Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities
UDC 517.5 In this paper, authors discover two interesting identities regarding Gauss–Jacobi and Hermite–Hadamard type integral inequalities. By using the first lemma as an auxiliary result, some new bounds with respect to Gauss–Jacobi type integral inequalities are established. Also, using the secon...
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| author | Kashuri, A. Ramosaçaj, M. Liko, R. Kashuri, Artion Ramosaçaj, Miftar Liko, Rozana Kashuri, A. Ramosaçaj, M. Liko, R. |
| author_facet | Kashuri, A. Ramosaçaj, M. Liko, R. Kashuri, Artion Ramosaçaj, Miftar Liko, Rozana Kashuri, A. Ramosaçaj, M. Liko, R. |
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| description | UDC 517.5
In this paper, authors discover two interesting identities regarding Gauss–Jacobi and Hermite–Hadamard type integral inequalities. By using the first lemma as an auxiliary result, some new bounds with respect to Gauss–Jacobi type integral inequalities are established. Also, using the second lemma, some new estimates with respect to Hermite–Hadamard type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from main results. Some applications to special means for different positive real numbers and new error estimates for the trapezoidal are provided as well. These results give us the generalizations, refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research. |
| doi_str_mv | 10.37863/umzh.v73i8.603 |
| first_indexed | 2026-03-24T02:03:15Z |
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DOI: 10.37863/umzh.v73i8.603
UDC 517.5
A. Kashuri, M. Ramosaçaj, R. Liko (Univ. Ismail Qemali, Vlora, Albania)
SOME NEW BOUNDS OF GAUSS – JACOBI
AND HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES
НОВI ГРАНИЦI ДЛЯ IНТЕГРАЛЬНИХ НЕРIВНОСТЕЙ
ТИПУ ГАУССА – ЯКОБI ТА ЕРМIТА – АДАМАРА
In this paper, authors discover two interesting identities regarding Gauss – Jacobi and Hermite – Hadamard type integral
inequalities. By using the first lemma as an auxiliary result, some new bounds with respect to Gauss – Jacobi type integral
inequalities are established. Also, using the second lemma, some new estimates with respect to Hermite – Hadamard type
integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be
deduced from main results. Some applications to special means for different positive real numbers and new error estimates
for the trapezoidal are provided as well. These results give us the generalizations, refinement and significant improvements
of the new and previous known results. The ideas and techniques of this paper may stimulate further research.
Знайдено двi цiкавi тотожностi для iнтегральних нерiвностей типу Гаусса – Якобi та Ермiта – Адамара. З викори-
станням першої леми як допомiжного результату встановлено деякi новi границi iнтегральних нерiвностей типу
Гаусса – Якобi. Далi, за допомогою другої леми та загальних дробових iнтегралiв отримано деякi новi границi iн-
тегральних нерiвностей типу Ермiта – Адамара. Зазначено, що з основних результатiв можна отримати деякi новi
випадки. Також запропоновано деякi застосування до спецiальних середнiх для рiзних додатних дiйсних чисел та
новi оцiнки похибок для методу трапецiї. Цi результати є узагальненням, уточненням та значним покращенням
нових та ранiше вiдомих результатiв. Iдеї та методи цiєї статтi мають стимулювати подальшi дослiдження.
1. Introduction. The following notations are used throughout this paper. We use I to denote an
interval on the real line \BbbR = ( - \infty ,+\infty ). For any subset K \subseteq \BbbR n, K\circ is the interior of K. The set
of integrable functions on the interval [a1, a2] is denoted by L[a1, a2].
The following inequality, named Hermite – Hadamard inequality, is one of the most famous in-
equalities in the literature for convex functions.
Theorem 1.1. Let f : I \subseteq \BbbR - \rightarrow \BbbR be a convex function on I and a1, a2 \in I with a1 < a2.
Then the following inequality holds:
f
\biggl(
a1 + a2
2
\biggr)
\leq 1
a2 - a1
a2\int
a1
f(x)dx \leq f(a1) + f(a2)
2
. (1.1)
This inequality (1.1) is also known as trapezium inequality.
The trapezium type inequality has remained an area of great interest due to its wide applications
in the field of mathematical analysis. For other recent results which generalize, improve and extend
the inequality (1.1) through various classes of convex functions interested readers are referred to
[1 – 33, 35, 37, 38]
The Gauss – Jacobi type quadrature formula has the following:
a2\int
a1
(x - a1)
p(a2 - x)qf(x)dx =
+\infty \sum
k=0
Bm,kf(\gamma k) +R \star
m| f | , (1.2)
c\bigcirc A. KASHURI, M. RAMOSAÇAJ, R. LIKO, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8 1067
1068 A. KASHURI, M. RAMOSAÇAJ, R. LIKO
for certain Bm,k, \gamma k and rest R \star
m| f | (see [34]).
Recently in [20], Liu obtained several integral inequalities for the left-hand side of (1.2). Also
in [28], Özdemir et al. established several integral inequalities concerning the left-hand side of (1.2)
via some kinds of convexity.
Let us recall some special functions and evoke some basic definitions as follows.
Definition 1.1. For k \in \BbbR + and x \in \BbbC , the k-gamma function is defined by
\Gamma k(x) = \mathrm{l}\mathrm{i}\mathrm{m}
n - \rightarrow \infty
n!kn(nk)
x
k
- 1
(x)n,k
.
Its integral representation is given by
\Gamma k(\alpha ) =
\infty \int
0
t\alpha - 1e -
tk
k dt. (1.3)
One can note that
\Gamma k(\alpha + k) = \alpha \Gamma k(\alpha ).
For k = 1, (1.3) gives integral representation of gamma function.
Definition 1.2 [24]. Let f \in L[a1, a2]. Then k-fractional integrals of order \alpha , k > 0 with
a1 \geq 0 are defined as
I\alpha ,k
a+1
f(x) =
1
k\Gamma k(\alpha )
x\int
a1
(x - t)
\alpha
k
- 1f(t)dt, x > a1,
and
I\alpha ,k
a - 2
f(x) =
1
k\Gamma k(\alpha )
a2\int
x
(t - x)
\alpha
k
- 1f(t)dt, a2 > x.
For k = 1, k-fractional integrals give Riemann – Liouville integrals.
Definition 1.3 [36]. A set S \subseteq \BbbR n is said to be invex set with respect to the mapping
\eta : S \times S - \rightarrow \BbbR n, if x+ t\eta (y, x) \in S for every x, y \in S and t \in [0, 1].
The invex set S is also termed an \eta -connected set.
Definition 1.4. Let S \subseteq \BbbR n be an invex set with respect to \eta : S \times S - \rightarrow \BbbR n. A function
f : S - \rightarrow [0,+\infty ) is said to be preinvex with respect to \eta , if , for every x, y \in S and t \in [0, 1],
f
\bigl(
x+ t\eta (y, x)
\bigr)
\leq (1 - t)f(x) + tf(y).
