In this paper, integral midpoint type inequalities involving Riemann-Liouville fractional integrals for tgs-convex functions are proved. Two different identities are utilized to get some new integral midpoint type inequalities. One identity is used to obtain\:inequalities for functions whose first derivatives are tgs-convex functions and another identity is used to obtain inequalities for the functions whose second derivatives are tgs-convex. Some numerical examples along with graphical representation are also included to demonstrate the effectiveness of the results. The results demonstrate that the newly established bounds offer significant improvements and tighter estimates compared to existing inequalities in the literature.
Keywords: convex function, integral midpoint inequality, Holder inequality, power mean inequality, Riemann-Liouville fractional integrals
1. Introduction
It has
been demonstrated that inequalities are the most effective construction
tools for a number of mathematical fields. Inequalities play a
fundamental role in the study of integral equations, differential
equations, optimization theory, and fractional calculus. Some recent
mathematical inequalities can be found in [1–3].
Convexity is a phrase that has gained much focus and developed into a
valuable origin of knowledge and inspiration. For enthusiastic readers,
the literature on convex analysis, convex functions, and their
applications can be found in [4] and tgs-convex function is available in [5]. In recent years, several
authors have investigated fractional integral inequalities for different
generalized convex functions, including h-convex, \((\alpha,s)\)-convex, and twice
differentiable functions. For example, recent contributions on
fractional inequalities and midpoint-type inequalities for generalized
convex functions can be found in [6,7].
Fractional integrals have an essential role in the theory of
inequalities and their applications. Mehreen et al. [8] presented the integral
identities related with integral midpoint inequality for tgs-convex
functions. Fahad et al. [9] obtain some inequalities for Caputo Fabrizio
fractional integrals involving \((\alpha,s)\) convex functions with
applications. Qaisar et al. obtained some integral midpoint
inequalities involving fractional integrals for convex functions [10].
In addition, inequalities associated with convexity have also been
extensively studied in the setting of time scales calculus, which
unifies continuous and discrete analysis. Several important
contributions include Ostrowski-type inequalities for functions whose
derivative modulus is relatively convex on time scales, generalized
Hölder and Minkowski inequalities on diamond-alpha time scales, and
multidimensional reverse Hölder inequalities on time scales [11–13]. These works
demonstrate the broad applicability of convexity-based inequalities in
both continuous and discrete frameworks.
Because of the numerous uses of integral midpoint type inequalities
[14,15] and fractional
calculus [16], the objective
of present work to investigate the integral midpoint type inequalities
involving fractional integrals. Although tgs-convex functions can be
viewed as a special case of h-convex functions, the analysis of
midpoint-type fractional inequalities for tgs-convex functions leads to
different estimates and sharper structures associated with the symmetry
of the tgs-condition. Moreover, the present work provides graphical
validations and numerical comparisons of the obtained inequalities,
which, to the best of our knowledge, have not been investigated
previously for this class of functions. Even in more generalized
settings such as those considered in [17,18], graphical illustrations and
numerical validations for tgs-convex fractional midpoint inequalities
are still unavailable.
The framework of the paper is as follows: After some preliminary
facts in §2, some integral midpoint type inequalities are
proved in §3 and §4, for these functions whose absolute values of
first or second derivatives are tgs-convex respectively. The graphical
representations of obtained results are given in §5. Moreover some
numerical examples are provided to make comparison of obtained results
with existing results.
2. Preliminaries
Jacques Hadamard and Charles Hermite derived the integral midpoint
type inequality, which is expressed as:
Definition 1 (Integral midpoint inequality). Let \(\zeta:I\subseteq\mathbb{R}\rightarrow\mathbb{R}\)
be a convex function and \(\iota,\kappa\in
I\) with \(\iota<\kappa\),
then the following double inequality holds: \[
\zeta\bigg(\frac{\iota+\kappa}{2}\bigg)\leq\frac{1}{\kappa-\iota}\int_{\iota}^{\kappa}\zeta(h)dh\leq\frac{\zeta(\iota)+\zeta(\kappa)}{2}.\tag{1}\]
If \(\zeta\) is concave, then (1) holds in reverse direction.
Definition 2 (\(h\)-convex function [19]). Let \(h:J\rightarrow{R}\) be a
non-negative function, \(h\neq0\). We
say that \(\zeta:I\rightarrow\mathbb{R}\) is an \(h\)-convex function, if \[
\zeta(\varrho\iota+(1-\varrho)\kappa)\leq
h(\varrho)\zeta(\iota)+h(1-\varrho)\zeta(\kappa),\tag{2}\] holds \(\forall \iota,\kappa\in I,
\varrho\in(0,1)\).
Definition 3 (tgs-convex function [19]). A function \(\zeta:I\subseteq\mathbb{R}\rightarrow\mathbb{R}\)
is tgs-convex, if \[
\zeta(\varrho \iota+(1-\varrho)\kappa)\leq
\varrho(1-\varrho)(\zeta(\iota)+\zeta(\kappa)),\tag{3}\] holds \(\forall\)\(\iota,\kappa\in I,\varrho\in(0,1).\)
If we take \(h(\varrho)=\varrho(1-\varrho)\) in (2), then (2) reduces to
(3). Namely, definition of tgs-convex
function may be regarded as special case of \(h\)-convex function, see [20].
Definition 4 (Beta function [16]). The classical Euler beta function, denoted by \(\beta(\iota, \kappa)\), is defined by \[\beta(\iota, \kappa) = \int_{0}^{1}
\varrho^{\iota-1} (1-\varrho)^{\kappa-1} \mathrm{d}\varrho,\tag{4}\] for
any \(\iota,\kappa>0\).
Sarikaya et al. [21] introduced the following integral midpoint type
inequalities involving Riemann-Liouville fractional integrals for the
class of classical convex functions:
Theorem 1. Suppose that \(\zeta:[\iota,\kappa]\rightarrow\mathbb{R}\)
is a positive function, where \(0\leq
\iota\leq \kappa\) with \(\zeta\in
L[\iota,\kappa]\). If \(\zeta\)
is a convex function on \([\iota,\kappa]\), then \[\zeta\bigg(\frac{\iota+\kappa}{2}\bigg)\leq\frac{\Gamma(\vartheta+1)}{2(\kappa-\iota)}[J_{\iota^+}^{\vartheta}\zeta(\kappa)+J_{\kappa^-}^{\vartheta}\zeta(\iota)]\leq\frac{\zeta(\iota)+\zeta(\kappa)}{2},\tag{5}\]
hold, where the left and right sided Riemann-Liouville fractional
integrals of the order \(\vartheta\in\mathbb{R_+}=[0,\infty)\) are
denoted by \(J_{\iota^+}^{\vartheta}\zeta\) and \(J_{\kappa^-}^{\vartheta}\zeta\): \[\begin{split}&(J_{\iota^+}^{\vartheta}\zeta)(h)=\frac{1}{\Gamma(\vartheta)}\int_{\iota}^{h}(h-\varrho)^{\vartheta-1}\zeta(\varrho)d\varrho;
\;\;\;0\leq \iota<h\leq \kappa,
\\&(J_{\kappa^-}^{\vartheta}\zeta)(h)=\frac{1}{\Gamma(\vartheta)}\int_{h}^{\kappa}(\varrho-h)^{\vartheta-1}\zeta(\varrho)d\varrho;
\;\;\;0\leq \iota<h\leq \kappa,\end{split}\] respectively and
\(\Gamma(\cdot)\) is classical Euler
Gamma function.
