On Hermite-Hadamard-type inequalities for second order differential inequalities with inverse-square potential

1.   Introduction

The double Hermite-Hadamard inequality ^{[13, 14]} can be stated as follows: If $u: I\to \mathbb{R}$ is convex on $I$ then for all $a_1, a_2\in I$ with $a_1 < a_2$ we have

Inequalities of type (1.1) have many applications in theoretical and applied mathematics, such as convex analysis, optimization, numerical integrations, etc. As it is mentioned above, (1.1) holds for convex functions $u: I\to \mathbb{R}$. A natural question is to ask whether it is possible to derive inequalities of type (1.1) for other classes of functions. Several results in this direction were obtained for different types of convex functions, such as $m$-convex functions ^{[7, 10, 25]}, $s$-convex functions ^{[11, 12, 17, 22, 27]}, $h$-convex functions ^{[5, 23, 24]}, logarithmically convex functions ^{[19, 20, 26]}, convex functions with respect to a pair of functions ^{[3, 4, 21, 28]} and convex functions on the coordinates ^{[6, 15, 18]}.

The study of Hermite-Hadamard-type inequalities for second-order differential inequalities was first considered in ^{[1, 2, 8, 9]}. Namely, in ^[8], Dragomir established inequalities of type (1.1) for the class of functions $u: I\to \mathbb{R}$ satisfying: For all $a, b\in I$ with $0 < b-a < \frac{\pi}{\rho}$

where $\rho > 0$ is a constant. A function $u$ satisfying the above condition is called trigonometrically $\rho$-convex. Moreover, in ^[1]- it was shown that, if $u\in C^2(I)$, then $u$ is trigonometrically $\rho$-convex if and only if $u$ satisfies the second order differential inequality

Similar results were obtained in ^[9] for the class of functions $u: I\to \mathbb{R}$ satisfying: For all $a, b\in I$ with $a < b$

where $\rho > 0$ is a constant. A function $u$ satisfying the above condition is called hyperbolic $\rho$-convex. On the other hand, it was proved in ^[2] that, if $u\in C^2(I)$, then $u$ is hyperbolic $\rho$-convex if and only if $u$ satisfies the second order differential inequality

Motivated by the above cited works, we extend in this paper the (right) Hermite-Hadamard inequality (1.1) to the class of functions $u\in C^2((0, \infty))$ satisfying second-order differential inequalities of the form

where $0 < k\leq \frac{1}{4}$ is a constant. Remark that in the limit case, $k=0$ (1.2) reduces to

which is equivalent to the convexity of $u$ on $(0, \infty)$. Remark also that, if $u\leq 0$ on $(0, \infty)$ and $u$ satisfies (1.2), then $u$ is convex on $(0, \infty)$. However, if $u$ satisfies (1.2) then $u$ is not necessarily convex. For instance, the function $u(x)=\sqrt{x}+x$ is concave on $(0, \infty)$ and

The rest of the paper is arranged as follows: In Section 2 we fix some notations and establish some lemmas that will be used later. In Section 3, we establish Hermite-Hadamard-type inequalities for the class of functions $u\in C^2((0, \infty))$ satisfying (1.2). We discuss separately the cases of $k=\frac{1}{4}$ and $0 < k < \frac{1}{4}$. The obtained results in Section 3 are extended to the two-dimensional case in Section 4.

2.   Auxiliary results

In this section, we establish some lemmas that will be useful in the proofs of our main results. We first fix some notations that will be used throughout this paper.

We denote by $M, N: (1, \infty)\to \mathbb{R}$ the functions defined by

For $0 < k\leq \frac{1}{4}$ and $0 < a < b$, we introduce the terms

Lemma 2.1. Let $0 < a < b$ and

Then the following holds:

${\rm{(i)}}$ $F(a)=F(b)=0$;

${\rm{(ii)}}$ $F'(a)=\sqrt{a}M\left(\frac{b}{a}\right)$, $F'(b)=\sqrt{b}M\left(\frac{b}{a}\right)$;

${\rm{(iii)}}$ $F(x)\geq 0$ for all $x\in [a, b]$.