Also, define a function \varphi : [0,+\infty ) - \rightarrow [0,+\infty ) satisfying the following conditions:
1\int
0
\varphi (t)
t
dt < +\infty , (1.4)
1
A
\leq \varphi (s)
\varphi (r)
\leq A for
1
2
\leq s
r
\leq 2, (1.5)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
SOME NEW BOUNDS OF GAUSS – JACOBI AND HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES 1069
\varphi (r)
r2
\leq B
\varphi (s)
s2
for s \leq r, (1.6)\bigm| \bigm| \bigm| \bigm| \varphi (r)r2
- \varphi (s)
s2
\bigm| \bigm| \bigm| \bigm| \leq C| r - s| \varphi (r)
r2
for
1
2
\leq s
r
\leq 2, (1.7)
where A,B,C > 0 are independent of r, s > 0. If \varphi (r)r\alpha is increasing for some \alpha \geq 0 and
\varphi (r)
r\beta
is decreasing for some \beta \geq 0, then \varphi satisfies (1.4) – (1.7) (see [31]). Therefore, we define the
following left- and right-hand sided generalized fractional integral operators, respectively, as follows:
a+1
I\varphi f(x) =
x\int
a1
\varphi (x - t)
x - t
f(t)dt, x > a1,
a - 2
I\varphi f(x) =
a2\int
x
\varphi (t - x)
t - x
f(t)dt, x < a2.
The most important feature of generalized fractional integrals is that they generalize some types of
fractional integrals such as Riemann – Liouville fractional integral, k-Riemann – Liouville fractional
integral, Katugampola fractional integrals, conformable fractional integral, Hadamard fractional inte-
grals etc. (see [30]).
Motivated by the above literatures, the main objective of this paper is to discover in Sections 2
and 3, two interesting identities and to established some new bounds regarding Gauss – Jacobi and
Hermite – Hadamard type integral inequalities. By using in Section 2 the first lemma as an auxiliary
result, some new bounds with respect to Gauss – Jacobi type integral inequalities will be given. Also,
by using in Section 3 the second lemma, some new estimates with respect to Hermite – Hadamard
type integral inequalities via general fractional integrals will be obtained. It is pointed out that some
new special cases will be deduced from main results. In Section 4, some applications to special
means for different positive real numbers and new error estimates for the trapezoidal will be given.
These results will give us the generalizations, refinement and significant improvements of the new
and previous known results. The ideas and techniques of this paper may stimulate further research.
2. Some new bounds of the quadrature formula of Gauss – Jacobi type. Throughout this
study, for brevity, we define
\Lambda \ast (t) =
t\int
0
\varphi (\eta (a2, a1)x)
x
dx < +\infty , \eta (a2, a1) > 0.
For establishing some new bounds integral inequalities for Gauss – Jacobi type, we need the following
lemma.
Lemma 2.1. Let P = [a1, a1 + \eta (a2, a1)] \subseteq \BbbR be an open invex subset. Assume that
f : P - \rightarrow \BbbR be a continuous mapping on P \circ with respect to \eta : P \times P - \rightarrow \BbbR for \eta (a2, a1) > 0.
Then, for any fixed p, q > 0, we have
a1+\eta (a2,a1)\int
a1
\biggl[
\Lambda \ast
\biggl(
x - a1
\eta (a2, a1)
\biggr) \biggr] p \biggl[
\Lambda \ast
\biggl(
a1 + \eta (a2, a1) - x
\eta (a2, a1)
\biggr) \biggr] q
f(x)dx =
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1070 A. KASHURI, M. RAMOSAÇAJ, R. LIKO
= \eta (a2, a1)
1\int
0
\bigl[
\Lambda \ast (t)
\bigr] p\bigl[
\Lambda \ast (1 - t)
\bigr] q
f(a1 + t\eta (a2, a1))dt. (2.1)
We denote
T p,q
f,\Lambda \ast (a1, a2) = \eta (a2, a1)
1\int
0
\bigl[
\Lambda \ast (t)
\bigr] p\bigl[
\Lambda \ast (1 - t)
\bigr] q
f(a1 + t\eta (a2, a1))dt. (2.2)
Proof. By using (2.2) and changing the variable x = a1 + t\eta (a2, a1), we have
T p,q
f,\Lambda \ast (a1, a2) = \eta (a2, a1) \times
\times
a1+\eta (a2,a1)\int
a1
\biggl[
\Lambda \ast
\biggl(
x - a1
\eta (a2, a1)
\biggr) \biggr] p \biggl[
\Lambda \ast
\biggl(
1 - x - a1
\eta (a2, a1)
\biggr) \biggr] q
f(x)
dx
\eta (a2, a1)
=
=
a1+\eta (a2,a1)\int
a1
\biggl[
\Lambda \ast
\biggl(
x - a1
\eta (a2, a1)
\biggr) \biggr] p \biggl[
\Lambda \ast
\biggl(
a1 + \eta (a2, a1) - x
\eta (a2, a1)
\biggr) \biggr] q
f(x)dx.
Lemma 2.1 is proved.
Corollary 2.1. Taking \eta (a2, a1) = a2 - a1 and \varphi (x) = x, in Lemma 2.1, we get the following
identity:
a2\int
a1
(x - a1)
p(a2 - x)qf(x)dx = (a2 - a1)
p+q+1
1\int
0
tp(1 - t)qf(a1 + t(a2 - a1))dt.
With the help of Lemma 2.1, we have the following results.
Theorem 2.1. Let P = [a1, a1 + \eta (a2, a1)] \subseteq \BbbR be an open invex subset. Assume that
f : P - \rightarrow \BbbR be a continuous mapping on P \circ with respect to \eta : P \times P - \rightarrow \BbbR for \eta (a2, a1) > 0. If
| f |
k
k - 1 is preinvex mapping on P for k > 1, then, for any fixed p, q > 0, we have
\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast (a1, a2)
\bigm| \bigm| \bigm| \leq \eta (a2, a1)
k
\sqrt{}
Ap,q
\Lambda \ast (k)
\Biggl[
| f(a1)|
k
k - 1 + | f(a2)|
k
k - 1
2
\Biggr] k - 1
k
,
where
Ap,q
\Lambda \ast (k) =
1\int
0
\bigl[
\Lambda \ast (t)
\bigr] kp\bigl[
\Lambda \ast (1 - t)
\bigr] kq
dt.
Proof. Since | f |
k
k - 1 is preinvex mapping on P, combining with Lemma 2.1, Hölder’s inequality
and properties of the modulus, we get
\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast (a1, a2)
\bigm| \bigm| \bigm| \leq \eta (a2, a1)
1\int
0
\bigl[
\Lambda \ast (t)
\bigr] p\bigl[
\Lambda \ast (1 - t)
\bigr] q| f(a1 + t\eta (a2, a1))| dt \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
SOME NEW BOUNDS OF GAUSS – JACOBI AND HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES 1071
\leq \eta (a2, a1)
\left[ 1\int
0
\bigl[
\Lambda \ast (t)
\bigr] kp\bigl[
\Lambda \ast (1 - t)
\bigr] kq
dt
\right]
1
k
\left[ 1\int
0
| f(a1 + t\eta (a2, a1))|
k
k - 1dt
\right]
k - 1
k
\leq
\leq \eta (a2, a1)
k
\sqrt{}
Ap,q
\Lambda \ast (k)
\left[ 1\int
0
\Bigl(
(1 - t)| f(a1)|
k
k - 1 + t| f(a2)|
k
k - 1
\Bigr)
dt
\right]
k - 1
k
=
= \eta (a2, a1)
k
\sqrt{}
Ap,q
\Lambda \ast (k)
\Biggl[
| f(a1)|
k
k - 1 + | f(a2)|
k
k - 1
2
\Biggr] k - 1
k
.
Theorem 2.1 is proved.
We point out some special cases of Theorem 2.1.