Theorem 2. Suppose that \(\zeta:[\iota,\kappa]\rightarrow\mathbb{R}\)
is a differentiable mapping on \((\iota,\kappa)\) with \(\iota<\kappa\). Let \(|\zeta’|\) is convex on \([\iota,\kappa]\), then \[\bigg|\frac{\zeta(\iota)+\zeta(\kappa)}{2}-\frac{\Gamma(\vartheta+1)}{2(\kappa-\iota)^{\vartheta}}
[J_{\iota^+}^{\vartheta}\zeta(\kappa)+J_{\kappa^-}^{\vartheta}\zeta(\iota)]\bigg|\leq\frac{\kappa-\iota}{2(\vartheta+1)}
\bigg(1-\frac{1}{2^{\vartheta}}\bigg)\big[\zeta'(\iota)+\zeta'(\kappa)\big],\tag{6}\]
holds.
Khan et al. [22]
and Dargomir et al. [23] presented the following lemmas listed
below:
Lemma 1. Consider \(\zeta:I\subseteq\mathbb{R}\rightarrow
\mathbb{R}\) is differentiable on \(I^o\), where \(\iota,\kappa \in I^o\) and \(\iota<\kappa\). If \(\zeta’ \in L[\iota,\kappa]\) and \(\vartheta>0\), then the equality \[\begin{aligned}
&\zeta(h)-\frac{\Gamma(\vartheta+1)}{2}\bigg[\frac{{J_{h^-}^\vartheta}\zeta(\iota)}{(h-\iota)^\vartheta}+\frac{{J_{h^+}^\vartheta}\zeta(\kappa)}{(\kappa-h)^\vartheta}\bigg]\\
&\quad=\frac{h-\iota}{2}\int_0^1\varrho^{\vartheta}\zeta'(\varrho
h+(1-\varrho)
\iota)d\varrho+\frac{h-\kappa}{2}\int_0^1\varrho^{\vartheta}\zeta'(\varrho
h+(1-\varrho)\kappa)d\varrho,
\end{aligned}\tag{7}\] holds for \(h\in(\iota,\kappa)\).
Lemma 2. Let \(\zeta:I\subseteq\rightarrow\mathbb{R}\) be
twice differentiable function on \(I^o.\) Assume that \(\iota,\kappa\in I^o\) , where \(\iota<\kappa\) and \(\zeta”\in L[\iota,\kappa],\) and
\(\vartheta>0\), then \[\begin{aligned}
&\frac{\zeta(\iota)+\zeta(\kappa)}{2}-\frac{\Gamma(\vartheta+1)}{2(\kappa-\iota)^{\vartheta}}
[J_{\iota^+}^{\vartheta}\zeta(\kappa)+J_{\kappa^-}^{\vartheta}\zeta(\iota)]\\&=
\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\int_{0}^{1}\varrho(1-\varrho^{\vartheta})[\zeta”(\varrho
\iota+(1-\varrho)\kappa)+\zeta”((1-\varrho)\iota+\varrho
\kappa)]d\varrho.
\end{aligned}\tag{8}\]
Furthermore, to simplify expressions, the following notations are
employed through out the next results: \[\Re=\zeta(h)-\frac{\Gamma(\vartheta+1)}{2}\bigg[\frac{{J_{h^-}^\vartheta}\zeta(\iota)}{(h-\iota)^\vartheta}
+\frac{{J_{h^+}^\vartheta}\zeta(\kappa)}{(\kappa-h)^\vartheta}\bigg],\]
and \[\mathfrak{J}=\frac{\zeta(\iota)+\zeta(\kappa)}{2}-\frac{\Gamma(\vartheta+1)}{2(\kappa-\iota)^{\vartheta}}
[J_{\iota^+}^{\vartheta}\zeta(\kappa)+J_{\kappa^-}^{\vartheta}\zeta(\iota)].\]
Theorem 3. Let
\(\zeta\) and \(\xi\) be real-valued functions on \([\iota, \kappa]\).
Hölder Inequality for Integrals [24]: If \(p > 1\) and \(\frac{1}{p} + \frac{1}{q} = 1\), and \(|\zeta|^p, |\xi|^q\) are integrable
functions on \([\iota, \kappa]\) with
\(q \geq 1\), then \[
\int_{\iota}^{\kappa} |\zeta(h)\xi(h)| \mathrm{d}h \leq
\left( \int_{\iota}^{\kappa} |\zeta(h)|^p \mathrm{d}h
\right)^{\frac{1}{p}} \left( \int_{\iota}^{\kappa} |\xi(h)|^q
\mathrm{d}h \right)^{\frac{1}{q}}.\tag{9}\]
Power-mean Integral Inequality [24]: If \(q \geq 1\) and \(|\zeta|, |\zeta||\xi|^q\) are integrable
functions on \([\iota, \kappa]\), then
\[
\int_{\iota}^{\kappa} |\zeta(h)\xi(h)| \mathrm{d}h \leq
\left( \int_{\iota}^{\kappa} |\zeta(h)| \mathrm{d}h
\right)^{1-\frac{1}{q}} \left( \int_{\iota}^{\kappa}
|\zeta(h)||\xi(h)|^q \mathrm{d}h
\right)^{\frac{1}{q}}.\tag{10}\]
3. Integral midpoint type inequalities involving fractional
integrals for differentiable functions
We need Lemma 1 to demonstrate the main
results related to integral midpoint type inequalities involving
fractional integrals.
Theorem 4. Suppose that \(\vartheta\geq1\) and \(\zeta:[\iota,\kappa]\rightarrow\mathbb{R}\)
is a positive valued integrable function, where \(0\leq \iota<h<\kappa\). If \(\zeta\) is a tgs-convex function on \([\iota,\kappa]\), then \[
\Gamma(\vartheta+1)\bigg[\frac{{J_{h^-}^\vartheta}\zeta(\iota)}{(h-\iota)^\vartheta}+\frac{{J_{h^+}^\vartheta}\zeta(\kappa)}{(\kappa-h)^\vartheta}\bigg]
\leq\frac{\vartheta}{(\vartheta+1)(\vartheta+2)}[(\zeta(h)+\zeta(\iota))+(\zeta(h)+\zeta(\kappa))],\tag{11}\]
holds.