Proof. (ⅰ) follows immediately from the definition of $F$. On the other hand, for all $x\in [a, b]$, we have

Taking $x=a$ (resp. $x=b$) in (2.2), we obtain (ⅱ). Furthermore, from (i) and Rolle's theorem, there exists at least $c\in (a, b)$ such that $F'(c)=0$. However, from (2.2), we have $F'(x)=0$ if and only if

Consequently, $c=\left(\frac{2}{3}B(a, b)\right)^{\frac{2}{3}}\in (a, b)$ and from (2.2) $F$ reaches its maximum value at $x=c$. Then due to (ⅰ) we deduce that $F(x)\geq 0$ for all $x\in [a, b]$, which proves (ⅲ). □

Lemma 2.2. Let $0 < a < b$ and

where $F$ is the function defined by (2.1). Then the following holds:

${\rm{(i)}}$ $f(a)=f(b)=0$;

${\rm{(ii)}}$ $f'(a)=\frac{4a}{9}M\left(\frac{b}{a}\right)$, $f'(b)=\frac{-4 b}{9}N\left(\frac{b}{a}\right)$;

${\rm{(iii)}}$ $f(x)\geq 0$ for all $x\in [a, b]$;

${\rm{(iv)}}$ $f''(x)+ \frac{1}{4x^2}f(x)=-1$ for all $x\in (a, b)$.

Proof. (ⅰ)–(ⅲ) are the immediate consequences of Lemma 2.1. (ⅳ) follows from the definition of $f$ and some elementary calculations. □

Lemma 2.3. Let $0 < k < \frac{1}{4}$, $0 < a < b$ and

Then the following holds:

${\rm{(i)}}$ $G(a)=G(b)=0$;

${\rm{(ii)}}$ $G'(a)=\lambda_ka^{\lambda_k-1}b^{\frac{3-\lambda_k}{2}}D_k(a, b)-\left(\frac{\lambda_k+3}{2}\right)a^{\frac{\lambda_k+1}{2}}$;

${\rm{(iii)}}$ $G'(b)=b^{\frac{\lambda_k+1}{2}}\left(\lambda_kD_k(a, b)-\left(\frac{\lambda_k+3}{2}\right)\right)$;

${\rm{(iv)}}$ $G(x)\geq 0$ for all $x\in [a, b]$.

Proof. (ⅰ) follows immediately from the definition of $G$. On the other hand, for all $x\in [a, b]$, we have

Taking $x=a$ (resp. $x=b$) in (2.5) we obtain (ⅱ) and (ⅲ). Furthermore, from (ⅰ) and Rolle's theorem, there exists at least $c\in (a, b)$ such that $G'(c)=0$. However, from (2.5), $G'(x)=0$ if and only if

Consequently, $c=\left(\frac{2\lambda_k D_k(a, b)}{\lambda_k+3}\right)^{\frac{2}{3-\lambda_k}}b\in (a, b)$ and from (2.5), $G$ reaches its maximum value at $x=c$. Then due to (ⅰ) we deduce that $G(x)\geq 0$ for all $x\in [a, b]$, which proves (ⅳ). □

Lemma 2.4. Let $0 < k < \frac{1}{4}$, $0 < a < b$ and

where $G$ is the function defined by (2.4). Then the following holds:

${\rm{(i)}}$ $g(a)=g(b)=0$;

${\rm{(ii)}}$ $g'(a)=\frac{3+\lambda_k}{2(k+2)}a\left(\frac{2\lambda_k}{3+\lambda_k}\left(\frac{b}{a}\right)^{\frac{3-\lambda_k}{2}}D_k(a, b)-1\right)$;

${\rm{(iii)}}$ $g'(b)=\frac{3+\lambda_k}{2(k+2)}b\left(\frac{2\lambda_k}{3+\lambda_k}D_k(a, b)-1\right)$;

${\rm{(iv)}}$ $g(x)\geq 0$ for all $x\in [a, b]$;

${\rm{(v)}}$ $g''(x)+ \frac{k}{x^2}g(x)=-1$ for all $x\in (a, b)$.

Proof. (ⅰ)–(ⅳ) are the immediate consequences of Lemma 2.3. (ⅴ) follows from the definition of $g$ and some elementary calculations. □

Lemma 2.5. We have

for all $s > 1$.