Corollary 2.2. Under the assumption of Theorem 2.1 with \varphi (t) = t, we get
\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast
1
(a1, a2)
\bigm| \bigm| \bigm| \leq \eta p+q+1(a2, a1)
k
\sqrt{}
\beta (kp+ 1, kq + 1)
\Biggl[
| f(a1)|
k
k - 1 + | f(a2)|
k
k - 1
2
\Biggr] k - 1
k
,
where \Lambda \ast
1 = \eta (a2, a1)t.
Corollary 2.3. Under the assumption of Theorem 2.1 with \varphi (t) =
t\alpha
\Gamma (\alpha )
, we have
\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast
2
(a1, a2)
\bigm| \bigm| \bigm| \leq \eta \alpha (p+q)+1(a2, a1)
\Gamma p+q(\alpha + 1)
k
\sqrt{}
\beta (\alpha kp+ 1, \alpha kq + 1) \times
\times
\Biggl[
| f(a1)|
k
k - 1 + | f(a2)|
k
k - 1
2
\Biggr] k - 1
k
,
where \Lambda \ast
2 =
\eta \alpha (a2, a1)
\Gamma (\alpha + 1)
t\alpha .
Corollary 2.4. Under the assumption of Theorem 2.1 with \varphi (t) =
t
\alpha
k1
k1\Gamma k1(\alpha )
, we obtain
\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast
3
(a1, a2)
\bigm| \bigm| \bigm| \leq \eta
\alpha
k1
(p+q)+1
(a2, a1)\Bigl[
k1\Gamma k1(\alpha + k1)
\Bigr] p+q
k
\sqrt{}
\beta
\biggl(
\alpha kp
k1
+ 1,
\alpha kq
k1
+ 1
\biggr)
\times
\times
\Biggl[
| f(a1)|
k
k - 1 + | f(a2)|
k
k - 1
2
\Biggr] k - 1
k
,
where \Lambda \ast
3 =
\eta
\alpha
k1 (a2, a1)
k1\Gamma k1(\alpha + k1)
t
\alpha
k1 .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1072 A. KASHURI, M. RAMOSAÇAJ, R. LIKO
Corollary 2.5. Under the assumption of Theorem 2.1 with \varphi (t) = t(a1 + \eta (a2, a1) - t)\alpha - 1 and
f(x) is symmetric to x = a1 +
\eta (a2, a1)
2
, we get
\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast
4
(a1, a2)
\bigm| \bigm| \bigm| \leq \eta
k - 1
k
(p+q)+1(a2, a1)
\alpha p+q
k
\sqrt{}
Cp,q(\alpha , k)
\Biggl[
| f(a1)|
k
k - 1 + | f(a2)|
k
k - 1
2
\Biggr] k - 1
k
,
where
Cp,q(\alpha , k) =
=
a1+\eta (a2,a1)\int
a1
\bigl[
(a1 + \eta (a2, a1))
\alpha - t\alpha
\bigr] kp \bigl[
(a1 + \eta (a2, a1))
\alpha - (2a1 + \eta (a2, a1) - t)\alpha
\bigr] kq
dt
and
\Lambda \ast
4 =
(a1 + \eta (a2, a1))
\alpha - (a1 + (1 - t)\eta (a2, a1))
\alpha
\alpha
.
Theorem 2.2. Let P = [a1, a1 + \eta (a2, a1)] \subseteq \BbbR be an open invex subset. Assume that
f : P - \rightarrow \BbbR be a continuous mapping on P \circ with respect to \eta : P \times P - \rightarrow \BbbR for \eta (a2, a1) > 0. If
| f | l is preinvex mapping on P for l \geq 1, then, for any fixed p, q > 0, we have\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast (a1, a2)
\bigm| \bigm| \bigm| \leq \eta (a2, a1)
\Bigl[
Ap,q
\Lambda \ast (1)
\Bigr] l - 1
l l
\sqrt{}
Bp,q
\Lambda \ast | f(a1)| l +Bq,p
\Lambda \ast | f(a2)| l,
where
Bp,q
\Lambda \ast =
1\int
0
(1 - t)
\bigl[
\Lambda \ast (t)
\bigr] p\bigl[
\Lambda \ast (1 - t)
\bigr] q
dt
and Ap,q
\Lambda \ast (1) is defined as in Theorem 2.1.
Proof. Since | f | l is preinvex mapping on P, combining with Lemma 2.1, the well-known power
mean inequality and properties of the modulus, we get
\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast (a1, a2)
\bigm| \bigm| \bigm| \leq \eta (a2, a1)
1\int
0
\bigl[
\Lambda \ast (t)
\bigr] p\bigl[
\Lambda \ast (1 - t)
\bigr] q| f(a1 + t\eta (a2, a1))| dt \leq
\leq \eta (a2, a1)
\left[ 1\int
0
\bigl[
\Lambda \ast (t)
\bigr] p\bigl[
\Lambda \ast (1 - t)
\bigr] q
dt
\right]
l - 1
l
\left[ 1\int
0
\bigl[
\Lambda \ast (t)
\bigr] p\bigl[
\Lambda \ast (1 - t)
\bigr] q| f(a1 + t\eta (a2, a1))| ldt
\right]
1
l
\leq
\leq \eta (a2, a1)
\Bigl[
Ap,q
\Lambda \ast (1)
\Bigr] l - 1
l
\left[ 1\int
0
\bigl[
\Lambda \ast (t)
\bigr] p\bigl[
\Lambda \ast (1 - t)
\bigr] q \Bigl(
(1 - t)| f(a1)| l + t| f(a2)| l
\Bigr)
dt
\right]
1
l
=
= \eta (a2, a1)
\Bigl[
Ap,q
\Lambda \ast (1)
\Bigr] l - 1
l l
\sqrt{}
Bp,q
\Lambda \ast | f(a1)| l +Bq,p
\Lambda \ast | f(a2)| l.
Theorem 2.2 is proved.
We point out some special cases of Theorem 2.2.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
SOME NEW BOUNDS OF GAUSS – JACOBI AND HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES 1073
Corollary 2.6. Under the assumption of Theorem 2.2 with \varphi (t) = t, we get\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast
1
(a1, a2)
\bigm| \bigm| \bigm| \leq \eta p+q+1(a2, a1)\beta
l - 1
l (p+ 1, q + 1) \times
\times l
\sqrt{}
\beta (p+ 1, q + 2)| f(a1)| l + \beta (q + 1, p+ 2)| f(a2)| l.
Corollary 2.7. Under the assumption of Theorem 2.2 with \varphi (t) =
t\alpha
\Gamma (\alpha )
, we have
\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast
2
(a1, a2)
\bigm| \bigm| \bigm| \leq \eta \alpha (p+q)+1(a2, a1)
\Gamma p+q(\alpha + 1)
\beta
l - 1
l (\alpha p+ 1, \alpha q + 1) \times
\times l
\sqrt{}
\beta (\alpha p+ 1, \alpha q + 2)| f(a1)| l + \beta (\alpha q + 1, \alpha p+ 2)| f(a2)| l.
Corollary 2.8. Under the assumption of Theorem 2.2 with \varphi (t) =
t
\alpha
k1
k1\Gamma k1(\alpha )
, we obtain
\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast
3
(a1, a2)
\bigm| \bigm| \bigm| \leq \eta
\alpha
k1
(p+q)+1
(a2, a1)\Bigl[
k1\Gamma k1(\alpha + k1)
\Bigr] p+q \beta
l - 1
l
\biggl(
p\alpha
k1
+ 1,
q\alpha
k1
+ 1
\biggr)
\times
\times l
\sqrt{}
\beta
\biggl(
p\alpha
k1
+ 1,
q\alpha
k1
+ 2
\biggr)
| f(a1)| l + \beta
\biggl(
q\alpha
k1
+ 1,
p\alpha
k1
+ 2
\biggr)
| f(a2)| l.