Proof. Applying tgs-convexity of \(\zeta\)\[
\zeta(\varrho h+(1-\varrho)\iota)+\zeta(\varrho
h+(1-\varrho)\kappa)\leq\varrho(1-\varrho)(\zeta(h)+\zeta(\iota))+\varrho(1-\varrho)(\zeta(h)+\zeta(\kappa)).\tag{12}\]
Multiply both sides of (12) by \(\varrho^{\vartheta-1}\) and integrate w.r.t
\(\varrho\) over \([0,1]\) to get \[\begin{aligned}
&\int_{0}^{1}\varrho^{\vartheta-1}\zeta(\varrho
h+(1-\varrho)\iota)d\varrho+\int_{0}^{1}\varrho^{\vartheta-1}\zeta(\varrho
h+(1-\varrho)\kappa)d\varrho
\\&\leq(\zeta(h)+\zeta(\iota))\int_{0}^{1}\varrho^{\vartheta}(1-\varrho)d\varrho+(\zeta(h)+\zeta(\kappa))\int_{0}^{1}\varrho^{\vartheta}(1-\varrho)d\varrho
\\&=\frac{1}{(\vartheta+1)(\vartheta+2)}[(\zeta(h)+\zeta(\iota))+(\zeta(h)+\zeta(\kappa))].
\end{aligned}\tag{13}\]
Now, substituting \(\ell=\varrho
h+(1-\varrho)\iota\) and \(\jmath=\varrho h+(1-\varrho)\kappa\) in
first and second terms on L.H.S of (13)\[
\int_{\iota}^{h}\bigg(\frac{\ell-\iota}{h-\iota}\bigg)^{\vartheta-1}\zeta(\ell)\frac{d\ell}{h-\iota}+\int_{\kappa}^{h}\bigg(\frac{\kappa-\jmath}{\kappa-h}\bigg)^{\vartheta-1}\zeta(\jmath)\frac{d\jmath}{h-\kappa}
\leq\frac{1}{(\vartheta+1)(\vartheta+2)}[(\zeta(h)+\zeta(\iota))+(\zeta(h)+\zeta(\kappa))].\]
Multiply both sides by \(\vartheta\)\[\frac{\vartheta}{(h-\iota)^{\vartheta}}\int_{\iota}^{h}(\ell-\iota)^{\vartheta-1}\zeta(\ell)d\ell+\frac{\vartheta}{(\kappa-h)^{\vartheta}}\int_{h}^{\kappa}(\kappa-\jmath)^{\vartheta-1}\zeta(\jmath)d\jmath\leq\frac{\vartheta}{(\vartheta+1)(\vartheta+2)}[(\zeta(h)+\zeta(\iota))+(\zeta(h)+\zeta(\kappa))].\] ◻
Theorem 5. Consider \(\zeta’\in
L[\iota,\kappa]\) such that \(|\zeta’|\) is tgs-convex on \([\iota,\kappa]\), then \[|\Re|\leq\beta(\vartheta+2,2)\bigg[\frac{h-\iota}{2}(|\zeta'(h)|+|\zeta'(\iota)|)+\frac{h-\kappa}{2}(|\zeta'(h)|+|\zeta'(\kappa)|)\bigg].\tag{14}\]
Proof. By using the Lemma 1\[|\Re|\leq\frac{h-\iota}{2}\int_{0}^{1}\varrho^{\vartheta}|\zeta'(\varrho
h+(1-\varrho)\iota)|d\varrho+\frac{h-\kappa}{2}\int_{0}^{1}\varrho^{\vartheta}|\zeta'(\varrho
h+(1-\varrho)\kappa)|d\varrho.\]
Since \(|\zeta’|\) is tgs-convex
\[\begin{aligned}
|\Re|\leq&\frac{h-\iota}{2}\bigg[(|\zeta'(h)|+|\zeta'(\iota)|)\int_{0}^{1}\varrho^{\vartheta+1}(1-\varrho)d\varrho\bigg]+\frac{h-\kappa}{2}\bigg[(|\zeta'(h)|+|\zeta'(\kappa)|)\int_{0}^{1}\varrho^{\vartheta+1}(1-\varrho)d\varrho\bigg],\\
|\Re|\leq&\beta(\vartheta+2,2)\bigg[\frac{h-\iota}{2}(|\zeta'(h)|+|\zeta'(\iota)|)+\frac{h-\kappa}{2}(|\zeta'(h)|+|\zeta'(\kappa)|)\bigg].
\end{aligned}\] ◻
Theorem 6. Assume \(\zeta’\in
L[\iota,\kappa]\) such that \(|\zeta’|^{q}\)\((q>1)\) is tgs-convex on \([\iota,\kappa]\), then \[\begin{aligned}
|\Re|\leq&\bigg(\frac{1}{(\vartheta+1)}\bigg)^{1-\frac{1}{q}}\bigg[\frac{h-\iota}{2}[|\zeta'(h)|^{q}\beta(\vartheta+2,2)
+|\zeta'(\iota)|^{q}\beta(\vartheta+2,2)]^{\frac{1}{q}}
\\&+\frac{h-\kappa}{2}[|\zeta'(h)|^{q}\beta(\vartheta+2,2)+|\zeta'(\kappa)|^{q}\beta(\vartheta+2,2)]^{\frac{1}{q}}\bigg].
\end{aligned}\tag{15}\]
Proof. By using Lemma 1\[
|\Re|\leq\frac{h-\iota}{2}\int_0^1\varrho^{\vartheta}|\zeta'(\varrho
h+(1-\varrho)
\iota)|d\varrho+\frac{h-\kappa}{2}\int_0^1\varrho^{\vartheta}|\zeta'(\varrho
h+(1-\varrho)\kappa)|d\varrho.\]
By utilizing the power-mean integral inequality (10), we
obtain \[\begin{aligned}
|\Re|\leq&\frac{h-\iota}{2}\bigg(\int_{0}^{1}\varrho^{\vartheta}d\varrho\bigg)^{1-\frac{1}{q}}\bigg[\int_{0}^{1}\varrho^{\vartheta}|\zeta'(\varrho
h+(1-\varrho)\iota)|^{q}d\varrho\bigg]^{\frac{1}{q}}
\\&+\frac{h-\kappa}{2}\bigg(\int_{0}^{1}\varrho^{\vartheta}d\varrho\bigg)^{1-\frac{1}{q}}\bigg[\int_{0}^{1}\varrho^{\vartheta}|\zeta'(\varrho
h+(1-\varrho)\kappa)|^{q}d\varrho\bigg]^{\frac{1}{q}}.
\end{aligned}\]
Since \(|\zeta’|^q\) is
tgs-convex \[\begin{aligned}
|\Re|\leq&\frac{h-\iota}{2}\bigg(\frac{1}{\vartheta+1}\bigg)^{1-{\frac{1}{q}}}\bigg[(|\zeta'(h)|^{q}+|\zeta'(\iota)|^{q})\int_{0}^{1}\varrho^{\vartheta+1}(1-\varrho)
d\varrho\bigg]^{\frac{1}{q}}\\&+\frac{h-\kappa}{2}\bigg(\frac{1}{\vartheta+1}\bigg)^{1-\frac{1}{q}}\bigg[(|\zeta'(h)|^{q}+|\zeta'(\kappa)|^{q})\int_{0}^{1}\varrho^{\vartheta+1}(1-\varrho)
d\varrho\bigg]^{\frac{1}{q}}.
