General Barnes–Godunova–Levin type inequalities for Sugeno integral

Information Sciences 181 (2011) 1072–1079

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General Barnes–Godunova–Levin type inequalities for Sugeno integral q Hamzeh Agahi a,b, H. Román-Flores c,⇑, A. Flores-Franulicˇ c a b c

Department of Statistics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Ave., Tehran 15914, Iran Statistical Research and Training Center, Tehran, Iran Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile

a r t i c l e

i n f o

Article history: Received 30 September 2009 Received in revised form 18 November 2010 Accepted 29 November 2010

Keywords: Sugeno integral Non-additive measure Barnes–Godunova–Levin’s inequality

a b s t r a c t Integral inequalities play important roles in classical probability and measure theory. Nonadditive measure is a generalization of additive probability measure. Sugeno’s integral is a useful tool in several theoretical and applied statistics which has been built on non-additive measure. For instance, in decision theory, the Sugeno integral is a median, which is indeed a qualitative counterpart to the averaging operation underlying expected utility. In this paper, Barnes–Godunova–Levin type inequalities for the Sugeno integral on abstract spaces are studied in a rather general form and, for this, we introduce some new technics for the treatment of concave functions in the Sugeno integration context. Also, several examples are given to illustrate the validity of this inequality. Moreover, a strengthened version of Barnes–Godunova–Levin type inequality for Sugeno integrals on a real interval based on a binary operation H is presented. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction In 1974, Sugeno [35] initiated research on non-additive measures and integrals. Non-additive measures and integrals can be used for modelling problems in nondeterministic environment. Sugeno’s integral is a useful tool in several theoretical and applied statistics. For instance, in decision theory, the Sugeno integral is a median, which is indeed a qualitative counterpart to the averaging operation underlying expected utility. The use of the Sugeno integral can be envisaged from two points of view: decision under uncertainty and multi-criteria decision-making [13]. Sugeno’s integral is analogous to Lebesgue integral which has been studied by many authors, including Pap [26], Ralescu and Adams [30] and, Wang and Klir [36], among others. The study of inequalities for Sugeno integral was initiated by Román-Flores et al. [14,31–34], and then followed by the authors [1,3–10,15,19–21,37]. In [33], Román-Flores, Flores-Franulicˇ and Chalco-Cano studied some properties of Sugeno integral for monotone functions, they also provided some Young type inequalities. Based on these results, Flores-Franulicˇ and Román-Flores [14] provided a Chebyshev type inequality for Sugeno integral of continuous and strictly monotone functions based on Lebesgue measure. Some other classical inequalities have also been generalized to Sugeno integral by them (see, for example [32,34]). Later on, Ouyang and Fang [20] generalized the main results of [33] to the case of monotone functions. Based on these results, Ouyang et al. further generalized the Sugeno Chebyshev type inequalities [14] to the case of arbitrary nonadditive measure-based Sugeno integral [19,21–24]. Recently, Agahi and Yaghoobi [1] proved a Minkowski type inequality for monotone functions and arbitrary non-additive measure-based Sugeno integral on real line, and then Agahi et al. [3,4]

q This work was supported by Conicyt-Chile through Project Fondecyt 1080438. Also the first author is partially supported by the Fuzzy Systems and Applications Center of Excellence, Shahid Bahonar University of Kerman, Kerman, Iran. ⇑ Corresponding author. E-mail address: [email protected] (H. Román-Flores).

