Dynamic complexity of a slow-fast predator-prey model with herd behavior

Ahmad Suleman; Rizwan Ahmed; Fehaid Salem Alshammari; Nehad Ali Shah; Ahmad Suleman; Rizwan Ahmed; Fehaid Salem Alshammari; Nehad Ali Shah

doi:10.3934/math.20231247

AIMS Mathematics

2023, Volume 8, Issue 10: 24446-24472. doi: 10.3934/math.20231247

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Research article Special Issues

Dynamic complexity of a slow-fast predator-prey model with herd behavior

1.
Department of Mathematics, Air University Multan Campus, Multan, Pakistan
2.
Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
3.
Department of Mechanical Engineering, Sejong University, Seoul 05006, South Korea
^† These authors contributed equally to this work and are co-first authors

Received: 12 June 2023 Revised: 23 July 2023 Accepted: 09 August 2023 Published: 17 August 2023
MSC : 39A28, 39A30

The complex dynamics of a slow-fast predator-prey interaction with herd behavior are examined in this work. We investigate the presence and stability of fixed points. By employing the bifurcation theory, it is shown that the model undergoes both a period-doubling and a Neimark-Sacker bifurcation at the interior fixed point. Under the influence of period-doubling and Neimark-Sacker bifurcations, chaos is controlled using the hybrid control approach. Moreover, numerical simulations are carried out to highlight the model's complexity and show how well they agree with analytical findings. Employing the slow-fast factor as the bifurcation parameter shows that the model goes through a Neimark-Sacker bifurcation for greater values of the slow-fast factor at the interior fixed point. This makes sense because if the slow-fast factor is large, the growth rates of the predator and its prey will be about identical, automatically causing the interior fixed point to become unstable owing to the predator's slow growth.

Keywords:

Citation: Ahmad Suleman, Rizwan Ahmed, Fehaid Salem Alshammari, Nehad Ali Shah. Dynamic complexity of a slow-fast predator-prey model with herd behavior[J]. AIMS Mathematics, 2023, 8(10): 24446-24472. doi: 10.3934/math.20231247