\end{aligned}\]
Theorem 7. Suppose \(\zeta’\in
L[\iota,\kappa]\) such that \(|\zeta’|^{q}\) is tgs-convex on \([\iota,\kappa]\), then \[|\Re|\leq\bigg(\frac{1}{\vartheta
p+1}\bigg)^{\frac{1}{p}}\bigg(\frac{1}{6}\bigg)\bigg[\frac{h-\iota}{2}[|\zeta'(h)|^{q}+|\zeta'(\iota)|^{q}]^{\frac{1}{q}}+
\frac{h-\kappa}{2}[|\zeta'(h)|^{q}+|\zeta'(\kappa)|^{q}]^{\frac{1}{q}}\bigg].\tag{16}\]
Proof. By using the Lemma 1\[
\begin{split}&|\Re|\leq\frac{h-\iota}{2}\int_0^1\varrho^{\vartheta}|\zeta'(\varrho
h+(1-\varrho)
\iota)|d\varrho+\frac{h-\kappa}{2}\int_0^1\varrho^{\vartheta}|\zeta'(\varrho
h+(1-\varrho)\kappa)|d\varrho.
\end{split}\]
According to Hölder inequality (9)\[
|\Re|\leq\frac{h-\iota}{2}\bigg(\int_{0}^{1}\varrho^{\vartheta
p}d\varrho\bigg)^{\frac{1}{p}}\bigg[\int_{0}^{1}|\zeta'(\varrho
h+(1-\varrho)\iota)|^{q}d\varrho\bigg]^{\frac{1}{q}}+\frac{h-\kappa}{2}\bigg(\int_{0}^{1}\varrho^{\vartheta
p}d\varrho\bigg)^{\frac{1}{p}}\bigg[\int_{0}^{1}
|\zeta'(\varrho
h+(1-\varrho)\kappa)|^{q}d\varrho\bigg]^{\frac{1}{q}}.\]
Since \(|\zeta’|^{q}\) is
tgs-convex \[\begin{aligned}
|\Re|\leq&\frac{h-\iota}{2}\bigg(\frac{1}{\vartheta
p+1}\bigg)^{\frac{1}{p}}\bigg[(|\zeta'(h)|^{q}+|\zeta'(\iota)|^{q})\int_{0}^{1}\varrho(1-\varrho)d\varrho\bigg]
^\frac{1}{q}\\&+\frac{h-\kappa}{2}\bigg(\frac{1}{\vartheta
p+1}\bigg)^{\frac{1}{p}}\bigg[(|\zeta'(h)|^{q}+|\zeta'(\kappa)|^{q})\int_{0}^{1}\varrho(1-\varrho)d\varrho\bigg]
^\frac{1}{q}.
\end{aligned}\]
4. Integral inequalities involving fractional integrals for
twice differentiable functions
Lemma 2 is used in this section to prove the
next main results.
Theorem 8. Assume that \(\zeta:I\subset\mathbb{R}\rightarrow\mathbb{R}\)
is twice differentiable function on \(I^o\) such that \(|\zeta”|\) is tgs-convex function
on I. Let \(\iota,\kappa\in I^o\) for
\(\iota<\kappa\) and \(|\zeta”|\in L[\iota,\kappa]\),
then \[
\begin{split}
|\mathfrak{J}|
\leq\frac{(\kappa-\iota)^{2}}{(\vartheta+1)}
&[\beta(3,\vartheta+2)(|\zeta”(\iota)|+|\zeta”(\kappa)|)].\end{split}\tag{17}\]
Proof. By using the Lemma 2\[\begin{aligned}
|\mathfrak{J}|&=
\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\int_{0}^{1}\varrho(1-\varrho^{\vartheta})[|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|+|\zeta”((1-\varrho)\iota+\varrho
\kappa)|]d\varrho\\
&\leq\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg[\int_{0}^{1}\varrho(1-\varrho^{\vartheta})[|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|]d\varrho+
\int_{0}^{1}\varrho(1-\varrho^{\vartheta})[|\zeta”((1-\varrho)\iota+\varrho
\kappa)|]d\varrho\bigg].
\end{aligned}\]
By applying the tgs-convexity of \(|\zeta”|\)\[\begin{aligned}
|\mathfrak{J}|\leq&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg[\int_{0}^{1}\varrho(1-\varrho^{\vartheta})\varrho(1-\varrho)(|\zeta”(\iota)|+|\zeta”(\kappa)|)d\varrho
+\int_{0}^{1}\varrho(1-\varrho^{\vartheta})\varrho(1-\varrho)(|\zeta”(\iota)|+|\zeta”(\kappa)|)d\varrho\bigg]
\\=&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg[\int_{0}^{1}\varrho^{2}(1-\varrho)(1-\varrho^{\vartheta})(|\zeta”(\iota)|
+|\zeta”(\kappa)|)d\varrho
\int_{0}^{1}\varrho^{2}(1-\varrho)(1-\varrho^{\vartheta})(|\zeta”(\iota)|+|\zeta”(\kappa)|)d\varrho\bigg].
\end{aligned}\]
Since\(\varrho^{\vartheta}\geq\varrho,\alpha\in(0,1]\)
and \(\varrho\in[0,1],\) we have \(-\varrho^{\vartheta}\leq\varrho\Rightarrow
1-\varrho^{\vartheta}\leq1-\varrho\leq(1-\varrho)^{\vartheta}.\)\[
|\mathfrak{J}|\leq\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg[\int_{0}^{1}\varrho^{2}(1-\varrho)^{\vartheta+1}(|\zeta”(\iota)|
+|\zeta”(\kappa)|)d\varrho+
\int_{0}^{1}\varrho^{2}(1-\varrho)^{\vartheta+1}(|\zeta”(\iota)|+|\zeta”(\kappa)|)d\varrho\bigg].\]
Theorem 9. Assume that \(\zeta:I\subset\mathbb{R}\rightarrow\mathbb{R}\)
is twice differentiable function on \(I^o.\) Let \(q>1\) such that \(|\zeta”|^q\) is tgs-convex
function on I. Let \(\iota,\kappa\in
I^o\), where \(\iota<\kappa\)
and \(\zeta”\in
L[\iota,\kappa]\), then \[
\begin{split}&|\mathfrak{J}|\leq\frac{(\kappa-\iota)^{2}}{(\vartheta+1)}[\beta(p+1,\vartheta
p+1)]^{\frac{1}{p}}\bigg[\frac{|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q}}{6}\bigg]^{\frac{1}{q}}\end{split},\tag{18}\]
holds, where \(\frac{1}{p}+\frac{1}{q}=1\).