0020-0255/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2010.11.029

H. Agahi et al. / Information Sciences 181 (2011) 1072–1079

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further generalized it to comonotone functions and arbitrary non-additive measure-base Sugeno integral on an arbitrary measurable space. Recently, the study of integral inequalities has been extended to a more general class of pseudo-integral operators (see [2,11,27]). Any integral inequality can be a very strong tool for applications. In particular, when we think of an integral operator as a predictive tool then an integral inequality can be very important in measuring and dimensioning such process. The classical Barnes–Godunova–Levin inequality [28,29] is one of the famous mathematical inequalities for concave functions. This inequality is an important tool for modern analysis. In general, due to the behavior of the Sugeno integral, the concept of concavity (or convexity) is not frequently used in the Sugeno integration context. Moreover, the most of the integral inequalities studied in the Sugeno integration context normally consider conditions such as monotonicity or comonotonicity. The aim of this paper is to study some general Barnes–Godunova–Levin type inequalities for Sugeno integrals of concave functions. We think that our results will be useful for those areas in which the classical Barnes–Godunova–Levin inequality plays a role whenever the environment is non-deterministic. The paper is organized as follows. Some necessary preliminaries are presented in Section 2. We address the essential problems in Section 3. In Section 4, we construct a strengthened version of Barnes–Godunova–Levin type inequality for Sugeno integrals on abstract spaces. Finally, a conclusion is given in Section 5. 2. Preliminaries In this section, we are going to review some well known results from the theory of non-additive measures, Sugeno’s integral. For details, we refer to [30,35,36]. As usual we denote by R the set of real numbers. Let X be a non-empty set, R be a r-algebra of subsets of X. Let N denote the set of all positive integers and Rþ denote ½0; þ1. Throughout this paper, we fix the measurable space ðX; RÞ, and all considered subsets are supposed to belong to R. Definition 1 (Ralescu and Adams [30]). A set function l : R ! Rþ is called a non-additive measure if the following properties are satisfied: (FM1)lð;Þ ¼ 0; (FM2)A  B implies lðAÞ 6 lðBÞ;  S (FM3)A1  A2     implies l 1 lðAn Þ; and n¼1 An ¼ limn!1 T (FM4)A1  A2    , and lðA1 Þ < þ1 imply l 1 n¼1 An ¼ limn!1 lðAn Þ. When

l is a non-additive measure, the triple ðX; R; lÞ then is called a non-additive measure space.

Let ðX; R; lÞ be a non-additive measure space, by Fl ðXÞ we denote the set of all nonnegative measurable functions f : X ! ½0; 1Þ with respect to l. In what follows, all considered functions belong to Fl ðXÞ. If f 2 Fl ðXÞ, we will denote the set fx 2 Xjf ðxÞ P ag by F a for a P 0. Clearly, F a is nonincreasing with respect to a, i.e., a 6 b implies F akFb. Definition 2 (Pap [26], Sugeno [35], Wang and Klir [36]). Let ðX; R; lÞ be a non-additive measure space and A 2 R. If f 2 Fl ðXÞ then the Sugeno integral of f on A, with respect to the non-additive measure l, is defined as

Z _ -- fdl ¼ ða ^ lðA \ F aÞÞ: A

aP0

When A ¼ X, then

Z Z _ -- fdl ¼ --fdl ¼ ða ^ lðF aÞÞ: X

aP0

It is well known that Sugeno integral is a type of nonlinear integral [18], i.e., for general case,

Z Z Z --ðaf þ bgÞdl ¼ a --fdl þ b --gdl does not hold. Some basic properties of Sugeno integral are summarized in [26,36], we cite some of them in the next theorem. Theorem 1 (Pap [26], Wang and Klir [36]). Let ðX; R; lÞ be a non-additive measure space, then R (i) --A fdl 6 lðAÞ; R (ii) --A kdl ¼ k ^ lðAÞ for all nonnegative constant k and A 2 R; R (iii) lðA \ F aÞ P a ) --A fdl P a; R (iv) lðA \ F aÞ 6 a ) --A fdl 6 a;

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R (v) If lðAÞ < 1, then lðA \ F aÞ P a () --A fdl P a; R R (vi) If f 6 g, then --fdl 6 --gdl. Before stating our main results we need a definition. Definition 3. Functions f ; g : X ! R are said to be comonotone if for all x; y 2 X,

ðf ðxÞ  f ðyÞÞðgðxÞ  gðyÞÞ P 0 and f and g are said to be countermonotone if for all x; y 2 X,

ðf ðxÞ  f ðyÞÞðgðxÞ  gðyÞÞ 6 0: The comonotonicity of functions f and g is equivalent to the nonexistence of points x; y 2 X such that f ðxÞ < f ðyÞ and gðxÞ > gðyÞ. Similarly, if f and g are countermonotone then f ðxÞ < f ðyÞ and gðxÞ < gðyÞ cannot happen. Also, it is clear that if f and g are comonotone then (see [19,25]) for any real numbers s; t either F s # Gt or F t # Gs . Observe that the concept of comonotonicity was first introduced in [12]. Now, our results can be stated as follows. 3. Barnes–Godunova–Levin type inequalities for Sugeno integrals The following is the classical Barnes–Godunova–Levin inequality [28,29]:

Z

b

!1p Z f ðxÞdx

a

!1q

b

p

q

g ðxÞdx

6 Bðp; qÞ

Z

a

where p; q > 1; Bðp; qÞ ¼

b

f ðxÞgðxÞdx;

ð1Þ

a 1þ11 q

6ðbaÞp 1

1

ð1þpÞp ð1þqÞq

and f ; g are nonnegative concave functions on ½a; b.

