Analysis of positive measure reducibility for quasi-periodic linear systems under Brjuno-Rüssmann condition

1.   Introduction

Several decades ago, classical calculus underwent a transformative phase, propelled by remarkable innovations. Researchers unanimously acknowledge the remarkable efficacy and accuracy of outcomes derived from fractional-order equations. Presently, fractional calculus finds widespread application across various domains, including chaos theory, simulation, and modeling. A myriad of elegant definitions and operators, such as Riemann, Caputo, Hadamard, Katugampola, Atangana-Baleanu, and many others, exemplify the beauty of fractional calculus ^{[1,2,3,4,5,6,7]}. For a comprehensive overview of the origins, advancements, and applications of fractional calculus, we direct the reader to the esteemed monographs ^[8,9] and compelling articles ^{[10,11,12,13,14,15]}. The Hermite-Hadamard inequality stands as a cornerstone in mathematical analysis, offering profound insights into the properties of integrable functions.

Convexity stands as a cornerstone in solving several problems in general and applied mathematics. Its robustness has led to the generalization and extension of convex functions and convex sets across numerous branches of mathematics, with many inequalities stemming from convexity theory present in the literature ^{[16,17,18,19,20]}. Among these inequalities, the Hermite-Hadamard (H-H) inequality shines as a strikingly useful result in the realm of mathematical inequalities ^{[21,22,23,24,25]}. This inequality holds pivotal significance due to its close connections with other notable inequalities such as the Hölder, Opial, Hardy, Minkowski, Ostrowski, and Young inequalities.

The H-H inequality, expressed as follows ^[26]:

holds for $\wp$ to be a convex function on the interval $\left[\varkappa_1, \varkappa_2 \right]$. This double inequality encapsulates profound insights into the behavior of convex functions over intervals, serving as a fundamental tool in various mathematical contexts.

In recent past several generalizations, refinements, and extensions of (1.1) are developed, which attracted the attention of a wide range of researchers both in applied and pure mathematics.

Suppose $\breve{A}\subseteq \mathbb{R}$ and $\wp :\breve{A}\rightarrow \mathbb{R}$ is a differentiable function on $\breve{A}^{\circ }$ (the interior of $\breve{A}$) such that $\varkappa_1, \varkappa_2 \in$ $\breve{A} ^{\circ }$ with $\varkappa_1 < \varkappa_2$. In this case, the well-known Ostrowski inequality ^[27] states that

for all $\nu \in \left[\varkappa_1, \varkappa_2 \right]$ if $\left\vert \wp ^{\prime }\left(\mu \right) \right\vert \leq S$ for all $\mu \in \left[\varkappa_1, \varkappa_2 \right].$

Ostrowski-type inequalities, which give error estimates for numerous quadrature rules, have important applications in numerical analysis. These disparities have been widened and applied to a wider range of disciplines in recent years.

Researchers in this intriguing field of study explore the applications of these variations in applied sciences and also examine the existence and uniqueness of solutions to fractional differential equations. By employing $K$-fractional integrals, the authors in ^[28], proposed several generalizations of Ostrowski-type estimations.

Definition 1.1. ^[29] Let $s\in \left[0, 1\right].$ A real valued function $\wp : \breve{A}\rightarrow \mathbb{R}$ is called $s$-type convex on $\breve{A}$ if

for all $\varkappa_1, \varkappa_2 \in \breve{A}$ and $\mu \in \left[0, 1\right].$

In ^[30], İşcan gave the definition of $n$-fractional polynomial convex functions as follows.

Definition 1.2. Let $n\in \mathbb{N}$. A non-negative function $\wp :\breve{A}\subseteq \mathbb{R}\rightarrow \mathbb{R}$ is called $n$-fractional polynomial convex ($FPC$) function if

for all $\varkappa_1, \varkappa_2 \in \breve{A}$ and $\mu \in \left[0, 1\right].$

Note that, every non-negative convex function is an $FPC$ function ^[30].

Now, we demonstrate key concepts related to the fractional integral, primarily originating from the work of Mubeen et al. ^[31].

Let $\alpha, R > 0,$ $\varkappa_1 < \varkappa_2,$ and $\wp \in L\left[\varkappa_1, \varkappa_2 \right].$ Then, the $K$-fractional integrals of order $\alpha$ are given by

and

where $\Gamma _{K}\left(\alpha \right)$ is the $K$-Gamma function ^[32] given by

Recall that

and for $K = 1,$ the $K$-fractional integrals coincide with the $RL$-fractional integrals.