Proof. By using the Lemma 2\[
\begin{split}&|\mathfrak{J}|\leq
\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\int_{0}^{1}\varrho(1-\varrho^{\vartheta})[|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|+|\zeta”((1-\varrho)\iota+\varrho
\kappa)|]d\varrho.\end{split}\]
According to Hölder inequality (9)\[\begin{aligned}
|\mathfrak{J}|\leq&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg[\bigg(\int_{0}^{1}\varrho^{p}(1-\varrho^{\vartheta})^{p}d\varrho\bigg)^{\frac{1}{p}}
\bigg(\int_{0}^{1}|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|^{q}d\varrho\bigg)^{\frac{1}{q}}\\&+\bigg(\int_{0}^{1}\varrho^{p}(1-\varrho^{\vartheta})^{p}d\varrho\bigg)^{\frac{1}{p}}
\bigg(\int_{0}^{1}|\zeta”((1-\varrho)\iota+\varrho
\kappa)|^{q}d\varrho\bigg)^{\frac{1}{q}}\bigg].
\end{aligned}\]
Since \(\varrho^{\vartheta}\geq\varrho,\vartheta\in(0,1]\)
and \(\varrho\in[0,1],\) we have \(-\varrho^{\vartheta}\leq\varrho\Rightarrow
1-\varrho^{\vartheta}\leq1-\varrho\leq(1-\varrho)^{\vartheta}.\)\[\begin{aligned}
|\mathfrak{J}|\leq&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg(\int_{0}^{1}\varrho^{p}(1-\varrho)^{\vartheta
p}d\varrho\bigg)^{\frac{1}{p}}
\bigg[\bigg(\int_{0}^{1}|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|^{q}d\varrho\bigg)^{\frac{1}{q}}+
\bigg(\int_{0}^{1}|\zeta”((1-\varrho)\iota+\varrho
\kappa)|^{q}d\varrho\bigg)^{\frac{1}{q}}\bigg].
\end{aligned}\]
Since \(|\zeta”|^q\) is
tgs-convex \[
\begin{split}|\mathfrak{J}|\leq&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg(\int_{0}^{1}\varrho^{p}(1-\varrho)^{\vartheta
p}d\varrho\bigg)^{\frac{1}{p}}
\bigg[\bigg(\int_{0}^{1}(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})\varrho(1-\varrho)d\varrho\bigg)^{\frac{1}{q}}\\&+
\bigg(\int_{0}^{1}(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})\varrho(1-\varrho)d\varrho\bigg)^{\frac{1}{q}}\bigg].\end{split}\]
Theorem 10. Let \(\zeta:I\subset\mathbb{R}\rightarrow\mathbb{R}\)
be differentiable on \(I^o.\) Suppose
that \(q\geq1\) such that \(|\zeta”|^{q}\) is tgs-convex
function on I. Assume that \(\iota,\kappa\in
I^o\), where \(\iota<\kappa\)
and \(\zeta”\in
L[\iota,\kappa]\), then \[
\begin{split}&|\mathfrak{J}|\leq\frac{\vartheta(\kappa-\iota)^{2}}{2(\vartheta+1)(\vartheta+2)}
\bigg(\frac{2(\vartheta+2)}{\vartheta}\bigg)^{\frac{1}{q}}\bigg(\beta(3,\vartheta+2)(|\zeta”(\iota)|^{q}+
|\zeta”(\kappa)|^{q})\bigg)^{\frac{1}{q}}.\end{split}\tag{19}\]
Proof. By applying Lemma 2\[
\begin{split}&|\mathfrak{J}|\leq\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\int_{0}^{1}\varrho(1-\varrho^{\vartheta})[|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|+|\zeta”((1-\varrho)\iota+\varrho
\kappa)|]d\varrho.\end{split}\]
Applying the power-mean integral inequality (10)\[\begin{aligned}
|\mathfrak{J}|\leq&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg[\bigg(\int_{0}^{1}\varrho(1-\varrho^{\vartheta})d\varrho\bigg)^{1-\frac{1}{q}}
\bigg(\int_{0}^{1}\varrho(1-\varrho^{\vartheta})|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|^{q}d\varrho\bigg)^{\frac{1}{q}}
\\&+\bigg(\int_{0}^{1}\varrho(1-\varrho^{\vartheta})d\varrho\bigg)^{1-\frac{1}{q}}\bigg(\int_{0}^{1}\varrho(1-\varrho^{\vartheta})
|\zeta”((1-\varrho \iota)+\varrho
\kappa))|^{q}d\varrho\bigg)^{\frac{1}{q}}\bigg]\\=&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg(\int_{0}^{1}\varrho(1-\varrho^{\vartheta})d\varrho\bigg)^{1-\frac{1}{q}}\bigg[\bigg(\int_{0}^{1}
\varrho(1-\varrho^{\vartheta})|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|^{q}d\varrho\bigg)^{\frac{1}{q}}\\&+
\bigg(\int_{0}^{1}\varrho(1-\varrho^{\vartheta})|\zeta”((1-\varrho)\iota+\varrho
\kappa)|^{q}d\varrho\bigg)^{\frac{1}{q}}\bigg].
\end{aligned}\]
Simplifying \[\int_{0}^{1}\varrho(1-\varrho^{\vartheta})d\varrho=\int_{0}^{1}(\varrho-\varrho^{\vartheta+1})d\varrho=\frac{\vartheta}{2(\vartheta+2)},\]
Since \(|\zeta”|^{q}\) is
tgs-convex \[\begin{split}|\mathfrak{J}|\leq&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg(\frac{\vartheta}{2(\vartheta+2)}\bigg)^{1-\frac{1}{q}}\bigg[\bigg(\int_{0}^{1}\varrho(1-\varrho^{\vartheta})
((|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})\varrho(1-\varrho))\bigg)^{\frac{1}{q}}d\varrho\\&+\bigg(\int_{0}^{1}\varrho(1-\varrho^{\vartheta})
((|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})\varrho(1-\varrho))\bigg)^{\frac{1}{q}}d\varrho\bigg)\bigg].\end{split}\]
Since \(\varrho^{\vartheta}\geq\varrho,\vartheta\in(0,1]\)
and \(\varrho\in[0,1],\text{we
have}\;-\varrho^{\vartheta}\leq\varrho\Rightarrow
1-\varrho^{\vartheta}\leq1-\varrho\leq(1-\varrho)^{\vartheta}.\)\[\begin{aligned}
|\mathfrak{J}|\leq&\frac{\vartheta(\kappa-\iota)^{2}}{4(\vartheta+1)(\vartheta+2)}\bigg(\frac{2(\vartheta+2)}{\vartheta}\bigg)^{\frac{1}{q}}
\bigg[\bigg(\int_{0}^{1}
\varrho^{2}(1-\varrho)^{\vartheta+1}(|\zeta”(\iota)|^q+|\zeta”(\kappa)|^{q})d\varrho\bigg)^{\frac{1}{q}}\\&+\bigg(\int_{0}^{1}
\varrho^{2}(1-\varrho)^{\vartheta+1}(|\zeta”(\iota)|^q+|\zeta”(\kappa)|^{q})d\varrho\bigg)^{\frac{1}{q}}\bigg]
\\=&\frac{\vartheta(\kappa-\iota)^{2}}{2(\vartheta+1)(\vartheta+2)}\bigg(\frac{2(\vartheta+2)}{\vartheta}\bigg)^{\frac{1}{q}}\bigg(\beta(3,\vartheta+2)(|\zeta”(\iota)|^{q}+
|\zeta”(\kappa)|^{q})\bigg)^{\frac{1}{q}}.
