On Some Inequalities with Higher Fractional Orders

Abstract

The novelty herein pertains to a class of fractional differential equations involving the Hadamard fractional derivative of higher order. Our investigation encompasses the fractional integral operator of a logarithmic function. The mathematical tools utilized in this study are derived from an important function, wherein its behavior in terms of maximum value facilitates the establishment of bounds necessary for proving the existence of solutions, specifically through Green’s function. Based on this, we endeavor to estimate the bounds of Green’s function as well as analyze its properties within the considered interval. This approach enables us to establish the Hartman–Wintner- and Lyapunov-type inequalities for a class of fractional Hadamard problems. Furthermore, we introduce a novel technique to determine the maximum value of Green’s function. Finally, we illustrate these findings through two applications.

1. Introduction

Fractional differential calculus has garnered considerable significance and has had a profound influence across various disciplines. Its development has been extensive, encompassing fields such as physics, oscillation theory, disconjugacy, eigenvalue problems, economics, and engineering sciences. Numerous studies have addressed solutions to both linear and nonlinear fractional differential equations (see [1,2,3,4,5,6] for key contributions in this domain). A pivotal result in this context is Lyapunov’s inequality, which establishes an integral relationship between the function A and the length of the interval. This inequality serves as a critical criterion for ensuring the existence of solutions to the second-order ordinary differential equation

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{w}^{\prime \prime }\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\hfill \\ w\left({\xi }_1\right)=w\left({\xi }_1\right)=0,\hfill \end{array}\right.\end{array}$$

(1)

where $A\in C\left([{\xi }_1,{\xi }_2]\right)$. In [7], Lyapunov’s result for Problem (1) is formulated by

$$\begin{array}{c}\hfill \begin{array}{c}{\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|ds>{\displaystyle \frac{4}{({\xi }_2-{\xi }_1)}},{\xi }_1<r<{\xi }_2.\hfill \end{array}\end{array}$$

(2)

This class of solution existence criteria, now widely referred to as Lyapunov-type inequalities, has proven to be remarkably useful across multiple domains. These inequalities play a fundamental role in eigenvalue problems, oscillation theory, and various applied mathematical contexts (for comprehensive references, see [2,8,9,10,11,12], and the extensive literature cited therein). These inequalities provide crucial connections between abstract existence theorems and practical problem-solving. The approach has inspired new techniques in fractional order analysis. For relevance sake, it continues to inform current research in fractional differential equations.

In [13], Ferreira established a Lyapunov-type inequality for the following Riemann–Liouville fractional boundary value problem

$$\begin{array}{cc}\hfill & {(}_{{\xi }_1}{D}^{l}w)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,1<l\le 2,\hfill \\ \hfill & w\left({\xi }_1\right)=w\left({\xi }_2\right)=0,\hfill \end{array}$$

where $A\in C\left([{\xi }_1,{\xi }_2]\right),$ admits a non-trivial solution provided that the following inequality is satisfied:

$$\begin{array}{c}\hfill {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|ds>\mathrm{\Gamma }\left(l\right){\left({\displaystyle \frac{4}{{\xi }_2-{\xi }_1}}\right)}^{l-1}.\end{array}$$

In reference [14], the author examined the same boundary value problem; however, the analysis was conducted using the Caputo fractional derivative operator instead of the Riemann–Liouville formulation:

$$\begin{array}{cc}\hfill & {(}_{{\xi }_1}^C{D}^{l}w)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,1<l\le 2,\hfill \\ \hfill & w\left({\xi }_1\right)=w\left({\xi }_2\right)=0,\hfill \end{array}$$

and showed the non-trivial solution exists if

$$\begin{array}{c}\hfill {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|ds>{\displaystyle \frac{\mathrm{\Gamma }\left(l\right)l^l}{{(l-1)}^{l-1}{({\xi }_2-{\xi }_1)}^{l-1}}}.\end{array}$$

Building upon these foundational works, numerous subsequent studies have emerged in this area. We particularly draw the reader’s attention to references [1,7,8,9,13,14,15], which represent key contributions to this evolving field. We mean by that that Dhar et al. [1] used the contraction mapping theorems to establish the unique solution and Lyapunov’s inequality for the Hadamard fractional differential equation subject to the Dirichlet conditions in $[a,b],$ rather than $[1,e]$ as investigated by Q. Ma et al. in [9]. Also, Laadjal, Z. et al. focused on a general interval $[a,b]$. However, the first and basic contribution in this field is due to Lyapunov [7] for the ordinary differential equation. So, we have basic contribution results for the ordinary differential equation in [7] $(\alpha =2),$ and for the Hadamard differential equations $(1<\alpha \le 2),$ in [8,9], and for $(3<\alpha \le 4),$ in [9].

For Caputo fractional differential equations with varied boundary conditions, an extensive body of literature exists; comprehensive treatments can be found in [2,14,16] and their cited references. A notable exception is the work of Wang et al. [17], who established a Lyapunov-type inequality for the following fractional boundary value problem:

$$\begin{array}{cc}\hfill & (D_{{\xi }_1}^{\alpha }w)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,m-1<l\le m,\hfill \\ \hfill & w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=\dots =w^{m-2}\left({\xi }_1\right)=0,w^{m-2}\left({\xi }_2\right)=0\hfill \end{array}$$

where m is an integer greater than or equal to $2,$ and $A\in C\left([a,b];\mathbb{R}\right)$.

In their significant contribution [17], they established novel results for a Riemann–Liouville fractional boundary value problem. Specifically, the authors developed a comprehensive framework for analyzing Lyapounov’s inequality and derived a new result by extending the previous work of [2,18]. Their work represents an important advancement in the study of the existence criteria of solutions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(D^{l}w)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\hfill \\ w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=w^{\prime \prime }\left({\xi }_1\right)=\dots =w^{(m-2)}\left({\xi }_1\right)=0,w^{(m-2)}\left({\xi }_2\right)=0,\hfill \end{array}\right.\end{array}$$

(3)

where $D^l$ stands for Riemann–Liouville operator, $2\le m-1<l\le m,$ and $A\in C\left([{\xi }_1,{\xi }_2]\right)$.

They proved that (3) admits a non-trivial solution if

$${\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|ds>{\displaystyle \frac{\mathrm{\Gamma }\left(l\right){(l-m+2)}^{l-m+2}}{(n-2){(l-m+1)}^{l-m+1}{({\xi }_2-{\xi }_1)}^{l-1}}}$$

(4)

is satisfied.

Our approach to deriving Lyapunov-type inequalities follows the established framework of constructing the corresponding Green function, investigating its fundamental properties, and analyzing its variational behavior to obtain the desired inequality. This methodology builds upon the work of Pourhadi and Mursaleen [16], who examined a novel Caputo-type fractional differential equation of the form

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{(}^{\mathbb{C}}{D}_{a^{+}}^{\alpha })y\left(t\right)+p\left(t\right)y^{\prime }\left(t\right)+q\left(t\right)y\left(t\right)=0,a<t<b,\hfill \\ y\left(a\right)=y^{\prime }\left(a\right)=y\left(b\right)=0,\hfill \end{array}\right.\end{array}$$

where $2<\alpha \le 3,$ and where p and q are two positive functions of class $C^1$ on $[a,b]$.

Their result covers a very good class of previous ones, in the literature, especially the fractional differential equation of order two and three. The method employed in finding the desired integral inequality is the same as the previous works cited in the literature as well as [8,9,11,12,13,14]. In this paper, we consider w a non-trivial solution of

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\hfill \\ w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=\dots =w^{(m-2)}\left({\xi }_1\right)=0,w^{(m-2)}\left({\xi }_2\right)=0,\hfill \end{array}\right.\end{array}$$

(5)

where $D_{{\xi }_1;log}^l$ stands for Hadamard derivative operator, ${\xi }_1$ and ${\xi }_2$ are two real positive numbers, $2\le m-1<l\le m,$ and $A\in C\left([{\xi }_1,{\xi }_2]\right)$.