Unfortunately, as we will see in the following example, in general, the classical Barnes–Godunova–Levin inequality is not valid for Sugeno integral. Example 1. Let f ; g be two real valued functions defined as f ðxÞ ¼ gðxÞ ¼ measure and p ¼ q ¼ 4 in (1). A straightforward calculus shows that

Z 100 Z 100 ðiÞ -- f 4 ðxÞdm ¼ -- g 4 ðxÞdm ¼ 0

0

_

½a ^ mð½0; 100 \ fx P agÞ ¼

a2½0;100

ffiffiffi p 4 x where x 2 ½0; 100. Let m be the Lebesgue

_

½a ^ ð100  aÞ ¼ 50;

a2½0;100

Z 100 _  _   pffiffiffi    a ^ m ½0; 100 \ x P a ¼ a ^ 100  a2 ¼ 9:512 5; ðiiÞ -- f ðxÞgðxÞdm ¼ 0

ðiiiÞ Bð4; 4Þ ¼

a2½0;10

a2½0;10

3 pffiffiffiffiffiffiffiffiffi 500 ¼ 0:268 33: 250

Therefore:

Z 100

14 Z 100

14 Z 100

-- f 4 ðxÞdm -- g 4 ðxÞdm ¼ 7:071 1 > Bð4; 4Þ -- f ðxÞgðxÞdm ¼ 2:5525 0

0

0

and, consequently, inequality (1) is not valid for Sugeno integral. The aim of this section is to show a Barnes–Godunova–Levin type inequality derived from (1) for the Sugeno integral. Theorem 2. If p; q 2 ð0; 1Þ and f ; g : ½0; 1 ! ½0; 1Þ are two concave functions and m be the Lebesgue measure on R. Then (a) If f ð0Þ < f ð1Þ and gð0Þ < gð1Þ, then

0 0 1 0 11 R 1p R 1q 1 p 1 q Z 1

1p Z 1

1q Z1 f dx  f ð0Þ g dx  gð0Þ 0 0 B B C B CC -- g q dx ^ @1  @ -- f p dx A ^ @1  AA 6 -- f ðxÞgðxÞdx: f ð1Þ  f ð0Þ gð1Þ  gð0Þ 0 0 0 (b) If f ð0Þ ¼ f ð1Þ and gð0Þ ¼ gð1Þ, then

Z 1

1p Z 1

1q Z1 p q -- f dx -- g dx ^ f ð0Þgð0Þ 6 -- f ðxÞgðxÞdx: 0

0

0

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(c) If f ð0Þ > f ð1Þ and gð0Þ > gð1Þ, then

0 0R 1 0R 11 1p 1q 1 p 1 q Z 1

1p Z 1

1q Z1 f dx  f ð0Þ g dx  gð0Þ B 0 C B 0 CC B -- g q dx ^ @ @ -- f p dx A^@ AA 6 -- f ðxÞgðxÞdx: f ð1Þ  f ð0Þ gð1Þ  gð0Þ 0 0 0 R1 R1 1 1 Proof. Let p; q 2 ð0; 1Þ; ð --0 f p dxÞp ¼ t 1 and ð --0 g q dxÞq ¼ t2 . Since f ; g : ½0; 1 ! ½0; 1Þ are two concave functions, for x 2 ½0; 1 we have

f ðxÞ ¼ f ðð1  xÞ  0 þ x  1Þ P ð1  xÞ  f ð0Þ þ x  f ð1Þ ¼ h1 ðxÞ; gðxÞ ¼ g ðð1  xÞ  0 þ x  1Þ P ð1  xÞ  gð0Þ þ x  gð1Þ ¼ h2 ðxÞ: (a) If f ð0Þ < f ð1Þ and gð0Þ < gð1Þ, then by Theorem 1 (vi) and the comonotonicity of h1 and h2 , we have