Now, we recall the concepts of the Euler's Beta function $\beta$ and hypergeometric function $_{2}F_{1}$, respectively.

and

where $\Gamma \left(\varkappa \right) = \int_{0}^{\infty }\mu ^{\alpha -1}e^{-\mu }d\mu$ is the Euler Gamma function ^[33,34].

Motivated by the aforementioned findings and existing literature, in Section 2, we will initially introduce the concept of a generalized $n$-fractional polynomial $s$-type convex function. Subsequently, in Section 3, we will establish a novel generalization of the H-H type inequality for the new class of functions. Moving forward to Section 4, we will obtain novel estimates for differentiable generalized $n$-fractional polynomial $s$-type convex functions. Notably, the results presented herein encompass $RL$-fractional integral inequalities and quadrature rules as special cases. The findings of our study prove beneficial in crafting fractals through iterative methodologies, an engaging research domain with implications for refining machine learning algorithms. Finally, Section 5 concludes with a brief conclusion.

2.   Definition of generalized $n$-fractional polynomial $s$-type convex functions

In this section, we introduce a new concept called the generalized $n$ -fractional polynomial $s$-type convex function and explore its fundamental algebraic properties.

Definition 2.1. Let $n\in \mathbb{N}$, $s\in \left[0, 1\right],$ $a_{u}\geq 0$ $\left(u = \overline{1, n} \right)$ such that $\sum_{u = 1}^{n}a_{u} > 0,$ $\breve{A}\subset \mathbb{ R}$ be an interval$.$ A non-negative function $\wp :\breve{A}\subset \mathbb{ R}\rightarrow \mathbb{R}$ is called a generalized $n$-fractional polynomial $s$ -type convex function if for every $\varkappa_1, \varkappa_2 \in \breve{A}$ and $\mu \in \left[0, 1\right]$,

We denote the class of all generalized $n$-fractional polynomial $s$-type convex functions by $GFPC-s$.

Example 2.2. Consider the function $\wp(x) = x^2$, and the parameters $s = 0.5$, $n = 2$, $a_1 = 1$, $a_2 = 2$, and $\mu = 0.5$. According to Definition 2.1,

For $\kappa_1 = 1$ and $\kappa_2 = 3$,

So, one has

According to Definition 1.2,

For $\kappa_1 = 1$ and $\kappa_2 = 3$,

So, one gets

Because the generalized $n$-fractional polynomial $s$-type convexity $(GFPC-s)$ is a generalization of the $n$-fractional polynomial convexity $(FPC)$, the resulting bounds extend those obtained for $FPC$.

Note that every $GFPC-s$ is an h-convex function with

Remark 2.3. For $s = 1$, Definition 2.1 reduces to the definition of generalized $n$ -fractional polynomial convex $(GFPC)$ functions.

Remark 2.4. For $s = 1$ and $a_{u} = 1$ $\left(u = \overline{1, n} \right)$, Definition 2.1 coincides with Definition 1.2.

Remark 2.5. For $s = 1$ and $n = 1$, Definition 2.1 coincides with the definition of classical convexity.

Remark 2.6. Every non-negative $n$-fractional polynomial convex function is a $GFPC-s$ function. It is clear from the inequalities

and

for all $n\in \mathbb{N}$, $s\in \left[0, 1\right],$ and $\mu \in \left[0, 1\right].$

Note that not every $GFPC-s$ function needs to be an $FPC$ function.

Remark 2.7. If $\wp$ is a $GFPC-s$ function, then $\wp$ is a non-negative function. Indeed, from the definition of $GFPC-s$ function, one can write

for all $\nu \in \breve{A}$ and $\mu \in \left[0, 1\right].$ Therefore, one has

Since

for all $\mu \in \left[0, 1\right],$ one obtains $\wp \left(\nu \right) \geq 0$ for all $\nu \in \breve{A}.$

3.   New generalizations of H-H type inequalities using generalized $n$-fractional polynomial $s$-type convex functions

Now, we obtain a new generalization of H-H inequality for the $GFPC-s$ function $\wp$.