We establish two principal results under the standing assumption that ${\xi }_1$ and ${\xi }_2$ are positive real constants. First, we derive a Hartman–Wintner-type inequality, followed by a Lyapunov-type inequality for Problem (5). The proofs of these results rely on several auxiliary lemmas. Our main theoretical advancement is presented in Theorem 1 below. This result significantly generalizes the previous work, particularly encompassing the fractional differential Equation (5) to the third-order m = 3, taking the form

$$\begin{array}{c}\hfill w^{\prime \prime \prime }+p\left(t\right)w=0,\end{array}$$

where p is a positive continuous function on $(0,+\infty ),$ as studied by Pani and Panigrahi [19]. The methodology we employ to obtain the integral inequality follows the established framework found in [1,10,13,14,20], while introducing crucial innovations in the Hadamard fractional setting. For readers interested in the broader context of Lyapunov-type inequalities and recent developments in this field, we particularly recommend the comprehensive surveys [20,21], which provide up-to-date coverage of this active research area.

The main contribution of this study is the generalization to a higher order, and the extension of the existing result in the sense of Riemann–Liouville [17] to the m-th order Hadamard fractional boundary value problem. In addition, we develop a new technique by constructing and analyzing the associated Green function based on the best maximum principle. Therefore, we establish a criterion for the existence of non-trivial solutions. It consists of

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\hfill \\ w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=w^{\prime \prime }\left({\xi }_1\right)=\dots =w^{(n-2)}\left({\xi }_1\right)=0,w^{(n-2)}\left({\xi }_2\right)=0,\hfill \end{array}\right.\end{array}$$

(6)

where $D_{{\xi }_1;log}^l$ stands for Hadamard derivative operator, and ${\xi }_1$ and ${\xi }_2$ are two real positive numbers, $2<m-1<l\le m,$ and $A\in C\left([{\xi }_1,{\xi }_2]\right)$.

In this paper, we establish two principal theorems under the standing assumption that ${\xi }_1$ and ${\xi }_2$ are positive real constants. First, we derive a Hartman–Wintner-type inequality, followed by a Lyapunov-type inequality for Problem (5). The proofs of these results employ several key lemmas that we develop in subsequent sections. Our primary theoretical advancement is presented in Theorem 1 below. This result represents a significant extension of previous work, particularly generalizing the boundary value Problem (3) studied in [17] involving the Riemann–Liouville operator to the Hadamard fractional derivative boundary value problem. We successfully establish the main result of this paper, stated in Theorem 1, along with its important corollaries. The proof technique combines innovative approaches with established methods from fractional calculus. Our work bridges an important gap in the literature by extending these inequalities to the Hadamard fractional context. The transition from Riemann–Liouville to Hadamard fractional operators introduces several mathematical challenges that we address through a careful analysis of Green’s function and its properties, a novel estimation technique for the associated Green function, and a development of the auxiliary lemmas for the Hadamard case. This extension not only broadens the theoretical framework but also opens new possibilities in solving other fractional boundary value problems with other derivative operators.

Motivated by the works in [14,17,22], this article establishes a novel Lyapunov-type and Hartman–Wintner inequalities for a class of Hadamard fractional boundary value problems. To the best of our knowledge, no existing results including those in [17,18,22] address the existence of non-trivial solutions for the Hadamard fractional Problem (5).

Theorem 1. 

Let $w\ne 0$ be the solution of

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\hfill \\ w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=w^{\prime \prime }\left({\xi }_1\right)=\dots =w^{(n-2)}\left({\xi }_1\right)=0,w^{(n-2)}\left({\xi }_2\right)=0,\hfill \end{array}\right.\end{array}$$

(7)

where $2\le m-1<l\le m,$ and $A\in C\left([{\xi }_1,{\xi }_2]\right)$. Then (7) admits a non-trivial solution w if

$${\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}>{\displaystyle \frac{\mathrm{\Gamma }\left(l\right){(l-m+2)}^{l-m+2}}{(m-2){(l-m+1)}^{l-m+1}{(log\left({\xi }_2\right)-log\left({\xi }_1\right))}^{m-2}}}$$

(8)

is satisfied.

The structure of this work is organized as follows. In Section 2, the foundations present essential definitions and preliminary lemmas required for our analysis. In Section 3, the main results focus on the bi-dimensional and multi-dimensional cases of Lyapunov-type inequalities. This section is divided into two parts, each addressing distinct classes of inequalities, a rigorous examination of sharpness in the derived inequalities, and a detailed analysis of the maximum principle and its critical role in obtaining sharp bounds. In Section 4, we deal with applications. We demonstrate the practical implications of the theoretical results in Section 3, from Hadamard to Riemann–Liouville fractional problems subject to some restriction of the domain, and we give two applications. In Section 5, we deal with the conclusion, summarize the findings, and discuss potential extensions or open problems. The following definitions and lemmas are fundamental to our work. For comprehensive treatments, we refer to the authoritative texts by Podlubny [23] and Kilbas et al. [24]. Let ${\xi }_1$ and ${\xi }_2$ be two real positive constants and $A\in C^m\left([{\xi }_1,{\xi }_2]\right)$ denotes the space of n times absolute continuous functions.

2. Definitions and Lemmas

In this section, we present definitions and lemmas that are required to be used in proving our main results.

Definition 1 

([23,24]). Let $f\in L^1(({\xi }_1,{\xi }_2);\mathbb{R})$. The Hadamard fractional integral of order $l>0$ of f is defined by

$$\begin{array}{cc}\hfill & I_{{\xi }_1^{+},log}^{l}f\left(r\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(l\right)}}{\int }_{{\xi }_1}^r{f\left(s\right)(log(r)-log(s\left)\right)}^{1-l}{\displaystyle \frac{ds}{s}},\hfill \\ \hfill & a.et\in [{\xi }_1,{\xi }_2].\hfill \end{array}$$

Definition 2 

([23,24]). Let $l>0$, and $m=\left[l\right]+1$. If $f\in A{C}^m\left([{\xi }_1,{\xi }_2]\right)$ then the Hadamard–Riemann fractional derivative of order l of f defined by

$$\begin{array}{c}\hfill D_{{\xi }_1;log}^{l}f\left(r\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(m-l)}}{\int }_{{\xi }_1}^r{\displaystyle \frac{f^{\left(m\right)}\left(s\right)}{{(logr-logs)}^{l-m+1}}}{\displaystyle \frac{ds}{s}}\end{array}$$

exists almost everywhere on $[{\xi }_1,{\xi }_2](\left[l\right]$ is the entire part of $l)$.

Definition 3 

([23,24]). Let $l>0$, and m be the smallest integer greater than or equal to l. Let $f:[{\xi }_1,{\xi }_2]\to \mathbb{R}$ be such that $\left({\displaystyle \frac{d^m}{d{r}^m}}\right)I_{a,log}^{n-l}f$ exists almost everywhere on $[{\xi }_1,{\xi }_2]$. The Hadamard–Riemann fractional derivative of order l of f is defined by

$$\begin{array}{ccc}\hfill D_{{\xi }_1;log}^{l}f\left(r\right)& =& {\displaystyle \frac{d^m}{d{r}^m}}{I}_{{\xi }_1,log}^{m-l}f\hfill \\ \hfill & =& {\displaystyle \frac{1}{\mathrm{\Gamma }(m-l)}}({\displaystyle \frac{d^m}{d{r}^m}}{\int }_{{\xi }_1}^{r}f\left(s\right)(log{\displaystyle \frac{r}{s}}){)}^{l-m+1}{\displaystyle \frac{ds}{s}},\hfill \\ & & fora.er\in [{\xi }_1,{\xi }_2].\hfill \end{array}$$

Lemma 1 

([22,23,24]). For $l>0$ and $A\in C({\xi }_1,{\xi }_2)$, the homogeneous fractional differential equation $\left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0$ has a solution

$$\begin{array}{cc}\hfill & f\left(r\right)=c_1{(log{\displaystyle \frac{r}{{\xi }_1}})}^{l-1}+c_2{(log{\displaystyle \frac{r}{{\xi }_1}})}^{l-2}\hfill \\ \hfill & +\dots +c_m{(log{\displaystyle \frac{r}{{\xi }_1}})}^{m-1}-{\int }_{{\xi }_1}^r{(log{\displaystyle \frac{r}{s}})}^{l-1}{\displaystyle \frac{ds}{s}},\hfill \end{array}$$

where $c_i\in \mathbb{R}$, $i=1,\dots ,m$, and $m=\left[l\right]+1$, (l non-integer).

