A Hadamard-type inequality for fuzzy integrals based on r-convex functions

Abstract

In this paper, it is shown that the Hadamard integral inequality for r-convex functions is not satisfied in the fuzzy context. Using the classical Hadamard integral inequality, we give an upper bound for the Sugeno integral of r-convex functions. In addition, we generalize the results related to the Hadamard integral inequality for Sugeno integral from 1-convex functions (ordinary convex functions) to r-convex functions. We present a geometric interpretation and some examples in the framework of the Lebesgue measure to illustrate the results.

1 Introduction

The process of combining several numerical values into a single representative one is called aggregation, and the numerical function performing this process is called an aggregation function, see Grabisch et al. (2009). Several methods for combining evidence produced by multiple information sources have been studied by different researchers and some synthesizing functions have been proposed. For example, arithmetic mean, geometric mean and median can be regarded as a basic class, because they are often used and very classic. However, these operators are not able to model an interaction between criteria. For having a representation of interaction phenomena between criteria, fuzzy integrals have been proposed by Sugeno (1974). The fuzzy integral is a nonlinear functional that is defined with respect to a fuzzy measure, which in turn is either a belief or a plausibility measure in the sense of Dempster–Shafer belief theory (Wierzchon 1982). The fuzzy integral combines objective evidence for a hypothesis with the system’s expectation of the importance of that evidence to the hypothesis, see Wang and Klir (1992) and Chen et al. (2014).

Two main classes of the fuzzy integrals are Choquet and Sugeno integrals. The properties and applications of the Sugeno integral have been studied by many authors. Ralescu and Adams (1980) studied several equivalent definitions of Sugeno integral. Román-Flores et al. (2007a, b) (Flores-Franulič and Román-Flores 2007; Román-Flores and Chalco-Cano 2006, 2007) studied the level-continuity of Sugeno integral, H-continuity of fuzzy measures and geometric inequalities for fuzzy measures and integrals, respectively. Wang and Klir (1992) had a general overview on fuzzy measurement and fuzzy integration theory.

Recently, many authors have studied the most well-known integral inequalities for Sugeno integral. Agahi et al. (2010a, b, 2011, 2012a, b) and Agahi and Eslami (2011) proved general Minkowski-type inequalities, general extensions of Chebyshev-type inequalities and general Barnes–Godunova–Levin-type inequalities for Sugeno integrals. Caballero and Sadarangani (2009, 2010a, b, c, 2011) proved Hermite–Hadamard-type inequalities, Chebyshev-type inequalities, Cauchy and Fritz Carlson’s type inequalities for Sugeno integral. Kaluszka et al. (2014) gave the necessary and sufficient conditions guaranteeing the validity of Chebyshev type inequalities for the generalized Sugeno integral in the case of functions belonging to a much wider class than the comonotone functions. Wu et al. (2010) proved two inequalities for the Sugeno integral on abstract spaces generalizing all previous Chebyshev’s inequalities.

Most of the integral inequalities studied in the Sugeno integration context normally consider conditions such as monotonicity or comonotonicity. In this paper, the main purpose is to consider the Sugeno integral for r-convex functions using the Hadamard integral inequality. The classical measure theory is closely connected with probability theory. A probability measure, as any other classical measure, is a set function that assigns 0 to the empty set and a nonnegative number to any other set, and that is additive. However, a probability measure requires, in addition, that 1 be assigned to the universal set in question. Hence, probability theory may be viewed as a part of classical measure theory. It is shown by Klir and Folger (1988) that classical (additive) probability measure can capture only one of several types of uncertainty that can clearly be recognized when the additivity property is abandoned. On the other hand, in many applications, assumptions about the convexity of a probability distribution allow just enough special structure to yield a workable theory. So, we believe that our results will be useful in non-additive probabilities.

The paper is organized as follows: Some necessary preliminaries and summarization of some previous known results are presented in Sect. 2. In Sect. 3, we deal with the upper bound of Sugeno integral for r-convex functions and give some examples. In Sect. 4, a geometric interpretation is presented to illustrate the results. Finally, a conclusion is given in Sect. 5.

