A new scheme of dispersion charts based on neoteric ranked set sampling

Tahir Abbas; Muhammad Riaz; Bushra Javed; Mu'azu Ramat Abujiya; Tahir Abbas; Muhammad Riaz; Bushra Javed; Mu'azu Ramat Abujiya

doi:10.3934/math.2023915

AIMS Mathematics

2023, Volume 8, Issue 8: 17996-18020. doi: 10.3934/math.2023915

Previous Article Next Article

Research article

A new scheme of dispersion charts based on neoteric ranked set sampling

1.
Department of Mathematics, College of Sciences, University of Sharjah, Sharjah 0000, United Arab Emirates
2.
Department of Mathematics, College of Computing and Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3.
Department of Statistics, Government College University Lahore, Lahore 54000, Pakistan
4.
Preparatory Year Mathematics Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received: 07 April 2023 Revised: 07 May 2023 Accepted: 14 May 2023 Published: 25 May 2023
MSC : 62F03, 62K05, 62P10, 62P30

There are certain areas of science and technology, such as agriculture, ecology, and environmental studies, that emphasize designing competent sampling strategies. The ranked set schemes, particularly the neoteric ranked set sampling (NRSS), are one method that meets such objectives. The NRSS provides plans that incorporates expert knowledge while choosing samples, which is beneficial. This study proposes a novel scheme for creating dispersion charts based on NRSS. The proposed scheme aims to improve the accuracy of dispersion charts by reducing the impact of outliers and non-normality in data sets. As a highly effective method in estimating population parameters, NRSS is used to select samples from the data set. The proposed dispersion charts are assessed based on individual performance measure criteria at shifts of different magnitudes. The dispersion charts created using this new scheme are compared with traditional dispersion charts, and the results demonstrate that the proposed scheme produces charts with higher accuracy and robustness. The study highlights the potential benefits of using NRSS-based dispersion charts in various fields, including quality control, environmental monitoring, and process control. An actual data application from a non-isothermal continuous stirred tank chemical reactor model further validates the simulation results.

Keywords:

Citation: Tahir Abbas, Muhammad Riaz, Bushra Javed, Mu'azu Ramat Abujiya. A new scheme of dispersion charts based on neoteric ranked set sampling[J]. AIMS Mathematics, 2023, 8(8): 17996-18020. doi: 10.3934/math.2023915

Related Papers:

[1]	Xiaogang Liu, Muhammad Ahsan, Zohaib Zahid, Shuili Ren . Fault-tolerant edge metric dimension of certain families of graphs. AIMS Mathematics, 2021, 6(2): 1140-1152. doi: 10.3934/math.2021069
[2]	Maryam Salem Alatawi, Ali Ahmad, Ali N. A. Koam, Sadia Husain, Muhammad Azeem . Computing vertex resolvability of benzenoid tripod structure. AIMS Mathematics, 2022, 7(4): 6971-6983. doi: 10.3934/math.2022387
[3]	Pradeep Singh, Sahil Sharma, Sunny Kumar Sharma, Vijay Kumar Bhat . Metric dimension and edge metric dimension of windmill graphs. AIMS Mathematics, 2021, 6(9): 9138-9153. doi: 10.3934/math.2021531
[4]	Chenggang Huo, Humera Bashir, Zohaib Zahid, Yu Ming Chu . On the 2-metric resolvability of graphs. AIMS Mathematics, 2020, 5(6): 6609-6619. doi: 10.3934/math.2020425
[5]	Meiqin Wei, Jun Yue, Xiaoyu zhu . On the edge metric dimension of graphs. AIMS Mathematics, 2020, 5(5): 4459-4465. doi: 10.3934/math.2020286
[6]	Lyimo Sygbert Mhagama, Muhammad Faisal Nadeem, Mohamad Nazri Husin . On the edge metric dimension of some classes of cacti. AIMS Mathematics, 2024, 9(6): 16422-16435. doi: 10.3934/math.2024795
[7]	Mohra Zayed, Ali Ahmad, Muhammad Faisal Nadeem, Muhammad Azeem . The comparative study of resolving parameters for a family of ladder networks. AIMS Mathematics, 2022, 7(9): 16569-16589. doi: 10.3934/math.2022908
[8]	Dalal Awadh Alrowaili, Uzma Ahmad, Saira Hameeed, Muhammad Javaid . Graphs with mixed metric dimension three and related algorithms. AIMS Mathematics, 2023, 8(7): 16708-16723. doi: 10.3934/math.2023854
[9]	Syed Ahtsham Ul Haq Bokhary, Zill-e-Shams, Abdul Ghaffar, Kottakkaran Sooppy Nisar . On the metric basis in wheels with consecutive missing spokes. AIMS Mathematics, 2020, 5(6): 6221-6232. doi: 10.3934/math.2020400
[10]	Ahmed Alamer, Hassan Zafar, Muhammad Javaid . Study of modified prism networks via fractional metric dimension. AIMS Mathematics, 2023, 8(5): 10864-10886. doi: 10.3934/math.2023551

Abstract

1. Introduction

The fractional derivatives with constant or variable order ^[3,9] are excellent mathematical tools for the description of memory and the hereditary properties of various processes and materials^[12,19]. In fractional calculus, these derivatives are defined through fractional integrals. There are several approaches to fractional derivatives including Riemann-Liouville ^[10,14,15], Caputo, Hadamard derivatives, ^[4,6,13,17].

Efforts have been dedicated to generalizations concerning mappings of bounded variation, absolute continuity, various classes of convex functions, and their extension to fractional calculus, involving Riemann-Liouville integrals and their generalizattions as referenced in ^[1,2,12,15].

In ^[8], the author proved some integral inequalities for functions whose $k$ th $(k\in\mathbb{N})$ derivatives are convex involving Caputo derivatives and obtain the following results for $a, \Delta\in I, a < \Delta, \, \alpha, \beta\in\mathbb{R},$ $\alpha, \beta\geq 1,$ and $\psi:I\to \mathbb{R}$ :

● If $\psi^{(k)}\, (k\in\mathbb{N})$ exists and is positive and convex, then

$\begin{align} &\Gamma(k-\alpha+1)^CD^{\alpha-1}_{a_{+}}\psi(\xi)+(-1)^k\Gamma(k-\beta+1)^CD^{\beta-1}_{\Delta-}\psi(\xi) \\ &\leq (\xi-a)^{k-\alpha+1}\dfrac{\psi^{(k)}(a)+\psi^{(k)}(\xi)}{2} +(\Delta-\xi)^{k-\beta+1}\dfrac{\psi^{(k)}(\Delta)+\psi^{(k)}(\xi)}{2}. \end{align}$

(1.1)

● If $\psi^{(k)}$ exists and is positive, convex and symmetric about $\frac{a +\Delta}{2},$ then

$\begin{align} & \frac{1}{2}\left(\dfrac{1}{k-\alpha+1}+\dfrac{1}{k-\beta+1}\right)\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right) \\ &\le \dfrac{\Gamma(k-\beta+1)^CD_{\Delta_-}^{\beta-1}\psi(\alpha)}{2(\Delta-a)^{k-\beta+1}}+(-1)^k\dfrac{\Gamma(k-\alpha+1) ^CD_{a_{+}}^{\alpha-1}\psi(\Delta)}{2(\Delta-a)^{k-\alpha+1}} \\ & \leq\dfrac{\psi^{(k)}(\Delta)+\psi^{(k)}(a)}{2}. \end{align}$

(1.2)

In ^[11], the authors gave a version of Hadamard's inequality using the Caputo derivative. In ^[7], the authors proved Hadamard inequalities for strongly $\alpha, m$ -convex functions via Caputo fractional derivatives. In this paper, we consider the Caputo derivatives of a real valued function $\psi$ whose derivatives $\psi^{(k)}\, (k \in \mathbb{N})$ are genaralized modified $h$ -convex. Some Caputo fractional versions of Hermite-Hadamard inequalities are obtained. From which particular cases are revealed, we have also established a new integral inequality between Caputo derivatives $^{C}D^{\alpha}_{.}\psi$ and the Riemann-Liouville integrals $R^{k-\alpha}_{.}(\psi^{(k)})^{2}.$ By deriving new differential inequalities in this context, we aim to extend the applicability of fractional calculus to problems involving generalized convex functions. These results have significance in various fields, including mathematics, physics, and engineering, where fractional calculus plays a crucial role in modeling complex phenomena with memory and long-range dependence.sts Our results generalize those cited in ^[8] and unify several classes of functions, like convex and $s$ -convex functions.

2. Preliminaries

This section deals with some definitions of convexity ^[2,5,8], generalized $h$ -convexity ^[20], fractional integrals and derivatives ^[6,18].

