Lag synchronization of complex-valued interval neural networks via distributed delayed impulsive control

Zhifeng Lu; Fei Wang; Yujuan Tian; Yaping Li; Zhifeng Lu; Fei Wang; Yujuan Tian; Yaping Li

doi:10.3934/math.2023277

AIMS Mathematics

2023, Volume 8, Issue 3: 5502-5521. doi: 10.3934/math.2023277

Previous Article Next Article

Research article

Lag synchronization of complex-valued interval neural networks via distributed delayed impulsive control

1.
School of Mathematics and Statistics, Shandong Normal University, Ji'nan, 250014, China
2.
School of Information Engineering, Shandong Management University, Ji'nan, 250357, China

Received: 21 October 2022 Revised: 27 November 2022 Accepted: 04 December 2022 Published: 19 December 2022
MSC : 34D06, 92B20, 93C27

This paper investigates the lag synchronization problem of complex-valued interval neural networks with both discrete and distributed time-varying delays under delayed impulsive control. A distributed delayed impulsive controller that depends on the accumulation of the states over a history time period is designed to guarantee the exponential lag synchronization between the drive and the response systems. By employing the complex Lyapunov method and a novel impulsive differential inequality technique, some delay-dependent synchronization criteria are established in terms of complex-valued linear matrix inequalities (LMIs). Finally, a numerical example is given to illustrate the effectiveness of the theoretical results.

Keywords:

Citation: Zhifeng Lu, Fei Wang, Yujuan Tian, Yaping Li. Lag synchronization of complex-valued interval neural networks via distributed delayed impulsive control[J]. AIMS Mathematics, 2023, 8(3): 5502-5521. doi: 10.3934/math.2023277

Related Papers:

[1]	Zhenshu Wen, Lijuan Shi . Exact explicit nonlinear wave solutions to a modified cKdV equation. AIMS Mathematics, 2020, 5(5): 4917-4930. doi: 10.3934/math.2020314
[2]	Abdulghani R. Alharbi . Traveling-wave and numerical solutions to a Novikov-Veselov system via the modified mathematical methods. AIMS Mathematics, 2023, 8(1): 1230-1250. doi: 10.3934/math.2023062
[3]	Naher Mohammed A. Alsafri, Hamad Zogan . Probing the diversity of kink solitons in nonlinear generalised Zakharov-Kuznetsov-Benjamin-Bona-Mahony dynamical model. AIMS Mathematics, 2024, 9(12): 34886-34905. doi: 10.3934/math.20241661
[4]	M. Ali Akbar, Norhashidah Hj. Mohd. Ali, M. Tarikul Islam . Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics. AIMS Mathematics, 2019, 4(3): 397-411. doi: 10.3934/math.2019.3.397
[5]	Maysaa Al-Qurashi, Saima Rashid, Fahd Jarad, Madeeha Tahir, Abdullah M. Alsharif . New computations for the two-mode version of the fractional Zakharov-Kuznetsov model in plasma fluid by means of the Shehu decomposition method. AIMS Mathematics, 2022, 7(2): 2044-2060. doi: 10.3934/math.2022117
[6]	Yunmei Zhao, Yinghui He, Huizhang Yang . The two variable (φ/φ, 1/φ)-expansion method for solving the time-fractional partial differential equations. AIMS Mathematics, 2020, 5(5): 4121-4135. doi: 10.3934/math.2020264
[7]	Naveed Iqbal, Muhammad Bilal Riaz, Meshari Alesemi, Taher S. Hassan, Ali M. Mahnashi, Ahmad Shafee . Reliable analysis for obtaining exact soliton solutions of (2+1)-dimensional Chaffee-Infante equation. AIMS Mathematics, 2024, 9(6): 16666-16686. doi: 10.3934/math.2024808
[8]	Hammad Alotaibi . Solitary waves of the generalized Zakharov equations via integration algorithms. AIMS Mathematics, 2024, 9(5): 12650-12677. doi: 10.3934/math.2024619
[9]	M. A. El-Shorbagy, Sonia Akram, Mati ur Rahman, Hossam A. Nabwey . Analysis of bifurcation, chaotic structures, lump and $M-W$ -shape soliton solutions to $(2+1)$ complex modified Korteweg-de-Vries system. AIMS Mathematics, 2024, 9(6): 16116-16145. doi: 10.3934/math.2024780
[10]	M. Hafiz Uddin, M. Ali Akbar, Md. Ashrafuzzaman Khan, Md. Abdul Haque . New exact solitary wave solutions to the space-time fractional differential equations with conformable derivative. AIMS Mathematics, 2019, 4(2): 199-214. doi: 10.3934/math.2019.2.199

Abstract

1. Introduction

The convexity of function is a classical concept, since it plays a fundamental role in mathematical programming theory, game theory, mathematical economics, variational science, optimal control theory and other fields, a new branch of mathematics, convex analysis, appeared in the 1960s. However, it has been noticed that the functions encountered in a large number of theoretical and practical problems in economics are not classical convex functions, therefore, in the past decades, the generalization of function convexity has attracted the attention of many scholars and aroused great interest, such as $h$ -convex functions ^[1,2,3,4,5], $\log$ -convex functions ^[6,7,8,9,10], $\log$ - $h$ -convex functions ^[11], and especially for coordinated convex ^[12]. Since 2001, various extensions and generalizations of integral inequalities for coordinated convex functions have been established in ^{[12,13,14,15,16,17]}.

On the other hand, calculation error has always been a troublesome problem in numerical analysis. In many problems, it is often to speculate the accuracy of calculation results or use high-precision operation as far as possible to ensure the accuracy of the results, because the accumulation of calculation errors may make the calculation results meaningless, interval analysis as a new important tool to solve uncertainty problems has attracted much attention and also has yielded fruitful results, we refer the reader to the papers ^[18,19]. It is worth notion that in recent decades, many authors have combined integral inequalities with interval-valued functions(IVFs) and obtained many excellent conclusions. In ^[20], Costa gave Opial-type inequalities for IVFs. In ^[21,22], Chalco-Cano investigated Ostrowski type inequalities for IVFs by using generalized Hukuhara derivative. In ^[23], Román-Flores derived the Minkowski type inequalities and Beckenbach's type inequalities for IVFs. Very recently, Zhao ^[5,24] established the Hermite-Hadamard type inequalities for interval-valued coordinated functions.

Motivated by these results, in the present paper, we introduce the concept of coordinated log- $h$ -convex for IVFs, and then present some new Jensen type inequalities and Hermite-Hadamard type inequalities for interval-valued coordinated functions. Also, we give some examples to illustrate our main results.

2. Preliminaries

Let $\mathbb{R}_{\mathcal{I}}$ the collection of all closed and bounded intervals of $\mathbb{R}$ . We use $\; \mathbb{R}_{\mathcal{I}}^{+}\;$ and $\; \mathbb{R}^{+}$ to represent the set of all positive intervals and the family of all positive real numbers respectively. The collection of all Riemann integrable real-valued functions on $[a, b]$ , IVFs on $[a, b]$ and IVFs on $\triangle = [a, b]\times[c, d]$ are denoted by $\mathcal{R}_{([a, b])}$ , $\mathcal{IR}_{([a, b])}$ and $\mathcal{ID}_{(\triangle)}$ . For more conceptions on IVFs, see ^[4,25]. Moreover, we have

Theorem 1. ^[4] Let $f:[a, b]\rightarrow \mathbb{R}_{\mathcal{I}}$ such that $f = [\underline{f}, \overline{f}]$ . Then $f\in\mathcal{IR}_{([a, b])}$ iff $\underline{f}$ , $\overline{f}\in\mathcal{R}_{([a, b])}$ and

$\begin{equation*} (\mathcal{IR})\int_{a}^{b}f(x)dx = \left[(\mathcal{R})\int_{a}^{b}\underline{f}(x)dx,(\mathcal{R})\int_{a}^{b}\overline{f}(x)dx\right]. \end{equation*}$

Theorem 2. ^[25] Let $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}$ . If $\mathcal{F}\in\mathcal{ID}_{(\triangle)}$ , then

$\begin{equation*} (\mathcal{ID})\iint_{\triangle} {\mathcal{F}}(x,y)dxdy = (\mathcal{IR})\int_{a}^{b}dx(\mathcal{IR})\int_{c}^{ d}\mathcal{{F}}(x,y)dy. \end{equation*}$

Definition 1. ^[26] Let $h:[0, 1]\rightarrow \mathbb{R}^{+}$ . We say that $f:[a, b]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ is interval $\log$ - $h$ -convex function or that $f\in SX(\log\text{-}h, [a, b], \mathbb{R}_{\mathcal{I}}^{+}),$ if for all $x, y\in[a, b]$ and $\vartheta\in[0, 1],$ we have

$\begin{eqnarray*} \label{hi} f(\vartheta x+(1-\vartheta)y)\supseteq [f(x)]^{h(\vartheta)}[f(y)]^{h(1-\vartheta)}. \end{eqnarray*}$

$h$ is called supermultiplicative if

$\begin{eqnarray} h(\vartheta\tau)\geq h(\vartheta)h(\tau) \end{eqnarray}$

(2.1)

for all $\vartheta, \tau\in[0, 1]$ . If " $\geq$ " in (2.1) is replaced with " $\leq$ ", then $h$ is called submultiplicative.