2 Preliminaries

In this section, we are going to review some well-known results from the theory of non-additive measures. Let X be a non-empty set and $\Sigma$ be a $\sigma$-algebra of subsets of X.

Definition 1

(Ralescu and Adams 1980) Suppose that $\mu : \Sigma \longrightarrow [0, \infty )$ is a set function. We say that $\mu$ is a fuzzy measure if it satisfies

• 1. $\mu (\emptyset )= 0$.

• 2. $E, F \in \Sigma$ and $E \subset F$ imply $\mu (E) \le \mu (F)$.

• 3. $E_n \in \Sigma$ $(n \in \mathbb {N})$, $E_1 \subset E_2 \subset \ldots ,$ imply $\lim _{n \rightarrow \infty } \mu (E_n)= \mu (\bigcup _{n= 1}^{\infty } E_n)$ (continuity from below).

• 4. $E_n \in \Sigma$ $(n \in \mathbb {N})$, $E_1 \supset E_2 \supset \ldots ,$ $\mu (E_1) < \infty$, imply $\lim _{n \rightarrow \infty } \mu (E_n)= \mu (\bigcap _{n= 1}^{\infty } E_n)$ (continuity from above).

The triple $(X, \Sigma , \mu )$ is called a fuzzy measure space.

Let $(X, \Sigma , \mu )$ be a fuzzy measure space. By $\mathcal {F}_{\mu }(X)$ we denote the set

$$\begin{aligned} \mathcal {F}_{\mu }(X)= \left\{ f: X \longrightarrow [0, \infty ) : f~ \text {is measurable with respect to}~ \Sigma \right\} . \end{aligned}$$

For $f \in \mathcal {F}_{\mu }(X)$ and $\alpha > 0$, we denote by $F_{\alpha }$ and $F_{\tilde{\alpha }}$ the following sets

$$\begin{aligned} F_{\alpha }= \{x \in X: f(x) \ge \alpha \}\,\,\text {and}\,\, F_{\tilde{\alpha }}= \{x \in X: f(x) > \alpha \}. \end{aligned}$$

Note that if $\alpha \le \beta$, then $F_{\beta } \subset F_{\alpha }$ and $F_{\tilde{\beta }} \subset F_{\tilde{\alpha }}$.

Definition 2

(Durante and Sempi 2005) A function $T: [0, 1] \times [0, 1] \longrightarrow [0, 1]$ is said to be a semicopula if, and only if, it satisfies the two following conditions:

• (T$_1$) $T(x, 1)= T(1, x)= x$    for any $x \in [0, 1]$.

• (T$_2$) For any $x_1, x_2, y_1, y_2 \in [0, 1]$ with $x_1 \le x_2$ and $y_1 \le y_2$, $T(x_1, y_1) \le T(x_2, y_2)$.

A semicopula T is a t-norm if

• (T$_3$) $T(x, y)= T(y, x)$    for  any $x, y \in [0, 1]$.

• (T$_4$) $T\big (T(x, y), z\big )= T\big (x, T(y ,z)\big ) \quad \mathrm{for\, any} x, y, z \in [0, 1]$.

A function $S: [0, 1] \times [0, 1] \longrightarrow [0, 1]$ is called a t-conorm (Klement et al. 2000), if there is a t-norm T such that $S(x, y)= 1- T(1- x, 1- y)$.

Example 1

The following functions are t-norms:

• 1. $T_M (x, y)= \min \{x, y\}.$

• 2. $T_P (x, y)= x \cdot y.$

• 3. $T_L (x, y)= \max \{x+ y- 1, 0\}.$

• 4. $T_D (x, y)=$ $\left\{ \begin{array}{ll} 0, &{} \quad \hbox {if} (x, y) \in [0, 1)^2; \\ \min (x, y), &{} \quad \hbox {otherwise.} \end{array} \right.$

Notice that if T is a t-norm, as an immediate consequence of (T$_1$), (T$_3$) and (T$_4$), the drastic product $T_D$ is the weakest, and the minimum $T_M$ is the strongest t-norm, i.e., for each t-norm T we have

$$\begin{aligned} T_D \le T \le T_M. \end{aligned}$$

(1)