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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]	E. Aulisa, S. R. J. Jang, Continuous-time predator-prey systems with allee effects in the prey, Math. Comput. Simul., 105 (2014), 1–16. https://doi.org/10.1016/j.matcom.2014.04.004 doi: 10.1016/j.matcom.2014.04.004
[2]	F. Wang, R. Yang, Y. Xie, J. Zhao, Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak allee effect on prey and fear effect on predator, AIMS Math., 8 (2023), 17719–17743. https://doi.org/10.3934/math.2023905 doi: 10.3934/math.2023905
[3]	S. S. Askar, On complex dynamics of differentiated products: cournot duopoly model under average profit maximization, Discrete Dyn. Nat. Soc., 2022 (2022), 1–14. https://doi.org/10.1155/2022/8677470 doi: 10.1155/2022/8677470
[4]	R. Ahmed, N. Ali, F. M. Rana, Analysis of a cournot-bertrand duoploy game with differentiated products: stability, bifurcation and control, Asian Res. J. Math., 18 (2022), 115–125. https://doi.org/10.9734/arjom/2022/v18i1030422 doi: 10.9734/arjom/2022/v18i1030422
[5]	N. A. Shah, A. A. Zafar, S. Akhtar, General solution for MHD-free convection flow over a vertical plate with ramped wall temperature and chemical reaction, Arab. J. Math., 7 (2018), 49–60. https://doi.org/10.1007/s40065-017-0187-z doi: 10.1007/s40065-017-0187-z
[6]	C. Fetecau, N. A. Shah, D. Vieru, General solutions for hydromagnetic free convection flow over an infinite plate with newtonian heating, mass diffusion and chemical reaction, Commun. Theor. Phys., 68 (2017), 768. https://doi.org/10.1088/0253-6102/68/6/768 doi: 10.1088/0253-6102/68/6/768
[7]	I. Khan, N. A. Shah, L. C. C. Dennis, A scientific report on heat transfer analysis in mixed convection flow of maxwell fluid over an oscillating vertical plate, Sci. Rep., 7 (2017), 40147. https://doi.org/10.1038/srep40147 doi: 10.1038/srep40147
[8]	A. J. Lotka, Elements of physical biology, Science Progress in the Twentieth Century (1919–1933), 21 (1926), 341–343.
[9]	V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926), 558–560. https://doi.org/10.1038/118558a0 doi: 10.1038/118558a0
[10]	H. Freedman, G. Wolkowicz, Predator-prey systems with group defence: the paradox of enrichment revisited, Bull. Math. Biol., 48 (1986), 493–508.
[11]	C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385–398. https://doi.org/10.4039/ent91385-7 doi: 10.4039/ent91385-7
[12]	P. H. Crowley, E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 8 (1989), 211–221. https://doi.org/10.2307/1467324 doi: 10.2307/1467324
[13]	J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331–340. https://doi.org/10.2307/3866 doi: 10.2307/3866
[14]	D. L. DeAngelis, R. A. Goldstein, R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881–892. https://doi.org/10.2307/1936298 doi: 10.2307/1936298
[15]	H. J. Alsakaji, S. Kundu, F. A. Rihan, Delay differential model of one-predator two-prey system with monod-haldane and holling type ii functional responses, Appl. Math. Comput., 397 (2021), 125919. https://doi.org/10.1016/j.amc.2020.125919 doi: 10.1016/j.amc.2020.125919
[16]	C. Arancibia-Ibarra, P. Aguirre, J. Flores, P. van Heijster, Bifurcation analysis of a predator-prey model with predator intraspecific interactions and ratio-dependent functional response, Appl. Math. Comput., 402 (2021), 126152. https://doi.org/10.1016/j.amc.2021.126152 doi: 10.1016/j.amc.2021.126152
[17]	X. Chen, X. Zhang, Dynamics of the predator-prey model with the sigmoid functional response, Stud. Appl. Math., 147 (2021), 300–318. https://doi.org/10.1111/sapm.12382 doi: 10.1111/sapm.12382
[18]	M. F. Elettreby, A. Khawagi, T. Nabil, Dynamics of a discrete prey-predator model with mixed functional response, Int. J. Bifurcat. Chaos, 29 (2019), 1950199. https://doi.org/10.1142/s0218127419501992 doi: 10.1142/s0218127419501992
[19]	P. Panja, Combine effects of square root functional response and prey refuge on predator-prey dynamics, Int. J. Model. Simul., 41 (2021), 426–433. https://doi.org/10.1080/02286203.2020.1772615 doi: 10.1080/02286203.2020.1772615
[20]	S. M. Sohel Rana, U. Kulsum, Bifurcation analysis and chaos control in a discrete-time predator-prey system of leslie type with simplified holling type iv functional response, Discrete Dyn. Nat. Soc., 2017 (2017), 9705985. https://doi.org/10.1155/2017/9705985 doi: 10.1155/2017/9705985
[21]	X. Han, C. Lei, Bifurcation and turing instability analysis for a space- and time-discrete predator-prey system with smith growth function, Chaos, Solitons Fract., 166 (2023), 112910. https://doi.org/10.1016/j.chaos.2022.112910 doi: 10.1016/j.chaos.2022.112910
[22]	V. Ajraldi, M. Pittavino, E. Venturino, Modeling herd behavior in population systems, Nonlinear Anal.: Real World Appl., 12 (2011), 2319–2338. https://doi.org/10.1016/j.nonrwa.2011.02.002 doi: 10.1016/j.nonrwa.2011.02.002
[23]	P. A. Braza, Predator-prey dynamics with square root functional responses, Nonlinear Anal.: Real World Appl., 13 (2012), 1837–1843. https://doi.org/10.1016/j.nonrwa.2011.12.014 doi: 10.1016/j.nonrwa.2011.12.014
[24]	M. G. Mortuja, M. K. Chaube, S. Kumar, Dynamic analysis of a predator-prey system with nonlinear prey harvesting and square root functional response, Chaos, Solitons Fract., 148 (2021), 111071. https://doi.org/10.1016/j.chaos.2021.111071 doi: 10.1016/j.chaos.2021.111071