3. Main Results

Our approach to establishing the main results follows a systematic strategy consisting of several steps. We first begin by transforming the fractional boundary value Problem (5) into its equivalent integral formulation. This representation allows us to employ tools from integral operator theory in our analysis. Next, the core of our proof relies on a detailed examination of the associated Green function $\left(GN\right)$, with a particular focus on determining its maximum value. We use the best maximum principle in this context. Building upon these foundations, we proceed to develop a class of Lyapunov-type inequalities, extend these to Hartman–Wintner inequalities through a careful application of the arithmetic–geometric mean inequality, the binomial identity, and the best maximum principles. Our investigation comprehensively addresses the problem across all fractional orders by considering two distinct cases: the lower-order case (1 < $l\le$ 2) and the higher-order case (3 < $l\le m$). Even though the first case is already investigated in [22], we quote it here for a full study and a complete proof. Several important aspects of our approach deserve emphasis: the operator, the specificity, and the connections to the existing literature. Indeed, our analysis focuses exclusively on the Hadamard fractional derivative operator. We deliberately exclude other fractional operators (Atangana–Baleanu, beta, conformable) to maintain focus on the Hadamard case only due to the variation of Green’s function. Notice that each fractional operator has its own merit in terms of research. The case (1 < $l\le$ 2) with $\lambda$-Hilfer fractional problem was treated in [22] for the particular case $\lambda :=log$. Our work extends the result in [17] to the Hadamard framework while providing a complete coverage of both situations. We unify the analysis of both fractional order ranges. The proof’s validity for arbitrary fractional order represents a significant advancement. For readers interested in broader developments, we recommend the comprehensive surveys by Tiryaki [20] and Ntouyas et al. [21], which cover the recent advances involving various fractional derivative operators.

Lemma 2. 

Let $2<m-1<l\le m,$$A\in C\left([{\xi }_1,{\xi }_2]\right)$. Then the unique $w\ne 0$ solution of

$$\begin{array}{cc}& \left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\end{array}$$

(9)

$$\begin{array}{cc}& w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=w^{\prime \prime }\left({\xi }_1\right)=\dots =w^{(m-2)}\left({\xi }_1\right)=0,w^{(m-2)}\left({\xi }_2\right)=0,\end{array}$$

(10)

is given by

$$\begin{array}{c}\hfill w\left(r\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(l\right)}}{\int }_{{\xi }_1}^{r}GN(r,s)w\left(s\right)A\left(s\right){\displaystyle \frac{ds}{s}}+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(l\right)}}{\int }_r^{{\xi }_2}GN(r,s)w\left(s\right)A\left(s\right){\displaystyle \frac{ds}{s}},\end{array}$$

where $GN$ is defined by

$$\mathrm{\Gamma }\left(l\right)GN(r,s)=\left\{\begin{array}{c}g(r,s)-{(log\left({\displaystyle \frac{r}{s}}\right))}^{l-1},{\xi }_1\le s\le r,\hfill \\ {(log\left({\displaystyle \frac{r}{{\xi }_1}}\right))}^{l-1}{\left({\displaystyle \frac{log\left(\frac{{\xi }_2}{s}\right)}{log\left(\frac{{\xi }_2}{{\xi }_1}\right)}}\right)}^{l-m+1}:=g(r,s),r\le s\le {\xi }_2.\hfill \end{array}\right.$$

(11)

Proof. 

We proceed as follows.

Building upon the framework established in [22,24], we derive the integral representation of solutions to the fractional differential Equation (5). This formulation emerges naturally from Lemma 1, yielding the following expression.

$$\begin{array}{cc}\hfill & w\left(r\right)=c_1{(log\left(r\right)-log\left({\xi }_1\right))}^{l-1}+c_2{(log\left(r\right)-log\left({\xi }_1\right))}^{l-2}\hfill \\ \hfill & +\dots +c_m{(log\left(r\right)-log\left({\xi }_1\right))}^{l-m}\hfill \\ \hfill & -{\displaystyle \frac{1}{\mathrm{\Gamma }\left(l\right)}}{\int }_{{\xi }_1}^r{(log\left(r\right)-log\left(s\right))}^{l-1}A\left(s\right)w\left(s\right){\displaystyle \frac{ds}{s}},\hfill \end{array}$$

where $c_1,$ $c_2,$ …, $c_m$ are real constants to be determined later.

Using the boundary conditions $w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=\dots =w^{(m-3)}\left({\xi }_1\right)=0,$ we find

$$\begin{array}{cc}\hfill & c_3=c_4=\dots c_n=0.\hfill \end{array}$$

By using successive partial differentiation of w with respect to $r,$ we obtain

$$\begin{array}{cc}\hfill & w^{m-2}\left(r\right)=c_1(l-1)\dots (l-m+1){(log{\displaystyle \frac{r}{{\xi }_1}})}^{l-m+1}\hfill \\ \hfill & +c_2(l-2)\dots (l-m+2){(log{\displaystyle \frac{r}{{\xi }_1}})}^{l-m+2}\hfill \\ \hfill & -{\displaystyle \frac{(l-1)\dots (l-m+2)}{\mathrm{\Gamma }\left(l\right)}}{\int }_{{\xi }_1}^{{\xi }_2}{(log{\displaystyle \frac{{\xi }_2}{s}})}^{l-m+1}A\left(s\right)w\left(s\right){\displaystyle \frac{ds}{s}}.\hfill \\ \hfill \end{array}$$

The boundary conditions $w^{(m-2)}\left({\xi }_1\right)=0$ and $w^{(m-2)}\left({\xi }_2\right)=0$ yield, respectively,

$$\begin{array}{cc}\hfill & c_2=0,\mathrm{and}\hfill \\ \hfill & c_1={\displaystyle \frac{1}{\mathrm{\Gamma }\left(l\right){(log\frac{{\xi }_2}{{\xi }_1})}^{(l-m+1)}}}{\int }_{{\xi }_1}^{{\xi }_2}{(log{\displaystyle \frac{{\xi }_2}{s}})}^{l-m+1}A\left(s\right)w\left(s\right){\displaystyle \frac{ds}{s}}.\hfill \\ \hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & w\left(r\right)={\displaystyle \frac{{(log\frac{r}{{\xi }_1})}^{l-1}}{\mathrm{\Gamma }\left(l\right){(log\frac{{\xi }_2}{{\xi }_1})}^{(l-m+1)}}}{\int }_{{\xi }_1}^{{\xi }_2}{(log{\displaystyle \frac{{\xi }_2}{s}})}^{l-m+1}A\left(s\right)w\left(s\right){\displaystyle \frac{ds}{s}}\hfill \\ \hfill & -{\displaystyle \frac{1}{\mathrm{\Gamma }\left(l\right)}}{\int }_{{\xi }_1}^r{(log{\displaystyle \frac{r}{s}})}^{l-1}A\left(s\right)w\left(s\right){\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

Thus, the proof of this lemma is completed. □

The necessary tools employed in establishing the result of this paper and its consequences are the construction of the function ($GN$) and the use of its properties as well as its maximum value. Our findings are represented and formulated in the next two theorems (Theorems 2 and 3). The results represented are, respectively, Hartman–Wintner and Lyapunov inequalities, classical and sharp. However, for Hartman–Wintner inequalities, they are represented in Corollaries 1 and 2 Our purpose is to split the bi-dimensional $(m=2),$ and multi-dimensional $(m\ge 3)$ cases separately to obtain a complete study of the fractional Hadamard value problem. So let us establish the bi-dimensional case first. It consists of the first situation, which involves the fractional order $1<l\le 2$. For completeness’ sake, we formulate and prove it in the next lemma.

The Bi-dimensional Case:

The following lemma is needed for both cases, the bi-dimensional and the multi-dimensional fractional Hadamard boundary value problems.

Lemma 3. 

Let $w\ne 0$ be the solution of the Hadamard fractional boundary value problem

$$\begin{array}{cc}& \left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,1<l\le 2,\end{array}$$

(12)

$$\begin{array}{cc}& w\left({\xi }_1\right)=w\left({\xi }_2\right)=0,\end{array}$$

(13)

where the function $A\in C\left([{\xi }_1,{\xi }_2]\right)$. Then $\left(GN\right)$ defined in (11) is positive and satisfies for all $(r,s)\in [{\xi }_1,{\xi }_2]\times [{\xi }_1,{\xi }_2]$

$$\begin{array}{c}\hfill 0\le GN(r,s)\le GN({\xi }_2,s),\end{array}$$

where $r,s\in [{\xi }_1,{\xi }_2]$.