Between the four basic t-norms we have these strict inequalities (see Klement et al. 2000, 2004):

$$\begin{aligned} T_D < T_L < T_P < T_M. \end{aligned}$$

Definition 3

(Pap 1995; Sugeno 1974; Wang and Klir 1992) Let $(X, \Sigma , \mu )$ be a fuzzy measure space, $f \in \mathcal {F}_{\mu }(X)$ and $A \in \Sigma$, then the Sugeno integral of f on A with respect to the fuzzy measure $\mu$ is defined by

where $\wedge$ is just the prototypical t-norm minimum and $\vee$ the prototypical t-conorm maximum. If $A= X$, then

The following properties of Sugeno integral are well known and can be found in Pap (1995) and Wang and Klir (1992).

Theorem 1

Let $(X, \Sigma , \mu )$ be a fuzzy measure space, $A, B \in \Sigma$ and $f, g \in \mathcal {F}_{\mu }(X)$ then

• (F$_1$) .

• (F$_2$) , k non-negative constant.

• (F$_3$) If f $\le g$ on A then .

• (F$_4$) If A $\subset B$ then .

By using the concept of semicopulas, García and Álvarez (1986) proposed the following family of fuzzy integrals.

Definition 4

Let T be a semicopula. Then the seminormed Sugeno integral of a function $f \in \mathcal {F}_{\mu }(X)$ over $A \in \Sigma$ with respect to T and the fuzzy measure $\mu$ is defined by

$$\begin{aligned} \int _{T, A} f \mathrm {d}\mu = \bigvee _{\alpha \in [0, 1]} T\left( \alpha , \mu (A\cap F_{\alpha }) \right) . \end{aligned}$$

It should be noted that the seminormed Sugeno integrals were independently introduced not only as (N) fuzzy integral due to Zhao (1981), but also as the weakest universal integrals (on [0, 1]) with respect to a given semicopula, see Klement et al. (2010).

Notice that the Sugeno integral of $f \in \mathcal {F}_{\mu }(X)$ over $A \in \Sigma$ is the seminormed Sugeno integral of f over $A \in \Sigma$ with respect to the semicopula $T_M$.

In virtue of (1),

(2)

Proposition 1

(García and Álvarez 1986) Let $(X, \Sigma , \mu )$ be a fuzzy measure space and T be a semicopula. Then

• 1. For any $A \in \Sigma$ and $f, g \in \mathcal {F}_{\mu }(X)$ with $f \le g$, we have

  $$\begin{aligned}\int _{T, A} f \mathrm {d}\mu \le \int _{T, A} g \mathrm {d}\mu .\end{aligned}$$

• 2. For $A, B \in \Sigma$ with $A \subset B$ and any $f \in \mathcal {F}_{\mu }(X)$,

  $$\begin{aligned}\int _{T, A} f \mathrm {d}\mu \le \int _{T, B} f \mathrm {d}\mu .\end{aligned}$$

The power mean $M_r (x, y; \lambda )$ of order r of positive numbers x, y is defined by

$$\begin{aligned}M_r (x, y; \lambda )= \left\{ \begin{array}{ll} \big (\lambda f(x)^{r}+ (1- \lambda )f(y)^{r}\big )^{\frac{1}{r}}, &{} \quad r \ne 0; \\ f(x)^{\lambda } f(y)^{1- \lambda }, &{} \quad r= 0. \end{array} \right. \end{aligned}$$

A positive function f is r-convex on a real interval [a, b] if for all $x, y \in [a, b]$ and $\lambda \in [0, 1]$ we have

$$\begin{aligned} f\left( \lambda x+ (1- \lambda )y \right) \le M_r (x, y; \lambda ). \end{aligned}$$