Theorem 3. ^[26] Let $\mathcal{F}:[a, b]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}, h\left(\frac{1}{2}\right)\not = 0.$ If $\mathcal{F}\in SX(\log\mathit{\text{-}}h, [a, b], \mathbb{R}_{\mathcal{I}}^{+})$ and $\mathcal{F}\in\mathcal{IR}_{([a, b])}$ , then

$\begin{eqnarray} \mathcal{F}\left(\frac{a+b}{2}\right)^{\frac{1}{2h\left(\frac{1}{2}\right)}}\supseteq \exp\left[\frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}(x)dx\right]\supseteq\left[\mathcal{F}(a)\mathcal{F}(b)\right]^{\int_{0}^{1}h(\vartheta)d\vartheta}. \end{eqnarray}$

(2.2)

Theorem 4. ^[27] Let $\mathcal{F}:[a, b]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}, h\left(\frac{1}{2}\right)\not = 0.$ If $\mathcal{F}\in SX(\log\mathit{\text{-}}h, [a, b], \mathbb{R}_{\mathcal{I}}^{+})$ and $\mathcal{F}\in\mathcal{IR}_{([a, b])}$ , then

$\begin{eqnarray} \begin{aligned} \left[\mathcal{F}\left(\frac{a+b}{2}\right)\right]^{\frac{1}{4h^{2}\left(\frac{1}{2}\right)}}&\supseteq\left[\mathcal{F}\left(\frac{3a+b}{4}\right)\mathcal{F}\left(\frac{a+3b}{4}\right)\right]^{\frac{1}{4h\left(\frac{1}{2}\right)}}\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; &\supseteq\left(\int_{a}^{b}\mathcal{F}(x)dx\right)^{\frac{1}{b-a}}\\ &\supseteq\left[\mathcal{F}(a)\mathcal{F}(b)\mathcal{F}^{2}\left(\frac{a+b}{2}\right)\right]^{\frac{1}{2}\int_{0}^{1}h(\vartheta)d\vartheta}\\ &\supseteq\left[\mathcal{F}(a)\mathcal{F}(b)\right]^{\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\int_{0}^{1}h(\vartheta)d\vartheta}.\\ \end{aligned} \end{eqnarray}$

(2.3)

3. Main results

In this section, we define the coordinated $\log$ - $h$ -convex for IVFs and prove some new Jensen type inequalities and Hermite-Hadamard type inequalities by using this new definition.

Definition 2. Let $h:[0, 1]\rightarrow \mathbb{R}^{+}$ . Then $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ is called a coordinated $\log$ - $h$ -convex IVFs on $\triangle$ if the partial mappings

$\begin{eqnarray*} &\mathcal{F}_{y}:[a,b]\rightarrow \mathbb{R}_{\mathcal{I}}^{+},\mathcal{F}_{y}(x) = \mathcal{F}(x,y),\\ &\mathcal{F}_{x}:[c,d]\rightarrow \mathbb{R}_{\mathcal{I}}^{+},\mathcal{F}_{x}(y) = \mathcal{F}(x,y) \end{eqnarray*}$

are $\log$ - $h$ -convex for all $y\in[c, d]$ and $x\in[a, b]$ . Then the set of all coordinated $\log$ - $h$ -convex IVFs on $\triangle$ is denoted by $SX(\log\text{-}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ .

Definition 3. Let $h:[0, 1]\rightarrow \mathbb{R}^{+}$ . Then $\mathcal{F}:\triangle\rightarrow \mathbb{R}^{+}$ is called a coordinated $\log$ - $h$ -convex function in $\triangle$ if for any $(x_{1}, y_{1}), (x_{2}, y_{2})\in\triangle$ and $\vartheta\in[0, 1]$ we have

$\begin{eqnarray} \begin{aligned} \mathcal{F}(\vartheta x_{1}+(1-\vartheta)x_{2},\vartheta y_{1}+(1-\vartheta)y_{2})\leq \left[\mathcal{F}( x_{1},y_{1})\right]^{h(\vartheta)}\left[\mathcal{F}(x_{2},y_{2})\right]^{h(1-\vartheta)}. \end{aligned} \end{eqnarray}$

(3.1)

The set of all $\log$ - $h$ -convex functions in $\triangle$ is denoted by $SX(\log\text{-}h, \triangle, \mathbb{R}^{+}).$ If inequality (3.1) is reversed, then $\mathcal{F}$ is said to be a coordinated $\log$ - $h$ -concave function, the set of all $\log$ - $h$ -concave functions in $\triangle$ is denoted by $SV(\log\text{-}h, \triangle, \mathbb{R}^{+}).$

Definition 4. Let $h:[0, 1]\rightarrow \mathbb{R}^{+}$ . Then $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ is called a coordinated $\log$ - $h$ -convex IVF in $\triangle$ if for any $(x_{1}, y_{1}), (x_{2}, y_{2})\in\triangle$ and $\vartheta\in[0, 1]$ we have

$\begin{eqnarray*} \mathcal{F}(\vartheta x_{1}+(1-\vartheta)x_{2},\vartheta y_{1}+(1-\vartheta)y_{2})\supseteq\left[\mathcal{F}( x_{1},y_{1})\right]^{h(\vartheta)}\left[\mathcal{F}(x_{2},y_{2})\right]^{h(1-\vartheta)}. \end{eqnarray*}$

The set of all $\log$ - $h$ -convex IVFs in $\triangle$ is denoted by $SX(\log\text{-}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+}).$

Theorem 5. Let $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ such that $\mathcal{F} = [\mathcal{\underline{F}}, \mathcal{\overline{F}}]$ . If $\mathcal{F}\in SX(\log\mathit{\text{-}}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ iff $\mathcal{\underline{F}}\in SX(\log\mathit{\text{-}}h, \triangle, \mathbb{R}^{+})$ and $\mathcal{\overline{F}}\in SV(\log\mathit{\text{-}}h, \triangle, \mathbb{R}^{+})$ .

Proof. The proof is completed by combining the Definitions 3 and 4 above and the Theorem 3.7 of ^[4].

Theorem 6. If $\mathcal{F}\in SX(\log\mathit{\text{-}}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ , then $\mathcal{F}\in SX(\log\mathit{\text{-}}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ .

Proof. Assume that $\mathcal{F}\in SX(\log\text{-}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ . Let $\mathcal{F}_{x}:[c, d]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}, \mathcal{F}_{x}(y) = \mathcal{F}(x, y).$ Then for all $\vartheta\in[0, 1]$ and $y_{1}, y_{2}\in[c, d]$ , we have

$\begin{eqnarray*} \begin{aligned} \mathcal{F}_{x}(\vartheta y_{1}+(1-\vartheta)y_{2})& = \mathcal{F}(x,\vartheta y_{1}+(1-\vartheta)y_{2})\\ &\supseteq \mathcal{F}(\vartheta x+(1-\vartheta)x,\vartheta y_{1}+(1-\vartheta)y_{2})\\ &\supseteq \left[\mathcal{F}( x,y_{1})\right]^{h(\vartheta)}\left[\mathcal{F}(x,y_{2})\right]^{h(1-\vartheta)}\\ & = \left[\mathcal{F}_{x}(y_{1})\right]^{h(\vartheta)}\left[\mathcal{F}_{x}(y_{2})\right]^{h(1-\vartheta)}. \end{aligned} \end{eqnarray*}$

Hence $\mathcal{F}_{x}(y) = \mathcal{F}(x, y)$ is $\log$ - $h$ -convex on $[c, d]$ . The fact that $\mathcal{F}_{y}(x) = \mathcal{F}(x, y)$ is $\log$ - $h$ -convex on $[a, b]$ goes likewise.