Theorem 3.1. Let $n\in \mathbb{N}$, $a_{u}\geq 0$ $\left(u = \overline{1, n}\right),$ $s\in \left[0, 1 \right]$, $\alpha \in \left[0, 1\right], K > 0,$ and $\wp :\breve{A} = \left[\varkappa_1, \varkappa_2 \right] \rightarrow \mathbb{R}$ be a $GFPC-s$ function such that $\wp \in L\left[\varkappa_1, \varkappa_2 \right].$ Then,

Proof. From the definition of the $GFPC-s$ function, one obtains

Multiplying both sides of the above inequality by $\mu ^{\frac{\alpha }{K} -1}$ and taking the integral with respect to $\mu \in \left[0, 1\right]$, one gets

So, one has

which completes the left-hand side of inequality (3.1). Now, we prove the right-hand side of inequality (3.1). Let $\mu \in \left[0, 1\right].$ From the definition of the $GFPC-s$ function, one obtains

and

Adding the above inequalities, one gets

Multiplying both sides of the above inequality by $\mu ^{\frac{\alpha }{K} -1},$ taking the integral with respect to $\mu \in \left[0, 1\right],$ and changing the variable of integration, one obtains

This completes the proof. □

Corollary 3.2. If one takes $s = 1$ in Theorem 3.1, one gets the H-H inequality for $GFPC$ functions with $K$-fractional integral operators:

Corollary 3.3. If one takes $K = 1$ in Corollary 3.2, one gets H-H inequality for $GFPC$ functions with $RL$-fractional integral operators:

Remark 3.4. If one takes $\alpha = 1$ and $n = 1$ in Corollary 3.3, then one gets the inequality (1.1).

4.   New generalizations of Ostrowski type inequalities using generalized $n$-fractional polynomial $s$-type convex functions

In this section, we find novel estimates that refine the Ostrowski-type inequalities for the functions whose first and second derivatives in absolute value at certain powers are $GFPC-s$. First, we give the following crucial lemma ^[35]:

Lemma 4.1. Let $\alpha \in \left[0, 1\right],$ $K > 0,$ $\varkappa_1 < \varkappa_2,$ and $\wp :\left[0, 1\right] \rightarrow \mathbb{R}$ be a differentiable function on $\breve{A}^{\circ }$ such that $\wp ^{\prime }\in L\left[\varkappa_1, \varkappa_2 \right].$ Then,

Theorem 4.2. Let $n\in \mathbb{N}$, $a_{u}\geq 0$ $\left(u = \overline{1, n}\right),$ $s\in \left[0, 1 \right]$, $\alpha, K > 0,$ $\varkappa_1 < \varkappa_2,$ and $\wp :\breve{A} = \left[\varkappa_1, \varkappa_2 \right] \rightarrow \mathbb{R}$ be a differentiable function on $\breve{A}^{\circ }$ such that $\wp ^{\prime }\in L\left[\varkappa_1, \varkappa_2 \right].$ Let $\left\vert \wp ^{\prime }\left(\nu \right) \right\vert$ be a $GFPC-s$ function on $\breve{A}$ with $\left\vert \wp ^{\prime }\left(\nu \right) \right\vert \leq S$ for all $\nu \in \left[\varkappa_1, \varkappa_2 \right].$ Then,

Proof. Using Lemma 4.1 and a property of the $GFPC-s$ function $\left\vert \wp ^{\prime }\right\vert,$ one has

□

Corollary 4.3. If one takes $s = 1$ in Theorem 3.1, one gets the following inequality for $GFPC$ functions with $K$-fractional integral operators:

Corollary 4.4. If one takes $K = 1$ in Corollary 4.3, one gets the following inequality for $GFPC$ functions with $RL$-integral operators:

Remark 4.5. If one takes $\alpha = 1$ and $n = 1$ in Corollary 4.4, then one gets inequality (1.2).