\end{aligned}\] ◻
Theorem 11. Let \(\zeta:I\subset\mathbb{R}\rightarrow\mathbb{R}\)
be differentiable on \(I^o\) such that
\(\zeta”\in L[\iota,\kappa]\)
for \(\iota,\kappa\in I^o\) and \(\iota<\kappa.\) Let \(|\zeta”|^{q}\) is tgs-convex on
\([\iota,\kappa]\) and \(q>1\), then \[
|\mathfrak{J}|\leq\frac{(\kappa-\iota)^{2}}{(\vartheta+1)}\bigg(\frac{1}{1+p}\bigg)^{\frac{1}{p}}\bigg[\bigg(\beta(2,\vartheta
q+2)(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})\bigg)^{\frac{1}{q}}\bigg],\tag{20}\]
holds, where \(\frac{1}{p}+\frac{1}{q}=1.\)
Proof. By using the Lemma 2\[
|\mathfrak{J}|\leq
\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg[\int_{0}^{1}\varrho(1-\varrho^{\vartheta})|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|d\varrho +
\int_{0}^{1}\varrho(1-\varrho^{\vartheta})|\zeta”((1-\varrho)\iota+\varrho
\kappa)|d\varrho\bigg].\]
Applying the Hölder inequality (9)\[\begin{split}|\mathfrak{J}|\leq&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg[\bigg(\int_{0}^{1}\varrho^{p}d\varrho\bigg)^{\frac{1}{p}}
\bigg(\int_{0}^{1}(1-\varrho^{\vartheta})^{q}
|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|^{q}d\varrho\bigg)^{\frac{1}{q}}\\&+\bigg(\int_{0}^{1}\varrho^{p}d\varrho\bigg)^{\frac{1}{p}}
\bigg(\int_{0}^{1}(1-\varrho^{\vartheta})^{q}|\zeta”((1-\varrho)\iota+\varrho
\kappa)|^{q}d\varrho\bigg)^{\frac{1}{q}}\bigg],\end{split}\]
Since \(|\zeta”|^{q}\) is
tgs-convex \[\begin{split}|\mathfrak{J}|\leq&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg[\bigg(\int_{0}^{1}\varrho^{p}d\varrho\bigg)^{\frac{1}{p}}\bigg(\int_{0}^{1}(1-\varrho^{\vartheta})^{q}\varrho(1-\varrho)
(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})\bigg)^{\frac{1}{q}}\\&+\bigg(\int_{0}^{1}\varrho^{p}d\varrho\bigg)^{\frac{1}{p}}
\bigg(\int_{0}^{1}(1-\varrho^{\vartheta})^{q}\varrho(1-\varrho)(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})d\varrho\bigg)^{\frac{1}{q}}\bigg].\end{split}\]
Since \(\varrho^{\vartheta}\geq\varrho,\vartheta\in(0,1]\)
and \(\varrho\in[0,1],\text{we
have}\;-\varrho^{\vartheta}\leq\varrho\Rightarrow
1-\varrho^{\vartheta}\leq1-\varrho\leq(1-\varrho)^{\vartheta}.\)\[\begin{split}|\mathfrak{J}|\leq&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg(\frac{1}{1+p}\bigg)^{\frac{1}{p}}\bigg[\bigg(\int_{0}^{1}\varrho(1-\varrho)^{\vartheta
q+1}
(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})d\varrho\bigg)^{\frac{1}{q}}\\&+\bigg(\int_{0}^{1}\varrho(1-\varrho)^{\vartheta
q+1}
(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})d\varrho\bigg)^{\frac{1}{q}}\bigg]\end{split}\]\[=\frac{(\kappa-\iota)^{2}}{(\vartheta+1)}\bigg(\frac{1}{1+p}\bigg)^{\frac{1}{p}}\bigg[\bigg(\beta(2,\vartheta
q+2)(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})\bigg)^{\frac{1}{q}}\bigg].\] ◻
Theorem 12. Suppose that \(\zeta:I\subseteq[0,\infty)\rightarrow\mathbb{R}\)
is differentiable on \(I^o\) such that
\(\zeta”\in L[\iota,\kappa]\)
for \(\iota,\kappa\in I^o\) with \(\iota<\kappa\). Let \(|\zeta”|^{q}\) is tgs-convex on
\([\iota,\kappa]\) also \(q\geq1\), then \[
|\mathfrak{J}|\leq\frac{(\kappa-\iota)^{2}}{(\vartheta+1)}\bigg(\frac{1}{2}\bigg)^{1-\frac{1}{q}}\bigg[\bigg(\beta(3,\vartheta
q+2)(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})\bigg)^{\frac{1}{q}}\bigg],\tag{21}\]
holds, where \(\frac{1}{p}+\frac{1}{q}=1.\)
Proof. By using the Lemma 2\[\begin{aligned}
|\mathfrak{J}|\leq&
\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg[\int_{0}^{1}\varrho(1-\varrho^{\vartheta})|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|d\varrho+
\int_{0}^{1}\varrho(1-\varrho^{\vartheta})|\zeta”((1-\varrho)\iota+\varrho
\kappa)|d\varrho\bigg]\\
&=\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg[\int_{0}^{1}\varrho^{1-\frac{1}{q}}\varrho^{\frac{1}{q}}
(1-\varrho^{\vartheta})[|\zeta”(\varrho
\iota+(1-\varrho)\kappa)|]d\varrho+
\int_{0}^{1}\varrho^{1-\frac{1}{q}}\varrho^{\frac{1}{q}}
(1-\varrho^{\vartheta})[|\zeta”((1-\varrho)\iota+\varrho
\kappa)|]d\varrho\bigg].