Z1 Z1 _ -- f ðxÞgðxÞdx P -- h1 ðxÞh2 ðxÞdx ¼ ða ^ mð½0; 1 \ fh1 ðxÞh2 ðxÞ P agÞÞ P t1 t2 ^ mð½0; 1 \ fh1 ðxÞh2 ðxÞ P t 1 t 2 gÞ 0

0

aP0

P t 1 t 2 ^ mð½0; 1 \ fh1 ðxÞ P t 1 g \ fh2 ðxÞ P t2 gÞ ¼ t1 t2 ^ mð½0; 1 \ fh1 ðxÞ P t1 gÞ ^ mð½0; 1 \ fh2 ðxÞ P t 2 gÞ ¼ ðt 1 t 2 ^ mð½0; 1 \ f½ð1  xÞ  f ð0Þ þ x  f ð1Þ P t 1 gÞ ^ mð½0; 1 \ f½ð1  xÞ  gð0Þ þ x  gð1Þ P t 2 gÞÞ







t1  f ð0Þ t 2  gð0Þ ^ m ½0; 1 \ x P ¼ t1 t2 ^ m ½0; 1 \ x P f ð1Þ  f ð0Þ gð1Þ  gð0Þ



t 1  f ð0Þ t 2  gð0Þ ¼ t1 t2 ^ 1  ^ 1 : f ð1Þ  f ð0Þ gð1Þ  gð0Þ (b) If f ð0Þ ¼ f ð1Þ and gð0Þ ¼ gð1Þ, then

f ðxÞ ¼ f ðð1  xÞ  0 þ x  1Þ P f ð0Þ ¼ h1 ðxÞ; gðxÞ ¼ g ðð1  xÞ  0 þ x  1Þ P gð0Þ ¼ h2 ðxÞ: Thus, by Theorem 1 (vi) and (ii) we have

Z1 Z1 Z1 -- f ðxÞgðxÞdx P -- h1 ðxÞh2 ðxÞdx ¼ -- f ð0Þgð0Þdx ¼ f ð0Þgð0Þ ^ 1: 0

0

0

R1 R1 1 1 Since ð --0 f p dxÞp ¼ t 1 and ð --0 g q dxÞq ¼ t 2 where p; q 2 ð0; 1Þ, Theorem 1(i) implies that t 1 6 1 and t 2 6 1. Therefore

Z1 -- f ðxÞgðxÞdx P f ð0Þgð0Þ ^ 1 P f ð0Þgð0Þ ^ t 1 t 2 : 0

(c) If f ð0Þ > f ð1Þ and gð0Þ > gð1Þ, then by Theorem 1 (vi) and the comonotonicity of h1 and h2 , we have

Z1 Z1 _ -- f ðxÞgðxÞdx P -- h1 ðxÞh2 ðxÞdx ¼ ða ^ mð½0; 1 \ fh1 ðxÞh2 ðxÞ P agÞÞ P t1 t2 ^ mð½0; 1 \ fh1 ðxÞh2 ðxÞ P t 1 t 2 gÞ 0

0

aP0

P t 1 t 2 ^ mð½0; 1 \ fh1 ðxÞ P t 1 g \ fh2 ðxÞ P t2 gÞ ¼ t1 t2 ^ mð½0; 1 \ fh1 ðxÞ P t1 gÞ ^ mð½0; 1 \ fh2 ðxÞ P t 2 gÞ ¼ ðt 1 t 2 ^ mð½0; 1 \ f½ð1  xÞ  f ð0Þ þ x  f ð1Þ P t 1 gÞ ^ mð½0; 1 \ f½ð1  xÞ  gð0Þ þ x  gð1Þ P t 2 gÞÞ







t 1  f ð0Þ t2  gð0Þ ^ m ½0; 1 \ x 6 ¼ t1 t2 ^ m ½0; 1 \ x 6 f ð1Þ  f ð0Þ gð1Þ  gð0Þ



t1  f ð0Þ t 2  gð0Þ ^ ¼ t1 t2 ^ f ð1Þ  f ð0Þ gð1Þ  gð0Þ and the proof is completed.

h

n o 1 f ð0Þ – ; and Remark 1. The last step in demonstration of Theorem 2 (part a) is valid whenever ½0; 1 \ x P f tð1Þf ð0Þ n o t 2 gð0Þ ½0; 1 \ x P gð1Þgð0Þ – ;. Analogously, the last step in demonstration of Theorem 2 (part c) is valid whenever n o n o t 2 gð0Þ 1 f ð0Þ ½0; 1 \ x 6 f tð1Þf – ; and ½0; 1 \ x 6 gð1Þgð0Þ – ;. ð0Þ Anyway, in any other case the inequality is trivially verified.