Let $I\subset\mathbb{R}$ be an interval and $h:[0, 1]\to (0, \infty), \, \psi:I \to (0, \infty)$ be two real valued functions, then

● $\psi$ is said to be $h$ -convex, if

$\begin{equation} \psi (\rho c + (1-\rho)d)\leq h(\rho) \psi (c)+h(1-\rho)\psi (d) \end{equation}$

(2.1)

holds for all $c, d \in I$ and $\rho\in (0, 1].$ If (2.1) is reversed, then $\psi$ is said to be $h$ -concave.

● The function $\psi$ is said to be modified $h$ -convex if

$\begin{equation} \psi (\rho c + (1-\rho)d)\leq h(\rho) \psi (c)+ (1- h(\rho))\psi (d). \end{equation}$

(2.2)

● The function $\psi$ is said to be generalized modified $h$ -convex if

$\begin{equation} \psi (\rho c + (1-\rho)d)\leq \psi (d)+ h(\rho) \theta(\psi (c),\psi (d)). \end{equation}$

(2.3)

Definition 2.1 (Additivity). ^[20] A continuous bifunction $\theta$ is said to be additive, if

$\theta(a_{1}, b_{1}) + \theta(a_{2}, b_{2}) = \theta(a_{1} + a_{2}, b_{1} + b_{2}),\quad \forall a_{1}, a_{2}, b_{1},b_{2 }\in\mathbb{R}.$

Definition 2.2 (Nonnegative homogeneity). ^[20] A continuous bifunction $\theta$ is said to be nonnegatively homogeneous if, for all $\lambda > 0,$

$\theta(\lambda a_{1},\lambda a_{2}) = \lambda \theta(a_{1}, a_{2}),\quad \forall a_1, a_{2} \in\mathbb{R}.$

Remark 2.1. For different functions $h, \theta$ one can obtain various classes of generalized modified convex functions:

● By taking in (2.1) $h(z) = z^{s} (0 < s\le 1),$ we have the definition of modified generalized $s$ -convex functions.

● If, we take $\theta(r, z) = r-z,$ then we obtain the definition of a modified $h$ -convex function.

Let $[a, \Delta]\, (-\infty < a < \Delta < +\infty)$ be a finite interval on the real axis $\mathbb{R}.$ For any function $\psi\in L_{1}([a, \Delta]),$ the Riemann-Liouville fractional integrals $R^{\alpha}_{a_{+}}$ and $R^{\alpha}_{\Delta_-}$ of order $\alpha\in\mathbb{R}$ $(\alpha > 0)$ of $\psi$ are defined by

$\begin{equation} R^{\alpha}_{a_{+}}\psi(s) = \frac{1}{\Gamma(\alpha)} \displaystyle {\int}_{a}^{s}(s-t)^{\alpha-1}\psi(t)dt ,\quad s > a \quad (left) \end{equation}$

(2.4)

and

$\begin{equation} R^{\alpha}_{\Delta_-}\psi(s) = \frac{1}{\Gamma(\alpha)}\displaystyle {\int}_{s}^{\Delta}(t-s)^{\alpha-1} \psi(t)dt ,\,\,\, s < \Delta \quad (right), \end{equation}$

(2.5)

respectively. Here $\Gamma(\alpha) = {\int}_{0}^{\infty}\, t^{\alpha-1}\, e^{-t}\, dt, \; \alpha > 0$ is the gamma function. We set $R^{0}_{a+}\psi = R^{0}_{\Delta-}\psi = \psi.$

Let $[a, \Delta]$ be a finite interval of the real line $\mathbb{R}$ . Let $\alpha > 0, k\in\mathbb{N},$ $k = [\alpha]+1$ and $\psi\in AC^{k}([a, \Delta])$ ( $AC^{k}([a, \Delta])$ means the space of complex-valued functions $\psi(x)$ which have continuous derivatives up to order $k-1$ on $[a, b]$ such that $\psi^{(k-1)}(x) \in AC([a, \Delta]):$ i.e., absolutely continuous) see Lemma 2.4 ^[18]. The left and right Caputo fractional derivatives of order $\alpha\, (\alpha\ge 0)$ of $\psi$ are given by the following formulas (see ^[1,4,10,13])

$^CD^{\alpha}_{a_{+}}\psi(\xi) = \dfrac{1}{\Gamma(k-\alpha)}\displaystyle {\int}_{a}^{\xi}\psi^{(k)}(t)(\xi-t)^{k-\alpha-1}dt,\qquad \xi > a$

and

$^CD^{\alpha}_{\Delta_-}\psi(\xi) = \dfrac{(-1)^k}{\Gamma(k-\alpha)}\displaystyle {\int}_{\xi}^{\Delta}\psi^{(k)}(t)(t-\xi)^{k-\alpha-1}dt,\qquad \xi < \Delta,$

respectively.

If $\alpha = k\in\mathbb{N},$ then

$^CD^{\alpha}_{a+}\psi(\xi) = \psi^{(k)}(\xi) \quad \mbox{and}\quad ^CD^{\alpha}_{\Delta-}\psi(\xi) = (-1)^k\, \psi^{(k)}(\xi).$

In particular, if $k = 1,$ $\alpha = 0,$ then

$^CD^{0}_{a_{+}}\psi(\xi) = ^CD^{0}_{\Delta_-}\psi(\xi) = \psi(\xi).$

Lemma 2.1. ^[16] The following formulas for Caputo fractional derivatives of order $\alpha > 0, k-1 < \alpha < k\, (k\in\mathbb{N})$ of a power function at $t = a$ and $t = b$ hold

$\begin{equation} ^CD^{\alpha}_{a_{+}}(t-a)^{p} = \dfrac{\Gamma(p+1)}{\Gamma(p-\alpha+1)}(t-a)^{p-\alpha},\,\qquad t > a \end{equation}$

(2.6)

and

$\begin{equation} ^CD^{\alpha}_{b_-}(b-t)^{p} = \dfrac{\Gamma(p+1)}{\Gamma(p-\alpha+1)}(b-t)^{p-\alpha}, \,\qquad t < b. \end{equation}$

(2.7)

Our objective in this work, is to prove some fractional integral inequalities for functions whose $k$ th $(k \in\mathbb{N})$ derivatives are generalized modified $h$ -convex functions involving the Caputo derivative operator.

3. Results

Theorem 3.1. Let $I$ be an interval of $\mathbb{R},$ $a, \Delta\in I, a < \Delta$ and $\alpha, \beta > 0,$ such that $k-1 < \alpha, \beta < k, k\in\mathbb{N}.$ Let $\psi:I\to\mathbb{R}$ be differentiable function. If, $\psi^{(k)}(k\in\mathbb{N})$ exists and is a positive generalized modified $h$ -convex function and $\theta$ is a continuous bifunction, then the following integral inequality

$\begin{eqnarray} \Gamma(k-\alpha+1)\,\left( ^CD^{\alpha-1}_{a_{+}}\psi\right) (\xi)+ (-1)^k\Gamma(k-\beta+1)\left( ^C D^{\beta-1}_{\Delta_-}\psi\right) (\xi) & &\\ \leq ( \Delta-\xi)^{k-\beta+1}\left[\psi^{(k)}(\xi)+\theta\,( \psi^{(k)}(\Delta),\psi^{(k)}(\xi))\displaystyle {\int}_{0}^{1}h(z)dz\right]& &\\ +\,(\xi-a)^{k-\alpha+1}\left[\psi^{(k)}(\xi)+ \theta\,( \psi^{(k)}(a),\psi^{(k)}(\xi))\displaystyle {\int}_{0}^{1}h(z)dz\right] \end{eqnarray}$

(3.1)

holds.