\end{aligned}\]
Since \(|\zeta”|^{q}\) is
the tgs-convex \[\begin{split}|\mathfrak{J}|\leq&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg(\int_{0}^{1}\varrho
d\varrho\bigg)^{1-\frac{1}{q}}\bigg[\bigg(\int_{0}^{1}
\varrho(1-\varrho^{\vartheta})^{q}(\varrho(1-\varrho)(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^q))d\varrho\bigg)^{\frac{1}{q}}\\&+
\bigg(\int_{0}^{1}
\varrho(1-\varrho^{\vartheta})^{q}(\varrho(1-\varrho)(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^q))d\varrho\bigg)^{\frac{1}{q}}\bigg].\end{split}\]
Since \(\varrho^{\vartheta}\geq\varrho,\vartheta\in(0,1]\)
and \(\varrho\in[0,1],\text{we
have}\;-\varrho^{\vartheta}\leq\varrho\Rightarrow
1-\varrho^{\vartheta}\leq1-\varrho\leq(1-\varrho)^{\vartheta}.\)\[\begin{split}|\mathfrak{J}|\leq&\frac{(\kappa-\iota)^{2}}{2(\vartheta+1)}\bigg(\frac{1}{2}\bigg)^{1-\frac{1}{q}}\bigg[\bigg(\int_{0}^{1}\varrho^{2}(1-\varrho)^{\vartheta
q+1}
(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})d\varrho\bigg)^{\frac{1}{q}}\\&+\bigg(\int_{0}^{1}\varrho^{2}(1-\varrho)^{\vartheta
q+1}
(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})d\varrho\bigg)^{\frac{1}{q}}\bigg]
\\=&\frac{(\kappa-\iota)^{2}}{(\vartheta+1)}\bigg(\frac{1}{2}\bigg)^{1-\frac{1}{q}}\bigg[\bigg(\beta(3,\vartheta
q+2)(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})\bigg)^{\frac{1}{q}}\bigg].\end{split}\] ◻
5. Numerical and graphical representation of inequalities
Some examples are constructed in this section for inequalities
appeared in §4.
Example 1. Let \(\vartheta=1\), in (17)
of Theorem 8, we get the following relation:
\[
\bigg|\frac{\zeta(\iota)+\zeta(\kappa)}{2}-\frac{1}{(\kappa-\iota)}
\int_{\iota}^{\kappa}\zeta(\varrho)d\varrho\bigg|\leq\frac{(\kappa-\iota)^{2}}{2}
\bigg[\frac{1}{30}(|\zeta”(\iota)|+|\zeta”(\kappa)|)\bigg].\tag{22}\]
Substitute \(\zeta(\varrho)=\frac{1}{\varrho}\), in (22)
to get \[\bigg|\frac{\frac{1}{\iota}+\frac{1}{\kappa}}{2}-\frac{1}{\kappa-\iota}\int_{\iota}^{\kappa}\frac{1}{\varrho}d\varrho\bigg|\leq
\frac{(\kappa-\iota)^{2}}{2}\bigg(\frac{\big|\frac{2}{\iota^3}\big|^{2}+\big|\frac{2}{\kappa^{3}}\big|^{2}}{30}\bigg).\]
Particularly, for \(\iota=1\) and
\(\kappa=3\) in (22) we get
\[\begin{aligned}
\bigg|\frac{2}{3}-\frac{1.098612}{2}\bigg|&\leq2\bigg(\frac{4.005487}{30}\bigg),\\
0.117364&\leq0.267032.
\end{aligned}\]
Similarly, let \(\vartheta=1\) in
[23, Theorem 1] then, we obtain: \[
\bigg|\frac{\zeta(\iota)+\zeta(\kappa)}{2}-\frac{1}{\kappa-\iota}
\int_{\iota}^{\kappa}\zeta(\varrho)d\varrho\bigg|
\leq\frac{(\kappa-\iota)^{2}}{2}\bigg[
\frac{1}{24}({|\zeta”(\iota)|+|\zeta”(\kappa)|})
\bigg].\tag{23}\]
Substitute \(\zeta(\varrho)=\frac{1}{\varrho}\), in (23) to get \[\bigg|\frac{\frac{1}{\iota}+\frac{1}{\kappa}}{2}-\frac{1}{\kappa-\iota}\int_{\iota}^{\kappa}\frac{1}{\varrho}d\varrho\bigg|\leq
\frac{(\kappa-\iota)^{2}}{2}\bigg(\frac{\big|\frac{2}{\iota^3}\big|^{2}+\big|\frac{2}{\kappa^{3}}\big|^{2}}{24}\bigg).\]
Particularly, for \(\iota=1\) and
\(\kappa=3\) in (23) we get
\[\bigg|\frac{2}{3}-\frac{1.098612}{2}\bigg|\leq2\bigg(\frac{4.005487}{24}\bigg),\]\[0.117364\leq 0.333790.\]
Figure 1. Graphical comparison of inequalities (22) and (23) as a function of \(\kappa\) where \(\iota=1\), \(\vartheta=1\), and \(\zeta(\varrho)=\frac{1}{\varrho}\).
Remark 1. The difference of bounds, appear in Theorem 8, is
0.267032 and difference of bounds, appeared in [23, Theorem 1], is 0.333790. Hence, our result is more
efficient.
Example 2. Let \(\vartheta=1\), in (18) of
Theorem 9, then we get the following relation:
\[
\bigg|\frac{\zeta(\iota)+\zeta(\kappa)}{2}-\frac{1}{\kappa-\iota}
\int_{\iota}^{\kappa}\zeta(\varrho)d\varrho\bigg|
\leq\frac{(\kappa-\iota)^{2}}{2}\bigg[\bigg(
\frac{|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q}}{6}\bigg)^{\frac{1}{q}}
\beta^{\frac{1}{p}}(p+1,p+1)\bigg].\tag{24}\]
Substitute \(\zeta(\varrho)=\frac{1}{\varrho}\), in (24) to get \[\bigg|\frac{\frac{1}{\iota}+\frac{1}{\kappa}}{2}-\frac{1}{\kappa-\iota}\int_{\iota}^{\kappa}\frac{1}{\varrho}d\varrho\bigg|\leq
\frac{(\kappa-\iota)^{2}}{2}\bigg[\bigg(\frac{\big|\frac{2}{\iota^3}\big|^{2}+\big|\frac{2}{\kappa^{3}}\big|^{2}}{6}\bigg)^\frac{1}{2}
\beta^\frac{1}{2}(3,3)\bigg].\]
Particularly, for \(\iota=1\) and
\(\kappa=3\) in (24) we get
\[\bigg|\frac{2}{3}-\frac{1.098612}{2}\bigg|\leq2\bigg(\frac{4.005486}{6}\bigg)^\frac{1}{2}\bigg(\frac{1}{30}\bigg)^\frac{1}{2},\]\[0.117364\leq 0.298346.\]
Similarly, if we choose \(\vartheta=1\) in Corollary 3.6 of [14] then, we get: \[
\bigg|\frac{\zeta(\iota)+\zeta(\kappa)}{2}-\frac{1}{\kappa-\iota}
\int_{\iota}^{\kappa}\zeta(\varrho)d\varrho\bigg|
\leq\frac{(\kappa-\iota)^{2}}{2}\bigg[\bigg(
\frac{|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q}}{s+1}\bigg)^{\frac{1}{q}}
\beta^{\frac{1}{p}}(p+1,p+1)\bigg].\tag{25}\]
Figure 2. Graphical comparison of inequalities (24) and (25) as a function of \(\kappa\) where \(\iota=1\), \(\vartheta=1\), and \(\zeta(\varrho)=\frac{1}{\varrho}\)
Substitute \(\zeta(\varrho)=\frac{1}{\varrho}\), \(p=q=2\) in (25) to get
\[\bigg|\frac{\frac{1}{\iota}+\frac{1}{\kappa}}{2}-\frac{1}{\kappa-\iota}\int_{\iota}^{\kappa}\frac{1}{\varrho}d\varrho\bigg|\leq
\frac{(\kappa-\iota)^{2}}{2}\bigg[\bigg(\frac{\big|\frac{2}{\iota^3}\big|^{2}+\big|\frac{2}{\kappa^{3}}\big|^{2}}{s+1}\bigg)^\frac{1}{2}
\beta^\frac{1}{2}(3,3)\bigg].\]
Particularly, for \(\iota=1\), \(\kappa=3\) and s=0.3 in (25) we get
\[\begin{aligned}
\bigg|\frac{2}{3}-\frac{1.098612}{2}\bigg|&\leq2\bigg(\frac{4.005486}{1.3}\bigg)^\frac{1}{2}\bigg(\frac{1}{30}\bigg)^\frac{1}{2},\\
0.117364&\leq 0.640950.