Corollary 2.9. Under the assumption of Theorem 2.2 with \varphi (t) = t(a1 + \eta (a2, a1) - t)\alpha - 1 and
f(x) is symmetric to x = a1 +
\eta (a2, a1)
2
, we get
\bigm| \bigm| \bigm| T p,q
f,\Lambda \ast
4
(a1, a2)
\bigm| \bigm| \bigm| \leq \eta (a2, a1)
\biggl[
Cp,q(\alpha , 1)
\alpha p+q
\biggr] l - 1
l
l
\sqrt{}
Dp,q| f(a1)| l +Dq,p| f(a2)| l,
where
Dp,q =
1
\alpha p+q\eta 2(a2, a1)
a1+\eta (a2,a1)\int
a1
(t - a1)
\bigl[
(a1 + \eta (a2, a1))
\alpha - t\alpha
\bigr] p \times
\times
\bigl[
(a1 + \eta (a2, a1))
\alpha - (2a1 + \eta (a2, a1) - t)\alpha
\bigr] q
dt.
3. Some new bounds of Hermite – Hadamard type via general fractional integral inequali-
ties.
Theorem 3.1. Let f : P = [a1, a1 + \eta (a2, a1)] - \rightarrow \BbbR be a preinvex function on P with
\eta (a2, a1) > 0. Then the following inequalities for generalized fractional integral hold:
f
\biggl(
a1 +
\eta (a2, a1)
2
\biggr)
\leq 1
2\Lambda \ast (1)
\Bigl[
a+1
I\varphi f(a1 + \eta (a2, a1)) + (a1+\eta (a2,a1)) - I\varphi f(a1)
\Bigr]
\leq
\leq f(a1) + f(a2)
2
. (3.1)
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1074 A. KASHURI, M. RAMOSAÇAJ, R. LIKO
Proof. For t \in [0, 1], let x = a1 + t\eta (a2, a1) and y = a1 + (1 - t)\eta (a2, a1). From preinvexity
of f, we get
f
\biggl(
a1 +
\eta (a2, a1)
2
\biggr)
= f
\biggl(
x+ y
2
\biggr)
\leq f(x) + f(y)
2
,
i.e.,
2f
\biggl(
a1 +
\eta (a2, a1)
2
\biggr)
\leq f(a1 + t\eta (a2, a1)) + f(a1 + (1 - t)\eta (a2, a1)). (3.2)
Multiplying both sides of (3.2) by
\varphi (\eta (a2, a1)t)
t
and integrating the resulting inequality with respect
to t over (0, 1], we obtain
2f
\biggl(
a1 +
\eta (a2, a1)
2
\biggr) 1\int
0
\varphi (\eta (a2, a1)t)
t
dt \leq
\leq
1\int
0
\varphi (\eta (a2, a1)t)
t
f(a1 + t\eta (a2, a1))dt+
1\int
0
\varphi (\eta (a2, a1)t)
t
f(a1 + (1 - t)\eta (a2, a1))dt.
Hence,
2f
\biggl(
a1 +
\eta (a2, a1)
2
\biggr) 1\int
0
\varphi (\eta (a2, a1)t)
t
dt \leq
\Bigl[
a+1
I\varphi f(a1 + \eta (a2, a1)) + (a1+\eta (a2,a1)) - I\varphi f(a1)
\Bigr]
.
So, the first inequality is proved.
To prove the other half of the inequality in (3.1), since f is preinvex, we have
f(a1 + t\eta (a2, a1)) + f(a1 + (1 - t)\eta (a2, a1)) \leq f(a1) + f(a2). (3.3)
Multiplying both sides of (3.3) by
\varphi (\eta (a2, a1)t)
t
and integrating the resulting inequality with respect
to t over (0, 1], we obtain
\Bigl[
a+1
I\varphi f(a1 + \eta (a2, a1)) + (a1+\eta (a2,a1)) - I\varphi f(a1)
\Bigr]
\leq
\bigl[
f(a1) + f(a2)
\bigr] 1\int
0
\varphi (\eta (a2, a1)t)
t
dt.
Therefore, the second inequality is proved.
Theorem 3.1 is proved.
We point out some special cases of Theorem 3.1.
Corollary 3.1. Taking \eta (a2, a1) = a2 - a1 in Theorem 3.1, we get Theorem 5 of [3].
Corollary 3.2. If in Theorem 3.1 we take \varphi (t) = t, then the inequalities (3.1) become the in-
equalities
f
\biggl(
a1 +
\eta (a2, a1)
2
\biggr)
\leq 1
2\eta (a2, a1)
\Bigl[
Ia+1
f(a1 + \eta (a2, a1)) + I(a1+\eta (a2,a1)) - f(a1)
\Bigr]
\leq
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SOME NEW BOUNDS OF GAUSS – JACOBI AND HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES 1075
\leq f(a1) + f(a2)
2
,
where Ia+1
f and Ia - 2
f are the classical Riemann integrals.
Corollary 3.3. If in Theorem 3.1 we choose \varphi (t) =
t\alpha
\Gamma (\alpha )
, then the inequalities (3.1) become the
inequalities
f
\biggl(
a1 +
\eta (a2, a1)
2
\biggr)
\leq \Gamma (\alpha + 1)
2\eta \alpha (a2, a1)
\Bigl[
J\alpha
a+1
f(a1 + \eta (a2, a1)) + J\alpha
(a1+\eta (a2,a1)) -
f(a1)
\Bigr]
\leq
\leq f(a1) + f(a2)
2
,
where J\alpha
a+1
f and J\alpha
a - 2
f are the fractional Riemann integrals.
Corollary 3.4. If in Theorem 3.1 we take \varphi (t) =
t
\alpha
k
k\Gamma k(\alpha )
, then the inequalities (3.1) become the
inequalities
f
\biggl(
a1 +
\eta (a2, a1)
2
\biggr)
\leq \Gamma k(\alpha + k)
2\eta
\alpha
k (a2, a1)
\Bigl[
I\alpha ,k
a+1
f(a1 + \eta (a2, a1)) + I\alpha ,k
(a1+\eta (a2,a1)) -
f(a1)
\Bigr]
\leq
\leq f(a1) + f(a2)
2
.
Corollary 3.5. If in Theorem 3.1 we choose \varphi (t) = t(a1 + \eta (a2, a1) - t)\alpha - 1 and f(x) is sym-
metric to x = a1 +
\eta (a2, a1)
2
, then the inequalities (3.1) become the inequalities
f
\biggl(
a1 +
\eta (a2, a1)
2
\biggr)
\leq \alpha
(a1 + \eta (a2, a1))\alpha - a\alpha 1
a1+\eta (a2,a1)\int
a1
f(t)d\alpha t \leq
f(a1) + f(a2)
2
.