Remark 1. The converse of Theorem 6 is not generally true. Let $h(\vartheta) = \vartheta$ and $\vartheta\in[0, 1]$ , $\triangle_{1} = \left[\frac{\pi}{4}, \frac{\pi}{2}\right]\times\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$ , and $\mathcal{F}:\triangle_{1}\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ be defined:

$\begin{eqnarray*} \mathcal{F}(x,y) = \left[e^{-sin x-sin y},64xy\right]. \end{eqnarray*}$

Obviously, we have that $\mathcal{F}\in SX(\log\text{-}ch, \triangle_{1}, \mathbb{R}_{\mathcal{I}}^{+})$ and $\mathcal{F}\not\in SX(\log\text{-}h, \triangle_{1}, \mathbb{R}_{\mathcal{I}}^{+}).$ Indeed, if $\left(\frac{\pi}{4}, \frac{\pi}{2}\right), \left(\frac{\pi}{2}, \frac{\pi}{4}\right)\in\triangle_{1}$ , we have

$\begin{eqnarray*} \mathcal{F}\left(\vartheta\frac{\pi}{4}+(1-\vartheta)\frac{\pi}{2},\vartheta\frac{\pi}{2}+(1-\vartheta)\frac{\pi}{4}\right) = \left[e^{-sin\frac{\vartheta\pi}{4}-sin\frac{(1-\vartheta)\pi}{2}},8\pi^{2}\vartheta(1-\vartheta)\right],\\ \left(\mathcal{F}\left(\frac{\pi}{4},\frac{\pi}{2}\right)\right)^{h(\vartheta)}\left(\mathcal{F}\left(\frac{\pi}{2},\frac{\pi}{4}\right)\right)^{h(1-\vartheta)} = \left[e^{\left(1-\frac{\sqrt{2}}{2}\right)\vartheta-1},2^{\vartheta+1}\pi\right]. \end{eqnarray*}$

If $\vartheta = 0,$ then

$\left[0,\frac{1}{e}\right]\not\supseteq\left[\frac{1}{e},2\pi\right].$

Thus, $\mathcal{F}\not\in SX(\log\text{-}h, \triangle_{1}, \mathbb{R}_{\mathcal{I}}^{+}).$

In the following, Jensen type inequalities for coordinated $\log$ - $h$ -convex functions in $\triangle$ is considered.

Theorem 7. Let $p_{i}\in\mathbb{R}^{+}, x_{i}\in[a, b], y_{i}\in[c, d], (i = 1, 2, ..., n), \mathcal{F}:\triangle\rightarrow \mathbb{R}^{+}.$ If $h$ is a nonnegative supermultiplicative function and $\mathcal{F}\in SX(\log\mathit{\text{-}}h, \triangle, \mathbb{R}^{+})$ , then

$\begin{eqnarray} \mathcal{F}\left(\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}x_{i},\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}y_{i}\right)\leq\prod\limits_{i = 1}^{n}[\mathcal{F}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{n}}\right)}, \end{eqnarray}$

(3.2)

where $\mathcal{P}_{n} = \sum\limits_{i = 1}^{n}p_{i}$ . If $h$ is a nonnegative submultiplicative function and $\mathcal{F}\in SV(\log\mathit{\text{-}}h, \triangle, \mathbb{R}^{+})$ , then $(3.2)$ is reversed.

Proof. If $n = 2,$ then from Definition 3, we have

$\begin{eqnarray*} \mathcal{F}\left(\frac{p_{1}}{\mathcal{P}_{2}}x_{1}+\frac{p_{2}}{\mathcal{P}_{2}}x_{2},\frac{p_{1}}{\mathcal{P}_{2}}y_{1}+\frac{p_{2}}{\mathcal{P}_{2}}y_{2}\right)\leq[\mathcal{F}(x_{1},y_{1})]^{h\left(\frac{{p}_{1}}{\mathcal{P}_{2}}\right)}[\mathcal{F}(x_{2},y_{2})]^{h\left(\frac{{p}_{2}}{\mathcal{P}_{2}}\right)}. \end{eqnarray*}$

Suppose (3.2) holds for $n = k$ , then

$\begin{eqnarray*} \mathcal{F}\left(\frac{1}{\mathcal{P}_{k}}\sum\limits_{i = 1}^{k}p_{i}x_{i},\frac{1}{\mathcal{P}_{k}}\sum\limits_{i = 1}^{k}p_{i}y_{i}\right)\leq\prod\limits_{i = 1}^{k}[\mathcal{F}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{k}}\right)}. \end{eqnarray*}$

Now, let us prove that (3.2) is valid when $n = k+1,$

$\begin{eqnarray*} \begin{aligned} &\mathcal{F}\left(\frac{1}{\mathcal{P}_{k+1}}\sum\limits_{i = 1}^{k+1}p_{i}x_{i},\frac{1}{\mathcal{P}_{k+1}}\sum\limits_{i = 1}^{k+1}p_{i}y_{i}\right)\\ & = \mathcal{F}\left(\frac{1}{\mathcal{P}_{k+1}}\sum\limits_{i = 1}^{k-1}p_{i}x_{i}+\frac{p_{k}+p_{k+1}}{\mathcal{P}_{k+1}}\left(\frac{p_{k}x_{k}}{p_{k}+p_{k+1}}+\frac{p_{k+1}x_{k+1}}{p_{k}+p_{k+1}}\right),\right.\\ &\left.\; \; \; \; \frac{1}{\mathcal{P}_{k+1}}\sum\limits_{i = 1}^{k-1}p_{i}y_{i}+\frac{p_{k}+p_{k+1}}{\mathcal{P}_{k+1}}\left(\frac{p_{k}y_{k}}{p_{k}+p_{k+1}}+\frac{p_{k+1}y_{k+1}}{p_{k}+p_{k+1}}\right)\right)\\ &\leq\left[\mathcal{F}\left(\frac{p_{k}x_{k}}{p_{k}+p_{k+1}}+\frac{p_{k+1}x_{k+1}}{p_{k}+p_{k+1}},\frac{p_{k}y_{k}}{p_{k}+p_{k+1}}+\frac{p_{k+1}y_{k+1}}{p_{k}+p_{k+1}}\right)\right]^{h\left(\frac{p_{k}+p_{k+1}}{\mathcal{P}_{k+1}}\right)}\prod\limits_{i = 1}^{k-1}[\mathcal{F}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{k+1}}\right)}\\ &\leq\left(\left[\mathcal{F}(x_{k},y_{k})\right]^{h\left(\frac{p_{k}}{p_{k}+p_{k+1}}\right)}\left[\mathcal{F}(x_{k+1},y_{k+1})\right]^{h\left(\frac{p_{k+1}}{p_{k}+p_{k+1}}\right)}\right)^{h\left(\frac{p_{k}+p_{k+1}}{\mathcal{P}_{k+1}}\right)}\prod\limits_{i = 1}^{k-1}[\mathcal{F}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{k+1}}\right)}\\ &\leq\left[\mathcal{F}(x_{k},y_{k})\right]^{h\left(\frac{p_{k}}{\mathcal{P}_{k+1}}\right)}\left[\mathcal{F}(x_{k+1},y_{k+1})\right]^{h\left(\frac{p_{k+1}}{\mathcal{P}_{k+1}}\right)}\prod\limits_{i = 1}^{k-1}[\mathcal{F}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{k+1}}\right)}\\ & = \prod\limits_{i = 1}^{k+1}[\mathcal{F}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{k+1}}\right)}. \end{aligned} \end{eqnarray*}$

This completes the proof.

Remark 2. If $h(\vartheta) = \vartheta,$ then the inequality (3.2) is the Jensen inequality for $\log$ -convex functions.

Now, we prove the Jensen inequality for $\log$ - $h$ -convex IVFs in $\triangle$ .

Theorem 8. Let $p_{i}\in\mathbb{R}^{+}, x_{i}\in[a, b], y_{i}\in[c, d], i = 1, 2, ..., n, \mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ such that $\mathcal{F} = [\underline{F}, \overline{F}]$ . If $h$ is a nonnegative supermultiplicative function and $\mathcal{F}\in SX(\log\mathit{\text{-}}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ , then

$\begin{eqnarray} \mathcal{F}\left(\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}x_{i},\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}y_{i}\right)\supseteq\prod\limits_{i = 1}^{n}[\mathcal{F}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{n}}\right)}, \end{eqnarray}$

(3.3)

where $\mathcal{P}_{n} = \sum\limits_{i = 1}^{n}p_{i}$ . If $\mathcal{F}\in SV(\log\mathit{\text{-}}h, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ , then $(3.3)$ is reversed.