Proof. For all $\xi\in[a, \Delta]$ and for all $t\in[a, \xi],$ we have

$\begin{equation} (\xi-t)^{k-\alpha}\leq(\xi-a)^{k-\alpha}, \end{equation}$

(3.2)

and

$t = \frac{\xi-t}{\xi-a}a +\frac{t-a}{\xi-a}\xi.$

Since $\psi^{(k)}$ is generalized modified $h$ -convex, (2.3) implies that

$\begin{equation} \psi^{(k)}(t)\leq \psi^{(k)}(\xi)+h\left( \frac{\xi-t}{\xi-a}\right)\theta\,(\psi^{(k)}(a),\psi^{(k)}(\xi)). \end{equation}$

(3.3)

Multiplying inequalities (3.2) and (3.3) on both side and integrating, we obtain

$\begin{align} & \displaystyle {\int}_{a}^{\xi}(\xi-t)^{k-\alpha}\, \psi^{(k)}(t)dt\\ &\leq \displaystyle {\int}_{a}^{\xi}(\xi-a)^{k-\alpha} \times \left[ \psi^{(k)}(\xi)+h\left( \dfrac{\xi-t}{\xi-a}\right)\theta\,(\psi^{(k)}(a),\psi^{(k)}(\xi)) \right] dt. \end{align}$

(3.4)

That is

$\begin{align} \Gamma(k-\alpha+1)\left(^CD^{\alpha-1}_{a_{+}}\psi\right)(\xi)\leq(\xi-a)^{k-\alpha+1} \times \left[ \psi^{(k)}(\xi) +\theta\,( \psi^{(k)}(a),\psi^{(k)}(\xi))\displaystyle {\int}_{0}^{1}h(z)dz \right]. \end{align}$

(3.5)

Let $\xi\in [a, \Delta], t\in[\xi, \Delta],$ thus

$\begin{equation} (t-\xi)^{k-\beta}\leq(\Delta-\xi)^{k-\beta}. \end{equation}$

(3.6)

We have

$t = \frac{t-\xi}{\Delta-\xi}\Delta +\frac{\Delta-t}{\Delta-\xi}\xi.$

Since $\psi^{(k)}$ is generalized modified $h$ -convex on $[\alpha, \Delta],$ then

$\begin{equation} \psi^{(k)}(t)\leq \psi^{(k)}(\xi)+ h\left( \dfrac{t-\xi}{\Delta-\xi}\right) \theta\,(\psi^{(k)}(\Delta),\psi^{(k)}(\xi)). \end{equation}$

(3.7)

Similarly, we obtain

$\begin{align} (-1)^k\Gamma(k-\beta+1)\left( ^C D^{\beta-1}_{\Delta_-}\psi\right) (\xi)\leq(\Delta-\xi)^{k-\beta+1} \times \left[ \psi^{(k)}(\xi) + \theta\,( \psi^{(k)}(\Delta),\psi^{(k)}(\xi))\displaystyle {\int}_{0}^{1}h(z)dz \right]. \end{align}$

(3.8)

Adding (3.5) and (3.8), the claim follows. □

Corollary 3.1. If, we set $\alpha = \beta$ in (3.1), then we obtain

$\begin{align*} &\Gamma(k-\alpha+1)\left[\left(^CD^{\alpha-1}_{a_{+}}\psi\right)(\xi)+(-1)^k\left(^CD^{\alpha-1}_{\Delta_-}\psi\right)(\xi)\right]\\ & \leq (\Delta-\xi)^{k-\alpha+1} \left[\psi^{(k)}(\xi)+\theta\left(\psi^{(k)}(\Delta),\psi^{(k)}(\xi)\right)\displaystyle {\int}_{0}^{1}h(z)dz\right]\\ & + (\xi-a)^{k-\alpha+1} \left[\psi^{(k)}(\xi)+ \theta\left( \psi^{(k)}(a),\psi^{(k)}(\xi)\right)\displaystyle {\int}_{0}^{1}h(z)dz \right]. \end{align*}$

Corollary 3.2. By setting $\theta\, (r, z) = r-z, h(t) = t^{s}, s\in[0, 1]$ in (3.1), we obtain

$\begin{align} &\Gamma(k-\alpha+1)\left(^CD^{\alpha-1}_{a_{+}}\psi\right)(\xi)+(-1)^k\Gamma(k-\beta+1)\left(^CD^{\beta-1}_{\Delta_-}\psi\right)(\xi)\\ &\leq \dfrac{(\Delta-\xi)^{k-\beta+1}\,\psi^{(k)}(\Delta)+(\xi-a)^{k-\alpha+1}\,\psi^{(k)}(a)}{s+1}\\ &+ \dfrac{(\xi-a)^{k-\alpha+1}+(\Delta-\xi)^{k-\beta+1}}{s+1}\, s\,\psi^{(k)}(\xi). \end{align}$

(3.9)

In particular, if $h(z) = z,$ then we have

$\begin{align} &\Gamma(k-\alpha+1)\left(^CD^{\alpha-1}_{\alpha_{+}}\psi\right)(\xi)+(-1)^k\Gamma(k-\beta+1)\left(^CD^{\beta-1}_{\Delta_-}\psi\right)(\xi)\\ &\leq \dfrac{(\Delta-\xi)^{k-\beta+1}\,\psi^{(k)}(\Delta)+(\xi-a)^{k-\alpha+1}\,\psi^{(k)}(a)}{2}\\ &+ \dfrac{(\xi-a)^{k-\alpha+1}+(\Delta-\xi)^{k-\beta+1}}{2}\,\psi^{(k)}(\xi). \end{align}$

(3.10)

Taking $\alpha = \beta$ in (3.10), we obtain

$\begin{align} &\Gamma(k-\alpha+1)\left[\left( ^CD^{\alpha-1}_{a_{+}}\psi\right) (\xi)+(-1)^k\left(^CD^{\alpha-1}_{\Delta_-}\psi\right)(\xi)\right]\\ &\leq \dfrac{(\Delta-\xi)^{k-\alpha+1}\,\psi^{(k)}(\Delta)+(\xi-a)^{k-\alpha+1}\psi^{(k)}(a)}{2}\\ &+ \dfrac{(\xi-a)^{k-\alpha+1}+(\Delta-\xi)^{k-\alpha+1}}{2}\,\psi^{(k)}(\xi). \end{align}$

(3.11)

Example 3.1. Let $\psi:[a, \Delta]\to [0, \infty),$ $\psi(\xi) = \frac{2}{(k+2)!}(\xi-a) ^{k+2},$ $a < \xi\le \Delta.$ Let $h:[0, 1]\to (0, \infty),$ $h(t)\ge t,$ $\theta(x, y) = 2x+y.$ We verify easly that $\psi^{(k)}(\xi) = (\xi-a)^{2}$ is generalized modified $h$ -convex on $[a, \Delta].$ From Corollary 3.1 and Lemma 2.1, we obtain

$\begin{equation} lhs : = \Gamma(k-\alpha+1)\left( ^CD^{\alpha-1}_{a_{+}}\psi\right) (\xi) = \frac{2(\xi-a)^{k-\alpha+3}}{(k-\alpha+1)(k-\alpha+2)(k-\alpha+3)}, \end{equation}$

(3.12)

and

$\begin{align} rhs: & = (\xi-a)^{k-\alpha+1} \left[(\xi-a)^{2} +( 0+(\xi-a)^{2})\displaystyle {\int}_{0}^{1}h(z)dz\right]\\ & = (\xi-a)^{k-\alpha+3}\left( 1+\displaystyle {\int}_{0}^{1}h(z)dz\right). \end{align}$

(3.13)

For the right derivative $\left(^CD^{\alpha-1}_{\Delta_-}\psi\right) (\xi),$ we consider the function $\psi(\xi) = \frac{2(\Delta-\xi) ^{k+2}}{(k+2)!},$ $a\le \xi < \Delta.$

$\begin{equation} (-1)^{k}\Gamma(k-\alpha+1)\left( ^CD^{\alpha-1}_{\Delta_-}\psi\right) (\xi) = \frac{2(\Delta-\xi)^{k-\alpha+3}}{(k-\alpha+1)(k-\alpha+2)(k-\alpha+3)} \end{equation}$

(3.14)

and

$\begin{equation} rhs: = (\Delta-\xi)^{k-\alpha+3}\left( 1+\displaystyle {\int}_{0}^{1}h(z)dz\right). \end{equation}$

(3.15)

Now let $I$ be an interval of $\mathbb{R},$ $a, \Delta\in I, (a < \Delta)$ and $\alpha, \beta > 0,$ such that $k-1 < \alpha, \beta < k, (k\in\mathbb{N}).$ Let $\psi:I\to\mathbb{R}.$ Assume that $|\psi^{(k+1)}|$ is generalized modified $h$ - convex on $[a, \Delta].$

It is clear that for all $\xi\in [a, \Delta], t\in [a, \xi],$ we have

$\begin{equation} (\xi-t)^{k-\alpha}\leq (\xi-a)^{k-\alpha},\qquad t\in[a,\xi]. \end{equation}$

(3.16)

Since $|\psi^{(k+1)}|$ is generalized modified $h$ -convex, we have for $t\in[a, \xi],$

$\begin{eqnarray} \qquad \qquad \textbf{Lhs = }-\left[|\psi^{(k+1)}(\xi)|+ \theta\left(|\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right) h\left( \dfrac{t-a}{\xi-a} \right) \right]\\ \leq |\psi^{(k+1)}(t)|\leq |\psi^{(k+1)}(\xi)|+ \theta\left(|\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right) h\left( \dfrac{t-a}{\xi-a} \right) = \textbf{Rhs}. \end{eqnarray}$