We have that 1-convex functions are ordinary convex functions. Moreover, 0-convex functions are simply log-convex functions. It will be convenient to invoke the generalized logarithmic mean $L_r(x, y)$ of order r of two positive numbers x, y, which is given by

$$\begin{aligned}L_r(x, y)= \left\{ \begin{array}{ll} \frac{r}{r+ 1}\cdot \frac{x^{r+ 1}- y^{r+ 1}}{x^{r}- y^{r}}, &{}\quad r \ne 0, -1,\quad x \ne y, \\ \frac{x- y}{\ln (x)- \ln (y)}, &{}\quad r= 0,\quad x \ne y, \\ x y \cdot \frac{\ln (x)- \ln (y)}{x- y}, &{}\quad r= -1,\quad x \ne y, \\ x, &{}\quad x= y. \end{array} \right. \end{aligned}$$

The following Hadamard inequality provides an upper bound for the mean value of an r-convex function $f: [a, b] \longrightarrow \mathbb {R}$, see (Gill et al. 1997):

$$\begin{aligned} \frac{1}{b- a} \int ^{b}_{a} f(x) \mathrm{d}x \le L_r\left( f(a), f(b)\right) . \end{aligned}$$

(3)

3 The main results

To simplify the calculation of the Sugeno integral, for a given $f \in \mathcal {F}^{\mu }(X)$ and $A \in \Sigma$, we write

$$\begin{aligned} \Gamma = \left\{ \alpha \mid \alpha \ge 0, \mu (A \cap F_{\alpha })> \mu (A \cap F_{\beta }) ~\text {for any} ~\beta >\alpha \right\} . \end{aligned}$$

It is easy to see that

Notice that if X is the set of real numbers ($X= \mathbb {R}$), $\Sigma$ is the Borel field and $\mu$ is the Lebesgue measure, then $(X, \Sigma , \mu )$ is a fuzzy measure space; but it should be noted that the Sugeno integral is not an extension of the Lebesgue integral.

The following example shows that the Hadamard integral inequality for r-convex functions is not valid in the fuzzy context.

Example 2

Let $\Sigma$ be the Borel field and $\mu$ be the Lebesgue measure on $\mathbb {R}$. Take the non-negative and $\frac{1}{2}$-convex function $f(x)= 2^{x- 2}$ on [0, 1]. We have

In this expression, $-1- \frac{\ln (\alpha )}{\ln (2)}$ is a decreasing continuous function of $\alpha$ when $\alpha \ge 0$. Hence, the supremum will be attained at the point which is one of the solutions of the equation

$$\begin{aligned} \alpha = -1- \frac{\ln (\alpha )}{\ln (2)}, \end{aligned}$$

that is, at $\alpha \approx 0.383$. So, we have

On the other hand, $L_{\frac{1}{2}}\big (f(0), f(1)\big ) \approx 0.368$. This proves that the Hadamard integral inequality (3) for r-convex functions is not satisfied in the fuzzy context.

In the sequel, we will establish an upper bound on Sugeno integral of r-convex functions ($r \ne 0$). Some specific examples will be given to illustrate the results.

Theorem 2

Let $([a, b], \Sigma , \mu )$ be a fuzzy measure space. Let $r> 0$ and $f: [a, b] \longrightarrow [0, \infty )$ be an r-convex function with $f(a)\ne f(b)$. If $f(b)> f(a)$, then

where $\Gamma = \big [f(a), f(b)\big )$. If $f(a)> f(b)$, then

where $\Gamma = \big [f(b), f(a)\big )$.

Proof

As f is r-convex, for $x \in [a, b]$ we have

$$\begin{aligned} f(x)&= f\left( \left( 1- \frac{x- a}{b- a}\right) a+ \frac{x- a}{b- a} b \right) \\&\le \left( \left( 1- \frac{x- a}{b- a}\right) f(a)^{r}+ \frac{x- a}{b- a} f(b)^{r}\right) ^{\frac{1}{r}}\\&= g(x). \end{aligned}$$

By (F$_3$) of Theorem 1 and Definition 3, we get

(4)

where G is the distribution function associated with g given by

$$\begin{aligned} G(\alpha )= \mu \big ([a, b] \cap \{g \ge \alpha \} \big ). \end{aligned}$$