Proof. By Theorem 5 and Theorem 7, we have

$\begin{eqnarray*} \mathcal{\underline{F}}\left(\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}x_{i},\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}y_{i}\right)\leq\prod\limits_{i = 1}^{n}[\mathcal{\underline{F}}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{n}}\right)} \end{eqnarray*}$

and

$\begin{eqnarray*} \mathcal{\overline{F}}\left(\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}x_{i},\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}y_{i}\right)\geq\prod\limits_{i = 1}^{n}[\mathcal{\overline{F}}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{n}}\right)}. \end{eqnarray*}$

Thus,

$\begin{eqnarray*} \begin{aligned} &\mathcal{F}\left(\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}x_{i},\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}y_{i}\right)\\ & = \left[\mathcal{\underline{F}}\left(\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}x_{i},\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}y_{i}\right),\mathcal{\overline{F}}\left(\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}x_{i},\frac{1}{\mathcal{P}_{n}}\sum\limits_{i = 1}^{n}p_{i}y_{i}\right)\right]\\ &\supseteq\left[\prod\limits_{i = 1}^{n}[\mathcal{\underline{F}}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{n}}\right)},\prod\limits_{i = 1}^{n}[\mathcal{\overline{F}}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{n}}\right)}\right]\\ & = \prod\limits_{i = 1}^{n}[\mathcal{F}(x_{i},y_{i})]^{h\left(\frac{{p}_{i}}{\mathcal{P}_{n}}\right)}. \end{aligned} \end{eqnarray*}$

This completes the proof.

Next, we prove the Hermite-Hadamard type inequalities for coordinated $\log$ - $h$ -convex IVFs.

Theorem 9. Let $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ and $h:[0, 1]\rightarrow \mathbb{R}^{+}$ be continuous. If $\mathcal{F}\in SX(\log\mathit{\text{-}}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ , then

$\begin{eqnarray} \begin{aligned} &\left[\mathcal{F}\left(\frac{a+b}{2},\frac{c+d}{2}\right)\right]^{\frac{1}{4h^{2}\left(\frac{1}{2}\right)}}\\ &\supseteq \exp\left[\frac{1}{4h\left(\frac{1}{2}\right)}\left(\frac{1}{2h\left(\frac{1}{2}\right)(b-a)}\int_{a}^{b}\ln\mathcal{F}\left(x,\frac{ c+d}{2}\right)dx\right.\right.\\ &\left.\left.\; \; \; \; +\frac{1}{2h\left(\frac{1}{2}\right)(d-c)}\int_{c}^{d}\ln\mathcal{F}\left(\frac{ a+b}{2},y\right)dy\right)\right]\\ &\supseteq \exp\left[\frac{1}{(b-a)(d-c)}\int_{a}^{b}\int_{c}^{d}\ln\mathcal{F}(x,y)dx dy\right]\\ &\supseteq \exp\left[\frac{1}{2}\int_{0}^{1}h(\vartheta)d\vartheta\left(\frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}(x,c)dx+\frac{1}{{\underset{-}-}a}\int_{a}^{b}\ln\mathcal{F}(x,d)dx\right.\right.\\ &\left.\left.\; \; \; \; +\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}(a,y)dy+\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}(b,y)dy\right)\right]\\ &\supseteq\left[\mathcal{F}(a,c)\mathcal{F}(a,d)\mathcal{F}(b,c)\mathcal{F}(b,d)\right]^{\left(\int_{0}^{1}h(\vartheta)d\vartheta\right)^{2}}.\\ \end{aligned} \end{eqnarray}$

(3.4)

Proof. Since $\mathcal{F}\in SX(\log\text{-}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ , we have

$\begin{eqnarray*} \begin{aligned} &\mathcal{F}_{x}\left(\frac{c+d}{2}\right)\\ & = \mathcal{F}_{x}\left(\frac{\vartheta c+(1-\vartheta)d+(1-\vartheta)c+\vartheta d}{2}\right)\\ &\supseteq\left[{\mathcal{F}_{x}(\vartheta c+(1-\vartheta)d)}\right]^{h\left(\frac{1}{2}\right)}\left[{\mathcal{F}_{x}((1-\vartheta) c+\vartheta d)}\right]^{h\left(\frac{1}{2}\right)}.\\ \end{aligned} \end{eqnarray*}$

That is,

$\begin{eqnarray*} \begin{aligned} &\ln\mathcal{F}_{x}\left(\frac{c+d}{2}\right)\\ &\supseteq h\left(\frac{1}{2}\right)\ln\left[{\mathcal{F}_{x}(\vartheta c+(1-\vartheta)d)}{\mathcal{F}_{x}((1-\vartheta) c+\vartheta d)}\right].\\ \end{aligned} \end{eqnarray*}$

Moreover, we have

$\begin{eqnarray*} \begin{aligned} &\frac{1}{h\left(\frac{1}{2}\right)}\ln\mathcal{F}_{x}\left(\frac{c+d}{2}\right)\\ &\supseteq \left[\int_{0}^{1}\ln{\mathcal{F}_{x}(\vartheta c+(1-\vartheta)d)}d\vartheta+\int_{0}^{1}\ln{\mathcal{F}_{x}((1-\vartheta)c+\vartheta d)}d\vartheta\right]\\ & = \left[\int_{0}^{1}\ln{\mathcal{\underline{F}}_{x}(\vartheta c+(1-\vartheta)d)}d\vartheta,\int_{0}^{1}\ln{\mathcal{\overline{F}}_{x}(\vartheta c+(1-\vartheta)d)}d\vartheta\right]\\ &\; \; \; \; +\left[\int_{0}^{1}\ln{\mathcal{\underline{F}}_{x}((1-\vartheta) c+\vartheta d)}d\vartheta,\int_{0}^{1}\ln{\mathcal{\overline{F}}_{x}((1-\vartheta) c+\vartheta d)}d\vartheta\right]\\ & = 2\left[\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{\underline{F}}_{x}(y)dy,\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{\overline{F}}_{x}(y)dy\right]\\ & = \frac{2}{d-c}\int_{c}^{d}\ln\mathcal{F}_{x}(y)dy. \end{aligned} \end{eqnarray*}$

Similarly, we get

$\begin{eqnarray*} \begin{aligned} \frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}_{x}(y)dy\supseteq \ln\left[\mathcal{F}_{x}(c)\mathcal{F}_{x}(d)\right]\int_{0}^{1}h(\vartheta)d\vartheta. \end{aligned} \end{eqnarray*}$

Then

$\begin{eqnarray*} \begin{aligned} \frac{1}{2h\left(\frac{1}{2}\right)}\ln\mathcal{F}_{x}\left(\frac{ c+d}{2}\right)\supseteq\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}_{x}(y)dy\supseteq \ln\left[\mathcal{F}_{x}(c)\mathcal{F}_{x}(d)\right]\int_{0}^{1}h(\vartheta)d\vartheta. \end{aligned} \end{eqnarray*}$

That is,

$\begin{eqnarray*} \begin{aligned} \frac{1}{2h\left(\frac{1}{2}\right)}\ln\mathcal{F}\left(x,\frac{c+d}{2}\right)\supseteq\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}(x,y)dy\supseteq \ln\left[\mathcal{F}(x,c)\mathcal{F}(x,d)\right]\int_{0}^{1}h(\vartheta)d\vartheta. \end{aligned} \end{eqnarray*}$

Integrating over $[a, b]$ , we have

$\begin{eqnarray*} \begin{aligned} &\frac{1}{2h\left(\frac{1}{2}\right)(b-a)}\int_{a}^{b}\ln\mathcal{F}\left(x,\frac{ c+d}{2}\right)dx\\&\supseteq\frac{1}{(b-a)(d-c)}\int_{a}^{b}\int_{c}^{d}\ln\mathcal{F}(x,y)dx dy\\&\supseteq \left[\frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}(x,c)dx+\frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}(x,d)dx\right]\int_{0}^{1}h(\vartheta)d\vartheta. \end{aligned} \end{eqnarray*}$

Similarly, we have

$\begin{eqnarray*} \begin{aligned} &\frac{1}{2h\left(\frac{1}{2}\right)(d-c)}\int_{c}^{d}\ln\mathcal{F}\left(\frac{ a+b}{2},y\right)dy\\&\supseteq\frac{1}{(b-a)(d-c)}\int_{a}^{b}\int_{c}^{d}\ln\mathcal{F}(x,y)dx dy\\&\supseteq \left[\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}(a,y)dy+\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}(b,y)dy\right]\int_{0}^{1}h(\vartheta)d\vartheta. \end{aligned} \end{eqnarray*}$