Corollary 3.6. If in Theorem 3.1 we take \varphi (t) =
t
\alpha
\mathrm{e}\mathrm{x}\mathrm{p}
\Bigl[ \biggl(
- 1 - \alpha
\alpha
\biggr)
t
\Bigr]
, \alpha \in (0, 1), then the
inequalities (3.1) become the inequalities
f
\biggl(
a1 +
\eta (a2, a1)
2
\biggr)
\leq 1 - \alpha
2(1 - \mathrm{e}\mathrm{x}\mathrm{p}( - D))
\Bigl[
\scrI \alpha
a+1
f(a1 + \eta (a2, a1)) + \scrI \alpha
(a1+\eta (a2,a1)) -
f(a1)
\Bigr]
\leq
\leq f(a1) + f(a2)
2
,
where \scrI \alpha
a+1
f and \scrI \alpha
a - 2
f are the right- and left-hand sided fractional integral operators with exponen-
tial kernel and D =
\biggl(
1 - \alpha
\alpha
\biggr)
\eta (a2, a1).
For establishing some new results regarding general fractional integrals we need to prove the
following lemma.
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1076 A. KASHURI, M. RAMOSAÇAJ, R. LIKO
Lemma 3.1. Let f : P = [a1, a1 + \eta (a2, a1)] - \rightarrow \BbbR be a differentiable mapping on (a1, a1 +
+ \eta (a2, a1)) with \eta (a2, a1) > 0. If f \prime \in L(P ), then the following identity for generalized fractional
integrals holds:
f(a1) + f(a1 + \eta (a2, a1))
2
- 1
2\Lambda \ast (1)
\Bigl[
a+1
I\varphi f(a1 + \eta (a2, a1)) + (a1+\eta (a2,a1)) - I\varphi f(a1)
\Bigr]
=
=
\eta (a2, a1)
2\Lambda \ast (1)
1\int
0
\Bigl[
\Lambda \ast (1 - t) - \Lambda \ast (t)
\Bigr]
f \prime (a1 + (1 - t)\eta (a2, a1))dt.
We denote
Hf,\Lambda \ast (a1, a2) =
\eta (a2, a1)
2\Lambda \ast (1)
1\int
0
\Bigl[
\Lambda \ast (1 - t) - \Lambda \ast (t)
\Bigr]
f \prime (a1 + (1 - t)\eta (a2, a1))dt. (3.4)
Proof. Integrating by parts (3.4) and changing the variable of integration, we have
Hf,\Lambda \ast (a1, a2) =
\eta (a2, a1)
2\Lambda \ast (1)
\times
\times
\Biggl\{ 1\int
0
\Lambda \ast (1 - t)f \prime (a1 + (1 - t)\eta (a2, a1))dt -
1\int
0
\Lambda \ast (t)f \prime (a1 + (1 - t)\eta (a2, a1))dt
\Biggr\}
=
=
\eta (a2, a1)
2\Lambda \ast (1)
\Biggl\{
- \Lambda \ast (1 - t)f(a1 + (1 - t)\eta (a2, a1))
\eta (a2, a1)
\bigm| \bigm| \bigm| \bigm| \bigm|
1
0
-
- 1
\eta (a2, a1)
1\int
0
\varphi (\eta (a2, a1)(1 - t))
1 - t
f(a1 + (1 - t)\eta (a2, a1))dt +
+
\Lambda \ast (t)f(a1 + (1 - t)\eta (a2, a1))
\eta (a2, a1)
\bigm| \bigm| \bigm| \bigm| \bigm|
1
0
- 1
\eta (a2, a1)
1\int
0
\varphi (\eta (a2, a1)t)
t
f(a1 + (1 - t)\eta (a2, a1))dt
\Biggr\}
=
=
\eta (a2, a1)
2\Lambda \ast (1)
\Biggl\{
\Lambda \ast (1)f(a1 + \eta (a2, a1))
\eta (a2, a1)
- 1
\eta (a2, a1)
(a1+\eta (a2,a1)) - I\varphi f(a1) +
+
\Lambda \ast (1)f(a1)
\eta (a2, a1)
- 1
\eta (a2, a1)
a+1
I\varphi f(a1 + \eta (a2, a1))
\Biggr\}
=
=
f(a1) + f(a1 + \eta (a2, a1))
2
- 1
2\Lambda \ast (1)
\Bigl[
a+1
I\varphi f(a1 + \eta (a2, a1)) + (a1+\eta (a2,a1)) - I\varphi f(a1)
\Bigr]
.
Lemma 3.1 is proved.
Remark 3.1. Taking \eta (a2, a1) = a2 - a1 in Lemma 3.1, we get Lemma 5 of [30].
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SOME NEW BOUNDS OF GAUSS – JACOBI AND HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES 1077
Theorem 3.2. Let f : P = [a1, a1 + \eta (a2, a1)] - \rightarrow \BbbR be a differentiable mapping on (a1, a1 +
+ \eta (a2, a1)) with \eta (a2, a1) > 0. If | f \prime | q is preinvex on P for q > 1 and p - 1 + q - 1 = 1, then the
following inequality for generalized fractional integrals holds:
\bigm| \bigm| Hf,\Lambda \ast (a1, a2)
\bigm| \bigm| \leq \eta (a2, a1)
2\Lambda \ast (1)
p
\sqrt{}
K\Lambda \ast (p)
q
\sqrt{}
| f \prime (a1)| q + | f \prime (a2)| q
2
,
where
K\Lambda \ast (p) =
1\int
0
\bigm| \bigm| \bigm| \Lambda \ast (1 - t) - \Lambda \ast (t)
\bigm| \bigm| \bigm| pdt.
Proof. From Lemma 3.1, preinvexity of | f \prime | q, Hölder’s inequality and properties of the modulus,
we have
\bigm| \bigm| Hf,\Lambda \ast (a1, a2)
\bigm| \bigm| \leq \eta (a2, a1)
2\Lambda \ast (1)
1\int
0
\bigm| \bigm| \Lambda \ast (1 - t) - \Lambda \ast (t)
\bigm| \bigm| \bigm| \bigm| f \prime (a1 + (1 - t)\eta (a2, a1))
\bigm| \bigm| dt \leq
\leq \eta (a2, a1)
2\Lambda \ast (1)
\left( 1\int
0
\bigm| \bigm| \Lambda \ast (1 - t) - \Lambda \ast (t)
\bigm| \bigm| pdt
\right)
1
p
\left( 1\int
0
\bigm| \bigm| f \prime (a1 + (1 - t)\eta (a2, a1))
\bigm| \bigm| qdt
\right)
1
q
\leq
\leq \eta (a2, a1)
2\Lambda \ast (1)
p
\sqrt{}
K\Lambda \ast (p)
\left( 1\int
0
\bigl(
(1 - t)
\bigm| \bigm| f \prime (a1)
\bigm| \bigm| q + t
\bigm| \bigm| f \prime (a2)
\bigm| \bigm| q\bigr) dt
\right)
1
q
=
=
\eta (a2, a1)
2\Lambda \ast (1)
p
\sqrt{}
K\Lambda \ast (p)
q
\sqrt{}
| f \prime (a1)| q + | f \prime (a2)| q
2
.
Theorem 3.2 is proved.
We point out some special cases of Theorem 3.2.
Corollary 3.7. Taking \eta (a2, a1) = a2 - a1 in Theorem 3.2, we get Theorem 7 of [30].
Corollary 3.8. Taking p = q = 2 in Theorem 3.2, we get
\bigm| \bigm| Hf,\Lambda \ast (a1, a2)
\bigm| \bigm| \leq \eta (a2, a1)
2\Lambda \ast (1)
\sqrt{}
K\Lambda \ast (2)
\sqrt{}
| f \prime (a1)| 2 + | f \prime (a2)| 2
2
.
Corollary 3.9. Taking \eta (a2, a1) = a2 - a1 and \varphi (t) = t in Theorem 3.2, we get Theorem 2.3 of
[7].