If $f(b)> f(a)$, then

$$\begin{aligned} G(\alpha )= & {} \mu \left( [a, b] \cap \left\{ \left( \left( 1- \frac{x- a}{b- a}\right) f(a)^{r}+ \frac{x- a}{b- a} f(b)^{r}\right) ^{\frac{1}{r}} \ge \alpha \right\} \right) \nonumber \\= & {} \mu \left( [a, b] \cap \left\{ x \ge (b- a)\frac{\alpha ^r - f(a)^r}{f(b)^r - f(a)^r}+ a \right\} \right) \nonumber \\= & {} \mu \left( \left[ (b- a)\frac{\alpha ^r - f(a)^r}{f(b)^r - f(a)^r}+ a, b\right] \right) . \end{aligned}$$

(5)

Thus, $\Gamma = \big [f(a), f(b)\big )$ and we only need to consider $\alpha \in \big [f(a), f(b)\big )$.

If $f(b)< f(a)$, then

$$\begin{aligned} G(\alpha )= & {} \mu \left( [a, b] \cap \left\{ \left( \left( 1- \frac{x- a}{b- a}\right) f(a)^{r}+ \frac{x- a}{b- a} f(b)^{r}\right) ^{\frac{1}{r}} \ge \alpha \right\} \right) \nonumber \\= & {} \mu \left( [a, b] \cap \left\{ x \le (b- a)\frac{\alpha ^r - f(a)^r}{f(b)^r - f(a)^r}+ a \right\} \right) \nonumber \\= & {} \mu \left( \left[ a, (b- a)\frac{\alpha ^r - f(a)^r}{f(b)^r - f(a)^r}+ a\right] \right) . \end{aligned}$$

(6)

Thus, $\Gamma = \big [f(b), f(a)\big )$ and we only need to consider $\alpha \in \big [f(b), f(a)\big )$.

Finally, the assertions of this theorem are true in view of (4), (5) and (6). $\square$

Remark 1

In the case $f(a)= f(b)$ in Theorem 2, we have $g(x)= f(a)$ and using (F$_2$) and (F$_3$) of Theorem 1, we get

Remark 2

In Theorem 2, if we suppose $r< 0$, then for $f(b)> f(a)$,

where $\Gamma = \big (f(a), f(b)\big ]$. If $f(a)> f(b)$, then

where $\Gamma = \big (f(b), f(a)\big ]$.

Corollary 1

Let $\Sigma$ be the Borel field and $\mu$ be the Lebesgue measure on $\mathbb {R}$. Let $r> 0$ and $f: [a, b] \longrightarrow [0, \infty )$ be an r-convex function with $f(a)\ne f(b)$. Then for $f(b)> f(a)$,

where $\beta$ is one of the solutions of the equation

$$\begin{aligned} (b- a)\alpha ^r+ (f(b)^r - f(a)^r) \alpha = (b- a) f(b)^r \end{aligned}$$

belonging to $\big [f(a), f(b)\big )$. If $f(b)< f(a)$, then

where $\beta$ is one of the solutions of the equation

$$\begin{aligned} (b- a)\alpha ^r+ (f(a)^r - f(b)^r) \alpha = (b- a) f(a)^r \end{aligned}$$

belonging to $\big [f(b), f(a)\big )$.

Proof

According to Theorem 2, we have

In the case $f(b)> f(a)$, $(b- a) \frac{f(b)^r- \alpha ^r}{f(b)^r - f(a)^r}$ is a decreasing continuous function of $\alpha$ when $\alpha \in \big [f(a), f(b)\big )$. Hence, the supremum will be attained at the point which is one of the solutions of the equation

$$\begin{aligned} \alpha = (b- a) \frac{f(b)^r- \alpha ^r}{f(b)^r - f(a)^r}, \end{aligned}$$

i.e.,

$$\begin{aligned} (b- a)\alpha ^r+ (f(b)^r - f(a)^r) \alpha = (b- a) f(b)^r. \end{aligned}$$

In the case $f(b)< f(a)$, $(b- a) \frac{f(a)^r- \alpha ^r}{f(a)^r - f(b)^r}$ is a decreasing continuous function of $\alpha$ when $\alpha \in \big [f(b), f(a)\big )$. Hence, the supremum will be attained at the point which is one of the solutions of the equation

$$\begin{aligned} \alpha = (b- a) \frac{f(a)^r- \alpha ^r}{f(a)^r - f(b)^r}, \end{aligned}$$

i.e.,

$$\begin{aligned} (b- a)\alpha ^r+ (f(a)^r - f(b)^r) \alpha = (b- a) f(a)^r. \end{aligned}$$