Finally, we obtain

$\begin{eqnarray*} \begin{aligned} &\frac{1}{4h^{2}\left(\frac{1}{2}\right)}\ln\mathcal{F}\left(\frac{ a+b}{2},\frac{c+d}{2}\right)\\ & = \frac{1}{4h\left(\frac{1}{2}\right)}\left[\frac{1}{2h\left(\frac{1}{2}\right)(b-a)}\int_{a}^{b}\ln\mathcal{F}\left(x,\frac{ c+d}{2}\right)dx+\frac{1}{2h\left(\frac{1}{2}\right)(d-c)}\int_{c}^{d}\ln\mathcal{F}\left(\frac{ a+b}{2},y\right)dy\right]\\ &\supseteq\frac{1}{(b-a)(d-c)}\int_{a}^{b}\int_{c}^{d}\ln\mathcal{F}(x,y)dx dy\\ &\supseteq \frac{1}{2} \int_{0}^{1}h(\vartheta)d\vartheta\left[\frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}(x,c)dx+\frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}(x,d)dx\right.\\ &\left.\; \; \; \; +\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}(a,y)dy+\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}(b,y)dy\right]\\ &\supseteq\frac{1}{2}\left(\int_{0}^{1}h(\vartheta)d\vartheta\right)^{2}\left[\ln\mathcal{F}(a,c)+\ln\mathcal{F}(a,d)+\ln\mathcal{F}(b,c)+\ln\mathcal{F}(b,d)\right.\\ &\left.\; \; \; \; +\ln\mathcal{F}(a,c)+\ln\mathcal{F}(a,d)+\ln\mathcal{F}(b,c)+\ln\mathcal{F}(b,d)\right]\\ &\supseteq\left(\int_{0}^{1}h(\vartheta)d\vartheta\right)^{2}\left[\ln\mathcal{F}(a,c)\mathcal{F}(a,d)\mathcal{F}(b,c)\mathcal{F}(b,d)\right].\\ \end{aligned} \end{eqnarray*}$

This concludes the proof.

Remark 3. If $\mathcal{\underline{F}} = \mathcal{\overline{F}}$ and $h(\vartheta) = \vartheta$ , then Theorem 9 reduces to Corollary 3.1 of ^[13].

Example 1. Let $[a, b] = [c, d] = [2, 3], h(\vartheta) = \vartheta$ . We define $\mathcal{F}:[2, 3]\times[2, 3]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ by

$\begin{eqnarray*} \mathcal{F}(x,y) = \left[\frac{1}{xy},e^{\sqrt{x}+\sqrt{y}}\right].\\ \end{eqnarray*}$

From Definition 2, $\mathcal{F}(x, y)\in SX(\log\text{-}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ .

Since

$\begin{eqnarray*} \begin{aligned} &\left[\mathcal{F}\left(\frac{a+b}{2},\frac{c+d}{2}\right)\right]^{\frac{1}{4h^{2}\left(\frac{1}{2}\right)}} = \left[\frac{4}{25},e^{\sqrt{10}}\right],\\ &\exp\left[\frac{1}{4h\left(\frac{1}{2}\right)}\left(\frac{1}{2h\left(\frac{1}{2}\right)(b-a)}\int_{a}^{b}\ln\mathcal{F}\left(x,\frac{ c+d}{2}\right)dx\right.\right.\\ &\left.\left.\; \; \; \; +\frac{1}{2h\left(\frac{1}{2}\right)(d-c)}\int_{c}^{d}\ln\mathcal{F}\left(\frac{ a+b}{2},y\right)dy\right)\right] = \left[\frac{8e}{135},e^{\frac{\sqrt{10}}{2}+2\sqrt{3}-\frac{4\sqrt{2}}{3}}\right],\\ \; \; \; \; \; \; \; &\exp\left[\frac{1}{(b-a)(d-c)}\int_{a}^{b}\int_{c}^{d}\ln\mathcal{F}(x,y)dx dy\right] = \left[\frac{16e^{2}}{729},e^{\frac{4}{3}(3\sqrt{3}-2\sqrt{2})}\right],\\ \; \; \; \; \; \; \; &\exp\left[\frac{1}{2}\int_{0}^{1}h(\vartheta)d\vartheta\left(\frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}(x,c)dx+\frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}(x,d)dx\right.\right.\\ &\left.\left.\; \; \; \; +\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}(a,y)dy+\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}(b,y)dy\right)\right] = \left[\frac{2\sqrt{6}e}{81},e^{\frac{15\sqrt{3}-5\sqrt{2}}{6}}\right],\\ \end{aligned} \end{eqnarray*}$

and

$\begin{eqnarray*} \left[\mathcal{F}(a,c)\mathcal{F}(a,d)\mathcal{F}(b,c)\mathcal{F}(b,d)\right]^{\left(\int_{0}^{1}h(\vartheta)d\vartheta\right)^{2}} = \left[\frac{1}{6},e^{\sqrt{2}+\sqrt{3}}\right].\\ \end{eqnarray*}$

It follows that

$\begin{eqnarray*} \left[\frac{4}{25},e^{\sqrt{10}}\right]\supseteq\left[\frac{8e}{135},e^{\frac{\sqrt{10}}{2}+2\sqrt{3}-\frac{4\sqrt{2}}{3}}\right]\supseteq\left[\frac{16e^{2}}{729},e^{\frac{4}{3}(3\sqrt{3}-2\sqrt{2})}\right]\supseteq\left[\frac{2\sqrt{6}e}{81},e^{\frac{15\sqrt{3}-5\sqrt{2}}{6}}\right]\supseteq\left[\frac{1}{6},e^{\sqrt{2}+\sqrt{3}}\right] \end{eqnarray*}$

and Theorem 9 is verified.

Theorem 10. Let $\mathcal{F}:\triangle\rightarrow \mathbb{R}_{\mathcal{I}}^{+}$ and $h:[0, 1]\rightarrow \mathbb{R}^{+}$ be continuous. If $\mathcal{F}\in SX(\log\mathit{\text{-}}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ , then

$\begin{eqnarray} \begin{aligned} &\left[\mathcal{F}\left(\frac{a+b}{2},\frac{c+d}{2}\right)\right]^{\frac{1}{4h^{3}\left(\frac{1}{2}\right)}}\\ &\supseteq \exp\left[{\frac{1}{4h^{2}\left(\frac{1}{2}\right)(b-a)}}\int_{a}^{b}\ln\left(\mathcal{F}\left(x,\frac{c+d}{2}\right)\right)dx\right.\\ &\left.\; \; \; \; +{\frac{1}{4h^{2}\left(\frac{1}{2}\right)(d-c)}}\int_{c}^{d}\ln\left(\mathcal{F}\left(\frac{a+b}{2},y\right)\right)dy\right]\\ &\supseteq \exp\left[{\frac{1}{4h\left(\frac{1}{2}\right)(b-a)}}\int_{a}^{b}\ln\left(\mathcal{F}\left(x,\frac{3c+d}{4}\right)\mathcal{F}\left(x,\frac{c+3d}{4}\right)\right)dx\right.\\ &\left.\; \; \; \; +{\frac{1}{4h\left(\frac{1}{2}\right)(d-c)}}\int_{c}^{d}\ln\left(\mathcal{F}\left(\frac{3a+b}{4},y\right)\mathcal{F}\left(\frac{a+3b}{4},y\right)\right)dy\right]\\ &\supseteq \exp\left[\frac{2}{(b-a)(d-c)}\int_{a}^{b}\int_{c}^{d}\ln\mathcal{F}(x,y)dx dy\right]\\ \end{aligned} \end{eqnarray}$

(3.5)