Corollary 3.10. Taking \eta (a2, a1) = a2 - a1 and \varphi (t) =
t\alpha
\Gamma (\alpha )
in Theorem 3.2, we get Theorem
8 of [27].
Corollary 3.11. Taking \eta (a2, a1) = a2 - a1 and \varphi (t) =
t
\alpha
k
k\Gamma k(\alpha )
in Theorem 3.2, we get Theorem
8 of [12].
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1078 A. KASHURI, M. RAMOSAÇAJ, R. LIKO
Corollary 3.12. Taking \eta (a2, a1) = a2 - a1, where \varphi (t) = t(a1+ \eta (a2, a1) - t)\alpha - 1 and f(x) is
symmetric to x = a1 +
\eta (a2, a1)
2
in Theorem 3.2, we get
\bigm| \bigm| Hf,\Lambda \ast
4
(a1, a2)
\bigm| \bigm| \leq q
\sqrt{}
\eta (a2, a1)
2
p
\surd
p\alpha + 1
\Bigl[
(a1 + \eta (a2, a1))
\alpha - a\alpha 1
\Bigr] \times
\times
p
\sqrt{}
ap\alpha +1
1 + (a1 + \eta (a2, a1))
p\alpha +1 - (2a1 + \eta (a2, a1))
p\alpha +1
2p\alpha
q
\sqrt{}
| f \prime (a1)| q + | f \prime (a2)| q
2
.
Theorem 3.3. Let f : P = [a1, a1 + \eta (a2, a1)] - \rightarrow \BbbR be a differentiable mapping on (a1, a1 +
+ \eta (a2, a1)) with \eta (a2, a1) > 0. If | f \prime | q is preinvex on P for q \geq 1, then the following inequality
for generalized fractional integrals holds:\bigm| \bigm| Hf,\Lambda \ast (a1, a2)
\bigm| \bigm| \leq \eta (a2, a1)
2\Lambda \ast (1)
\Bigl[
K\Lambda \ast (1)
\Bigr] 1 - 1
q q
\sqrt{}
K\Lambda \ast
q
\sqrt{}
| f \prime (a1)| q + | f \prime (a2)| q,
where
K\Lambda \ast =
1\int
0
t
\bigm| \bigm| \Lambda \ast (1 - t) - \Lambda \ast (t)
\bigm| \bigm| dt
and K\Lambda \ast (1) is defined as in Theorem 3.2.
Proof. From Lemma 3.1, preinvexity of | f \prime | q, the well-known power mean inequality and
properties of the modulus, we have
\bigm| \bigm| Hf,\Lambda \ast (a1, a2)
\bigm| \bigm| \leq \eta (a2, a1)
2\Lambda \ast (1)
1\int
0
\bigm| \bigm| \Lambda \ast (1 - t) - \Lambda \ast (t)
\bigm| \bigm| \bigm| \bigm| f \prime (a1 + (1 - t)\eta (a2, a1))
\bigm| \bigm| dt \leq
\leq \eta (a2, a1)
2\Lambda \ast (1)
\left( 1\int
0
\bigm| \bigm| \Lambda \ast (1 - t) - \Lambda \ast (t)
\bigm| \bigm| dt
\right) 1 - 1
q
\times
\times
\left( 1\int
0
\bigm| \bigm| \Lambda \ast (1 - t) - \Lambda \ast (t)
\bigm| \bigm| \bigm| \bigm| f \prime (a1 + (1 - t)\eta (a2, a1))
\bigm| \bigm| qdt
\right)
1
q
\leq
\leq \eta (a2, a1)
2\Lambda \ast (1)
\Bigl[
K\Lambda \ast (1)
\Bigr] 1 - 1
q
\left( 1\int
0
\bigm| \bigm| \Lambda \ast (1 - t) - \Lambda \ast (t)
\bigm| \bigm| \bigl( (1 - t)
\bigm| \bigm| f \prime (a1)
\bigm| \bigm| q + t
\bigm| \bigm| f \prime (a2)
\bigm| \bigm| q\bigr) dt
\right)
1
q
=
=
\eta (a2, a1)
2\Lambda \ast (1)
\Bigl[
K\Lambda \ast (1)
\Bigr] 1 - 1
q q
\sqrt{}
K\Lambda \ast
q
\sqrt{}
| f \prime (a1)| q + | f \prime (a2)| q.
Theorem 3.3 is proved.
We point out some special cases of Theorem 3.3.
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SOME NEW BOUNDS OF GAUSS – JACOBI AND HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES 1079
Corollary 3.13. Taking q = 1 in Theorem 3.3, we get\bigm| \bigm| Hf,\Lambda \ast (a1, a2)
\bigm| \bigm| \leq \eta (a2, a1)
2\Lambda \ast (1)
K\Lambda \ast
\Bigl[
| f \prime (a1)| + | f \prime (a2)|
\Bigr]
.
Corollary 3.14. Under the assumption of Theorem 3.3 with \varphi (t) = t, we have\bigm| \bigm| Hf,\Lambda \ast
1
(a1, a2)
\bigm| \bigm| \leq \eta (a2, a1)
2
2+ 1
q
q
\sqrt{}
| f \prime (a1)| q + | f \prime (a2)| q.
Corollary 3.15. Under the assumption of Theorem 3.3 with \varphi (t) = t\alpha
\Gamma (\alpha ) , we obtain
\bigm| \bigm| Hf,\Lambda \ast
2
(a1, a2)
\bigm| \bigm| \leq \biggl( 2\alpha - 1
2\alpha +1
\biggr)
q
\sqrt{}
\Gamma (\alpha + 1)
\Gamma (\alpha + 2)
\eta (a2, a1)
q
\sqrt{}
| f \prime (a1)| q + | f \prime (a2)| q.
Corollary 3.16. Under the assumption of Theorem 3.3 with \varphi (t) =
t
\alpha
k1
k1\Gamma k1(\alpha )
, we get
\bigm| \bigm| Hf,\Lambda \ast
3
(a1, a2)
\bigm| \bigm| \leq \Biggl( 2
\alpha
k1 - 1
2
\alpha
k1
+1
\Biggr)
q
\sqrt{}
\Gamma k1(\alpha + k1)
\Gamma k1(\alpha + k1 + 1)
\eta (a2, a1)
q
\sqrt{}
| f \prime (a1)| q + | f \prime (a2)| q.