$\square$

Remark 3

In Corollary 1, if we suppose $r< 0$, then for $f(b)> f(a)$,

where $\beta$ is one of the solutions of the equation

$$\begin{aligned} (b- a)\alpha ^r+ (f(a)^r - f(b)^r) \alpha = (b- a) f(a)^r \end{aligned}$$

belonging to $\big (f(a), f(b)\big ]$. If $f(b)< f(a)$, then

where $\beta$ is one of the solutions of the equation

$$\begin{aligned} (b- a)\alpha ^r+ (f(b)^r - f(a)^r) \alpha = (b- a) f(b)^r \end{aligned}$$

belonging to $\big (f(b), f(a)\big ]$.

Example 3

Let $\Sigma$ be the Borel field and $\mu$ be the Lebesgue measure on $\mathbb {R}$. Consider the non-negative and $\frac{1}{2}$-convex function $f(x)= e^{x^2 - 1}$ on [0, 1]. As $f(0)= 1/e$, $f(1)= 1$, using Corollary 1 we can get the following estimate:

In this expression, $\frac{1- \sqrt{\alpha }}{1 - 1/\sqrt{e}}$ is a decreasing continuous function of $\alpha$ when $\alpha \in \big [1/e, 1\big )$. We put

$$\begin{aligned} \beta =\bigvee _{\alpha \in \big [1/e, 1\big )} \left( \alpha \wedge \frac{1- \sqrt{\alpha }}{1 - 1/\sqrt{e}}\right) . \end{aligned}$$

So, $\beta$ is one of the solutions of the equation

$$\begin{aligned} \sqrt{\alpha }+ \left( 1 - 1/\sqrt{e}\right) \alpha = 1 \end{aligned}$$

belonging to $\big [1/e, 1\big )$, that is, $\beta \approx 0.590$. Consequently, we have

Example 4

Let $\Sigma$ be the Borel field and $\mu$ be the Lebesgue measure on $\mathbb {R}$. Consider the non-negative and $\frac{1}{2}$-convex function $f(x)= (3/2)^{x}- x$ on [0, 2]. As $f(0)= 1$, $f(2)= 1/4$, by Corollary 1 we may approximate the upper bound of the Sugeno integral of f on [0, 2] by

In this expression, $4(1- \sqrt{\alpha })$ is a decreasing continuous function of $\alpha$ when $\alpha \in \big [1/4, 1\big )$. We put

$$\begin{aligned} \beta = \bigvee _{\alpha \in \big [1/4, 1\big )} \left( \alpha \wedge 4(1- \sqrt{\alpha })\right) . \end{aligned}$$

So, $\beta$ is one of the solutions of the equation

$$\begin{aligned} 2 \sqrt{\alpha }+ (1 - 1/2) \alpha = 2 \end{aligned}$$

belonging to $\big [1/4, 1\big )$, that is, $\beta \approx 0.686$. Therefore,

It should be noted that the exact solution of cannot be easily calculated. But surely the exact solution is less than or equal to 0.686.

In the following proposition, we deal with the seminormed fuzzy integral on [0, 1] based on the semicopula $T_P$. It should be noted that $T_P$-based integrals are standardly known as Shilkret integral, see Pap (1995).

Proposition 2

Let $\Sigma$ be the Borel field and $\mu$ be the Lebesgue measure on $\mathbb {R}$. Let $r > 0$ and $f: [0, 1] \longrightarrow [0, 1]$ be an r-convex function with $f(0)\ne f(1)$. Then

$$\begin{aligned} \int _{T_P, [0, 1]} f \mathrm {d}\mu \le \left\{ \begin{array}{ll} \frac{r}{f(1)^r - f(0)^r} \left( \frac{f(1)}{(r+ 1)^{\frac{1}{r}}}\right) ^{r+ 1}, &{} f(1)> f(0), \\ \frac{r}{f(0)^r - f(1)^r} \left( \frac{f(0)}{(r+ 1)^{\frac{1}{r}}}\right) ^{r+ 1}, &{} f(1)< f(0). \end{array} \right. \end{aligned}$$