Theorem 4.6. Let $n\in \mathbb{N}$, $a_{u}\geq 0$ $\left(u = \overline{1, n}\right),$ $s\in \left[0, 1 \right]$, $\alpha, K > 0,$ $\varkappa_1 < \varkappa_2,$ $q > 1,$ and $\wp :\breve{A} = \left[\varkappa_1, \varkappa_2 \right] \rightarrow \mathbb{R}$ be a differentiable function on $\breve{A}^{\circ }$ such that $\wp ^{\prime }\in L\left[\varkappa_1, \varkappa_2 \right].$ Let $\left\vert \wp ^{\prime }\left(\nu \right) \right\vert ^{q}$ be a $GFPC-s$ function on $\breve{A}$ with $\left\vert \wp ^{\prime }\left(\nu \right) \right\vert \leq S$ for all $\nu \in \left[\varkappa_1, \varkappa_2 \right].$ Then,

Proof. Using Lemma 4.1, a property of the $GFPC-s$ function $\left\vert \wp ^{\prime }\right\vert ^{q},$ and the power mean inequality, one has

Thus, the proof is completed. □

Corollary 4.7. If one takes $s = 1$ in Theorem 4.6, one gets the following inequality for $GFPC$ functions with $K$-fractional integral operators:

Corollary 4.8. If one takes $K = 1$ in Corollary 4.7, one gets the following inequality for $GFPC$ functions with $RL$-integral operators:

Theorem 4.9. Let $n\in \mathbb{N}$, $a_{u}\geq 0$ $\left(u = \overline{1, n}\right),$ $s\in \left[0, 1 \right]$, $\alpha, K > 0,$ $\varkappa_1 < \varkappa_2,$ $t, q > 1$ with $\frac{1}{t}+ \frac{1}{q} = 1,$ and $\wp :\breve{A} = \left[\varkappa_1, \varkappa_2 \right] \rightarrow \mathbb{R}$ be a differentiable function on $\breve{A}^{\circ }$ such that $\wp ^{\prime }\in L\left[\varkappa_1, \varkappa_2 \right].$ Let $\left\vert \wp ^{\prime }\left(\nu \right) \right\vert ^{q}$ be a $GFPC-s$ function on $\breve{A}$ with $\left\vert \wp ^{\prime }\left(\nu \right) \right\vert \leq S$ for all $\nu \in \left[\varkappa_1, \varkappa_2 \right].$ Then,

Proof. Using Lemma 4.1, a property of the $GFPC-s$ function $\left\vert \wp ^{\prime }\right\vert ^{q},$ and the Hölder inequality, one has

□

Corollary 4.10. If one takes $s = 1$ in Theorem 4.9, one gets the following inequality for $GFPC$ functions with $K$-fractional integral operators:

Corollary 4.11. If one takes $K = 1$ in Corollary 4.10, one gets the following inequality for $GFPC$ functions with $RL$-integral operators:

Now, we establish new Ostrowski type inequalities for twice differentiable functions. First, we give the following lemma, which will be used in what follows ^[36].

Lemma 4.12. Let $\alpha, K > 0,$ $\varkappa_1 < \varkappa_2,$ $\breve{A} = \left[\varkappa_1, \varkappa_2 \right],$ and $\wp :\breve{A}\rightarrow \mathbb{R}$ be a twice differentiable function on $\breve{A}^{\circ }$ such that $\wp ^{\prime \prime }\in L\left[\varkappa_1, \varkappa_2 \right].$ Then,

holds for all $\nu \in \left[\varkappa_1, \varkappa_2 \right]$ and $\kappa \in \left[0, 1\right].$

Theorem 4.13. Let $n\in \mathbb{N}$, $a_{u}\geq 0$ $\left(u = \overline{1, n}\right),$ $s\in \left[0, 1 \right]$, $\alpha, K > 0,$ $\varkappa_1 < \varkappa_2,$ and $\wp :\breve{A} = \left[\varkappa_1, \varkappa_2 \right] \rightarrow \mathbb{R}$ be a twice differentiable function on $\breve{A}^{\circ }$ such that $\wp ^{\prime \prime }\in L\left[\varkappa_1, \varkappa_2 \right].$ Let $\left\vert \wp ^{\prime \prime }\left(\nu \right) \right\vert$ be a $GFPC-s$ function on $\breve{A}.$ Then,

Proof. Using Lemma 4.12, a property of the $GFPC-s$ function $\left\vert \wp ^{\prime \prime }\right\vert,$ and the property of modulus, one has

So, the proof is completed. □

Corollary 4.14. If one takes $s = 1$ in Theorem 4.13, one gets the following inequality for $GFPC$ functions with $K$-fractional integral operators:

Corollary 4.15. If one takes $K = 1$ in Corollary 4.14, one gets the following inequality for $GFPC$ functions with $RL$-integral operators:

Theorem 4.16. Let $n\in \mathbb{N}$, $a_{u}\geq 0$ $\left(u = \overline{1, n}\right),$ $s\in \left[0, 1 \right]$, $\alpha, K > 0,$ $\varkappa_1 < \varkappa_2,$ $q > 1,$ and $\wp :\breve{A} = \left[\varkappa_1, \varkappa_2 \right] \rightarrow \mathbb{R}$ be a twice differentiable function on $\breve{A}^{\circ }$ such that $\wp ^{\prime \prime }\in L\left[\varkappa_1, \varkappa_2 \right].$ Let $\left\vert \wp ^{\prime \prime }\left(\nu \right) \right\vert ^{q}$ be a $GFPC-s$ function on $\breve{A}.$ Then for every $\nu \in \left[\varkappa_1, \varkappa_2 \right],$ one has

where

Proof. Using Lemma 4.12, a property of the $GFPC-s$ function $\left\vert \wp ^{\prime \prime }\right\vert ^{q},$ and the power mean inequality, one has

□

Corollary 4.17. If one takes $s = 1$ in Theorem 4.16, one gets the following inequality for generalized $n$-fractional polynomial convex functions with $K$-fractional integral operators:

for all $\nu \in \left[\varkappa_1, \varkappa_2 \right],$ where

Corollary 4.18. If one takes $K = 1$ in Corollary 4.17, one gets the following inequality for $GFPC$ functions with $RL$-integral operators:

for all $\nu \in \left[\varkappa_1, \varkappa_2 \right],$ where

Theorem 4.19. Let $n\in \mathbb{N}$, $a_{u}\geq 0$ $\left(u = \overline{1, n}\right),$ $s\in \left[0, 1 \right]$, $\alpha, K > 0,$ $\varkappa_1 < \varkappa_2,$ $t, q > 1$ with $\frac{1}{t}+ \frac{1}{q} = 1,$ and $\wp :\breve{A} = \left[\varkappa_1, \varkappa_2 \right] \rightarrow \mathbb{R}$ be a twice differentiable function on $\breve{A} ^{\circ }$ such that $\wp ^{\prime \prime }\in L\left[\varkappa_1, \varkappa_2 \right].$ Let $\left\vert \wp ^{\prime \prime }\left(\nu \right) \right\vert ^{q}$ be a $GFPC-s$ function on $\breve{A}.$ Then

Proof. Using Lemma 4.12, a property of the $GFPC-s$ function $\left\vert \wp ^{\prime \prime }\right\vert ^{q},$ and the Hölder inequality, one has

□

Corollary 4.20. If one takes $s = 1$ in Theorem 4.19, one gets the following inequality for $GFPC$ functions with $K$-fractional integral operators:

Corollary 4.21. If one takes $K = 1$ in Corollary 4.20, one gets the following inequality for $GFPC$ functions with $RL$-integral operators:

5.   Conclusions

We introduced the concept of $n$-fractional polynomial $s$-type convex functions and investigated their related properties. Algebraic relationships between such functions and other kinds of convex functions were explored. Several novel variants of the well known H-H and Ostrowski-type inequalities were established using the newly defined class of functions and $K$-fractional integral operators. Considering the introduced class and employing fractional operators, we have derived new refinements of the Ostrowski-type inequalities. Several special cases of our results were discussed. For some special cases, the definition and results of generalized $n$-fractional polynomial $s$-type convex functions reduce to a novel definition and new results for the class of convex functions, called generalized $n$-fractional polynomial convex functions. The results obtained from the future plan are even more exhilarating compared to the results available in the literature.

Author contributions

Serap Özcan: Conceptualization, Formal Analysis, Investigation, Writing-Original Draft, Writing-Review & Editing, Supervision; Saad Ihsan Butt: Conceptualization, Formal Analysis, Investigation, Methodology, Writing-Original Draft, Writing-Review & Editing, Supervision; Sanja Tipurić-Spužević: Conceptualization, Formal Analysis, Methodology, Software, Writing-Review & Editing, Funding Acquisition; Bandar Bin Mohsin: Conceptualization, Formal Analysis, Software, Writing-Review & Editing. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research is supported by Researcher Supporting Project number (RSP2024R158), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

All authors declare no conflicts of interest in this paper.