Corollary 3.17. Under the assumption of Theorem 3.3 with \varphi (t) = t(a1 + \eta (a2, a1) - t)\alpha - 1 and
f(x) is symmetric to x = a1 +
\eta (a2, a1)
2
, we have
\bigm| \bigm| Hf,\Lambda \ast
4
(a1, a2)
\bigm| \bigm| \leq \eta (a2, a1)
2\Lambda \ast (1)
\Bigl[
K\Lambda \ast (1)
\Bigr] 1 - 1
q q
\sqrt{}
K\Lambda \ast
q
\sqrt{}
| f \prime (a1)| q + | f \prime (a2)| q,
where
\Lambda \ast (1) =
(a1 + \eta (a2, a1))
\alpha
\alpha
,
K\Lambda \ast (1) =
2
\alpha
\Biggl[
(a1 + \eta (a2, a1))
\alpha +1 - 2
\biggl(
a1 +
\eta (a2, a1)
2
\biggr) \alpha +1
+ a\alpha +1
1
\Biggr]
,
K\Lambda \ast =
1
\alpha
\bigl[
F11 - F12 + F21 - F22
\bigr]
,
and
F11 =
1
\eta 2(a2, a1)
\Biggl\{
(a1 + \eta (a2, a1))
\alpha + 1
\Biggl[
(a1 + \eta (a2, a1))
\alpha +1 -
\biggl(
a1 +
\eta (a2, a1)
2
\biggr) \alpha +1
\Biggr]
-
- 1
\alpha + 2
\Biggl[
(a1 + \eta (a2, a1))
\alpha +2 -
\biggl(
a1 +
\eta (a2, a1)
2
\biggr) \alpha +2
\Biggr] \Biggr\}
,
F12 =
1
\eta 2(a2, a1)
\Biggl\{
1
\alpha + 2
\Biggl[ \biggl(
a1 +
\eta (a2, a1)
2
\biggr) \alpha +2
- a\alpha +2
1
\Biggr]
-
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1080 A. KASHURI, M. RAMOSAÇAJ, R. LIKO
- a1
\alpha + 1
\Biggl[ \biggl(
a1 +
\eta (a2, a1)
2
\biggr) \alpha +1
- a\alpha +1
1
\Biggr] \Biggr\}
,
F21 =
1
\eta 2(a2, a1)
\Biggl\{
1
\alpha + 2
\Biggl[
(a1 + \eta (a2, a1))
\alpha +2 -
\biggl(
a1 +
\eta (a2, a1)
2
\biggr) \alpha +2
\Biggr]
-
- a1
\alpha + 1
\Biggl[
(a1 + \eta (a2, a1))
\alpha +1 -
\biggl(
a1 +
\eta (a2, a1)
2
\biggr) \alpha +1
\Biggr] \Biggr\}
,
F22 =
1
\eta 2(a2, a1)
\Biggl\{
(a1 + \eta (a2, a1))
\alpha + 1
\Biggl[ \biggl(
a1 +
\eta (a2, a1)
2
\biggr) \alpha +1
- a\alpha +1
1
\Biggr]
-
- 1
\alpha + 2
\Bigl[
(a1 + \eta (a2, a1))
\alpha +2 - a\alpha +2
1
\Bigr] \Biggr\}
.
4. Applications. Consider the following special means for different real numbers \alpha , \beta and
\alpha \beta \not = 0 as follows:
(1) the arithmetic mean
A(\alpha , \beta ) =
\alpha + \beta
2
,
(2) the harmonic mean
H(\alpha , \beta ) =
2
1
\alpha
+
1
\beta
,
(3) the logarithmic mean
L(\alpha , \beta ) =
\beta - \alpha
\mathrm{l}\mathrm{n} | \beta | - \mathrm{l}\mathrm{n} | \alpha |
,
(4) the generalized log-mean
Ln(\alpha , \beta ) =
\biggl[
\beta n+1 - \alpha n+1
(n+ 1)(\beta - \alpha )
\biggr] 1
n
, n \in \BbbZ \setminus \{ - 1, 0\} .
Now, by using the theory results in Section 3, we give some applications to special means for
different real numbers.
Proposition 4.1. Let a1, a2 \in \BbbR \setminus \{ 0\} , where a1 < a2 and \eta (a2, a1) > 0. Then, for n \in
\in \BbbZ \setminus \{ - 1, 0\} , where q > 1 and p - 1 + q - 1 = 1, the following inequality holds:\bigm| \bigm| \bigm| A (an1 , (a1 + \eta (a2, a1))
n) - Ln (a1, a1 + \eta (a2, a1))
\bigm| \bigm| \bigm| \leq | n|
2
\eta (a2, a1)
p
\surd
p+ 1
\times
\times q
\sqrt{}
A
\bigl(
| a1| q(n - 1), | a2| q(n - 1)
\bigr)
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
SOME NEW BOUNDS OF GAUSS – JACOBI AND HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES 1081
Proof. Applying Theorem 3.2 for f(x) = xn and \varphi (t) = t, one can obtain the result immedi-
ately.
Proposition 4.2. Let a1, a2 \in \BbbR \setminus \{ 0\} , where a1 < a2 and \eta (a2, a1) > 0. Then, for q > 1 and
p - 1 + q - 1 = 1, the following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| 1
H (a1, a1 + \eta (a2, a1))
- 1
L (a1, a1 + \eta (a2, a1))
\bigm| \bigm| \bigm| \bigm| \bigm| \leq \eta (a2, a1)
2 p
\surd
p+ 1
1
q
\sqrt{}
H
\Bigl(
a2q1 , a2q2
\Bigr) .
Proof. Applying Theorem 3.2 for f(x) =
1
x
and \varphi (t) = t, one can obtain the result immedi-
ately.
Proposition 4.3. Let a1, a2 \in \BbbR \setminus \{ 0\} , where a1 < a2 and \eta (a2, a1) > 0. Then, for n \in
\in \BbbZ \setminus \{ - 1, 0\} and q \geq 1, the following inequality holds:\bigm| \bigm| \bigm| A (an1 , (a1 + \eta (a2, a1))
n) - Ln (a1, a1 + \eta (a2, a1))
\bigm| \bigm| \bigm| \leq | n|
2
2+ 1
q
\eta (a2, a1) \times
\times q
\sqrt{}
A
\bigl(
| a1| q(n - 1), | a2| q(n - 1)
\bigr)
.
Proof. Applying Theorem 3.3 for f(x) = xn and \varphi (t) = t, one can obtain the result immedi-
ately.
Proposition 4.4. Let a1, a2 \in \BbbR \setminus \{ 0\} , where a1 < a2 and \eta (a2, a1) > 0. Then, for q \geq 1, the
following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| 1
H (a1, a1 + \eta (a2, a1))
- 1
L (a1, a1 + \eta (a2, a1))
\bigm| \bigm| \bigm| \bigm| \bigm| \leq \eta (a2, a1)
2
2+ 1
q
1
q
\sqrt{}
H
\Bigl(
a2q1 , a2q2
\Bigr) .
Proof. Applying Theorem 3.3 for f(x) =
1
x
and \varphi (t) = t, one can obtain the result immedi-
ately.
Remark 4.1. Applying our Theorems 3.2 and 3.3 for appropriate choices of function \varphi (t) =
=
t\alpha
\Gamma (\alpha )
,
t
\alpha
k1
k1\Gamma k1(\alpha )
; \varphi (t) = t(a1 + \eta (a2, a1) - t)\alpha - 1, where f(x) is symmetric to x = a1 +
+
\eta (a2, a1)
2
and \varphi (t) =
t
\alpha
\mathrm{e}\mathrm{x}\mathrm{p}
\Bigl[ \biggl(
- 1 - \alpha
\alpha
\biggr)
t
\Bigr]
for \alpha \in (0, 1), such that | f \prime | q to be preinvex, we can
deduce some new general fractional integral inequalities using special means. We omit their proofs
and the details are left to the interested reader.
Remark 4.2. Also, in Remark 4.1, if we choose \eta (a2, a1) = a2 - a1, we can deduce some new
fascinating general fractional integral inequalities for convex functions using special means. The
details are left to the interested reader.
Next, we provide some new error estimates for the trapezoidal formula.
Let Q be the partition of the points a1 = x0 < x1 < . . . < xn = a2 of the interval [a1, a2]. Let
consider the quadrature formula
a2\int
a1
f(x)dx = T (f,Q) + E(f,Q),
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1082 A. KASHURI, M. RAMOSAÇAJ, R. LIKO
where
T (f,Q) =
n - 1\sum
i=0
f(xi) + f(xi+1)
2
(xi+1 - xi)
is the trapezoidal version and E(f,Q) is denote their associated approximation error.
