General form of Chebyshev type inequality for generalized Sugeno integral
aa r X i v : . [ m a t h . G M ] F e b General form of Chebyshev type inequalityfor generalized Sugeno integral
Michał Boczek, Anton Hovana, Ondrej Hutník ∗ Abstract
We prove a general form of Chebyshev type inequality for generalized upper Sugenointegral in the form of necessary and sufficient condition. A key role in our considerationsis played by the class of m -positively dependent functions which includes comonotonefunctions as a proper subclass. As a consequence, we state an equivalent condition forChebyshev type inequality to be true for all comonotone functions and any monotonemeasure. Our results generalize many others obtained in the framework of q-integral,seminormed fuzzy integral and Sugeno integral on the real half-line. Some further con-sequences of these results are obtained, among others Chebyshev type inequality for anyfunctions. We also point out some flaws in existing results and provide their improvements. In many practical investigations, it is necessary to bound one quantity by another. The classicalinequalities are very useful for this purpose. For instance, the classical Chebyshev integralinequality gives a lower bound for Lebesgue integral of product of two functions in terms ofproduct of their Lebesgue integrals. The best result in the additive setting is as follows, see [3]: If ( X, A ) is a measurable space, then two A -measurable real functions f and g defined on X satisfy the inequality Z f g d P > Z f d P Z g d P (1) for any probability measure P if and only if functions f, g are comonotone. It is well-known that the classical integral inequalities (including the Chebyshev one) neednot hold in general when replacing in (1) the probability measure by a non-additive measure andthe additive (Lebesgue) integral by a non-additive integral. Nowadays, there is a huge numberof papers dealing with inequalities for various non-additive integrals, see e.g. [1, 2, 13, 15, 20]. Inthis paper, we aim to provide a general form of the Chebyshev type inequality (necessary as well ∗ Mathematics Subject Classification (2010):
Primary 28A25, 28E10, Secondary 91B06, 60E05
Key words and phrases:
Aggregation function, Monotone measure, Chebyshev inequality, Generalized Sugenointegral, Q-integral, Positively dependent functions.
1s sufficient conditions) for the generalized upper Sugeno integral for m -positively dependentfunctions, as well as for any comonotone functions and monotone measure, see Section 3. Weshow that our results generalize many results from the literature, see Section 4. Moreover, weimprove some statements including the most recent results on Chebyshev type inequality forq-integral from [20] (see Sections 4.1 and 4.2). Finally, in Section 5 we present Chebyshev typeinequality for all functions under some mild assumptions on a monotone measure. We alsopresent several examples demonstrating these results. Let ( X, A ) be a measurable space, where A is a σ -algebra of subsets of a non-empty set X. Fora given measurable space ( X, A ) we denote the set of all A -measurable functions f : X → [0 , ¯ y ] for some ¯ y ∈ (0 , ∞ ] by F ¯ y ( X, A ) . We also consider the set M ( X, A ) of all monotone (or, non-additive ) measures , i.e., set functions m : A → [0 , ∞ ] satisfying m ( A ) m ( B ) whenever A ⊂ B with the boundary condition m ( ∅ ) = 0 and m ( X ) > . If m ( X ) = 1 , then m is called a capacity and M X, A ) denotes the set of all capacities. Let m ( A ∩ D ) = { m ( A ∩ D ) : A ∈ A} for a fixed D ∈ A and m ∈ M ( X, A ) . To shorten the notation, in the case D = X we denote by m ( A ) therange of m .A binary operation ◦ : [0 , ¯ y ] → [0 , ¯ y ] is called a fusion function . Pre-aggregation functionsand aggregation functions are the most important examples of fusion functions (see [5]). Wesay that a function ◦ : Y × Y → [0 , ∞ ] is non-decreasing if a ◦ a b ◦ b whenever a i b i , where a i , b i ∈ Y i ⊂ [0 , ∞ ] for i = 1 , . A non-decreasing fusion function ◦ : [0 , ¯ y ] → [0 , ¯ y ] is left-continuous if it is left-continuous with respect to each coordinate. With ¯ y = ∞ the importantfusion functions are M( a, b ) = a ∧ b and Π( a, b ) = ab (under the convention · ∞ = ∞ · ),where a ∧ b = min( a, b ) . Both of them are semicopulas for ¯ y = 1 as well. Recall that a semicopula (also called a t -seminorm ) S : [0 , → [0 , is a non-decreasing fusion function such that S( a,
1) = S(1 , a ) = a (see [4, 15]). From it follows that S( x,
0) = 0 = S(0 , x ) for all x ∈ [0 , and S( x, y ) x ∧ y for all x, y ∈ [0 , . Another example is the Łukasiewicz semicopula definedby W( a, b ) = ( a + b − ∨ , where a ∨ b = max( a, b ) . The set of all semicopulas S will bedenoted by S . The generalized (upper) Sugeno integral of f ∈ F ¯ y ( X, A ) on D ∈ A with respect to m ∈ M ( X, A ) is defined by I ◦ ,D ( m, f ) := sup t ∈ [0 , ¯ y ] (cid:8) t ◦ m ( D ∩ { f > t } ) (cid:9) , (2)where ◦ : [0 , ¯ y ] × m ( A∩ D ) → [0 , ∞ ] is a non-decreasing function and { f > t } = { x ∈ X : f ( x ) > t } (see [13]). To simplify the notation for D = X we write I ◦ ( m, f ) := sup t ∈ [0 , ¯ y ] (cid:8) t ◦ m ( { f > t } ) (cid:9) . (3)Note that I ◦ ( m, h D ) = I ◦ ,D ( m, h ) for all h ∈ F ¯ y ( X, A ) and m ∈ M ( X, A ) whenever ◦ m ( X ) =0 ◦ m ( D ) , where A denotes the indicator function of the set A. Replacing the operation ◦ in2he formula (3) with M , Π , W and S ∈ S we get the Sugeno integral I M [19], Shilkret integral I Π [17], opposite-Sugeno integral I W [10] and seminormed fuzzy integral I S [18], respectively.When investigating various inequalities for non-additive integrals, one usually has to restrictthe class of measurable functions, or the class of monotone measures under consideration. Inconnection with the Chebyshev type inequalities, the class of m -positively dependent functionswill play the key role. From now on, h | A denotes the restriction of function h to set A. Definition 2.1.