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Example 2. Let f ; g be two real valued functions defined as f ðxÞ ¼ gðxÞ ¼ and p ¼ q ¼ 2. A straightforward calculus shows that

pffiffiffi x where x 2 ½0; 1. Let m be the Lebesgue measure

Z1 Z1 Z1 _ _ ½a ^ mð½0; 1 \ fx P agÞ ¼ ½a ^ ð1  aÞ ¼ 0:5; -- g 2 ðxÞdm ¼ -- f 2 ðxÞdm ¼ -- f ðxÞgðxÞdm ¼ 0

0

0

a2½0;1

0

1 0 1 R 12 R 12 1 2 1 2 g dm  gð0Þ f dm  f ð0Þ 0 0 B C B C @1  A ¼ @1  A ¼ 0:292 89; gð1Þ  gð0Þ f ð1Þ  f ð0Þ

a2½0;1

Therefore:

0 0 1 0 11 R 12 R 12 1 2 1 2 Z 1

12 Z 1

12 Z1 f dm  f ð0Þ g dm  gð0Þ 0 0 B B C B CC -- g 2 dm ^ @1  @ -- f 2 dm A ^ @1  AA ¼ 0:29289 6 -- f ðxÞgðxÞdm ¼ 0:5: f ð1Þ  f ð0Þ gð1Þ  gð0Þ 0 0 0

Now, we will prove the general case of the Theorem 2. Theorem 3. Let p; q 2 ð0; 1Þ and f ; g : ½a; b ! ½0; 1Þ be two concave functions and m be the Lebesgue measure on R. Then (a) If f ðaÞ < f ðbÞ and gðaÞ < gðbÞ, then

0

0 1 0 11 R 1p R 1q !1p Z !1q b b Zb b --a f p dx ðb  aÞ þ af ðbÞ  bf ðaÞC B --a g q dx ðb  aÞ þ ag ðbÞ  bgðaÞCC B B -- g q dx ^ @b  @ -- f p dx A ^ @b  AA f ðbÞ  f ðaÞ g ðbÞ  gðaÞ a a Zb 6 -- f ðxÞgðxÞdx: a

(b) If f ðaÞ ¼ f ðbÞ and gðaÞ ¼ gðbÞ, then q p !pþq !pþq Zb Zb Zb p q ^ f ðaÞgðaÞ 6 -- f ðxÞgðxÞdx: -- f dx -- g dx

a

a

a

(c) If f ðaÞ > f ðbÞ and gðaÞ > gðbÞ, then

0

0R 1 0R 11 1p 1q !1p Z !1q b p b q Zb b f dx ð b  a Þ þ af ð b Þ  bf ðaÞ g dx ð b  a Þ þ agðbÞ  bgðaÞ a a B C B CC B -- g q dx ^ @  aA ^ @  aAA @ -- f p dx f ðbÞ  f ðaÞ gðbÞ  gðaÞ a a Zb 6 -- f ðxÞgðxÞdx: a

R 1p R 1q b b Proof. Let p; q 2 ð0; 1Þ; --a f p dx ¼ t 1 and --a g q dx ¼ t2 . Since f ; g : ½a; b ! ½0; 1Þ are two concave functions, for x 2 ½a; b we have

f ðxÞ ¼ f

 x  a  x  a x  a x  a 1 aþ b P 1  f ðaÞ þ  f ðbÞ ¼ h1 ðxÞ; ba ba ba ba

gðxÞ ¼ g



1

x  a  x  a x  a x  a aþ b P 1  gðaÞ þ  gðbÞ ¼ h2 ðxÞ: ba ba ba ba

We will prove (a) and (b), the other case is similar. (a) If f ðaÞ < f ðbÞ and gðaÞ < gðbÞ, then by Theorem 1(vi) and the comonotonicity of h1 and h2 , we have