Proof. 

We first consider $GN(r,s)$ for $r\le s$ given by

$$\begin{array}{cc}\hfill & \mathrm{\Gamma }\left(l\right)GN(r,s)={(log{\displaystyle \frac{r}{{\xi }_1}})}^{l-1}{\left({\displaystyle \frac{log\frac{{\xi }_2}{s}}{log\frac{{\xi }_2}{{\xi }_1}}}\right)}^{l-m+1}.\hfill \end{array}$$

Since the function g defined in (11) is a positive function in r, it is clear that $\left(GN\right)$ is positive too. However, for $s\le r,$$\left(GN\right)$ is given by

$$\begin{array}{c}\hfill \mathrm{\Gamma }\left(l\right)GN(r,s)={(log\left({\displaystyle \frac{r}{{\xi }_1}}\right))}^{l-1}{\left({\displaystyle \frac{log\left(\frac{{\xi }_2}{s}\right)}{log\left(\frac{\xi 2}{\xi 1}\right)}}\right)}^{l-m+1}-{(log\left({\displaystyle \frac{r}{s}}\right))}^{l-1}.\end{array}$$

Let us rewrite $log\left({\displaystyle \frac{r}{s}}\right){)}^{l-1}$ as follows:

$$\begin{array}{cc}\hfill & {(log\left({\displaystyle \frac{r}{s}}\right))}^{l-1}={\left({\displaystyle \frac{log\left(\frac{r}{{\xi }_1}\right)}{log\left(\frac{{\xi }_2}{\xi 1}\right)}}\right)}^{l-1}{\left(log\left({\displaystyle \frac{{\xi }_2}{{\xi }_1}}\right)-{\displaystyle \frac{log\left(\frac{s}{{\xi }_1}\right)log\frac{{\xi }_2}{{\xi }_1}}{log\left(\frac{r}{{\xi }_1}\right)}}\right)}^{l-1}\hfill \\ \hfill & ={\left({\displaystyle \frac{log\left(\frac{r}{{\xi }_1}\right)}{log\left(\frac{\xi 2}{\xi 1}\right)}}\right)}^{l-1}{\left(log{\xi }_2-\left(log{\xi }_1+{\displaystyle \frac{log\left(\frac{s}{{\xi }_1}\right)log\frac{{\xi }_2}{{\xi }_1}}{log\left(\frac{r}{{\xi }_1}\right)}}\right)\right)}^{l-1}.\hfill \end{array}$$

Now due to the following fact

$$\begin{array}{ccc}\hfill (log\left({\xi }_1\right)+{\displaystyle \frac{log\left(\frac{s}{{\xi }_1}\right)log\left(\frac{{\xi }_2}{{\xi }_1}\right)}{log\left(\frac{r}{{\xi }_1}\right)}})& \ge & log\left(s\right),\hfill \end{array}$$

we get

$$\begin{array}{ccc}\hfill {\displaystyle \frac{log\left({\xi }_1\right)log\left(\frac{r}{{\xi }_1}\right)+log\left(\frac{s}{{\xi }_1}\right)log\left(\frac{\xi 2}{\xi 1}\right)}{log\left(\frac{r}{{\xi }_1}\right)}}& \ge & log\left(s\right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill log\left({\xi }_1\right)log\left({\displaystyle \frac{r}{{\xi }_1}}\right)+log\left(s\right)log\left({\displaystyle \frac{{\xi }_2}{r}}\right)& \ge 0& \hfill \end{array}$$

if and only if $s\ge {\xi }_1$, where it is true that $\left(GN\right)$ is positive for $r\le s$.

Indeed, it is sufficient to observe that $GN(r,s)$ may be rewritten as follows:

$$\begin{array}{cc}\hfill & \mathrm{\Gamma }\left(l\right)GN(r,s):={(log{\displaystyle \frac{r}{{\xi }_1}})}^{l-1}\left\{\left(1-{\left({\displaystyle \frac{log\frac{s}{{\xi }_1}}{log\frac{{\xi }_2}{{\xi }_1}}}\right)}^{l-m+1}\right)-\left(1-{\left({\displaystyle \frac{log\frac{s}{{\xi }_1}}{log\frac{r}{{\xi }_1}}}\right)}^{l-1}\right)\right\}\hfill \\ \hfill & \ge {(log{\displaystyle \frac{r}{{\xi }_1}})}^{l-1}\left\{\left(1-{\left({\displaystyle \frac{log\frac{s}{{\xi }_1}}{log\frac{{\xi }_2}{{\xi }_1}}}\right)}^{l-m+1}\right)-\left(1-{\left({\displaystyle \frac{(log\frac{s}{{\xi }_1})}{(log\frac{{\xi }_2}{{\xi }_1})}}\right)}^{l-1}\right)\right\}\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

(14)

Thus, $GN$ is positive for all $r,s\in [{\xi }_1,{\xi }_2]$ since the function g defined in (11) is obviously positive.

Now in order to show that

$$\begin{array}{c}\hfill GN(r,s)\le GN({\xi }_2,s),\end{array}$$

we make differentiation of $\left(GN\right)$ defined in (11) with respect to r for fixed s in $({\xi }_1,{\xi }_2),$ where $r\le s$, we get

$$\begin{array}{ccc}\hfill \mathrm{\Gamma }\left(l\right){\left(GN\right)}_r& =& (l-1){(log{\displaystyle \frac{r}{{\xi }_1}})}^{l-2}{\left({\displaystyle \frac{log\frac{{\xi }_2}{s}}{log\frac{{\xi }_2}{{\xi }_1}}}\right)}^{l-n+1}\hfill \\ \hfill & -& (l-1){(log{\displaystyle \frac{r}{{\xi }_1}})}^{l-2}.\hfill \end{array}$$

(15)

Similarly to above, we rewrite the expression ${(log\left(r\right)-log\left(s\right))}^{l-2}$ as

$$\begin{array}{ccc}\hfill {(log\left(r\right)-log\left(s\right))}^{l-2}& =& {\left((log{\displaystyle \frac{r}{{\xi }_1}})+(log{\displaystyle \frac{{\xi }_1}{s}})\right)}^{l-2}\hfill \\ \hfill & =& {\left({\displaystyle \frac{log\frac{r}{s}}{log\frac{{\xi }_2}{{\xi }_1}}}\right)}^{l-2}\times \hfill \\ \hfill & & {[log\left({\xi }_2\right)-\left((log\left({\xi }_1\right)+{\displaystyle \frac{log\frac{s}{{\xi }_1}}{log\frac{r}{{\xi }_1}}}\right)]}^{l-2}.\hfill \\ \hfill \end{array}$$

(16)

Now inserting (16) into (15), one may observe that ${\left(GN\right)}_r$ is non-negative for $r\le s$ and therefore $\left(GN\right)$ is a non-decreasing function on this interval.

For the case $s\le r$, one may see that the derivative function ${\left(GN\right)}_r$ is positive and consequently the function $GN$ is a non-decreasing function.

Hence, $\left(GN\right)$ is a non-decreasing function in r. Therefore

$$\begin{array}{c}\hfill 0<\left(GN\right)(r,s)\le \left(GN\right)({\xi }_2,s).\end{array}$$

Evaluating $GN(r,s)$ defined in (11) at $r={\xi }_2,$ we get

$$\begin{array}{cc}\hfill & GN({\xi }_2,s)\mathrm{\Gamma }\left(l\right)={(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}})}^{l-1}{\left({\displaystyle \frac{log\frac{{\xi }_2}{s}}{log\frac{{\xi }_2}{{\xi }_1}}}\right)}^{l-m+1}\hfill \\ \hfill \end{array}$$

where $r,s\in [{\xi }_1,{\xi }_2]$.