$\begin{eqnarray*} \begin{aligned}&\supseteq \exp\left[{\frac{1}{2(b-a)}}\int_{a}^{b}\ln\left(\mathcal{F}(x,c)\mathcal{F}(x,d)\mathcal{F}^{2}\left(x,frac{c+d}{2}\right)\right)dx\int_{0}^{1}h(\vartheta)d\vartheta\right.\\ &\left.\; \; \; \; +{\frac{1}{2(d-c)}}\int_{c}^{d}\ln\left(\mathcal{F}(a,y)\mathcal{F}(b,y)\mathcal{F}^{2}\left(\frac{a+b}{2},y\right)\right)dy\int_{0}^{1}h(\vartheta)d\vartheta\right]\\ &\supseteq \exp\left[{\left(\frac{1}{2}+h\left(\frac{1}{2}\right)\right)\frac{1}{b-a}\int_{a}^{b}\ln\left[\mathcal{F}(x,c)\mathcal{F}(x,d)\right]dx\int_{0}^{1}h(\vartheta)d\vartheta}\right.\\ &\left.\; \; \; \; +{\left(\frac{1}{2}+h\left(\frac{1}{2}\right)\right)\frac{1}{d-c}\int_{c}^{d}\ln\left[\mathcal{F}(a,y)\mathcal{F}(b,y)\right]dy\int_{0}^{1}h(\vartheta)d\vartheta}\right]\\ &\supseteq\left[\mathcal{F}(a,c)\mathcal{F}(a,d)\mathcal{F}(b,c)\mathcal{F}(b,d)\mathcal{F}\left(\frac{a+b}{2},c\right)\mathcal{F}\left(\frac{a+b}{2},d\right)\right.\\ &\left.\; \; \; \; \times\mathcal{F}\left(a,\frac{c+d}{2}\right)\mathcal{F}\left(b,\frac{c+d}{2}\right)\right]^{\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\left(\int_{0}^{1}h(\vartheta)d\vartheta\right)^{2}}\\ &\supseteq\left[\mathcal{F}(a,c)\mathcal{F}(a,d)\mathcal{F}(b,c)\mathcal{F}(b,d)\right]^{2\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]^{2}\left(\int_{0}^{1}h(\vartheta)d\vartheta\right)^{2}}. \end{aligned} \end{eqnarray*}$

Proof. Since $\mathcal{F}\in SX(\log\text{-}ch, \triangle, \mathbb{R}_{\mathcal{I}}^{+})$ , by using Theorem 6 and (2.3), we have

$\begin{eqnarray*} \begin{aligned} &{\frac{1}{4h^{2}\left(\frac{1}{2}\right)}}\ln\left[\mathcal{F}_{y}\left(\frac{a+b}{2}\right)\right]\\ &\supseteq{\frac{1}{4h\left(\frac{1}{2}\right)}}\ln\left[\mathcal{F}_{y}\left(\frac{3a+b}{4}\right)\mathcal{F}_{y}\left(\frac{a+3b}{4}\right)\right]\\ &\supseteq\frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}_{y}(x)dx\\ &\supseteq{\frac{1}{2}}\ln\left[\mathcal{F}_{y}(a)\mathcal{F}_{y}(b)\mathcal{F}_{y}^{2}\left(\frac{a+b}{2}\right)\right]\int_{0}^{1}h(\vartheta)d\vartheta\\ &\supseteq{\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\ln\left[\mathcal{F}_{y}(a)\mathcal{F}_{y}(b)\right]\int_{0}^{1}h(\vartheta)d\vartheta}.\\ \end{aligned} \end{eqnarray*}$

That is,

$\begin{eqnarray*} \begin{aligned} &{\frac{1}{4h^{2}\left(\frac{1}{2}\right)}}\ln\left[\mathcal{F}\left(\frac{a+b}{2},y\right)\right]\\ &\supseteq{\frac{1}{4h\left(\frac{1}{2}\right)}}\ln\left[\mathcal{F}\left(\frac{3a+b}{4},y\right)\mathcal{F}\left(\frac{a+3b}{4},y\right)\right]\\ &\supseteq\frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}(x,y)dx\\ &\supseteq{\frac{1}{2}}\ln\left[\mathcal{F}(a,y)\mathcal{F}(b,y)\mathcal{F}^{2}\left(\frac{a+b}{2},y\right)\right]\int_{0}^{1}h(\vartheta)d\vartheta\\ &\supseteq{\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\ln\left[\mathcal{F}(a,y)\mathcal{F}(b,y)\right]\int_{0}^{1}h(\vartheta)d\vartheta}.\\ \end{aligned} \end{eqnarray*}$

Moreover, we have

$\begin{eqnarray*} \begin{aligned} &{\frac{1}{4h^{2}\left(\frac{1}{2}\right)(d-c)}}\int_{c}^{d}\ln\left[\mathcal{F}\left(\frac{a+b}{2},y\right)\right]dy\\ &\supseteq{\frac{1}{4h\left(\frac{1}{2}\right)(d-c)}}\int_{c}^{d}\ln\left[\mathcal{F}\left(\frac{3a+b}{4},y\right)\mathcal{F}\left(\frac{a+3b}{4},y\right)\right]dy\\ &\supseteq\frac{1}{(b-a)(d-c)}\int_{a}^{b}\int_{c}^{d}\ln\mathcal{F}(x,y)dx dy\\ &\supseteq{\frac{1}{2(d-c)}}\int_{c}^{d}\ln\left[\mathcal{F}(a,y)\mathcal{F}(b,y)\mathcal{F}^{2}\left(\frac{a+b}{2},y\right)\right]dy\int_{0}^{1}h(\vartheta)d\vartheta\\ &\supseteq{\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\frac{1}{d-c}\int_{c}^{d}\ln\left[\mathcal{F}(a,y)\mathcal{F}(b,y)\right]dy\int_{0}^{1}h(\vartheta)d\vartheta}.\\ \end{aligned} \end{eqnarray*}$

Similarly, we have

We also from (2.2),

$\begin{eqnarray*} \frac{1}{2h\left(\frac{1}{2}\right)}\ln\mathcal{F}\left(\frac{a+b}{2},\frac{c+d}{2}\right)\supseteq\frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}\left(x,\frac{c+d}{2}\right)dx,\\ \frac{1}{2h\left(\frac{1}{2}\right)}\ln\mathcal{F}\left(\frac{a+b}{2},\frac{c+d}{2}\right)\supseteq\frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}\left(\frac{a+b}{2},y\right)dy. \end{eqnarray*}$

Again from (2.3),

$\begin{eqnarray*} \frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}\left(x,c\right)dx&\; \supseteq{\frac{1}{2}}\ln\left[\mathcal{F}(a,c)\mathcal{F}(b,c)\mathcal{F}^{2}\left(\frac{a+b}{2},c\right)\right]\int_{0}^{1}h(\vartheta)d\vartheta\\ &\supseteq{\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\ln\left[\mathcal{F}(a,c)\mathcal{F}(b,c)\right]\int_{0}^{1}h(\vartheta)d\vartheta},\\ \frac{1}{b-a}\int_{a}^{b}\ln\mathcal{F}\left(x,d\right)ds&\; \supseteq{\frac{1}{2}}\ln\left[\mathcal{F}(a,d)\mathcal{F}(b,d)\mathcal{F}^{2}\left(\frac{a+b}{2},d\right)\right]\int_{0}^{1}h(\vartheta)d\vartheta\\ &\supseteq{\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\ln\left[\mathcal{F}(a,d)\mathcal{F}(b,d)\right]\int_{0}^{1}h(\vartheta)d\vartheta},\\ \frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}\left(a,y\right)dy&\; \supseteq{\frac{1}{2}}\ln\left[\mathcal{F}(a,c)\mathcal{F}(a,d)\mathcal{F}^{2}\left(a,\frac{c+d}{2}\right)\right]\int_{0}^{1}h(\vartheta)d\vartheta\\ &\supseteq{\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\ln\left[\mathcal{F}(a,c)\mathcal{F}(a,d)\right]\int_{0}^{1}h(\vartheta)d\vartheta},\\ \frac{1}{d-c}\int_{c}^{d}\ln\mathcal{F}\left(b,y\right)dy&\; \supseteq{\frac{1}{2}}\ln\left[\mathcal{F}(b,c)\mathcal{F}(b,d)\mathcal{F}^{2}\left(b,\frac{c+d}{2}\right)\right]\int_{0}^{1}h(\vartheta)d\vartheta\\ &\supseteq{\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\ln\left[\mathcal{F}(b,c)\mathcal{F}(b,d)\right]\int_{0}^{1}h(\vartheta)d\vartheta}\\ \end{eqnarray*}$

and proof is completed.