(3.17)

Multiplying (3.16) by the Rhs of inequality (3.17) and integrating the resulting inequality over $[a, \xi],$ we obtain

$\begin{align} \displaystyle {\int}_{a}^{\xi}(\xi-t)^{k-\alpha}\,\psi^{(k+1)}(t)dt \leq (\xi-a)^{k-\alpha} \left( |\psi^{(k+1)}(\xi)|+\theta\left( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right)\displaystyle {\int}_{0}^{1}h(z)dz \right), \end{align}$

(3.18)

by integration by parts, we have

$\begin{align*} \displaystyle {\int}_{a}^{\xi}(\xi-t)^{k-\alpha}\,\psi^{(k+1)}(t)dt & = \psi^{(k)}(t)(\xi-t)^{k-\alpha}\;|_a^\xi +(k-\alpha)\displaystyle {\int}_{a}^{\xi}(\xi-t)^{k-\alpha-1}\,\psi^{(k)}(t)dt\\ & = \Gamma(k-\alpha+1)\left(^CD^{\alpha}_{a+}\psi\right)(\xi) -\psi^{(k)}(a)(\xi-a)^{k-\alpha}. \end{align*}$

Hence

$\begin{align} & \Gamma(k-\alpha+1)\left( ^CD^{\alpha}_{a_{+}}\psi\right) (\xi)-\psi^{(k)}(a)(\xi-a)^{k-\alpha} \\ &\leq\left(\psi^{(k+1)}(\xi)+\theta\left( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right) \displaystyle {\int}_{0}^{1}h(z)dz \right)(\xi-a)^{k-\alpha}. \end{align}$

(3.19)

In a similar way, if we proceed with the Lhs of (3.17) as we did for the Rhs, it follows that

$\begin{align} \psi^{(k)}(a)(\xi-a)^{k-\alpha}-\Gamma(k-\alpha+1)&\left( ^CD^{\alpha}_{a_{+}}\psi\right) (\xi)\\ &\leq \left(|\psi^{(k+1)}(\xi)|+\theta\left(|\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)| \right)\displaystyle {\int}_{0}^{1}h(z)dz \right)(\xi-a)^{k-\alpha}. \end{align}$

(3.20)

From (3.19) and (3.20), we obtain

$\begin{align} |\Gamma(k-\alpha+1)\left( ^CD^{\alpha}_{a_{+}}\psi\right) (\xi)-\psi^{(k)}&(a)(\xi-a)^{k-\alpha}|\\ &\leq \left(|\psi^{(k+1)}(\xi)|+\theta\left( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right) \displaystyle {\int}_{0}^{1}h(z)dz \right)(\xi-a)^{k-\alpha}. \end{align}$

(3.21)

Doing the same for $t\in [\xi, \Delta]$ and $\beta > 0, k-1 < \beta < k,$ and taking into acount that $|\psi^{(k+1)}|$ is generalized modified $h$ -convex, we have

$\begin{align} \textbf{Lhs = } -&\left[ |\psi^{(k+1)}(\xi)|+ \theta\left(|\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right) h\left( \dfrac{t-\xi}{\Delta-\xi}\right) \right] \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \leq \,\psi^{(k+1)}(t)\,\le \psi^{(k+1)}(\xi)+ \theta\left(|\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right) h\left( \dfrac{t-\xi}{\Delta-\xi}\right) = \textbf{Rhs}. \end{align}$

(3.22)

Hence

$\begin{align} &|\Gamma(k-\beta+1)\left( ^CD^{\beta}_{\Delta_-}\psi\right) (\xi)-\psi^{(k)}(\Delta)(\Delta-\xi)^{k-\beta}|\\ &\leq (\Delta-\xi)^{k-\beta} \times\left[|\psi^{(k+1)}(\xi)|+ \theta\left( |\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right) \displaystyle {\int}_{0}^{1}h(z)dz \right]. \end{align}$

(3.23)

Combine (3.21) and (3.23) via triangular inequality, and we obtain the double inequality

$\begin{eqnarray} \left|\Gamma(k-\alpha+1)\left( ^CD^{\alpha}_{a_{+}}\psi\right) (\xi)+\Gamma(k-\beta+1)\left( ^CD^{\beta}_{\Delta_-}\psi\right) (\xi)\right.\\ \left. -\left(\psi^{(k)}(a)(\xi-a)^{k-\alpha}+\psi^{(k)}(\Delta)(\Delta-\xi)^{k-\beta}\right)\right|& &\\ \le (\Delta-\xi)^{k-\beta}\left[|\psi^{(k+1)}(\xi)|+ \theta\left( |\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right)\displaystyle {\int}_{0}^{1}h(z)dz \right]& &\\ +\, (\xi-a)^{k-\alpha} \left[|\psi^{(k+1)}(\xi)|+\theta( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|) \displaystyle {\int}_{0}^{1}h(z)dz \right]. \end{eqnarray}$

(3.24)

Which leads to the following result:

Theorem 3.2. Let $I$ be an interval of $\mathbb{R},$ $a, \Delta\in I\, (a < \Delta)$ and $\alpha, \beta > 0,$ such that $k-1 < \alpha, \beta < k,$ $(k\in\mathbb{N})$ . Let $\psi:I\to\mathbb{R}$ be a function such that $\psi\in AC^{k+1}.$ Assume that $|\psi^{(k+1)}|$ is a generalized modified $h$ -convex function and $\theta$ a continuous bifunction, then

$\begin{eqnarray} \left|\Gamma(k-\alpha+1)\left( ^CD^{\alpha}_{a_{+}}\psi\right) (\xi)+\Gamma(k-\beta+1)\left( ^CD^{\beta}_{\Delta_-}\psi\right) (\xi)\right.\\ \left. -\left(\psi^{(k)}(a)(\xi-a)^{k-\alpha}+\psi^{(k)}(\Delta)(\Delta-\xi)^{k-\beta}\right)\right| & &\\ \le (\Delta-\xi)^{k-\beta}\left[|\psi^{(k+1)}(\xi)|+ \theta\left( |\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right)\displaystyle {\int}_{0}^{1}h(z)dz \right] & &\\ +\, (\xi-a)^{k-\alpha} \left[|\psi^{(k+1)}(\xi)|+\theta\left( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right) \displaystyle {\int}_{0}^{1}h(z)dz \right] \end{eqnarray}$

(3.25)

holds.

As a consequences, we have

Corollary 3.3. If in (3.25), we set $\alpha = \beta,$ then

$\begin{align} &\left|\Gamma(k-\alpha+1)(^CD^{\alpha}_{a_{+}}\psi(\xi)+^CD^{\alpha}_{\Delta_-}\psi(\xi)) - \left(\psi^{(k)}(a)(\xi-a)^{k-\alpha}+\psi^{(k)}(\Delta) (\Delta-\xi)^{k-\alpha}\right)\right| \\ &\le (\Delta-\xi)^{k-\alpha}\left[|\psi^{(k+1)}(\xi)|+ \theta\left( |\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right)\displaystyle {\int}_{0}^{1}h(z)dz \right] \\ &+\,(\xi-a)^{k-\alpha}\left[|\psi^{(k+1)}(\xi)|+\theta\left( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right)\displaystyle {\int}_{0}^{1}h(z)dz\right] \end{align}$

(3.26)

holds.