To start with the bi-dimensional case, let us set $m=3,$ and consider $2<l\le 2;$ we then obtain

$$\begin{array}{cc}\hfill & GN({\xi }_2,s)\mathrm{\Gamma }\left(l\right)={(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}})}^{l-1}{\left({\displaystyle \frac{log\frac{{\xi }_2}{s}}{log\frac{{\xi }_2}{{\xi }_1}}}\right)}^{l-2}.\hfill \\ \hfill \end{array}$$

To this end, the Hartman–Wintner and Lyapunov inequalities are respectively derived by

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}{(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}})}^{m-2}{\left(log{\displaystyle \frac{{\xi }_2}{s}}\right)}^{l-m+1}A\left(s\right){\displaystyle \frac{ds}{s}}\ge \mathrm{\Gamma }\left(l\right),\hfill \\ \hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ \hfill & \ge {\displaystyle \frac{\mathrm{\Gamma }\left(l\right)}{{\left(log\frac{{\xi }_2}{{\xi }_1}\right)}^{l-1}}}.\hfill \end{array}$$

To invoke the second situation, we provide the fractional Hadamard Problem (5) with the order $3<l\le m$. □

The Multi-dimensional Case:

The next theorem deals with the multi-dimensional case where we look at the Green function’s lower bound. The estimate of the Green function that we wish to obtain will be derived using the best maximum principle. Accordingly, the sharp Lyapunov inequality will be achieved.

Theorem 2.  

(Hartman–Wintner Inequality)

Let $w\ne 0$ be the solution of

$$\begin{array}{cc}\hfill & \left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\hfill \\ \hfill & w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=\dots =w^{(m-2)}\left({\xi }_1\right)=0,w^{(m-2)}\left({\xi }_2\right)=0,\hfill \end{array}$$

(17)

where $2<m-1<l\le m,$ and $A\in C\left([{\xi }_1,{\xi }_2]\right)$. Then the following inequality is satisfied:

$$\begin{array}{ccc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}{(log{\displaystyle \frac{{\xi }_2}{s}})}^{l-m+1}(log{\displaystyle \frac{s}{{\xi }_1}})\hfill & \hfill {\sum }_{i=1}^{m-2}{C}_{m-2}^i{(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}})}^{m-2-i}{(log{\displaystyle \frac{{\xi }_2}{s}})}^i\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\\ \hfill & \ge \mathrm{\Gamma }\left(l\right),\hfill \end{array}$$

where $r,s\in [{\xi }_1,{\xi }_2]$.

Proof. 

Let $E:=C\left([{\xi }_1,{\xi }_2]\right)$ be equipped with the norm

$$\begin{array}{c}\hfill \left|\right|w\left|\right|=\underset{r\in [{\xi }_1,{\xi }_2]}{max}\left|w\left(r\right)\right|.\end{array}$$

Recall that

$$\begin{array}{c}\hfill w\left(r\right)={\int }_{{\xi }_1}^{{\xi }_2}G(r,s)A\left(s\right)w\left(s\right){\displaystyle \frac{ds}{s}}.\end{array}$$

Since w is non-trivial, it follows that

$$\begin{array}{cc}\hfill & \left|w\left(r\right)\right|\le {\int }_{{\xi }_1}^{{\xi }_2}\left|GN(r,s)\right|\left|A\left(s\right)\right|\left|w\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ \hfill & \le \underset{r\in [{\xi }_1,{\xi }_2]}{max}\{{\int }_{{\xi }_1}^{{\xi }_2}\left|GN(r,s)\right|\left|A\left(s\right)\right|\left|w\left(s\right)\right|{\displaystyle \frac{ds}{s}}\}\hfill \\ \hfill & \le \underset{r\in [{\xi }_1,{\xi }_2]}{max}\{{\int }_{{\xi }_1}^{{\xi }_2}\left|GN(r,s)\right|\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\}\parallel w\parallel .\hfill \end{array}$$

Hence

$$\begin{array}{cc}\hfill & 1\le {\int }_{{\xi }_1}^{{\xi }_2}\underset{t\in [{\xi }_1,{\xi }_2]}{max}\left|GN(r,s)\right|\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

In addition, in view of Lemma 3, it yields

$$\begin{array}{cc}\hfill & 1\le \left({\int }_{{\xi }_1}^{{\xi }_2}\left|GN({\xi }_2,s)\right|ds\right)\parallel A\parallel .\hfill \end{array}$$

According to the binomial identity, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\left(log\frac{{\xi }_2}{{\xi }_1}\right)}^{l-1}{\left(log\frac{{\xi }_2}{s}\right)}^{l-m+1}}{{\left(log\frac{{\xi }_2}{{\xi }_1}\right)}^{l-m+1}}}-{\left(log{\displaystyle \frac{{\xi }_2}{s}}\right)}^{l-1}\hfill \\ \hfill & =(log\left(s\right)-log\left({\xi }_1\right))\times \hfill \\ \hfill & {\sum }_{i=1}^{m-2}{C}_{m-2}^i{(log\left(s\right)-log\left({\xi }_1\right))}^{i-1}{(log\left({\xi }_2\right)-log\left({\xi }_1\right))}^{n-2-i},\hfill \end{array}$$

where $r,s\in [{\xi }_1,{\xi }_2],$ and $C_{m-2}^i={\displaystyle \frac{i!}{(m-2)!(i-m+2)!}}$.

Therefore, we get the Hartman–Wintner-type inequality for (16)

$$\begin{array}{ccc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}{(log{\displaystyle \frac{{\xi }_2}{s}})}^{l-m+1}(log{\displaystyle \frac{s}{{\xi }_1}})\hfill & \hfill {\sum }_{i=1}^{m-2}{C}_{m-2}^i{(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}})}^{m-2-i}{(log{\displaystyle \frac{{\xi }_2}{s}})}^i\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\\ \hfill & \ge \mathrm{\Gamma }\left(l\right),\hfill \end{array}$$

□

Corollary 1. 

(Hartman–Wintner Inequality)

$$\begin{array}{cc}\hfill & \left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\hfill \\ \hfill & w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=\dots =w^{(m-2)}\left({\xi }_1\right)=0,w^{(m-2)}\left({\xi }_2\right)=0,\hfill \end{array}$$

(18)

where $2\le m-1<l\le m,$ and $A\in C\left([{\xi }_1,{\xi }_2]\right)$. Then the following inequality is satisfied:

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}(log{\displaystyle \frac{s}{{\xi }_1}}){(log{\displaystyle \frac{{\xi }_2}{s}})}^{l-m+1}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ \hfill & \ge {\displaystyle \frac{\mathrm{\Gamma }\left(l\right)}{(m-2){(log\frac{{\xi }_2}{{\xi }_1})}^{l-3}}}.\hfill \end{array}$$

Proof. 

Apply Theorem 2 to conclude that

$$\begin{array}{ccc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}{(log{\displaystyle \frac{{\xi }_2}{s}})}^{l-m+1}(log{\displaystyle \frac{s}{{\xi }_1}})\hfill & \hfill {\sum }_{i=1}^{m-2}{C}_{m-2}^i{(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}})}^{m-2-i}{(log{\displaystyle \frac{{\xi }_2}{s}})}^i\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\\ \hfill & \ge \mathrm{\Gamma }\left(l\right),\hfill \end{array}$$

Notice that

$$\begin{array}{cc}\hfill & {\sum }_{i=1}^{m-2}{C}_{m-2}^i{(log{\displaystyle \frac{s}{{\xi }_1}})}^{i-1}{(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}})}^{m-2-i}\hfill \\ \hfill & \ge {\sum }_{i=1}^{m-2}{C}_{m-2}^i{(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}})}^{i-1}{(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}})}^{m-2-i}\hfill \\ \hfill & \ge {\sum }_{i=1}^{m-2}{C}_{m-2}^i{\left(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}}\right)}^{m-3}\hfill \\ \hfill & =(2^{m-2}-1){\left(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}}\right)}^{m-3}\hfill \\ \hfill & \ge (m-2){\left(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}}\right)}^{m-3},\hfill \end{array}$$

since $(2^{m-2}-1)\ge (m-2)$ for all $m\ge 2$ and with equality if and only if $m=2,$ and $m=3$.