Example 2. Furthermore, by Example 1, we have

and

$\begin{eqnarray*} \left[\mathcal{F}(a,c)\mathcal{F}(a,d)\mathcal{F}(b,c)\mathcal{F}(b,d)\right]^{2\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]^{2}\left(\int_{0}^{1}h(\theta)d\theta\right)^{2}} = \left[\frac{1}{36},e^{2\sqrt{3}+2\sqrt{2}}\right]. \end{eqnarray*}$

It follows that

$\begin{eqnarray*} \begin{aligned} &\left[\frac{16}{625},e^{2\sqrt{10}}\right]\supseteq\left[\frac{64e^{2}}{18225},e^{\frac{4(3\sqrt{3}-2\sqrt{2})+3\sqrt{10}}{3}}\right]\\ &\supseteq\left[\frac{256e^{2}}{72171},e^{\frac{4(3\sqrt{3}-2\sqrt{2})}{3}+\frac{3+\sqrt{11}}{2}}\right]\supseteq\left[\frac{256e^{4}}{531441},e^{\frac{8(3\sqrt{3}-2\sqrt{2})}{3}}\right]\\ &\supseteq\left[\frac{16\sqrt{6}e^{2}}{10935},e^{\frac{12\sqrt{3}-8\sqrt{2}+3\sqrt{10}}{6}}\right]\supseteq\left[\frac{8e^{2}}{2187},e^{\frac{15\sqrt{3}-5\sqrt{2}}{3}}\right]\\ &\supseteq\left[\frac{\sqrt{6}}{90},e^{\frac{3\sqrt{3}+3\sqrt{2}+\sqrt{10}}{2}}\right]\supseteq\left[\frac{1}{36},e^{2\sqrt{3}+2\sqrt{2}}\right] \end{aligned} \end{eqnarray*}$

and Theorem 10 is verified.

4. Conclusions

We introduced the coordinated log- $h$ -convexity for interval-valued functions, some Jensen type inequalities and Hermite-Hadamard type inequalities are proved. Our results generalize some known inequalities and will be useful in developing the theory of interval integral inequalities and interval convex analysis. The next step in the research direction investigated inequalities for fuzzy-interval-valued functions, and some applications in interval nonlinear programming.

Acknowledgments

The first author was supported in part by the Key Projects of Educational Commission of Hubei Province of China (D20192501), the Natural Science Foundation of Jiangsu Province (BK20180500) and the National Key Research and Development Program of China (2018YFC1508100).

Conflict of interest

The authors declare no conflict of interest.