Z b Z b _ -- f ðxÞgðxÞdx P -- h1 ðxÞh2 ðxÞdx ¼ ða ^ mð½a; b \ fh1 ðxÞh2 ðxÞ P agÞÞ P t 1 t2 ^ mð½a; b \ fh1 ðxÞh2 ðxÞ P t1 t 2 gÞ a

a

aP0

P t 1 t 2 ^ mð½a; b \ fh1 ðxÞ P t 1 g \ fh2 ðxÞ P t 2 gÞ ¼ t1 t2 ^ mð½a; b \ fh1 ðxÞ P t1 gÞ ^ mð½a; b \ fh2 ðxÞ P t2 gÞ x  a   n o x  a  f ðaÞ þ  f ðbÞ P t 1 ¼ t1 t 2 ^ m ½a; b \ 1  ba  ba  n o x  a x  a  gðaÞ þ  gðbÞ P t 2 ^ m ½a; b \ 1  b ba

a 





t 1 ðb  aÞ þ af ðbÞ  bf ðaÞ t 2 ðb  aÞ þ agðbÞ  bgðaÞ ^ m ½a; b \ x P ¼ t1 t2 ^ m ½a; b \ x P f ðbÞ 

f ðaÞ

gðbÞ  gðaÞ t 1 ðb  aÞ þ af ðbÞ  bf ðaÞ t2 ðb  aÞ þ agðbÞ  bgðaÞ ^ b ; ¼ t1 t2 ^ b  f ðbÞ  f ðaÞ gðbÞ  gðaÞ

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(b) If f ðaÞ ¼ f ðbÞ and gðaÞ ¼ gðbÞ, then h1 ðxÞ ¼ f ðaÞ and h2 ðxÞ ¼ gðaÞ. Thus, by Theorem 1(vi) and (ii) we have

Zb Zb Zb -- f ðxÞgðxÞdx P -- h1 ðxÞh2 ðxÞdx ¼ -- f ðaÞgðaÞdx ¼ f ðaÞgðaÞ ^ ðb  aÞ: a

a

a

Rb Rb 1 1 1 1 If ð --a f p dxÞp ¼ t 1 and ð --a g q dxÞq ¼ t 2 where p; q 2 ð0; 1Þ, Theorem 1(i) implies that t 1 6 ðb  aÞp and t 2 6 ðb  aÞq , which im1 1 1

plies that b  a P ðt1 t 2 Þpþq and, consequently

Zb 1 1 1 -- f ðxÞgðxÞdx P f ðaÞgðaÞ ^ ðb  aÞ P f ðaÞgðaÞ ^ ðt 1 t 2 Þpþq a

and the proof is completed.

h

Example 3. Let f ; g be two real valued functions defined as f ðxÞ ¼ lnð1 þ xÞ and gðxÞ ¼ lnð1 þ xÞ2 where x 2 ½0; 4. Let m be the Lebesgue measure and p ¼ q ¼ 1. A straightforward calculus shows that

Z4 ðiÞ -- f ðxÞdm ¼ 0

_

½a ^ mð½0; 4 \ flnð1 þ xÞ P agÞ ¼

a2½0;ln5

Z4 ðiiÞ -- gðxÞdm ¼ 0

_

_

½a ^ mð½0; 4 \ f2 lnð1 þ xÞ P agÞ ¼

a2½0;2ln5

Z4 ðiiiÞ -- f ðxÞgðxÞdm ¼ 0

½a ^ ð5  ea Þ ¼ 1:306 6;

a2½0;ln5

_

h



1

_

h

a ^ 5  e2a

i

¼ 2:117 4;

a2½0;2ln5

_

2

½a ^ mð½0; 4 \ f2ðln ð1 þ xÞÞ P agÞ ¼

a2½0;2ðln5Þ2 



1

a ^ 5  e2

pffiffipffiffi i 2 a

¼ 2:1677:

a2½0;2ðln5Þ2 

Therefore:

R R 0 0 1 0 11 4 4 Z 4

Z 4

4 --0 fdm þ 0:f ð4Þ  4f ð0Þ 4 --0 gdm þ 0:gðbÞ  4gð0Þ @ -- fdm A ^ @4  AA -- gdm ^ @4  f ð4Þ  f ð0Þ gð4Þ  gð0Þ 0 0