Proof

For an r-convex function $f: [0, 1] \longrightarrow [0, 1]$ with $f(0)\ne f(1)$, according to Proposition 1 and Corollary 1 with semicopula $T_P$, we have

$$\begin{aligned}&\int _{T_P, [0, 1]} f \mathrm {d}\mu \\&\le \left\{ \begin{array}{ll} \bigvee _{\alpha \in (0, 1]} \left( \alpha \cdot \frac{f(1)^r- \alpha ^r}{\big (f(1)^r - f(0)^r\big )}\right) , &{} f(1)> f(0) \\ \bigvee _{\alpha \in (0, 1]} \left( \alpha \cdot \frac{f(0)^r- \alpha ^r}{\big (f(0)^r - f(1)^r\big )}\right) , &{} f(1)< f(0) \end{array} \right. \\&= \left\{ \begin{array}{ll} \frac{r}{f(1)^r - f(0)^r} \left( \frac{f(1)}{(r+ 1)^{\frac{1}{r}}}\right) ^{r+ 1}, &{} f(1)> f(0), \\ \frac{r}{f(0)^r - f(1)^r} \left( \frac{f(0)}{(r+ 1)^{\frac{1}{r}}}\right) ^{r+ 1}, &{} f(1)< f(0). \end{array} \right. \end{aligned}$$

$\square$

Example 5

Let $\Sigma$ be the Borel field and $\mu$ be the Lebesgue measure on $\mathbb {R}$ and consider the non-negative and $\frac{1}{2}$-convex function $f(x)= 3^{-x}$ on [0, 1]. As $f(0)= 1$ and $f(1)= 1/3$, using Proposition 2, we can get the following estimate:

$$\begin{aligned} \int _{T_P, [0, 1]} 3^{-x} \mathrm {d}\mu \le \frac{1/2}{1 - \sqrt{1/3}} \left( \frac{1}{(3/2)^2}\right) ^{3/2} \approx 0.351. \end{aligned}$$

4 Geometric interpretation

Let $\Sigma$ be the Borel field and $\mu$ be the Lebesgue measure on $\mathbb {R}$. If $f: A \subseteq \mathbb {R} \longrightarrow [0, \infty )$ is a continuous function, then the geometric significance of is the edge’s length of the largest square between the curve of f(x) and the x-axis.

In Example 3, for the real $\frac{1}{2}$-convex function $f(x)= e^{x^2 - 1}$ on [0, 1], there exists the real function

$$\begin{aligned} g(x)= \left( (1- x) \sqrt{1/e}+ x \right) ^2 \end{aligned}$$

such that

(7)

Geometric interpretation of (7) is shown in Fig. 1. The lengths of the lines 1 and 2 are the solutions of the integrals in left and right hand sides of (7), respectively. We have a similar geometric interpretation for Example 4 (Fig. 2)

Fig. 1

Geometric interpretation of Example 3

Fig. 2

Geometric interpretation of Example 4

In Example 5, for the real $\frac{1}{2}$-convex function $f(x)= 3^{-x}$ on [0, 1], we have

$$\begin{aligned} \int _{T_P, [0, 1]} 3^{-x} \mathrm {d}\mu \le 0.351. \end{aligned}$$

The length of the line 1 in Fig. 3 is the solution of belonging to the interval (0.5, 0.6). This shows that [cf. the inequality (2)]

Fig. 3

Geometric interpretation of Example 5

5 Conclusion

The Hadamard integral inequality is the first fundamental result for r-convex functions defined on an interval of real numbers with a natural geometrical interpretation and a loose number of applications for particular inequalities. In this paper, we established the Hadamard integral inequality for the Sugeno integral based on r-convex functions which is a useful tool to approximate unsolvable integrals of this kind. In addition, there are numerous applications of Sugeno integral, and thus the study of Hadamard and similar inequalities for Sugeno integral is an important and interesting topic for the further research.