Proof. We have

where

Remark that $\lim\limits_{s\to 1^+}\varphi(s)=0$. On the other hand, for all $s > 1$ we have

which shows that $\varphi$ is a nondecreasing function. Consequently, we have $\varphi(s)\geq 0$, which implies that $M(s)\geq 0$. Similarly, we have

where

Remark that $\lim\limits_{s\to 1^+}\psi(s)=0$. We also have

which shows that $\psi$ is a nondecreasing function. Consequently, we have $\psi(s)\geq 0$, which implies that $N(s)\geq 0$. □

Lemma 2.6. Let $0 < k < \frac{1}{4}$ and $0 < a < b$. Then the following holds:

and

Proof. We have

where

Observe that $\gamma(1)=0$. On the other hand, for all $s\geq 1$, we have

which shows that $\gamma$ is a nondecreasing function. Consequently, we have $\gamma(s)\geq 0$ for all $s\geq 1$, which implies that $\gamma\left(\frac{b}{a}\right)\geq 0$. Then by (2.9), we obtain (2.7). On the other hand, we have

where

Remark that $\eta(1)=0$, and for all $0 < s\leq 1$, we have

which shows that $\gamma$ is a decreasing function. Consequently, we have $\eta\left(\frac{a}{b}\right)\geq \eta(1)=0$. Then by (2.10) we obtain (2.8). □

3.   Hermite-Hadamard-type inequalities

For $0 < k\leq \frac{1}{4}$, we consider the class of real-valued functions

Remark that, in the limit case $k=0$, we have

In this section we establish Hermite-Hadamard-type inequalities for functions $u\in C_k^{2, +}((0, \infty))$.

We first consider the case $k=\frac{1}{4}$.

Theorem 3.1. If $u\in C_{1/4}^{2, +}((0, \infty))$, then for all $a, b\in \mathbb{R}$ with $0 < a < b$, we have

Proof. Let $u\in C_{1/4}^{2, +}((0, \infty))$ and $0 < a < b$. By Lemma 2.2-(ⅳ), we have

where $f$ is the function defined by (2.1). Integrations by parts give us that

On the other hand, from Lemma 2.2-(ⅰ) we know that $f(a)=f(b)=0$. Consequently, we have

From (3.2) and (3.3), we deduce that

Since $f\geq 0$ by Lemma 2.2-(ⅲ) and $u\in C_{1/4}^{2, +}((0, \infty))$, then

Hence, it holds that

Finally, using Lemma 2.2-(ⅱ), we obtain (3.1). □

We now consider case $0 < k < \frac{1}{4}$.

Theorem 3.2. Let $0 < k < \frac{1}{4}$. If $u\in C_{k}^{2, +}((0, \infty))$, then for all $a, b\in \mathbb{R}$ with $0 < a < b$, we have

Proof. Let $u\in C_{k}^{2, +}((0, \infty))$ and $0 < a < b$. We shall follow the same approach used in the proof of Theorem 3.1. Namely, by Lemma 2.4-(v), we have

where $g$ is the function defined by (2.6). Integrating by parts, we get

which implies by Lemma 2.4-(ⅰ) that

In view of (3.5) and (3.6), we have

Since $g\geq 0$ by Lemma 2.4-(ⅳ) and $u\in C_{k}^{2, +}((0, \infty))$, then

Consequently, we get

Finally, using Lemma 2.4-(ⅱ)-(ⅲ), we obtain (3.4). □

Remark 3.1. Remark that in the limit case $k=0$, (3.4) reduces to the right Hermite-Hadamard inequality for convex functions

4.   The two-dimensional case

We now consider the class of real-valued functions

where $0 < k\leq \frac{1}{4}$ and $(0, \infty)^2=(0, \infty)\times (0, \infty)$. Notice that in the limit case $k=0$, $C_0^{2, +}((0, \infty)^2)$ is the class of functions $u\in C^2((0, \infty)^2)$ such that $u$ is convex on the coordinates in $(0, \infty)^2$ (see Dragomir ^[6]). In this section, we extend some of the obtained results in ^[6] to the class of functions $C_k^{2, +}((0, \infty)^2)$.

We first consider the case $k=\frac{1}{4}$.

Theorem 4.1. If $u\in C_{1/4}^{2, +}((0, \infty)^2)$, then for all $a, b, c, d\in \mathbb{R}$ with $0 < a < b$ and $0 < c < d$ we have

Proof. Let $u\in C_{1/4}^{2, +}((0, \infty)^2)$ and $a, b, c, d\in \mathbb{R}$ with $0 < a < b$ and $0 < c < d$. By the definition of $C_{1/4}^{2, +}((0, \infty)^2)$, the function $u(x, \cdot): y\mapsto u(x, \cdot)(y)=u(x, y)$ belongs to $C_{1/4}^{2, +}((0, \infty))$ for all $x > 0$. Then by Theorem 3.1 we have

for all $x > 0$. Integrating the above-mentioned inequality over $(a, b)$ we get

Similarly, by the definition of $C_{1/4}^{2, +}((0, \infty)^2)$, the function $u(\cdot, y): x\mapsto u(\cdot, y)(x)=u(x, y)$ belongs to $C_{1/4}^{2, +}((0, \infty))$ for all $y > 0$. Then by Theorem 3.1 we have

for all $y > 0$. Integrating the above inequality over $(c, d)$, we obtain

Summing (4.2) and (4.3), we obtain

which proves the first inequality in (4.1). On the other hand, since the function $u(\cdot, c): x\mapsto u(\cdot, c)(x)=u(x, c)$ belongs to $C_{1/4}^{2, +}((0, \infty))$, then by Theorem 3.1 we have

