mathrm{m}\mathrm{a}\mathrm{x} \{ hk : k = 0, 1, . . . , n - 1\} and define the sum
T (f, u,\Delta n, \xi ) :=
n - 1\sum
k=0
\bigl\{
f
\bigl(
eixk+1
\bigr)
u(xk+1) - f
\bigl(
eixk
\bigr)
u(xk)
\bigr\}
-
- 1
2
n - 1\sum
k=0
\bigl\{ \bigl[
f
\bigl(
eixk+1
\bigr)
- f
\bigl(
eixk
\bigr) \bigr]
[u(\xi k) + u(xk + xk+1 - \xi k)]
\bigr\}
.
Then the following theorem holds.
Theorem 3.1. Let f and u be as in Theorem 2.1. Then
b\int
a
f
\bigl(
eit
\bigr)
du(t) = T (f, u,\Delta n, \xi ) +R(f, u,\Delta n, \xi ),
where T (f, u,\Delta n, \xi ) is defined as above and the remainder term R(f, u,\Delta n, \xi ) satisfies
| R(f, u,\Delta n, \xi )| \leq 2rH \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
v(h)
2
\biggr) b\bigvee
a
(u) \leq Hvr(h)
b\bigvee
a
(u).
Proof. Since v(h) \leq \pi , then application of Corollary 2.3 to the interval [xi, xi+1] , i =
= 0, 1, . . . , n - 1, for intermediate points \xi k, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \bigl(
eixk+1
\bigr)
u(xk+1) - f
\bigl(
eixk
\bigr)
u(xk) -
-
\Biggl[
f
\bigl(
eixk+1
\bigr)
- f
\bigl(
eixk
\bigr)
2
\Biggr] \bigl[
u(\xi k) + u(xk + xk+1 - \xi k)
\bigr]
-
xk+1\int
xk
f
\bigl(
eit
\bigr)
du(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2rH \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
hk
2
\biggr) xk+1\bigvee
xk
(u) (3.1)
for all k \in \{ 0, 1, . . . , n - 1\} .
Summing the inequality (3.1) over k from 0 to n - 1 and using the generalized triangle inequality,
we get
| R(f, u,\Delta n, \xi )| \leq 2rH
n - 1\sum
k=0
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
hk
2
\biggr) xk+1\bigvee
xk
(u) \leq
\leq 2rH \mathrm{m}\mathrm{a}\mathrm{x}
k\in \{ 0,1,..,n - 1\}
\mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
hk
2
\biggr) n - 1\sum
k=0
xk+1\bigvee
xk
(u) \leq
\leq 2rH \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
v(h)
2
\biggr) b\bigvee
a
(u) \leq Hvr(h)
b\bigvee
a
(u),
which completes the proof.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1632 H. BUDAK
Remark 3.1. Choosing \xi k = xk in Theorem 3.1, we obtain
b\int
a
f
\bigl(
eit
\bigr)
du(t) = T (f, u,\Delta n) +R(f, u,\Delta n)
and
| R(f, u,\Delta n, \xi )| \leq 2rH \mathrm{s}\mathrm{i}\mathrm{n}r
\biggl(
v(h)
2
\biggr) b\bigvee
a
(u) \leq Hvr(h)
b\bigvee
a
(u)
with the sum
T (f, u,\Delta n) :=
1
2
n - 1\sum
k=0
\bigl\{ \bigl[
f
\bigl(
eixk+1
\bigr)
+ f
\bigl(
eixk
\bigr) \bigr]
[uxk+1) - u(xk)]
\bigr\}
given by Dragomir in [11].
References
1. H. Budak, M. Z. Sarikaya, On generalization of Dragomir’s inequalities, Turkish J. Analysis and Number Theory, 5,
№ 5, 191 – 196 (2017).
2. H. Budak, M. Z. Sarikaya, A companion of Ostrowski type inequalities for mappings of bounded variation and some
applications, Trans. A. Razmadze Math. Inst., 171, № 2, 136 – 143 (2017).
3. H. Budak, M. Z. Sarikaya, A. Qayyum, New refinements and applications of Ostrowski type inequalities for mappings
whose nth derivatives are of bounded variation, TWMS J. Appl. and Eng. Math. (to appear).
4. H. Budak, M. Z. Sarikaya, A. Qayyum, Improvement in companion of Ostrowski type inequalities for mappings whose
first derivatives are of bounded variation and application, Filomat, 31, № 16, 5305 – 5314 (2017).