Hence, we obtain

$$\begin{array}{cc}\hfill & \mathrm{\Gamma }\left(l\right)\ge {\int }_{{\xi }_1}^{{\xi }_2}(log{\displaystyle \frac{s}{{\xi }_1}}){(log{\displaystyle \frac{{\xi }_2}{s}})}^{l-m+1}\times \hfill \\ \hfill & {\sum }_{i=1}^{m-2}{C}_{m-2}^i{(log{\displaystyle \frac{s}{{\xi }_1}})}^{i-1}{(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}})}^{m-2-i}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ \hfill & \ge {\int }_{{\xi }_1}^{{\xi }_2}(log{\displaystyle \frac{s}{{\xi }_1}}){(log{\displaystyle \frac{{\xi }_2}{s}})}^{l-m+1}(m-2){\left(log{\displaystyle \frac{{\xi }_2}{{\xi }_1}}\right)}^{m-3}{\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

Equivalently, we have

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}(log{\displaystyle \frac{s}{{\xi }_1}}){(log{\displaystyle \frac{{\xi }_2}{s}})}^{l-m+1}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ \hfill & \ge {\displaystyle \frac{\mathrm{\Gamma }\left(l\right)}{(m-2){\left(log\frac{{\xi }_2}{{\xi }_1}\right)}^{m-3}}},\hfill \end{array}$$

which achieves the proof of this corollary. □

Remark 1. 

A direct consequence of Corollary 1 gives

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge {\displaystyle \frac{\mathrm{\Gamma }\left(l\right)}{(m-2){(log\frac{{\xi }_2}{{\xi }_1})}^{l-2}}}.\hfill \end{array}$$

When the domain $[{\xi }_1,{\xi }_2]$ is restricted to $I:=\{{\xi }_1,{\xi }_2\in \mathbf{R}:{\xi }_1,{\xi }_2>1\},$ we retrieve the result in [17] Corollary 3.5 (in the sense of Riemann–Liouville derivative operator). In addition, with l = 4, it is reduced to Corollary 2.5, and Corollary 2.4 [18] for both Hartman–Wintner and Lyapunov inequalities, respectively.

An immediate application of Corollary 1 leads to the following new result.

Corollary 2. 

Let $w\ne 0$ be the solution of

$$\begin{array}{cc}\hfill & \left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\hfill \\ \hfill & w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=\dots =w^{(m-2)}\left({\xi }_1\right)=0,w^{(m-2)}\left({\xi }_2\right)=0,\hfill \end{array}$$

(19)

where $2<m-1<l\le m,$ and $A\in C\left([{\xi }_1,{\xi }_2]\right)$. Then the following inequality is satisfied:

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge {\displaystyle \frac{\mathrm{\Gamma }\left(l\right)}{(m-2)}}{{\xi }_1}^{(m-1)}{{\xi }_2}^{(m-3)},\hfill \end{array}$$

provided that ${\displaystyle \frac{s}{{\xi }_1}}>1,$ and ${\displaystyle \frac{{\xi }_2}{{\xi }_1}}>1$.

Proof. 

It is evident that the function f defined by $f\left(x\right):=logx-x$ is decreasing for $x>1$. In light of ${\displaystyle \frac{s}{{\xi }_1}}>1,$ and ${\displaystyle \frac{{\xi }_2}{{\xi }_1}}>1,$ and the use of

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left(log{\displaystyle \frac{s}{{\xi }_1}}\right){\left(log{\displaystyle \frac{{\xi }_2}{s}}\right)}^{l-m+1}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge {\displaystyle \frac{\mathrm{\Gamma }\left(l\right)}{(m-2){\left(log\frac{{\xi }_2}{{\xi }_1}\right)}^{m-3}}},\hfill \end{array}$$

one deduces that

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge {\displaystyle \frac{\mathrm{\Gamma }\left(l\right)}{(m-2)}}{{\xi }_1}^{(l-2)}{{\xi }_2}^{(l-2)}.\hfill \end{array}$$

□

Remark 2. 

It is easy to see that $log{\xi }_2-log{\xi }_1\le {\xi }_2-{\xi }_{1}for{\xi }_2>{\xi }_1>1,$ so it holds that

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}(s-{\xi }_1){({\xi }_2-s)}^{l-m+1}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge {\displaystyle \frac{\mathrm{\Gamma }\left(l\right)}{(m-2){({\xi }_2-{\xi }_1)}^{m-3}}}.\hfill \end{array}$$

Therefore, on this restricted domain $I,$ Theorem 2 reduces to Theorem 3.3 in [17], and also with l = 4 Theorem 2 reduces to Corollary 2.7 in [18]. Furthermore, the use of the arithmetic–geometric–harmonic inequality

$$\begin{array}{c}\hfill ({\xi }_2-s)(s-{\xi }_1)\le {\displaystyle \frac{{({\xi }_2-{\xi }_1)}^2}{4}}\end{array}$$

conducts us to

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|ds\ge \left({\displaystyle \frac{4}{m-2}}\right){\displaystyle \frac{\mathrm{\Gamma }\left(l\right)}{{({\xi }_2-{\xi }_1)}^{l-1}}}.\hfill \end{array}$$

The Hartman–Wintner inequality established for the Hadamard fractional Problem (18) yields, as a special case, the classical Hartman–Wintner inequality for Riemann–Liouville operator on some restricted bounded interval $[{\xi }_1,{\xi }_2],$ where ${\xi }_2>{\xi }_1>1$.

Now we focus on how to get Lyapunov-type inequalities based on a thorough analysis of $\left(GN\right)$ and its properties. The backbone of this investigation is the maximum value of $\left(GN\right)$. For this matter, we distinguish two kinds of Lyapunov inequalities (sharp and not sharp). For sharp ones, the maximum value of $\left(GN\right)$, named maximal value, is the best maximum for which the sharp constant is reached (as an example, the number 4 in the classical case ${\int }_{{\xi }_1}^{{\xi }_2}A\left(s\right)ds>{\displaystyle \frac{4}{({\xi }_2-{\xi }_1)}})$. So depending on each fractional problem, one could avoid difficulties on how to obtain the best maximum value of the corresponding Green function. Although the construction of such functions is possible, without maximum principles, the difficulty remains and we are unable to overcome it. Regardless of the fraction order in consideration for each situation, some researchers resort to reducing some terms of $GN(b,s)$ or $GN(s,s)$ (depending on the study of the variation of the $\left(GN\right)$) by finding the estimates of bounds and getting the maximum value, which does not lead to Lyapunov’s inequality being sharp. For a simple comparison, we refer the reader to the literature of this field. For clarity’s sake, it appears that it deserves a thorough analysis. To do so, we start by investigating the first kind of Lyapunov inequality for the fractional Problem (20) below. Our approach to establishing the Lyapunov-type inequalities centers on a comprehensive analysis of Green’s function $\left(GN\right)$ and its fundamental properties. The key to this investigation lies in determining the extremal values of $\left(GN\right),$ which naturally leads to two distinct classes of inequalities. The sharp inequalities are attained when the maximal value of $\left(GN\right)$ yields optimal constants (e.g., the constant 4 in the classical case [7]):

$$\begin{array}{c}\hfill {\int }_{{\xi }_1}^{{\xi }_2}A\left(s\right)ds>{\displaystyle \frac{4}{{({\xi }_2-{\xi }_1)}^2}}.\end{array}$$

The sharp inequality result is based on the derivation process, which involves the precise estimation of $\left(GN\right)$’s extrema, and a careful analysis of the function’s variational properties.

First Kind of Lyapunov inequality

Corollary 3. 

(Sharp Lyapunov inequality)

Let $w\ne 0$ be the solution of

$$\begin{array}{cc}\hfill & \left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\hfill \\ \hfill & w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=\dots =w^{(m-2)}\left({\xi }_1\right)=0,w^{(m-2)}\left({\xi }_2\right)=0,\hfill \end{array}$$

(20)

where $2<m-1<l\le m,$ and $A\in C\left([{\xi }_1,{\xi }_2]\right)$. Then the following inequality is satisfied:

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge {\displaystyle \frac{\mathrm{\Gamma }\left(l\right)}{\left[(l-m+1)\right][(m-l)log\left({\xi }_2\right)-\frac{1}{l-m}log\left({\xi }_1\right)](m-2){(log\left({\xi }_2\right)-log\left({\xi }_1\right))}^{m-3}}}.\hfill \end{array}$$

Proof. 