Multiplying the above inequality by $cM\left(\frac{d}{c}\right)$ and using Lemma 2.5, we obtain

Similarly, we have

which implies after multiplication by $dN\left(\frac{d}{c}\right)$ (notice that by Lemma 2.5 we have $N(s)\geq 0$ for all $s > 1$) that

In the same manner, we obtain

and

Finally, the second inequality in (4.1) follows from (4.4)–(4.7). □

We next consider case $0 < k < \frac{1}{4}$.

Theorem 4.2. Let $0 < k < \frac{1}{4}$. If $u\in C_{k}^{2, +}((0, \infty)^2)$, then for all $a, b, c, d\in \mathbb{R}$ with $0 < a < b$ and $0 < c < d$ we have

Proof. Let $u\in C_{k}^{2, +}((0, \infty)^2)$ and $a, b, c, d\in \mathbb{R}$ with $0 < a < b$ and $0 < c < d$. By Theorem 3.2, we have

for all $x > 0$. Integrating the above inequality over $(a, b)$, we obtain

Similarly, by Theorem 3.1, we have

for all $y > 0$. Integrating the above inequality over $(c, d)$, we obtain

Summing (4.9) and (4.10), we obtain

which proves the first inequality in (4.8). We now use Theorem 3.2 to get

Multiplying the above inequality by $c\left(\left(\frac{d}{c}\right)^{\frac{3-\lambda_k}{2}}\frac{2\lambda_k}{\lambda_k+3}D_k(c, d)-1\right)$, it follows from Lemma 2.6 (see (2.7)) that

Using Theorem 3.2, we obtain

Multiplying the above inequality by $d\left(1-\frac{2\lambda_k}{\lambda_k+3}D_k(c, d)\right)$, it follows from Lemma 2.6 (see (2.8)) that

Similarly, we have

and

Finally the second inequality in (4.8) follows from (4.11)–(4.14). □

Remark 4.1 In the limit case $k=0$, we have

In this case, (4.8) reduces to

The above inequalities were derived previously by Dragomir ^[6] when $u$ is convex on the coordinates.

5.   Conclusions

Hermite-Hadamard-type inequalities are very useful in many branches of pure and applied mathematics. Such inequalities were established by many authors for different kinds of convex functions. The class of functions $u\in C^2((0, \infty))$ satisfying second-order differential inequalities of the form

is investigated in this paper. Namely, Hermite-Hadamard-type inequalities are established for this class of functions in both cases $k=\frac{1}{4}$ (see Theorem 3.1) and $0 < k < \frac{1}{4}$ (see Theorem 3.2). Next, the obtained results are extended to the two-dimensional case by considering the class of functions $u=u(x, y)\in C^2((0, \infty)^2)$ satisfying the system of second order differential inequalities

Hermite-Hadamard-type inequalities are derived for this class of functions in the case $k=\frac{1}{4}$ (see Theorem 4.1) and the case $0 < k < \frac{1}{4}$ (see Theorem 4.2).

An interesting question consists of studying the class of functions $u\in C^2(\Omega)$ satisfying second-order differential inequalities of the form

where $\Omega\subset \mathbb{R}^n\backslash\{0\}$ and $\Delta$ is the Laplacian operator ($\Delta =\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2}$). Notice that in the limit case $k=0$, the above class of functions reduces to the class of subharmonic functions (see, e.g., ^[16]).

Author contributions

Hassen Aydi: Conceptualization, methodology, investigation, formal analysis, writing review and editing; Bessem Samet: Conceptualization, methodology, validation, investigation, writing original draft preparation; Manuel De La Sen: methodology, validation, formal analysis, investigation, writing review and 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

B. Samet is supported by Researchers Supporting Project number (RSP2024R4), King Saud University, Riyadh, Saudi Arabia. M. De La Sen is supported by the project: Basque Government IT1555-22.

Conflict of interest

The authors declare no conflict of interest.