Corollary 3.4. By taking $\theta(z, r) = z-r, h(t) = t^{s}, s\in [0, 1]$ in (3.26), we obtain

$\begin{align} |\Gamma(k-\alpha+&1)[\left(^CD^{\alpha}_{a_{+}}\psi\right)(\xi)+\left(^CD^{\alpha}_{\Delta_-}\psi\right)(\xi)]- \left( \psi^{(k)}(a)(\xi-a)^{k-\alpha}+\psi^{(k)}(\Delta)(\Delta-\xi)^{k-\alpha}\right) |\\& \leq \dfrac{s \left((\xi-a)^{k-\alpha}+(\Delta-\xi)^{k-\alpha}\right)|\psi^{(k+1)}(\xi)|}{s+1} +\dfrac{(\xi-a)^{k-\alpha}|\psi^{(k+1)}(a)|+(\Delta-\xi)^{k-\alpha}|\psi^{(k+1)}(\Delta)|}{s+1}. \end{align}$

(3.27)

In particular for $s = 1,$ we have

$\begin{align} |\Gamma(k-\alpha+&1)[\left(^CD^{\alpha}_{a_{+}}\psi\right)(\xi)+\left(^CD^{\alpha}_{\Delta_-}\psi\right)(\xi)] - \left( \psi^{(k)}(a)(\xi-a)^{k-\alpha}+\psi^{(k)}(\Delta)(\Delta-\xi)^{k-\alpha}\right) | \\ &\leq \dfrac{\left((\xi-a)^{k-\alpha}+(\Delta-\xi)^{k-\alpha}\right)|\psi^{(k+1)}(\xi)|}{2} +\, \dfrac{(\xi-a)^{k-\alpha}|\psi^{(k+1)}(a)|+(\Delta-\xi)^{k-\alpha}|\psi^{(k+1)}(\Delta)|}{2}. \end{align}$

(3.28)

Example 3.2. Let $\psi, h, \theta$ as in the Example 3.1. We verify easily that $\psi^{(k+1)}(\xi) = 2(\xi-a)$ is generalized modified $h$ -convex on $[a, \Delta].$ From Corollary 3.3 and Lemma 2.1, we obtain

$\begin{equation} lhs : = \Gamma(k-\alpha+1)^CD^{\alpha}_{a_{+}}\psi(\xi) = \frac{2(\xi-a)^{k-\alpha+2}}{(k-\alpha+1)(k-\alpha+2)}, \end{equation}$

(3.29)

and

$\begin{align} \label{aex01} rhs: = (\xi-a)^{k-\alpha} \left[2(\xi-a) +( 0+2(\xi-a))\displaystyle {\int}_{0}^{1}h(z)dz\right] = 2\,(\xi-a)^{k-\alpha+1}\left( 1+ \displaystyle {\int}_{0}^{1}h(z)dz\right). \end{align}$

For the right derivative $^CD^{\alpha}_{\Delta_-}\psi(\xi),$ we have

$\begin{equation} lhs: = (-1)^{k}\Gamma(k-\alpha+1)^CD^{\alpha}_{\Delta_-}\psi(\xi) = \frac{2(\Delta-\xi)^{k-\alpha+2}}{(k-\alpha+1)(k-\alpha+2)} \end{equation}$

(3.30)

and

$\begin{equation} rhs: = 2\,(\Delta-\xi)^{k-\alpha+1}\left( 1+\displaystyle {\int}_{0}^{1}h(z)dz\right). \end{equation}$

(3.31)

Now suppose that $\psi : [a, \Delta]\to (0, \infty)$ is a generalized modified $h$ -convex function and symmetric about $\dfrac{a+\Delta}{2},$ then for all $\xi\in[a, \Delta]$ the inequality

$\begin{equation} \psi\left( \dfrac{a+\Delta}{2}\right) \leq \psi(\xi)\left( 1+h\left( \dfrac{1}{2}\right)\theta(1,1)\right) \quad \end{equation}$

(3.32)

is valid. Here $\theta$ is assumed to be nonnegatively homogeneous. Indeed, set

$r = a\dfrac{\xi-a}{\Delta-a}+\Delta\dfrac{\Delta-\xi}{\Delta-a},\quad z = \Delta\dfrac{\xi-a}{\Delta-a}+\alpha\dfrac{\Delta-\xi}{\Delta-a}.$

Hence

$\dfrac{a+\Delta}{2} = \dfrac{r}{2}+\dfrac{z}{2}.$

Since $\psi$ is generalized modified $h$ -convex, symmetric about $\frac{a+\Delta}{2},$ and the bifunction $\theta$ is assumed to be nonnegatively homogeneous, it results in

$\begin{eqnarray*} \psi\left(\dfrac{a+\Delta}{2} \right) & = & \psi\left( \frac{r}{2}+\frac{z}{2}\right)\\ &\leq&\psi(z)+ h\left( \dfrac{1}{2}\right)\theta(\psi(r),\psi(z))\\ & = & \psi(\xi)+h\left( \dfrac{1}{2}\right)\theta(\psi(\xi),\psi(\xi)) \\ & = &\psi(\xi)\left( 1+h\left( \dfrac{1}{2}\right)\theta(1,1)\right). \end{eqnarray*}$

Theorem 3.3. Let $I$ be an interval of $\mathbb{R},$ $a, \Delta\in I$ $(a < \Delta)$ and $\alpha, \beta\geq 1,$ $k-1 < \alpha, \beta < k, k\in\mathbb{N}$ . Let $\psi:I\to \mathbb{R}$ be a real valued function such that $\psi\in AC^{k}.$ If $\psi^{(k)}$ is a positive, generalized modified $h$ -convex and symmetric about $\frac{a+\Delta}{2}$ and furthermore the bifunction $\theta$ is nonnegatively homogeneous, then the following inequality holds

$\begin{eqnarray} N_{\theta}^{-1} \left\lbrace\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\beta+1}+\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\alpha+1}\right\rbrace \leq \dfrac{\Gamma(k-\beta+1)\left(^CD_{\Delta_-}^{\beta-1}\psi\right)(a)}{(\Delta-a)^{k-\beta+1}}+ \dfrac{\Gamma(k-\alpha+1)\left(^CD_{a_{+}}^{\alpha-1}\psi\right)(\Delta )}{(\Delta-a)^{k-\alpha+1}}\\ \leq \psi^{(k)}(\Delta)+\psi^{(k)}(a)+ \left[\theta( \psi^{(k)}(\Delta),\psi^{(k)}(a))+ \theta(\psi^{(k)}(a),\psi^{(k)}(\Delta))\right]\displaystyle {\int}_{0}^{1}h(z)dz. \end{eqnarray}$

(3.33)

If, furthermore, $\theta$ is additive, then

$\begin{align} N_{\theta}^{-1} \left\lbrace\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\beta+1}+\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\alpha+1}\right\rbrace & \leq \dfrac{\Gamma(k-\beta+1)\left(^CD_{\Delta_-}^{\beta-1}\psi\right)(\alpha)}{(\Delta-a)^{k-\beta+1}}+ \dfrac{\Gamma(k-\alpha+1)\left(^CD_{a_{+}}^{\alpha-1}\psi\right)(\Delta )} {(\Delta-a)^{k-\alpha+1}} \\ &\leq M_{\theta} (\psi^{(k)}(\Delta)+\psi^{(k)}(a)) \end{align}$

(3.34)

holds. Here

$N_{\theta} = 1+h\left( \dfrac{1}{2}\right)\theta(1,1),\; M_{\theta} = 1+\theta(1,1)\displaystyle {\int}_{0}^{1}h(z)dz.$

Proof. For all $\xi\in [a, \Delta], k-1 < \alpha < k,$ we have $\xi = \frac{\Delta-\xi}{\Delta-a}a +\frac{\xi-a}{\Delta-a}\Delta$ and

$\begin{equation} (\xi-\alpha)^{k-\alpha}\leq(\Delta-a)^{k-\alpha} \end{equation}$

(3.35)

and $\psi^{(k)}$ satisfies

$\begin{equation} \psi^{(k)}(\xi)\leq \psi^{(k)}(a)+h\left( \dfrac{\xi-a}{\Delta-a}\right) \theta\,(\psi^{(k)} (\Delta),\psi^{(k)}(a)). \end{equation}$

(3.36)

Multiplying (3.35) and (3.36) and proceeding as above, we obtain

$\begin{align} &\Gamma(k-\alpha+1)\left(^CD_{\Delta-}^{\alpha-1}\psi\right)(a) \\ &\leq \left[ \psi^{(k)}(a)+ \theta\left( \psi^{(k)}(\Delta),\psi^{(k)}(a)\right) \displaystyle {\int}_{0}^{1}h(z)dz\right] \times\,(\Delta-\alpha)^{k-\alpha+1}. \end{align}$

(3.37)

Also, we have for $\xi\in [a, \Delta], k-1 < \beta < k,$

$\begin{equation} (\Delta -\xi)^{k-\beta}\leq (\Delta-a)^{k-\beta} \end{equation}$

(3.38)

and

$\begin{equation} \psi^{(k)}(\xi)\leq \psi^{(k)}(\Delta)+h\left( \dfrac{\Delta-\xi}{\Delta-a}\right) \theta(\psi^{(k)} (a),\psi^{(k)}(\Delta)) . \end{equation}$

(3.39)

Multiplying (3.39) and (3.38) and integrating over $[a, \Delta],$ we get

$\begin{align} \Gamma(k-\beta+1)\left(^CD_{\Delta_-}^{\beta-1}\psi\right)(a)\leq \left[\psi^{(k)}(\Delta)+\theta\left( \psi^{(k)}(a),\psi^{(k)}(\Delta)\right) \displaystyle {\int}_{0}^{1}h(z)dz\right] (\Delta-a)^{k-\beta+1}. \end{align}$

(3.40)