5. P. Cerone, S. S. Dragomir, C. E. M. Pearce, A generalized trapezoid inequality for functions of bounded variation,
Turkish J. Math., 24, № 2, 147 – 163 (2000).
6. S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc., 60,
№ 3, 495 – 508 (1999).
7. S. S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Math.
Inequal. and Appl., 4, № 1, 59 – 66 (2001).
8. S. S. Dragomir, A companion of Ostrowski’s inequality for functions of bounded variation and applications, Int. J.
Nonlinear Anal. and Appl., 5, № 1, 89 – 97 (2014).
9. S. S. Dragomir, Generalized trapezoid type inequalities for complex functions defined on unit circle with applications
for unitary operators in Hilbert spaces, Mediterr. J. Math., 12, № 3, 573 – 591 (2015).
10. S. S. Dragomir, Ostrowski’s type inequalities for complex functions defined on unit circle with applications for unitary
operators in Hilbert spaces, Arch. Math., 51, № 4, 233 – 254 (2015).
11. S. S. Dragomir, Trapezoid type inequalities for complex functions defined on unit circle with applications for unitary
operators in Hilbert spaces, Georgian Math. J, 23, № 2, 199 – 210 (2016).
12. Z. Liu, Some companion of an Ostrowski type inequality and application, JIPAM, 10, № 2, Article 52 (2009), 12 p.
13. K.-L. Tseng, G.-S. Yang, S. S. Dragomir, Generalizations of weighted trapezoidal inequality for mappings of bounded
variation and their applications, Math. and Comput. Modelling, 40, № 1-2, 77 – 84 (2004).
14. K.-L. Tseng, Improvements of some inequalities of Ostrowski type and their applications, Taiwan. J. Math. 12, № 9,
2427 – 2441 (2008).
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Received 07.01.18
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
|
| id | umjimathkievua-article-6035 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:35Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f6/4c92317c1f4e78a96f5de3b052bb7bf6.pdf |
| spelling | umjimathkievua-article-60352025-03-31T08:49:28Z New generalized trapezoid type inequalities for complex functions defined on unit circle and applications New generalized trapezoid type inequalities for complex functions defined on unit circle and applications Budak, H. Budak, H. Function of bounded variation trapezoid type inequalities H¨older type condition Riemann-Stieltjes integral Function of bounded variation trapezoid type inequalities H¨older type condition Riemann-Stieltjes integral UDC 517.5 We establish new generalized trapezoid type inequalities for complex functions defined on unit circle via the function of bounded variation and the functions satisfying H¨older type condition. Using these results, quadrature rule formula is also provided. UDC 517.5 Новi узагальненi нерiвностi типу трапецiї для комплексних функцiй на одиничному колi та їх застосуванняЗа допомогою функцій обмеженої варіації та функцій, які задовольняють умову типу Гельдера, отримано нові узагальнені нерівності типу трапеції для комплексних функцій на одиничному колі. З цих результатів також виведено квадратурну формулу. Institute of Mathematics, NAS of Ukraine 2020-12-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6035 10.37863/umzh.v72i12.6035 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 12 (2020); 1621-1632 Український математичний журнал; Том 72 № 12 (2020); 1621-1632 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6035/8871 |
| spellingShingle | Budak, H. Budak, H. New generalized trapezoid type inequalities for complex functions defined on unit circle and applications |
| title | New generalized trapezoid type inequalities for complex functions defined on unit circle and applications |
| title_alt | New generalized trapezoid type inequalities for complex functions defined on unit circle and applications |
| title_full | New generalized trapezoid type inequalities for complex functions defined on unit circle and applications |
| title_fullStr | New generalized trapezoid type inequalities for complex functions defined on unit circle and applications |
| title_full_unstemmed | New generalized trapezoid type inequalities for complex functions defined on unit circle and applications |
| title_short | New generalized trapezoid type inequalities for complex functions defined on unit circle and applications |
| title_sort | new generalized trapezoid type inequalities for complex functions defined on unit circle and applications |
| topic_facet | Function of bounded variation trapezoid type inequalities H¨older type condition Riemann-Stieltjes integral Function of bounded variation trapezoid type inequalities H¨older type condition Riemann-Stieltjes integral |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6035 |
| work_keys_str_mv | AT budakh newgeneralizedtrapezoidtypeinequalitiesforcomplexfunctionsdefinedonunitcircleandapplications AT budakh newgeneralizedtrapezoidtypeinequalitiesforcomplexfunctionsdefinedonunitcircleandapplications |