Let us fix s in $({\xi }_1,{\xi }_2)$ and consider

$$\begin{array}{c}\hfill {\Phi }_{log}\left(s\right):={\left(log\left({\xi }_2\right)-log\left(s\right)\right)}^{l-m+1}\left(log\left(s\right)-log\left({\xi }_1\right)\right).\end{array}$$

For $s={\xi }_1,$ or $s={\xi }_2,$${\Phi }_{log}\left(s\right):=0$ and therefore it is not of interest.

Making the differentiation of ${\Phi }_{log}$ with respect to $s,$ we get

$$\begin{array}{cc}\hfill & {\Phi }_{log}^{\prime }\left(s\right)=\left({\displaystyle \frac{-1}{s}}\right)[-\left(log\left({\xi }_2\right)-log\left(s\right)\right)+(l-m+1)\left(log\left(s\right)-log\left({\xi }_1\right)\right)]=0,\hfill \end{array}$$

and the corresponding root takes the form

$$\begin{array}{cc}\hfill & s=s\ast ={\displaystyle \frac{{{\xi }_2}^{\left(\frac{l-m+1}{l-m}\right)}}{{{\xi }_1}^{\frac{1}{l-m}}}}.\hfill \end{array}$$

Therefore

$$\begin{array}{ccc}\hfill \underset{{\xi }_1\le s\le {\xi }_2}{max}{\Phi }_{log}\left(s\right)& =& {\Phi }_{log}(s\ast )\hfill \\ & =& \left((m-l)log\left({\xi }_2\right)-{\displaystyle \frac{1}{l-m}}log\left({\xi }_1\right)\right)(l-m+1)log\left({\displaystyle \frac{{\xi }_2}{{\xi }_1}}\right).\hfill \end{array}$$

(21)

Now using the result of Corollary 1 we find

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge {\displaystyle \frac{\mathrm{\Gamma }\left(l\right)}{\left[(l-m+1)\right][(m-l)log\left({\xi }_2\right)-\frac{1}{l-m}log\left({\xi }_1\right)](m-2){(log\left({\xi }_2\right)-log\left({\xi }_1\right))}^{m-2}}}.\hfill \end{array}$$

Thus, the proof of Corollary 3 is completed. □

Next, let us define the function $\Psi$ by: $\Psi \left(s\right):=GN({\xi }_2,s)\mathrm{\Gamma }\left(l\right)$.

The function $\Psi$ takes the following form:

$$\begin{array}{cc}\hfill \Psi \left(s\right):& =GN({\xi }_2,s)\mathrm{\Gamma }\left(l\right)\hfill \\ \hfill & ={\displaystyle \frac{{(log\left({\xi }_2\right)-log\left({\xi }_1\right))}^{l-1}{(log\left({\xi }_2\right)-log\left(s\right))}^{l-m+1}}{{(log\left({\xi }_2\right)-log\left({\xi }_1\right))}^{l-m+1}}}\hfill \\ \hfill & -{(log\left({\xi }_2\right)-log\left(s\right))}^{l-1}\hfill \\ \hfill & ={(log\left({\xi }_2\right)-log\left({\xi }_1\right))}^{m-2}{(log\left({\xi }_2\right)-log\left(s\right))}^{l-m+1}\hfill \\ \hfill & ={(log\left({\xi }_2\right)-log\left(s\right))}^{l-m+1}[{(log\left({\xi }_2\right)-log\left({\xi }_1\right))}^{m-2}-{(log\left({\xi }_2\right)-log\left(s\right))}^{m-2}].\hfill \end{array}$$

Applying the binomial formula, it becomes

$$\begin{array}{cc}\hfill \Psi \left(s\right):& =GN({\xi }_2,s)\mathrm{\Gamma }\left(l\right)\hfill \\ \hfill & ={(log\left({\xi }_2\right)-log\left(s\right))}^{l-m+1}(log\left(s\right)-log\left({\xi }_1\right)){\sum }_{i=1}^{m-2}{\left(log{\xi }_2-log{\xi }_1\right)}^{m-2-i}{(log\left({\xi }_2\right)-log\left(s\right))}^{i-1}.\hfill \end{array}$$

We start with the second kind of Lyapunov inequality.

Second kind of Lyapunov inequaliy

Theorem 3. 

(Sharp Lyapunov inequality)

Let $w\ne 0$ be the solution of

$$\begin{array}{cc}\hfill & \left(D_{a;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\hfill \\ \hfill & w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=\dots =w^{(m-2)}\left({\xi }_1\right)=0,w^{(m-2)}\left({\xi }_2\right)=0,\hfill \end{array}$$

where $2<m-1<l\le m,$ and $A\in C\left([{\xi }_1,{\xi }_2]\right)$. Then the following inequality is satisfied:

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge \mathrm{\Gamma }\left(l\right){\displaystyle \frac{{(l-m+1)}^{(l-m+1)}}{{(l-m+2)}^{(l-m+2)}}}{\displaystyle \frac{1}{(m-2){(log{\xi }_2-log{\xi }_1)}^{l-1}}}.\hfill \end{array}$$

Proof. 

Making the differentiation of $\Psi$ with respect to s, we find

$$\begin{array}{cc}\hfill & {\Psi }^{\prime }\left(s\right):=-{\displaystyle \frac{1}{s}}[(l-m+1){(log\left({\xi }_2\right)-log\left(s\right))}^{l-m}{(log\left({\xi }_2\right)-log\left({\xi }_1\right))}^{m-2}\hfill \\ \hfill & -(l-1){(log\left({\xi }_2\right)-log\left(s\right))}^{l-2}]\hfill \end{array}$$

The equation ${\Psi }^{\prime }\left(s\right):=0$ leads to

$$\begin{array}{cc}\hfill & {\Psi }^{\prime }\left(s\right):={\left({\displaystyle \frac{l-m+1}{l-1}}\right)}^{{\displaystyle \frac{1}{m-2}}}(log\left({\xi }_2\right)-log\left({\xi }_1\right))=(log\left({\xi }_2\right)-log\left(s\right))=0,\hfill \end{array}$$

and therefore the unique solution s is given by

$$\begin{array}{c}\hfill s=s^{\ast }={\xi }_1^{\left({\displaystyle \frac{l-m+1}{l-m+2}}\right)}{{\xi }_2}^{\left({\displaystyle \frac{1}{l-m+2}}\right)}.\end{array}$$

Hence, we deduce that the maximum of $\Psi$ is attained at $s=s^{\ast }$ and consequently

$$\begin{array}{cc}\hfill \Psi \left(s^{\ast }\right):& ={\displaystyle \frac{{(l-m+1)}^{(}l-m+1)}{{(l-m+2)}^{(}l-m+2)}}{(log\left({\xi }_2\right)-log\left({\xi }_1\right))}^{l-m+2},\hfill \end{array}$$

where $2<m-1<l\le m,$ and $A\in C\left([{\xi }_1,{\xi }_2]\right)$.

And the proof of Theorem 3 is completed. □

Our analysis reveals a fundamental progression in the quality of estimate from Corollary 3 and Theorem 3. Each subsequent corollary improves the bound on $\left(GN\right)$’s maximum value. The progression culminates in Corollary 3 achieving sharpness. Corollary 3 provides the exact maximal value of $\left(GN\right)$. The optimal constant in the resulting inequality is achieved. While all derived inequalities are mathematically valid, the hierarchy demonstrates: the critical dependence on precise Green function analysis, how maximum value determination directly impacts inequality sharpness, and the non-trivial relationship between derivative operator type and optimal constants.

Remark 3. 

Notice that the arithmetic–geometric–harmonic inequality involving the logarithmic function takes the following form:

$$\begin{array}{c}\hfill (log{\displaystyle \frac{s}{{\xi }_1}})(log{\displaystyle \frac{{\xi }_2}{s}})<{\left({\displaystyle \frac{log\frac{{\xi }_2}{{\xi }_1}}{2}}\right)}^2.\end{array}$$

This inequality results from the classical inequality by changes of variables.

Remark 4. 