References

[1]	L. O. Chua, L. Yang, Cellular neural networks: Applications, IEEE Trans. Circ. Syst., 35 (1988), 1273–1290. http://dx.doi.org/10.1109/31.7601 doi: 10.1109/31.7601
[2]	J. Hu, G. Sui, X. Lv, X. Li, Fixed-time control of delayed neural networks with impulsive perturbations, Nonlinear Anal.-Model., 23 (2018), 904–920. http://dx.doi.org/10.15388/NA.2018.6.6 doi: 10.15388/NA.2018.6.6
[3]	G. Stamov, E. Gospodinova, I. Stamova, Practical exponential stability with respect to h-manifolds of discontinuous delayed cohen-grossberg neural networks with variable impulsive perturbations, Math. Model. Control, 1 (2021), 26–34. http://dx.doi.org/10.3934/mmc.2021003 doi: 10.3934/mmc.2021003
[4]	M. Ceylan, R. Ceylan, Y. Özbay, S. Kara, Application of complex discrete wavelet transform in classification of doppler signals using complex-valued artificial neural network, Artif. Intell. Med., 44 (2008), 65–76. http://dx.doi.org/10.1016/j.artmed.2008.05.003 doi: 10.1016/j.artmed.2008.05.003
[5]	A. Hirose, Complex-valued neural networks. Springer Science & Business Media, 2012. http://doi.org/10.1007/978-3-642-27632-3
[6]	M. E. Valle, Complex-valued recurrent correlation neural networks, IEEE Trans. Neural Netw. Learn. Syst., 25 (2014), 1600–1612. http://dx.doi.org/10.1109/TNNLS.2014.2341013 doi: 10.1109/TNNLS.2014.2341013
[7]	X. Li, X. Yang, S. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica, 103 (2019), 135–140. http://dx.doi.org/10.1016/j.automatica.2019.01.031 doi: 10.1016/j.automatica.2019.01.031
[8]	X. Liu, S. Zhong, Stability analysis of delayed switched cascade nonlinear systems with uniform switching signals, Math. Model. Control, 1 (2021), 90–101. http://dx.doi.org/10.3934/mmc.2021007 doi: 10.3934/mmc.2021007
[9]	Q. Song, Q. Yu, Z. Zhao, Y. Liu, F. E. Alsaadi, Boundedness and global robust stability analysis of delayed complex-valued neural networks with interval parameter uncertainties, Neural Netw., 103 (2018), 55–62. http://doi.org/10.1016/j.neunet.2018.03.008 doi: 10.1016/j.neunet.2018.03.008
[10]	M. Liu, Z. Li, H. Jiang, C. Hu, Z. Yu, Exponential synchronization of complex-valued neural networks via average impulsive interval strategy, Neural Process. Lett., 52 (2020), 1377–1394. http://doi.org/10.1007/s11063-020-10309-5 doi: 10.1007/s11063-020-10309-5
[11]	W. Zhang, X. Yang, C. Xu, J. Feng, C. Li, Finite-time synchronization of discontinuous neural networks with delays and mismatched parameters, IEEE Trans. Neural Netw. Learn. Syst., 29 (2018), 3761–3771. http://doi.org/10.1109/TNNLS.2017.2740431 doi: 10.1109/TNNLS.2017.2740431
[12]	J. Cao, L. Li, Cluster synchronization in an array of hybrid coupled neural networks with delay, Neural Netw., 22 (2009), 335–342. http://doi.org/10.1016/j.neunet.2009.03.006 doi: 10.1016/j.neunet.2009.03.006
[13]	R. Kumar, S. Sarkar, S. Das, J. Cao, Projective synchronization of delayed neural networks with mismatched parameters and impulsive effects, IEEE Trans. Neural Netw. Learn. Syst., 31 (2020), 1211–1221. http://doi.org/10.1109/TNNLS.2019.2919560 doi: 10.1109/TNNLS.2019.2919560
[14]	Y. Yang, J. Cao, Exponential lag synchronization of a class of chaotic delayed neural networks with impulsive effects, Physica A., 386 (2007), 492–502. http://doi.org/10.1016/j.physa.2007.07.049 doi: 10.1016/j.physa.2007.07.049
[15]	L. Wang, Y. Shen, G. Zhang, Synchronization of a class of switched neural networks with time-varying delays via nonlinear feedback control, IEEE Trans. Cybern., 46 (2016), 2300–2310. http://doi.org/10.1109/TCYB.2015.2475277 doi: 10.1109/TCYB.2015.2475277
[16]	H. Ren, Z. Peng, Y. Gu, Fixed-time synchronization of stochastic memristor-based neural networks with adaptive control, Neural Netw., 130 (2020), 165–175. http://doi.org/10.1016/j.neunet.2020.07.002 doi: 10.1016/j.neunet.2020.07.002
[17]	G. Zhang, X. Lin, X. Zhang, Exponential stabilization of neutral-type neural networks with mixed interval time-varying delays by intermittent control: a ccl approach, Circ. Syst. Signal Pr., 33 (2014), 371–391. http://doi.org/10.1007/s00034-013-9651-y doi: 10.1007/s00034-013-9651-y
[18]	X. Li, S. Song, Research on synchronization of chaotic delayed neural networks with stochastic perturbation using impulsive control method, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3892–3900. http://dx.doi.org/10.1016/j.cnsns.2013.12.012 doi: 10.1016/j.cnsns.2013.12.012
[19]	Q. Tang, J. Jian, Matrix measure based exponential stabilization for complex-valued inertial neural networks with time-varying delays using impulsive control, Neurocomputing, 273 (2018), 251–259. http://dx.doi.org/10.1016/j.neucom.2017.08.009 doi: 10.1016/j.neucom.2017.08.009
[20]	X. Li, X. Yang, J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica, 117 (2020), 108981. http://doi.org/10.1016/j.automatica.2020.108981 doi: 10.1016/j.automatica.2020.108981
[21]	X. Li, J. Fang, H. Li, Master–slave exponential synchronization of delayed complex-valued memristor-based neural networks via impulsive control, Neural Netw., 93 (2017), 165–175. http://dx.doi.org/10.1016/j.neunet.2017.05.008 doi: 10.1016/j.neunet.2017.05.008
[22]	Y. Kan, J. Lu, J. Qiu, J. Kurths, Exponential synchronization of time-varying delayed complex-valued neural networks under hybrid impulsive controllers, Neural Netw., 114 (2019), 157–163. https://doi.org/10.1016/j.neunet.2019.02.006 doi: 10.1016/j.neunet.2019.02.006
[23]	L. Li and G. Mu, Synchronization of coupled complex-valued impulsive neural networks with time delays, Neural Process. Lett., 50 (2019), 2515–2527. http://doi.org/10.1007/s11063-019-10028-6 doi: 10.1007/s11063-019-10028-6
[24]	X. Li, S. Song, J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Autom. Control, 64 (2019), 4024–4034. http://doi.org/10.1109/TAC.2019.2905271 doi: 10.1109/TAC.2019.2905271
[25]	B. Jiang, Y. Lou, J. Lu, Input-to-state stability of delayed systems with bounded-delay impulses, Math. Model. Control, 2 (2022), 44–54. http://doi.org/10.3934/mmc.2022006 doi: 10.3934/mmc.2022006
[26]	Y. Shen, X. Liu, Event-based master–slave synchronization of complex-valued neural networks via pinning impulsive control, Neural Netw., 145 (2022), 374–385. http://doi.org/10.1016/j.neunet.2021.10.025 doi: 10.1016/j.neunet.2021.10.025
[27]	L. Zhang, X. Yang, C. Xu, J. Feng, Exponential synchronization of complex-valued complex networks with time-varying delays and stochastic perturbations via time-delayed impulsive control, Appl. Math. Comput., 306 (2017), 22–30. http://dx.doi.org/10.1016/j.amc.2017.02.004 doi: 10.1016/j.amc.2017.02.004
[28]	H. S. Hurd, J. B. Kaneene, J. W. Lloyd, A stochastic distributed-delay model of disease processes in dynamic populations, Prev. Vet. Med., 16 (1993), 21–29. http://dx.doi.org/10.1016/0167-5877(93)90005-E doi: 10.1016/0167-5877(93)90005-E
[29]	D. Zennaro, A. Ahmad, L. Vangelista, E. Serpedin, H. Nounou, M. Nounou, Network-wide clock synchronization via message passing with exponentially distributed link delays, IEEE Trans. Commun., 61 (2013), 2012–2024. http://dx.doi.org/10.1109/TCOMM.2013.021913.120595 doi: 10.1109/TCOMM.2013.021913.120595
[30]	G. Samanta, Permanence and extinction of a nonautonomous hiv/aids epidemic model with distributed time delay, Nonlinear Anal.-Real., 12 (2011), 1163–1177. http://dx.doi.org/10.1016/j.nonrwa.2010.09.010 doi: 10.1016/j.nonrwa.2010.09.010
[31]	H. Li, C. Li, D. Ouyang, S. K. Nguang, Z. He, Observer-based dissipativity control for T-S fuzzy neural networks with distributed time-varying delays, IEEE Trans. Cybern., 51 (2021), 5248–5258. http://dx.doi.org/10.1109/TCYB.2020.2977682 doi: 10.1109/TCYB.2020.2977682
[32]	X. Yang, Q. Song, J. Liang, B. He, Finite-time synchronization of coupled discontinuous neural networks with mixed delays and nonidentical perturbations, J. Frankl. Inst., 352 (2015), 4382–4406. http://dx.doi.org/10.1016/j.jfranklin.2015.07.001 doi: 10.1016/j.jfranklin.2015.07.001
[33]	L. Wang, H. He, Z. Zeng, Global synchronization of fuzzy memristive neural networks with discrete and distributed delays, IEEE Trans. Fuzzy Syst., 28 (2020), 2022–2034. http://dx.doi.org/10.1109/TFUZZ.2019.2930032 doi: 10.1109/TFUZZ.2019.2930032
[34]	H. Li, L. Zhang, X. Zhang, J. Yu, A switched integral-based event-triggered control of uncertain nonlinear time-delay system with actuator saturation, IEEE Trans. Cybern., 52 (2022), 11335–11347. http://dx.doi.org/10.1109/TCYB.2021.3085735 doi: 10.1109/TCYB.2021.3085735
[35]	X. Li, Global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type, Appl. Math. Comput., 215 (2010), 4370–4384. http://dx.doi.org/10.1016/j.amc.2009.12.068 doi: 10.1016/j.amc.2009.12.068
[36]	Z. Guo, J. Wang, Z. Yan, A systematic method for analyzing robust stability of interval neural networks with time-delays based on stability criteria, Neural Netw., 54 (2014), 112–122. http://dx.doi.org/10.1016/j.neunet.2014.03.002 doi: 10.1016/j.neunet.2014.03.002
[37]	Z. Xu, X. Li, P. Duan, Synchronization of complex networks with time-varying delay of unknown bound via delayed impulsive control, Neural Netw., 125 (2020), 224–232. http://doi.org/10.1016/j.neunet.2020.02.003 doi: 10.1016/j.neunet.2020.02.003
[38]	A. Abdurahman, H. Jiang, Z. Teng, Exponential lag synchronization for memristor-based neural networks with mixed time delays via hybrid switching control, J. Frankl. Inst., 353 (2016), 2859–2880. http://dx.doi.org/10.1016/j.jfranklin.2016.05.022 doi: 10.1016/j.jfranklin.2016.05.022
[39]	Z. Xu, D. Peng, X. Li, Synchronization of chaotic neural networks with time delay via distributed delayed impulsive control, Neural Netw., 118 (2019), 332–337. http://doi.org/10.1016/j.neunet.2019.07.002 doi: 10.1016/j.neunet.2019.07.002
[40]	W. Gong, J. Liang, J. Cao, Global $\mu$ -stability of complex-valued delayed neural networks with leakage delay, Neurocomputing, 168 (2015), 135–144. http://dx.doi.org/10.1016/j.neucom.2015.06.006 doi: 10.1016/j.neucom.2015.06.006
[41]	T. Yu, J. Cao, L. Rutkowski, Y.-P. Luo, Finite-time synchronization of complex-valued memristive-based neural networks via hybrid control, IEEE Trans. Neural Netw. Learn. Syst., 33 (2022), 3938–3947. http://doi.org/10.1109/TNNLS.2021.3054967 doi: 10.1109/TNNLS.2021.3054967
[42]	L. Li, X. Shi, J. Liang, Synchronization of impulsive coupled complex-valued neural networks with delay: the matrix measure method, Neural Netw., 117 (2019), 285–294. http://doi.org/10.1016/j.neunet.2019.05.024 doi: 10.1016/j.neunet.2019.05.024

This article has been cited by:

1.	Hajar F. Ismael, Haci Mehmet Baskonus, Hasan Bulut, Wei Gao, Instability modulation and novel optical soliton solutions to the Gerdjikov–Ivanov equation with M-fractional, 2023, 55, 0306-8919, 10.1007/s11082-023-04581-7
2.	Nauman Raza, Abdel-Haleem Abdel-Aty, Traveling wave structures and analysis of bifurcation and chaos theory for Biswas–Milovic Model in conjunction with Kudryshov’s law of refractive index, 2023, 287, 00304026, 171085, 10.1016/j.ijleo.2023.171085
3.	Hakima Khudher Ahmed, Hajar Farhan Ismael, Optical soliton solutions for the nonlinear Schrödinger equation with higher-order dispersion arise in nonlinear optics, 2024, 99, 0031-8949, 105276, 10.1088/1402-4896/ad78c3
4.	Nirman Bhowmike, Zia Ur Rehman, Zarmeena Naz, Muhammad Zahid, Sultan Shoaib, Yasar Amin, Non-linear electromagnetic wave dynamics: Investigating periodic and quasi-periodic behavior in complex engineering systems, 2024, 184, 09600779, 114984, 10.1016/j.chaos.2024.114984
5.	Saumya Ranjan Jena, Itishree Sahu, A novel approach for numerical treatment of traveling wave solution of ion acoustic waves as a fractional nonlinear evolution equation on Shehu transform environment, 2023, 98, 0031-8949, 085231, 10.1088/1402-4896/ace6de
6.	Ri Zhang, Muhammad Shakeel, Nasser Bin Turki, Nehad Ali Shah, Sayed M Tag, Novel analytical technique for mathematical model representing communication signals: A new travelling wave solutions, 2023, 51, 22113797, 106576, 10.1016/j.rinp.2023.106576
7.	Exact Solutions of Beta-Fractional Fokas-Lenells Equation via Sine-Cosine Method, 2023, 16, 20710216, 10.14529/mmp230201

Reader Comments

Your name:*

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