Adding (3.37) and (3.40), we obtain

$\begin{align} &\dfrac{\Gamma(k-\beta+1)\left(^CD_{\Delta_-}^{\beta-1}\psi\right)(\alpha)}{(\Delta-a)^{k-\beta+1}}+ \dfrac{\Gamma(k-\alpha+1)\left(^CD_{a_{+}}^{\alpha-1}\psi\right)(\Delta )}{(\Delta-a)^{k-\alpha+1}} \end{align}$

(3.41)

$\begin{align} & \leq\psi^{(k)}(\Delta)+\psi^{(k)}(a) +\,\left[\theta(\psi^{(k)}(\Delta),\psi^{(k)}(a))+\theta(\psi^{(k)}(a),\psi^{(k)}(\Delta))\right]\displaystyle {\int}_{0}^{1}h(z)dz. \end{align}$

(3.42)

Set $N_{\theta} = 1+h\left(\dfrac{1}{2}\right)\theta(1, 1),$ thus (3.32) is written as

$\begin{equation} \psi^{(k)}\left(\dfrac{a+\Delta}{2}\right) \leq N_{\theta}\,\psi^{(k)}(\xi), \quad \xi\in[a,\Delta]. \end{equation}$

(3.43)

Multiplying by $(\xi-a)^{k-\alpha}$ on both sides of (3.43) and integrating the result over $[a, \Delta],$ it results that

$\begin{equation} N_{\theta}^{-1}\,\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\alpha+1} \leq \dfrac{\Gamma(k-\alpha+1)\left(^CD_{\Delta_-}^{\alpha-1}\psi\right)(a)}{(\Delta-a)^{k-\alpha+1}}. \end{equation}$

(3.44)

Multiplying (3.43) by $(\Delta-\xi)^{k-\beta},$ and integrating over $[a, \Delta],$ we obtain

$\begin{equation} N_{\theta}^{-1}\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\beta+1}\leq \dfrac{\Gamma(k-\beta+1)\left(^CD_{a_{+}}^{\beta-1}\psi\right)(\Delta )}{(\Delta-a)^{k-\beta+1}}. \end{equation}$

(3.45)

Adding (3.44) and (3.45), we obtain the first inequality. By combining the resulting inequality with (3.41), we obtain (3.33). Using the fact that $\theta$ is additive and nonnegatively homogeneous (3.34) results. That proves the claim. □

Corollary 3.5. By taking $\alpha = \beta$ in (3.33), then

$\begin{eqnarray} N_{\theta}^{-1}\, \dfrac{2\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\alpha+1}&\leq& \dfrac{\Gamma(k-\alpha+1)\left(^CD_{a_{+}}^{\alpha-1} \psi(\Delta) +^CD_{\Delta_-}^{\alpha-1}\psi(a)\right)}{(\Delta-a)^{k-\alpha+1}}\\ &\leq& \psi^{(k)}(\Delta)+\psi^{(k)}(a)\\ &+& \left[\theta(\psi^{(k)}(\Delta),\psi^{(k)}(a))+\theta(\psi^{(k)}(a),\psi^{(k)}(\Delta))\right]\displaystyle {\int}_{0}^{1}h(z)dz \end{eqnarray}$

(3.46)

holds.

If, $\theta$ is additive, then

$\begin{eqnarray} \dfrac{2N_{\theta}^{-1}\,\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)} {k-\alpha+1} &\leq&\dfrac{\Gamma(k-\alpha+1)\left(^CD_{a_{+}}^{\alpha-1}\psi(\Delta) + ^CD_{\Delta_-}^{\alpha-1}\psi(a)\right)}{(\Delta-a)^{k-\alpha+1}} \\ &\leq& M_{\theta}\,(\psi^{(k)}(\Delta)+\psi^{(k)}(a)). \end{eqnarray}$

(3.47)

Corollary 3.6. By setting $h(t) = t^{s}, s\in[0, 1]$ in (3.47), it results that

$\begin{align*} &\dfrac{2^{s}\psi^{(k)}\left(\frac{a+\Delta}{2}\right)}{(2^{s}+\theta(1,1))(k-\alpha+1)} \\ &\leq \dfrac{\Gamma (k-\alpha+1)\left[ \left( ^CD_{\Delta_-}^{\alpha+1}\psi\right) (a)+ \left( ^CD_{a_{+}}^{\alpha+1}\psi\right)(\Delta)\right] }{(\Delta-a)^{k-\alpha+1}} \\& \leq \dfrac{\psi^{(k)}(a)+\psi^{(k)}(\Delta)}{s+1}(s+1+\theta(1,1)). \end{align*}$

In particular, if $h(t) = t,$ then

$\begin{align*} & \dfrac{2\psi^{(k)}\left(\frac{a+\Delta}{2}\right)}{(2+\theta(1,1))(k-\alpha+1)} \\ & \leq \dfrac{\Gamma (k-\alpha+1)\left[ \left(^CD_{\Delta_-}^{\alpha+1}\psi\right) (a)+ \left( ^CD_{a_{+}}^{\alpha+1}\psi\right)(\Delta)\right] }{(\Delta-a)^{k-\alpha+1}} \\& \leq \dfrac{\psi^{(k)}(a)+\psi^{(k)}(\Delta)}{2}(2+\theta(1,1)). \end{align*}$

Theorem 3.4. Let $\psi\in AC^{k}(a, \Delta),$ $k\in N; k-1 < \alpha < k.$ Assume that $\psi^{(k)}$ is positive, generalized modified $h$ -convex on $[a, \Delta]$ and symmetric to $\dfrac{a+\Delta}{2}.$ Assume that $\theta$ is nonnegatively homogeneous. Then

$\begin{align} \frac{\psi^{(k)}\left( \dfrac{a+\Delta}{2}\right)}{ 1+h(\frac{1}{2})\theta(1,1)} \left[ \left( ^CD_{\Delta_-}^{\alpha}\psi\right)(a)+ \left( ^CD_{a_{+}}^{\alpha}\psi\right)(\Delta)\right] \le R_{\Delta-}^{k-\alpha}(\psi^{(k)})^{2}( a)+ R_{a+}^{k-\alpha}(\psi^{(k)})^{2}(\Delta) \end{align}$

(3.48)

holds. Where $R_{.}^{k-\alpha}$ is the Riemann-Liouville integral operator of order $k-\alpha.$

Proof. Since $\psi^{(k)}$ is generalized modified $h$ -convex and $\theta$ is nonnegatively homogeneous, then we have for $\mu\in [0, 1]$

$\begin{align} \psi^{(k)}\left( \dfrac{a+\Delta}{2}\right) & = \psi^{(k)}\left(\dfrac{\mu \Delta+(1-\mu )a + \mu a+(1-\mu)\Delta}{2}\right)\\ &\le \psi^{(k)}(\mu \Delta+(1-\mu )a )+ h(\frac{1}{2})\,\theta\,(\psi^{(k)}(\mu a+(1-\mu )\Delta ),\psi^{(k)}(\mu \Delta+(1-\mu )a)) \\ & = (\psi^{(k)})^{2}(\mu \Delta+(1-\mu )a )\left[ 1+h\left( \frac{1}{2}\right) \theta(1,1)\right] . \end{align}$

(3.49)

Multiplying (3.49) by $\mu^{k-\alpha-1}\psi^{(k)}(\mu \Delta+(1-\mu)a)$ and integrating over $[0, 1],$ with respect to $\mu,$ we obtain

$\begin{eqnarray} \psi^{(k)}\left( \dfrac{a+\Delta}{2}\right)\displaystyle {\int}_{0}^{1}\mu^{k-\alpha-1}\psi^{(k)}( \mu \Delta+(1-\mu )a )d\mu = \frac{\psi^{(k)}\left( \frac{a+\Delta}{2}\right)}{(\Delta-a)^{k-\alpha}}\Gamma(k-\alpha)(^CD_{a_{+}}^{\alpha}\psi)(\Delta), \end{eqnarray}$

and

$\begin{align*} & \left[ 1+h\left( \frac{1}{2}\right)\, \theta(1,1)\right]\displaystyle {\int}_{0}^{1}\mu^{k-\alpha-1} (\psi^{(k)})^{2}(\mu \Delta+(1-\mu )a ) d\mu\\ & = \frac{ 1+h(\frac{1}{2})\theta(1,1)}{(\Delta-a)^{k-\alpha}}\displaystyle {\int}_{a}^{\Delta}(x-a)^{k-\alpha-1}(\psi^{(k)})^{2}(x )dx\\ & = \left[ 1+h(\frac{1}{2})\theta(1,1)\right]\frac{\Gamma(k-\alpha)}{(\Delta-a)^{k-\alpha}} R_{a_{+}}^{k-\alpha}(\psi^{(k)})^{2}(\Delta). \end{align*}$