For the same argument as above, since the logarithmic function is monotone, the following inequality is satisfied:

$$\begin{array}{c}\hfill (log{\displaystyle \frac{s}{{\xi }_1}})(log{\displaystyle \frac{{\xi }_2}{s}})\le {\displaystyle \frac{{({\xi }_2-{\xi }_1)}^2}{4}},\end{array}$$

for ${\xi }_1,{\xi }_2>1$. With a slight modification of the domain $[{\xi }_1,{\xi }_2],$ one gets a novel Lyapunov inequality for the Hadamard fractional value Problem (20).

Corollary 4. 

Let $w\ne 0$ be the solution of

$$\begin{array}{cc}\hfill & \left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,{\xi }_1<r<{\xi }_2,\hfill \\ \hfill & w\left({\xi }_1\right)=w^{\prime }\left({\xi }_1\right)=\dots =w^{(m-2)}\left({\xi }_1\right)=0,w^{(m-2)}\left({\xi }_2\right)=0,\hfill \end{array}$$

where $2<m-1<l\le m,$${\xi }_2>{\xi }_1>1,$ and $A\in C\left([{\xi }_1,{\xi }_2]\right)$.

Then the following inequality is satisfied:

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge \mathrm{\Gamma }\left(l\right){\displaystyle \frac{{(l-m+1)}^{(l-m+1)}}{{(l-m+2)}^{(l-m+2)}}}{\displaystyle \frac{1}{(m-2){({\xi }_2-{\xi }_1)}^{l-1}}}.\hfill \end{array}$$

Proof. 

Based on Theorem 3 and Remark 4, the proof is straightforward.

The next example illustrates the sharp Lyapunov inequality for the Hadamard fractional value problem in the interval $[{\xi }_1,{\xi }_2]$, and when ${\xi }_1>1,{\xi }_2>1,$ we obtain the sharp one for the Riemann–Liouville fractional value problem. □

Example 1. 

For m = 3, the conclusion of Corollary 4 takes the form

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge \mathrm{\Gamma }\left(l\right){\displaystyle \frac{{(l-m+1)}^{(l-m+1)}}{{(l-m+2)}^{(l-m+2)}}}{\displaystyle \frac{1}{(m-2){(log{\xi }_2-log{\xi }_1)}^{l-1}}},\hfill \end{array}$$

where $2<l\le 3,$ and ${\xi }_2>{\xi }_1>1$.

In [17], with $2<\alpha \le 3$, the authors obtained the following result:

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge \mathrm{\Gamma }\left(l\right){\displaystyle \frac{{(l-2)}^{(l-2)}}{{(l-1)}^{(l-1)}}}{\displaystyle \frac{1}{{(log{\xi }_2-log{\xi }_1)}^{l-1}}}.\hfill \end{array}$$

Remark 5. 

For ${\xi }_1,{\xi }_2>1,$ we obtain Corollary 3.9 investigated in [17].

4. Applications

In this section, we give two applications of the above results.

For this first application, we illustrate the result in Corollary 4.

Indeed, let $w\ne 0$ be the solution of the following Hadamard fractional problem:

$$\begin{array}{cc}\hfill & \left(D_{{\xi }_1;log}^{l}w\right)\left(r\right)+A\left(r\right)w\left(r\right)=0,e<r<\pi ,\hfill \\ \hfill & w\left(e\right)=w^{\prime }\left(e\right)=w^{\prime \prime }\left(e\right)=0,w^{\prime \prime }\left(\pi \right)=0,\hfill \end{array}$$

(22)

where $l=4.5,$ and $A\in C\left(\right[e,\pi \left]\right)$.

Corollary 5. 

Let $w\ne 0$ be the solution of Problem (22), then

$$\begin{array}{cc}\hfill & {\int }_e^{\pi }\left|A\left(s\right)\right|ds\ge {\displaystyle \frac{3}{4{(\pi -e)}^2}}.\hfill \end{array}$$

Proof. 

It is sufficient to apply Corollary 4 with ${\xi }_1=e,{\xi }_2=\pi ,$ $l=2.5$, and $m=3$. We obtain

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge \mathrm{\Gamma }\left(2.5\right){\displaystyle \frac{{\left(0.5\right)}^{\left(0.5\right)}}{{\left(1.5\right)}^{\left(1.5\right)}}}{\displaystyle \frac{1}{\left(1\right){({\xi }_2-{\xi }_1)}^{1.5}}}=1.84.\hfill \end{array}$$

The second application is devoted to the illustration of Theorem 3. We consider the following Hadamard fractional problem where we look to estimate the lower bound of the eigenvalue $\lambda$.

$$\begin{array}{cc}\hfill & \left(D_{1.5;log}^{l}w\right)\left(r\right)+\lambda w\left(r\right)=0,1.5<r\le 2,\hfill \\ \hfill & w\left(1.5\right)=w^{\prime }\left(1.5\right)=\dots =w^{{(}^{\prime \prime })}\left(1.5\right)=0,w^{{(}^{\prime \prime })}\left(2\right)=0.\hfill \end{array}$$

(23)

We say that $\lambda$ is an eigenvalue of the following Hadamard fractional problem if and only if (23) admits at least a solution $w_{\ast }\ne 0,$$w_{\ast }\in C^{m-2}\left([a,b]\right)\cap A{C}^l\left([1.5,2]\right),$ where $m\ge 3,$$3<l\le 4$. □

Corollary 6. 

If λ is an eigenvalue of (23), then

$$\begin{array}{c}\hfill \lambda \ge 2.63\sqrt{\pi }.\end{array}$$

We will show that there exists a non-trivial solution to the following Hadamard fractional Problem (23) if

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge \mathrm{\Gamma }\left(l\right){\displaystyle \frac{{\left(0.5\right)}^{\left(0.5\right)}}{{\left(1.5\right)}^{\left(1.5\right)}}}{\displaystyle \frac{1}{\left(2\right){(log2-log1.5)}^{2.5}}}=1.062\hfill \end{array}$$

is satisfied.

Proof. 

We apply the conclusion of Theorem 3:

$$\begin{array}{cc}\hfill & {\int }_{{\xi }_1}^{{\xi }_2}\left|A\left(s\right)\right|{\displaystyle \frac{ds}{s}}\ge \mathrm{\Gamma }\left(l\right){\displaystyle \frac{{\left(\frac{l-m+1}{l-1}\right)}^{\left(\frac{l-m+1}{m-2}\right)}}{{(log{\xi }_2-log{\xi }_1)}^{m-2}}}\left({\displaystyle \frac{m-2}{l-1}}\right),\hfill \end{array}$$

with $m=4,$ $l=3.5,$ ${\xi }_1=1.5,$ and ${\xi }_2=2$ and the proof will be straightforward. □

5. Conclusions

In the last decades, the class of Lyapunov-type inequalities associated with different fractional Hadamard boundary value problems has received special attention from many researchers. In this article, we got Lyapunov-type and Hartman–Wintner inequalities for the Hadamard fractional boundary value problem. Our investigation is based on the arithmetic–harmonic–geometric inequality and the best maximum principle as well. An essential role in this study is the best estimates of the Green function based on the best maximum principle. When we turn to the literature focusing on this subject, a deep consultation of the previous work leads us to conclude that the published results in this field are not always sharp. This is why not all extensions involving the different types of fractional derivative operators, in particular for the Hadamard case, are possible. The new results that we established here are valid for any fractional order for the Hadamard case, whereas for the Riemann–Liouville (resp. Caputo) case, by restricting the domain in consideration to $[{\xi }_1,{\xi }_2]$ where ${\xi }_1>1,$ and ${\xi }_2>1,$ we were able to find the particular cases for the Riemann–Liouville operators investigated in [17,18]. As an open question, this fractional Hadamard boundary value problem may be extended to the $\Psi$-fractional derivative operator, the derivative of a function with respect to another function. To this end, it seems that once Green’s function is constructed, one may always consider $G(t,s)$ for $t\le s$ only and look to the best maximum principle of $g_2(s,s)$. As an example, if we return to the two pioneer papers of R. A. Ferreira, we retrieve the same result for the maximum of Green’s function without considering the case $s\le t$ where $G(t,s)=g_1(t,s)$. Frequently, the main difficulty is in getting the maximum of G, in this case, with limited success in overcoming it. This is why, by comparison, $|G_1(t,s)|\le |G_2(t,s)|$ overcomes the obstacle and gets the desired result.