Proposition 4.5. Let f : [a1, a2] - \rightarrow \BbbR be a differentiable function on (a1, a2), where a1 < a2.
If | f \prime | q is convex on [a1, a2] for q > 1 and
1
p
+
1
q
= 1, then the following inequality holds:
\bigm| \bigm| E(f,Q)
\bigm| \bigm| \leq 1
2
q+1
q p
\surd
p+ 1
n - 1\sum
i=0
(xi+1 - xi)
2 q
\sqrt{}
| f \prime (xi)| q + | f \prime (xi+1)| q.
Proof. Applying Theorem 3.2 for \eta (a2, a1) = a2 - a1 and \varphi (t) = t on the subintervals
[xi, xi+1], i = 0, . . . , n - 1, of the partition Q, we have\bigm| \bigm| \bigm| \bigm| \bigm| f(xi) + f(xi+1)
2
- 1
xi+1 - xi
xi+1\int
xi
f(x)dx
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (xi+1 - xi)
2 p
\surd
p+ 1
\biggl[
| f \prime (xi)| q + | f \prime (xi+1)| q
2
\biggr] 1
q
. (4.1)
Hence from (4.1), we get
\bigm| \bigm| E(f,Q)
\bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \bigm|
a2\int
a1
f(x)dx - T (f,Q)
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \bigm|
n - 1\sum
i=0
\Biggl\{ xi+1\int
xi
f(x)dx - f(xi) + f(xi+1)
2
(xi+1 - xi)
\Biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
n - 1\sum
i=0
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl\{ xi+1\int
xi
f(x)dx - f(xi) + f(xi+1)
2
(xi+1 - xi)
\Biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
2
q+1
q p
\surd
p+ 1
n - 1\sum
i=0
(xi+1 - xi)
2 q
\sqrt{}
| f \prime (xi)| q + | f \prime (xi+1)| q.
Proposition 4.5 is proved.
Proposition 4.6. Let f : [a1, a2] - \rightarrow \BbbR be a differentiable function on (a1, a2), where a1 < a2.
If | f \prime | q is convex on [a1, a2] for q \geq 1, then the following inequality holds:
\bigm| \bigm| E(f,Q)
\bigm| \bigm| \leq 1
2
2+ 1
q
n - 1\sum
i=0
(xi+1 - xi)
2 q
\sqrt{}
| f \prime (xi)| q + | f \prime (xi+1)| q.
Proof is analogous as to that of Proposition 4.5 but use Theorem 3.3 for \eta (a2, a1) = a2 - a1
and \varphi (t) = t.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
SOME NEW BOUNDS OF GAUSS – JACOBI AND HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES 1083
Remark 4.3. Applying Theorems 3.2 and 3.3 for appropriate choices of function \varphi (t) =
t\alpha
\Gamma (\alpha )
,
t
\alpha
k1
k1\Gamma k1(\alpha )
; \varphi (t) = t(a2 - t)\alpha - 1, where f(x) is symmetric to x =
a1 + a2
2
and
\varphi (t) =
t
\alpha
\mathrm{e}\mathrm{x}\mathrm{p}
\Bigl[ \biggl(
- 1 - \alpha
\alpha
\biggr)
t
\Bigr]
for \alpha \in (0, 1), such that | f \prime | q to be convex, we can deduce some new general fractional integral
inequalities using above ideas and techniques. We omit their proofs and the details are left to the
interested reader.
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ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
|
| id | umjimathkievua-article-603 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:15Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/82/fa326ead22df3ce9982a80f248dda682.pdf |
| spelling | umjimathkievua-article-6032025-03-31T08:47:35Z Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities Some new bounds of Gauss-Jacobi and Hermite-Hadamard type integral inequalities Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities Kashuri, A. Ramosaçaj, M. Liko, R. Kashuri, Artion Ramosaçaj, Miftar Liko, Rozana Kashuri, A. Ramosaçaj, M. Liko, R. Hermite-Hadamard inequality H¨older’s inequality power mean inequality general fractional integrals UDC 517.5 In this paper, authors discover two interesting identities regarding Gauss–Jacobi and Hermite–Hadamard type integral inequalities. By using the first lemma as an auxiliary result, some new bounds with respect to Gauss–Jacobi type integral inequalities are established. Also, using the second lemma, some new estimates with respect to Hermite–Hadamard type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from main results. Some applications to special means for different positive real numbers and new error estimates for the trapezoidal are provided as well. These results give us the generalizations, refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research. УДК 517.5 Нові границі для інтегральних нерівностей типу Гаусса–Якобі та Ерміта–Адамара Знайдено дві цікаві тотожності для інтегральних нерівностей типу Гаусса–Якобі та Ерміта–Адамара. З використанням першої леми як допоміжного результату встановлено деякі нові границі інтегральних нерівностей типу Гаусса–Якобі. Далі, за допомогою другої леми та загальних дробових інтегралів отримано деякі нові границі інтегральних нерівностей типу Ерміта–Адамара. Зазначено, що з основних результатів можна отримати деякі нові випадки. Також запропоновано деякі застосування до спеціальних середніх для різних додатних дійсних чисел та нові оцінки похибок для методу трапеції. Ці результати є узагальненням, уточненням та значним покращенням нових та раніше відомих результатів. Ідеї та методи цієї статті мають стимулювати подальші дослідження. Institute of Mathematics, NAS of Ukraine 2021-08-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/603 10.37863/umzh.v73i8.603 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 8 (2021); 1067 - 1084 Український математичний журнал; Том 73 № 8 (2021); 1067 - 1084 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/603/9095 Copyright (c) 2021 Artion Kashuri, Miftar Ramosacaj, Rozana Liko |
| spellingShingle | Kashuri, A. Ramosaçaj, M. Liko, R. Kashuri, Artion Ramosaçaj, Miftar Liko, Rozana Kashuri, A. Ramosaçaj, M. Liko, R. Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities |
| title | Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities |
| title_alt | Some new bounds of Gauss-Jacobi and Hermite-Hadamard type integral inequalities Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities |
| title_full | Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities |
| title_fullStr | Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities |
| title_full_unstemmed | Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities |
| title_short | Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities |
| title_sort | some new bounds оf gauss – jacobi аnd hermite – hadamard type integral inequalities |
| topic_facet | Hermite-Hadamard inequality H¨older’s inequality power mean inequality general fractional integrals |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/603 |
| work_keys_str_mv | AT kashuria somenewboundsofgaussjacobiandhermitehadamardtypeintegralinequalities AT ramosacajm somenewboundsofgaussjacobiandhermitehadamardtypeintegralinequalities AT likor somenewboundsofgaussjacobiandhermitehadamardtypeintegralinequalities AT kashuriartion somenewboundsofgaussjacobiandhermitehadamardtypeintegralinequalities AT ramosacajmiftar somenewboundsofgaussjacobiandhermitehadamardtypeintegralinequalities AT likorozana somenewboundsofgaussjacobiandhermitehadamardtypeintegralinequalities AT kashuria somenewboundsofgaussjacobiandhermitehadamardtypeintegralinequalities AT ramosacajm somenewboundsofgaussjacobiandhermitehadamardtypeintegralinequalities AT likor somenewboundsofgaussjacobiandhermitehadamardtypeintegralinequalities |