\end{aligned}\]
Remark 2. The difference of bounds, appear in Theorem 9, is
0.298346 and difference of bounds, appeared in Corollary 3.6 in [14], is 0.640950. Hence, present
result is more efficient.
Example 3. Let \(\vartheta=1\), in (19)
of Theorem 10, then we get the following relation:
\[
\bigg|\frac{\zeta(\iota)+\zeta(\kappa)}{2}-\frac{1}{\kappa-\iota}\int_{\iota}^{\kappa}\zeta(\varrho)d\varrho\bigg|\leq
\frac{(\kappa-\iota)^{2}}{2}\bigg(\frac{1}{6}\bigg)^{1-\frac{1}{q}}\bigg[\frac{1}{30}(|\zeta”(\iota)|^{q}
+|\zeta”(\kappa)|^{q})\bigg]^{\frac{1}{q}}.\tag{26}\]
Substitute \(\zeta(\varrho)=\frac{1}{\varrho}\), and
\(q=2\) in (26) to get
\[\bigg|\frac{\frac{1}{\iota}+\frac{1}{\kappa}}{2}-\frac{1}{\kappa-\iota}\int_{\iota}^{\kappa}\frac{1}{\varrho}d\varrho\bigg|
\leq\frac{(\kappa-\iota)^{2}}{2}\bigg(\frac{1}{6}\bigg)^{1-\frac{1}{2}}\bigg(\frac{\big|\frac{2}{\iota^{3}}\big|^{2}
+\big|\frac{2}{\kappa^{3}}\big|^{2}}{30}\bigg)^\frac{1}{2}.\]
Particularly, for \(\iota=1\) and
\(\kappa=3\) in (26) we get
\[\bigg|\frac{2}{3}-\frac{1.098612}{2}\bigg|
\leq2\bigg(\frac{1}{6}\bigg)^\frac{1}{2}\bigg(\frac{4.005486}{30}\bigg)^\frac{1}{2},\]\[0.117364\leq 0.298346.\]
Figure 3. Graphical comparison of inequalities (26) and (27) as a function of \(\kappa\) where \(\iota=1\), \(\vartheta=1\), and \(\zeta(\varrho)=\frac{1}{\varrho}\).
Similarly, let \(\alpha=m=1\) in
[14, Theorem 3.4] then, we get: \[\begin{aligned}
\bigg|\frac{\zeta(\iota)+\zeta(\kappa)}{2}-\frac{1}{\kappa-\iota}\int_{\iota}^{\kappa}\zeta(\varrho)d\varrho\bigg|\leq
\frac{(\kappa-\iota)^{2}}{2}\bigg(\frac{1}{6}\bigg)^{1-\frac{1}{q}}\bigg[\frac{1}{2}(|\zeta”(\iota)|^{q}
+|\zeta”(\kappa)|^{q})\bigg]^{\frac{1}{q}}.
\end{aligned}\tag{27}\]
Substitute \(\zeta(\varrho)=\frac{1}{\varrho}\) and
\(q=2\) in (27) to get
\[\bigg|\frac{\frac{1}{\iota}+\frac{1}{\kappa}}{2}-\frac{1}{\kappa-\iota}\int_{\iota}^{\kappa}\frac{1}{\varrho}d\varrho\bigg|
\leq\frac{(\kappa-\iota)^{2}}{2}\bigg(\frac{1}{6}\bigg)^{1-\frac{1}{2}}\bigg(\frac{\big|\frac{2}{\iota^{3}}\big|^{2}
+\big|\frac{2}{\kappa^{3}}\big|^{2}}{2}\bigg)^\frac{1}{2}.\]
Particularly, for \(\iota=1\) and
\(\kappa=3\) in (27) we get
\[\bigg|\frac{2}{3}-\frac{1.098612}{2}\bigg|
\leq2\bigg(\frac{1}{6}\bigg)^\frac{1}{2}\bigg(\frac{4.005486}{2}\bigg)^\frac{1}{2},\]\[0.117364\leq 1.155491.\]
Remark 3. The difference of bounds, appear in Theorem 10, is
0.298346 and difference of bounds, appeared in Theorem 3.4 in [14], is 1.155491. Hence, present
result is more efficient.
Corollary 1. For \(\vartheta=1\), the inequality
(20) becomes: \[\begin{aligned}
\bigg|\frac{\zeta(\iota)+\zeta(\kappa)}{2}-\frac{1}{\kappa-\iota}\int_{\iota}^{\kappa}\zeta(\varrho)d\varrho\bigg|
\leq\frac{(\kappa-\iota)^{2}}{2}\bigg(\frac{1}{1+p}\bigg)^{\frac{1}{p}}
\bigg[\bigg(\beta(2,q+2)(|\zeta”(\iota)|^{q}+|\zeta”(\kappa)|^{q})\bigg)^{\frac{1}{q}}\bigg].
\end{aligned}\tag{28}\]
Figure 4. Graphical representation of inequality (28). The
inequality’s left side is shown in red colour, while its right side is
shown in green colour. The right most part of inequality when we remove
the absolute value from left hand side is shown in blue
colour.
6. Conclusion
In this paper, the authors have generated integral midpoint type
inequalities involving tgs-convex functions and Riemann-Liouville
fractional integrals. The numerical and graphical representation of
inequalities in §5, show that present results are more efficient
as compared with the ones that already exist in [14,23]. In similar fashion, present inequalities
can be discussed with different kinds of integral operators, including
quantum integral operators, generalized fractional integral operators,
delta or nabla integral operators in combination with tgs convexity.
Moreover, some other companion inequalities including Jensen, Hardy and
Fejér type can also be investigated by using the same tools, which are
utilized in present work.
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