4  1:306 6  4 lnð1Þ 4  2:117 4  4 lnð1Þ ¼ 2:766 6 ^ 4  ^ 4 ¼ 2:766 6 ^ 0:752 66 ^ 1:368 8 ¼ 0:752 66 lnð5Þ  lnð1Þ 2 lnð5Þ  lnð1Þ Z4 6 2:1677 ¼ -- f ðxÞgðxÞdx: 0

4. A -based B–G–L-inequality In this section, we provide a strengthened version of Barnes–Godunova–Levin type inequality for Sugeno integrals on a real interval based on a binary operation H (see [16,17]), where H : ½0; 1Þ2 ! ½0; 1Þ is continuous and nondecreasing in both arguments. Theorem 4. Let p; q 2 ð0; 1Þ and f ; g : ½a; b ! ½0; 1Þ be two concave functions and m be the Lebesgue measure on R. If the binary operation H : ½0; 1Þ2 ! ½0; 1Þ is continuous and nondecreasing in both arguments, then (a) If f ðaÞ < f ðbÞ and gðaÞ < gðbÞ, then

0 02 1 0 11 R 1p R 1q !1p !1q 3 b p b q Zb Zb f dx ðb  aÞ þ af ðbÞ  bf ðaÞ g dx ðb  aÞ þ agðbÞ  bgðaÞ a a B4 B C B CC -- f p dx H -- g q dx 5 ^ @b  @ A ^ @b  AA f ðbÞ  f ðaÞ gðbÞ  gðaÞ a a Zb 6 -- f ðxÞHgðxÞdx: a

(b) If f ðaÞ ¼ f ðbÞ and gðaÞ ¼ gðbÞ, then q p !pþq !pþq Zb Zb Zb p q -- f dx H -- g dx ^ ðf ðaÞHgðaÞÞ 6 -- f ðxÞHgðxÞdx:

a

a

a

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(c) If f ðaÞ > f ðbÞ and gðaÞ > gðbÞ, then

02 0R 1 0R 11 1p 1q !1p !1q 3 b b Zb Zb --a f p dx ðb  aÞ þ af ðbÞ  bf ðaÞ --a g q dx ðb  aÞ þ agðbÞ  bgðaÞ B4 B C B CC -- f p dx H -- g q dx 5 ^ @  aA ^ @  aAA @ f ðbÞ  f ðaÞ gðbÞ  gðaÞ a a Zb 6 -- f ðxÞHgðxÞdx: a

Proof. The proof is similar to that of Theorem 3. h Example 4. Consider the binary operation H ¼ þ. Then H is continuous and nondecreasing in both arguments and, taking f and g as in Example 2, we have



 Z1 _ _  pffiffiffi 1 ðiÞ -- f ðxÞ þ gðxÞdm ¼ ½a ^ mð½0; 1 \ f2 x P agÞ ¼ a ^ 1  a2 ¼ 0:828 43: 4 0 a2½0;2 a2½0;2 Z1 Z1 ðiiÞ -- g 2 ðxÞdm ¼ -- f 2 ðxÞdm ¼ 0:5; 0

0

0

1 0 1 R 1p R 12 b p 1 2 ð Þ f dx b  a þ af ðbÞ  bf ðaÞ f dm  f ð0Þ a 0 B C B C ðiiiÞ @b  A ¼ @1  A ¼ 0:292 89: f ðbÞ  f ðaÞ f ð1Þ  f ð0Þ Therefore:

02 1 0 11 R 12 R 12 3 0 1 1 Z 1

12 Z 1

12 --0 f 2 dm  f ð0ÞC B --0 g 2 dm  gð0ÞCC B B4 -- f 2 dx þ -- g 2 dx 5 ^ @1  @ A ^ @1  AA ¼ 1:414 2 ^ 0:292 89 f ð1Þ  f ð0Þ gð1Þ  gð0Þ 0 0 Z1 ¼ 0:292 89 6 -- f ðxÞ þ gðxÞdm ¼ 0:828 43: 0

5. Conclusion In this paper, we have investigated some Barnes–Godunova–Levin type inequalities for Sugeno integral of concave functions defined on a real interval ½a; b (Section 3). Also, a strengthened version of Barnes–Godunova–Levin type inequality for Sugeno integrals on a real interval based on a binary operation H is presented (Section 4). In the future research, we will continue exploring other integral inequalities for nonadditive measures and integrals. Acknowledgements The authors are grateful to the referees and Area Editor of the paper for their critical reading of the manuscript and many valuable recommendations for improvements. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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