Hence

$\begin{equation} \frac{\psi^{(k)}\left( \frac{a+\Delta}{2}\right)}{ 1+h(\frac{1}{2})\theta(1,1)} (^{C}D_{a_{+}}^{\alpha}f)(\Delta) \le R_{a_{+}}^{k-\alpha}(\psi^{(k)})^{2}(\Delta). \end{equation}$

(3.50)

And similarly

$\begin{equation} \psi^{(k)}\left( \dfrac{a+\Delta}{2}\right) \le \psi^{(k)}(\mu a +(1-\mu )\Delta )\left[ 1+h\left( \frac{1}{2}\right) \theta(1,1)\right] \end{equation}$

(3.51)

by multiplying (3.51) by $\mu^{k-\alpha-1}\psi^{(k)}(\mu a+(1-\mu)\Delta),$ integration yields to

$\begin{eqnarray} \psi^{(k)}\left( \frac{a+\Delta}{2}\right)\displaystyle {\int}_{0}^{1}\mu^{k-\alpha-1}\psi^{(k)}(\mu a+(1-\mu )\Delta )d\mu = \frac{\psi^{(k)}\left( \frac{a+\Delta}{2}\right)} {(\Delta-a)^{k-\alpha}}\,\Gamma(k-\alpha)(^{C}D_{\Delta_-} ^{\alpha}\psi)(a) \end{eqnarray}$

and

$\begin{align} &\left[1+h\left( \frac{1}{2}\right)\, \theta(1,1)\right]\displaystyle {\int}_{0}^{1}\mu^{k-\alpha-1}(\psi^{(k)})^{2}(\mu a +(1-\mu )\Delta ) d\mu\\ & = \left[1+h\left( \frac{1}{2}\right) \theta(1,1)\right]\frac{\Gamma(k-\alpha)}{(\Delta-a)^{k-\alpha}} R_{\Delta_-}^{k-\alpha}(\psi^{(k)})^{2}(a), \end{align}$

(3.52)

it results that

$\begin{equation} \frac{\psi^{(k)}\left( \frac{a+\Delta}{2}\right)}{ 1+h(\frac{1}{2})\,\theta(1,1)}\, (^{C}D_{\Delta-}^{\alpha}\psi)(a) \le \,R_{\Delta_-}^{k-\alpha}(\psi^{(k)})^{2}(a). \end{equation}$

(3.53)

By adding (3.50) and (3.53), we get (3.48). That proves the claim. □

Corollary 3.7. Under the same assumptions as Theorem 3.4, if $h(t) = t^{s}, s\in [0, 1],$ then

$\begin{eqnarray} \label{eca1} \frac{2^{s}\,\psi^{(k)}\left( \dfrac{a+\Delta}{2}\right)}{ 2^{s}+\theta(1,1)} \left[ \left( ^CD_{\Delta-}^{\alpha}\psi\right)(a)+ \left( ^CD_{a_{+}}^{\alpha}\psi\right)(\Delta)\right] \le R_{\Delta_-}^{k-\alpha}(\psi^{(k)})^{2}(a)+ R_{a+}^{k-\alpha}(\psi^{(k)})^{2}(\Delta). \end{eqnarray}$

If $\theta(u, v) = -\theta(v, u)$ , then

$\begin{equation} \psi^{(k)}\left( \dfrac{a+\Delta}{2}\right) \left(^CD_{\Delta_-}^{\alpha}\psi(a) +^CD_{a_{+}}^{\alpha}\psi(\Delta) \right) \le R_{\Delta_-}^{k-\alpha}(\psi^{(k)})^{2}(\alpha)+ R_{a_{+}}^{k-\alpha}(\psi^{(k)})^{2}(\Delta) \end{equation}$

(3.54)

is valid.

4. Conclusions

In this work, we have established some estimates including once the derivatives of Caputo and another time the integrals of Riemann-Liouville and the derivatives of Caputo for a function whose derivative order $k$ th $(k \in \mathbb{N})$ is generalized modified $h$ -convex and symmetrical in the middle. Estimates of consequences for special classes of convex functions and $s$ -convex functions in $[0, 1]$ were obtained. The estimates we have just made are compared to those presented in the results ^[8].

Future research could focus on extending these results to variable order or other types of convex functions or exploring inequalities for functions that do not necessarily have symmetry. Furthermore, the application of derived inequalities to concrete problems in applied mathematics, physics, or engineering could still validate the practical significance of our theoretical contributions. Taking these limitations into account could lead to a more complete understanding and wider applicability of fractional inequalities.

Author contributions

HB: conceptualization, writing original draft preparation, writing review and editing, supervision; MSS: conceptualization, writing original draft preparation, writing review and editing, supervision; HG: conceptualization, writing review and editing, supervision; UFG: funding, writing review and editing. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The work of U.F.-G. was supported by the government of the Basque Country for the ELKARTEK24/78 and ELKARTEK24/26 research programs, respectively.

Conflict of interest

The authors declare no competing interests.