[25]	D. Pal, P. Santra, G. S. Mahapatra, Predator-prey dynamical behavior and stability analysis with square root functional response, Int. J. Appl. Comput. Math., 3 (2017), 1833–1845. https://doi.org/10.1007/s40819-016-0200-9 doi: 10.1007/s40819-016-0200-9
[26]	S. M. Salman, A. M. Yousef, A. A. Elsadany, Stability, bifurcation analysis and chaos control of a discrete predator-prey system with square root functional response, Chaos, Solitons Fract., 93 (2016), 20–31. https://doi.org/10.1016/j.chaos.2016.09.020 doi: 10.1016/j.chaos.2016.09.020
[27]	N. C. Stenseth, W. Falck, O. N. Bjornstad, C. J. Krebs, Population regulation in snowshoe hare and canadian lynx: Asymmetric food web configurations between hareandlynx, Proceedings of the National Academy of Sciences, 94 (1997), 5147–5152. https://doi.org/10.1073/pnas.94.10.5147 doi: 10.1073/pnas.94.10.5147
[28]	G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347–386. https://doi.org/10.1007/s00285-009-0266-7 doi: 10.1007/s00285-009-0266-7
[29]	S. Rinaldi, S. Muratori, Slow-fast limit cycles in predator-prey models, Ecol. Model., 61 (1992), 287–308. https://doi.org/10.1016/0304-3800(92)90023-8 doi: 10.1016/0304-3800(92)90023-8
[30]	W. Liu, D. Cai, Bifurcation, chaos analysis and control in a discrete-time predator-prey system, Adv. Differ. Equ., 2019 (2019), 11. https://doi.org/10.1186/s13662-019-1950-6 doi: 10.1186/s13662-019-1950-6
[31]	A. Q. Khan, I. Ahmad, H. S. Alayachi, M. S. M. Noorani, A. Khaliq, Discrete-time predator-prey model with flip bifurcation and chaos control, Math. Biosci. Eng., 17 (2020), 5944–5960. https://doi.org/10.3934/mbe.2020317 doi: 10.3934/mbe.2020317
[32]	Z. AlSharawi, S. Pal, N. Pal, J. Chattopadhyay, A discrete-time model with non-monotonic functional response and strong allee effect in prey, J. Differ. Equ. Appl., 26 (2020), 404–431. https://doi.org/10.1080/10236198.2020.1739276 doi: 10.1080/10236198.2020.1739276
[33]	R. Ahmed, A. Ahmad, N. Ali, Stability analysis and neimark-sacker bifurcation of a nonstandard finite difference scheme for lotka-volterra prey-predator model, Commun. Math. Biol. Neurosci., 2022 (2022), 61. https://doi.org/10.28919/cmbn/7534 doi: 10.28919/cmbn/7534
[34]	P. A. Naik, Z. Eskandari, M. Yavuz, J. Zu, Complex dynamics of a discrete-time Bazykin-Berezovskaya prey-predator model with a strong Allee effect, J. Comput. Appl. Math., 413 (2022), 114401. https://doi.org/10.1016/j.cam.2022.114401 doi: 10.1016/j.cam.2022.114401
[35]	S. M. Sohel Rana, Dynamics and chaos control in a discrete-time ratio-dependent holling-tanner model, J. Egypt. Math. Soc., 27 (2019), 48. https://doi.org/10.1186/s42787-019-0055-4 doi: 10.1186/s42787-019-0055-4
[36]	P. Baydemir, H. Merdan, E. Karaoglu, G. Sucu, Complex dynamics of a discrete-time prey-predator system with leslie type: stability, bifurcation analyses and chaos, Int. J. Bifurcat. Chaos, 30 (2020), 2050149. https://doi.org/10.1142/s0218127420501497 doi: 10.1142/s0218127420501497
[37]	M. Zhao, C. Li, J. Wang, Complex dynamic behaviors of a discrete-time predator-prey system, J. Appl. Anal. Comput., 7 (2017), 478–500. https://doi.org/10.11948/2017030 doi: 10.11948/2017030
[38]	A. A. Khabyah, R. Ahmed, M. S. Akram, S. Akhtar, Stability, bifurcation, and chaos control in a discrete predator-prey model with strong allee effect, AIMS Math.., 8 (2023), 8060–8081. https://doi.org/10.3934/math.2023408 doi: 10.3934/math.2023408
[39]	S. Akhtar, R. Ahmed, M. Batool, N. A. Shah, J. D. Chung, Stability, bifurcation and chaos control of a discretized leslie prey-predator model, Chaos, Solitons Fract., 152 (2021), 111345. https://doi.org/10.1016/j.chaos.2021.111345 doi: 10.1016/j.chaos.2021.111345
[40]	A. Tassaddiq, M. S. Shabbir, Q. Din, H. Naaz, Discretization, bifurcation, and control for a class of predator-prey interactions, Fractal Fract., 6 (2022), 31. https://doi.org/10.3390/fractalfract6010031 doi: 10.3390/fractalfract6010031
[41]	Z. Zhu, Y. Chen, Z. Li, F. Chen, Stability and bifurcation in a Leslie-Gower predator-prey model with allee effect, Int. J. Bifurcat. Chaos, 32 (2022), 2250040. https://doi.org/10.1142/s0218127422500407 doi: 10.1142/s0218127422500407
[42]	C. Lei, X. Han, W. Wang, Bifurcation analysis and chaos control of a discrete-time prey-predator model with fear factor, Math. Biosci. Eng., 19 (2022), 6659–6679. https://doi.org/10.3934/mbe.2022313 doi: 10.3934/mbe.2022313
[43]	X. Han, C. Lei, Stability, bifurcation analysis and pattern formation for a nonlinear discrete predator-prey system, Chaos, Solitons Fract., 173 (2023), 113710. https://doi.org/10.1016/j.chaos.2023.113710 doi: 10.1016/j.chaos.2023.113710
[44]	A. C. J. Luo, Regularity and complexity in dynamical systems, Springer, 2012. https://doi.org/10.1007/978-1-4614-1524-4
[45]	S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Springer, 2003. https://doi.org/10.1007/b97481
[46]	J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer, 1983. https://doi.org/10.1007/978-1-4612-1140-2
[47]	R. Ahmed, M. S. Yazdani, Complex dynamics of a discrete-time model with prey refuge and holling type-ii functional response, J. Math. Comput. Sci., 12 (2022), 113. https://doi.org/10.28919/jmcs/7205 doi: 10.28919/jmcs/7205
[48]	X. S. Luo, G. Chen, B. H. Wang, J. Q. Fang, Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems, Chaos, Solitons Fract., 18 (2003), 775–783. https://doi.org/10.1016/s0960-0779(03)00028-6 doi: 10.1016/s0960-0779(03)00028-6

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