References

[1]	W. Shewhart, Some applications of statistical methods to the analysis of physical and engineering data, Bell System Technical Journal, 3 (1924), 43–87. https://doi.org/10.1002/j.1538-7305.1924.tb01347.x doi: 10.1002/j.1538-7305.1924.tb01347.x
[2]	E. Page, Continuous inspection schemes, Biometrika, 41 (1954), 100–115. https://doi.org/10.2307/2333009 doi: 10.2307/2333009
[3]	S. Roberts, Control charts tests based on geometric moving average, Technometrics, 42 (2000), 97–101. https://doi.org/10.2307/1271439 doi: 10.2307/1271439
[4]	T. Abbas, B. Zaman, A. Atir, M. Riaz, S. Abbasi, On improved dispersion control charts under ranked set schemes for normal and non-normal processes, Qual. Reliab. Eng. Int., 35 (2019), 1313–1341. https://doi.org/10.1002/qre.2531 doi: 10.1002/qre.2531
[5]	M. Schoonhoven, M. Riaz, R. Does, Design and analysis of control charts for standard deviation with estimated parameters, J. Qual. Technol., 43 (2011), 307–333. https://doi.org/10.1080/00224065.2011.11917867 doi: 10.1080/00224065.2011.11917867
[6]	N. Abbas, M. Riaz, R. Does, CS-EWMA chart for monitoring process dispersion, Qual. Reliab. Eng. Int., 29 (2013), 653–663. https://doi.org/10.1002/qre.1414 doi: 10.1002/qre.1414
[7]	H. Nazir, M. Riaz, R. Does, Robust CUSUM control charting for process dispersion, Qual. Reliab. Eng. Int., 31 (2015), 369–379. https://doi.org/10.1002/qre.1596 doi: 10.1002/qre.1596
[8]	C. Acosta-Mejia, J. Pigniatiello, B. Rao, A comparison of control charting procedures for monitoring process dispersion, ⅡE Transactions, 31 (1999), 569–579. https://doi.org/10.1023/A:1007606524244 doi: 10.1023/A:1007606524244
[9]	D. Wolfe, Ranked set sampling, WIREs Comput. Stat., 2 (2010), 460–466. https://doi.org/10.1002/wics.92 doi: 10.1002/wics.92
[10]	G. McIntyre, A method for unbiased selective sampling, using ranked sets, Aust. J. Agr. Res., 3 (1952), 385–390. https://doi.org/10.1071/AR9520385 doi: 10.1071/AR9520385
[11]	H. Muttlak, W. Al-Sabah, Statistical quality control based on ranked set sampling, J. Appl. Stat., 30 (2003), 1055–1078. https://doi.org/10.1080/0266476032000076173 doi: 10.1080/0266476032000076173
[12]	S. Abbasi, Location charts based on ranked set sampling for normal and non-normal processes, Qual. Reliab. Eng. Int., 35 (2019), 1603–1620. https://doi.org/10.1002/qre.2463 doi: 10.1002/qre.2463
[13]	M. Abujiya, H. Muttlak, Quality control chart for the mean using double ranked set sampling, J. Appl. Stat., 31 (2004), 1185–1201. https://doi.org/10.1080/0266476042000285549 doi: 10.1080/0266476042000285549
[14]	A. Al-Nasser, M. Al-Rawwash, A control chart based on ranked set, Journal of Applied Science, 7 (2007), 1936–1941. https://doi.org/10.3923/jas.2007.1936.1941 doi: 10.3923/jas.2007.1936.1941
[15]	A. Al-Omari, A. Haq, A new sampling method for estimating the population mean, J. Stat. Comput. Sim., 89 (2019), 1973–1985. https://doi.org/10.1080/00949655.2019.1604710 doi: 10.1080/00949655.2019.1604710
[16]	M. Abujiya, M. Lee, M. Riaz, New EWMA S2 control charts for monitoring of process dispersion, Sci. Iran., 24 (2017), 378–389. https://doi.org/10.24200/sci.2017.4041 doi: 10.24200/sci.2017.4041
[17]	E. Zamanzade, A. Al-Omari, New ranked set sampling for estimating the population mean and variance, Hacet. J. Math. Stat., 45 (2016), 1891–1905. https://doi.org/10.15672/HJMS.20159213166 doi: 10.15672/HJMS.20159213166
[18]	T. Nawaz, M. Raza, D. Han, A new approach to design efficient univariate control charts to monitor the process mean, Qual. Reliab. Eng. Int., 34 (2018), 1732–1751. https://doi.org/10.1002/qre.2366 doi: 10.1002/qre.2366
[19]	N. Koyuncu, D. Karagöz, New mean charts for bivariate asymmetric distributions using different ranked set sampling designs, Qual. Technol. Quant. M., 15 (2018), 602–621. https://doi.org/10.1080/16843703.2017.1321220 doi: 10.1080/16843703.2017.1321220
[20]	N. Koyuncu, D. Karagoz, Neoteric ranked set sampling for robust $\bar{X}$ and R control charts, Soft Comput., 24 (2020), 17195–17204. https://doi.org/10.1007/s00500-020-05012-5 doi: 10.1007/s00500-020-05012-5
[21]	T. Nawaz, D. Han, Neoteric ranked set sampling based combined Shewhart-CUSUM and Shewhart-EWMA control charts for monitoring the process location, Eur. J. Industrial Eng., 14 (2020), 649–683. https://doi.org/10.1504/EJIE.2020.109913 doi: 10.1504/EJIE.2020.109913
[22]	G. Da Silva, C. Taconeli, W. Zeviani, I. Nascimento, Performance of Shewhart control charts based on neoteric ranked set sampling to monitor the process mean for normal and non-normal processes, Chil. J. Stat., 10 (2019), 131–154.
[23]	R. Putri, M. Mashuri, Irhamah, The comparison of exponentially weighted moving variance and double moving average-S control charts based on neoteric ranked set sampling, J. Phys.: Conf. Ser., 1538 (2020), 012056. https://doi.org/10.1088/1742-6596/1538/1/012056
[24]	T. Abbas, T. Mahmood, M. Riaz, M. Abid, Improved linear profiling methods under classical and Bayesian setups: an application to chemical gas sensors, Chemometr. Intell. Lab., 196 (2020), 103908. https://doi.org/10.1016/j.chemolab.2019.103908 doi: 10.1016/j.chemolab.2019.103908
[25]	S. Hussain, T. Mahmood, M. Riaz, H. Nazir, A new approach to design median control charts for location monitoring, Commun. Stat.-Simul. Comput., 51 (2022), 3553–3577. https://doi.org/10.1080/03610918.2020.1716245 doi: 10.1080/03610918.2020.1716245
[26]	J. Jain, Quality control and total quality management, New Delhi: McGraw-Hill education Pvt Limited, 2001.
[27]	N. Adegoke, S. Abbasi, A. Smith, M. Anderson, M. Pawley, A multivariate homogeneously weighted moving average control chart, IEEE Access, 7 (2019), 9586–9597. https://doi.org/10.1109/ACCESS.2019.2891988 doi: 10.1109/ACCESS.2019.2891988
[28]	T. Dell, J. Clutter, Ranked set sampling theory with order statistics background, Biometrics, 28 (1972), 545–555. https://doi.org/10.2307/2556166 doi: 10.2307/2556166
[29]	K. Takahasi, K. Wakimoto, On unbiased estimates of population mean based on sample stratified by means of ordering, Ann. Inst. Stat. Math., 20 (1968), 1–31. https://doi.org/10.1007/BF02911622 doi: 10.1007/BF02911622
[30]	S. Stokes, Estimation of variance using judgment ordered ranked set samples, Biometrics, 36 (1980), 35–42. https://doi.org/10.2307/2530493 doi: 10.2307/2530493
[31]	H. Muttlak, Median ranked set sampling, J. Appl. Stat. Sci., 6 (1997), 245–255.
[32]	H. Samawi, M. Ahmed, W. Abu-Dayyeh, Estimating the population mean using extreme ranked set sampling, Biometrical J., 38 (1996), 577–586. https://doi.org/10.1002/bimj.4710380506 doi: 10.1002/bimj.4710380506
[33]	M. Abid, H. Nazir, M. Riaz, Z. Lin, Use of ranked set sampling in nonparametric control charts, J. Chin. Inst. Eng., 39 (2016), 627–636. https://doi.org/10.1080/02533839.2016.1152165 doi: 10.1080/02533839.2016.1152165
[34]	M. Abujiya, New cumulative sum control chart for monitoring Poisson processes, IEEE Access, 5 (2017), 14298–14308. https://doi.org/10.1109/ACCESS.2017.2733520 doi: 10.1109/ACCESS.2017.2733520
[35]	S. Hussain, T. Mahmood, M. Riaz, H. Nazir, A new approach to design median control charts for location monitoring, Commun. Stat.-Simul. Comput., 51 (2020), 3553–3577. https://doi.org/10.1080/03610918.2020.1716245 doi: 10.1080/03610918.2020.1716245
[36]	S. Abbasi, A. Miller, On proper choice of variability control chart for normal and non-normal processes, Qual. Reliab. Eng. Int., 28 (2012), 279–296. https://doi.org/10.1002/qre.1244 doi: 10.1002/qre.1244
[37]	B. Zaman, N. Abbas, M. Riaz, M. Lee, Mixed CUSUM-EWMA chart for monitoring process dispersion, Int. J. Adv. Manuf. Technol., 86 (2016), 3025–3039. https://doi.org/10.1007/s00170-016-8411-0 doi: 10.1007/s00170-016-8411-0
[38]	C. Sim, W. Wong, R-charts for the exponential, Laplace and logistic processes, Stat. Pap., 44 (2003), 535–554. https://doi.org/10.1007/BF02926009 doi: 10.1007/BF02926009
[39]	A. Haq, M. Khoo, M. Lee, S. Abbasi, Enhanced adaptive multivariate EWMA and CUSUM charts for process mean, J. Stat. Comput. Sim., 91 (2021), 2361–2382. https://doi.org/10.1080/00949655.2021.1894564 doi: 10.1080/00949655.2021.1894564
[40]	T. Mahmood, M. Riaz, A. Iqbal, K. Mulenga, An improved statistical approach to compare means, AIMS Mathematics, 8 (2023), 4596–4629. https://doi.org/10.3934/math.2023227 doi: 10.3934/math.2023227
[41]	Z. Chen, M. Peng, L. Xi, A new procedure for unit root to long-memory process change-point monitoring, AIMS Mathematics, 7 (2022), 6467–6477. https://doi.org/10.3934/math.2022360 doi: 10.3934/math.2022360
[42]	M. Aslam, K. Khan, M. Albassam, L. Ahmad, Moving average control chart under neutrosophic statistics, AIMS Mathematics, 8 (2023), 7083–7096. https://doi.org/10.3934/math.2023357 doi: 10.3934/math.2023357
[43]	T. Abbas, F. Rafique, T. Mahmood, M. Riaz, Efficient phase Ⅱ monitoring methods for linear profiles under the random effect model, IEEE Access, 7 (2019), 148278–148296. https://doi.org/10.1109/ACCESS.2019.2946211 doi: 10.1109/ACCESS.2019.2946211
[44]	T. Abbas, B. Zaman, A. Atir, M. Riaz, S. Abbasi, On improved dispersion control charts under ranked set schemes for normal and non-normal processes, Qual. Reliab. Eng. Int., 35 (2019), 1313–1341. https://doi.org/10.1002/qre.2531 doi: 10.1002/qre.2531
[45]	S. Ali, A predictive Bayesian approach to EWMA and CUSUM charts for time-between-events monitoring, J. Stat. Comput. Sim., 90 (2020), 3025–3050. https://doi.org/10.1080/00949655.2020.1793987 doi: 10.1080/00949655.2020.1793987
[46]	T. Marlin, Process control: controling processes and control systems for dynamic performance, 2 Eds., New York: McGraw-Hill Book Company, 2000.
[47]	X. Shi, Y. Lv, Z. Fei, J. Ling, A multivariable statistical process monitoring method based on multiscale analysis and principal curves, Int. J. Innov. Comput. I., 9 (2013), 1781–1800.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)