Let f, g ∈ F k ( X, A ) , m ∈ M ( X, A ) and A, B ∈ A with k > . Functions f | A and g | B are called m -positively dependent with respect to an operator △ : m ( A ) × m ( A ) → m ( A ) , if m (cid:0) A ∩ B ∩ { f > α } ∩ { g > β } (cid:1) > m ( A ∩ { f > α } ) △ m ( B ∩ { g > β } ) (4)holds for all α, β ∈ [0 , k ] . The concept of m -positively dependent functions was introduced by Kaluszka et al. [13]as a natural generalization of positive dependency defined for the first time by Lehmann [14]in probability theory. These ideas coincide for △ = · and probability measure m = P . Also,comonotone functions on A are included here being m -positively dependent with respect to △ = ∧ with B = A and any m ∈ M ( X, A ) . Recall that two functions f, g : X → [0 , ∞ ] are called comonotone on D ⊂ X if ( f ( x ) − f ( y ))( g ( x ) − g ( y )) > for all x, y ∈ D. Now, we providea few examples of m -positively dependent functions. Example 2.1.
All functions f | A , g | B ∈ F ∞ ( X, A ) are m -positively dependent with respect to any △ whenever m ∈ M ( X, A ) satisfies the condition m ( C ∩ D ) > m ( C ) △ m ( D ) for all C, D ∈ A . For instance, if m ∈ M ( X, A ) is minitive , i.e., m ( C ∩ D ) = m ( C ) ∧ m ( D ) , with m ( X ) , thenany functions f | A , g | B ∈ F ∞ ( X, A ) are m -positively dependent for any △ ∈ S (e.g. M or Π ). Example 2.2.
Let X = { ω , ω } and m ∈ M X, A ) be such that m ( { ω } ) = p, where p ∈ (0 , .Assume that m ( { f > t } ) = m ( { g > t } ) = { } ( t ) + p (0 , ( t ) for all t > . Then f | X and g | X are m -positively dependent functions with respect to △ such that △6 · . Example 2.3.
Let m ∈ M X, A ) and △ = W . Functions f | A , g | B ∈ F ∞ ( X, A ) are m -positivelydependent if m (cid:0) A ∩ B ∩ { f > α } ∩ { g > β } (cid:1) > (cid:0) m ( A ∩ { f > α } ) + m ( B ∩ { g > β } ) − (cid:1) + for any α, β ∈ [0 , ∞ ] , where ( a ) + = max( a, . The above inequality can be rewritten as follows m d (cid:0) ( A ∩ { f > α } ) c ∪ ( B ∩ { g > β } ) c (cid:1) m d (cid:0) ( A ∩ { f > α } ) c (cid:1) + m d (cid:0) ( B ∩ { g > β } ) c (cid:1) for all α, β, where C c = X \ C and m d ( C ) = 1 − m ( C c ) is a dual capacity. If m d is subadditive ,then all functions f | A and g | B are m -positively dependent with respect to △ . For instance,all functions are m -positively dependent whenever m is a supermodular capacity . Note that A monotone measure m is subadditive , if m ( A ∪ B ) m ( A ) + m ( B ) for all A, B ∈ A . A monotone measure m is supermodular , if m ( A ∪ B ) + m ( A ∩ B ) > m ( A ) + m ( B ) for all A, B ∈ A . m ( B ) = h ( P ( B )) for all B ∈ A is supermodular. Here, P is a probability measure and h : [0 , → [0 , is an increasing and convex function such that h (0) = 0 and h (1) = 1 , see [7, p. 17]. Distorted probabilities play the key role in cumulativeprospect theory [11] and insurance [12]. Example 2.4.
Let a △ b = a ⊗ G b := b { a> − b } (the so-called Gödel conjunction). Then allfunctions f | A , g | A ∈ F ∞ ( X, A ) are m -positively dependent for all m ∈ M X, A ) such that m ( A ) ⊂ [0 , . .Further examples of m -positively dependent functions can be found in [13]. When consid-ering equality instead of inequality in (4) with △ ∈ S , m ∈ M X, A ) and A = B = X, we mayobtain a connection between monotone measures and semicopulas. Under some quite commonassumptions, one can derive an analogous result to Sklar’s theorem (see [9, Theorem 2.2.1]), i.e.,every semicopula links the survival function of the capacity to its marginal survival functions(see [9, Theorem 8.3.3] and [16, Theorem 9]). Using the concept of m -positive dependency, we give a new general form of Chebyshev typeinequality for integral I ◦ . The following lemma will be useful in the proof of our main result.Hereafter, we use the convention sup ∅ = 0 . Lemma 3.1.
Let ¯ y ∈ (0 , ∞ ] and c, k ∈ (0 , ¯ y ] . Assume that g : [0 , c ] → [0 , ¯ y ] and h : [0 , ¯ y ] → [0 , ¯ y ] , where g is a non-decreasing function. If g is left-continuous and sup y ∈ [0 ,k ] h ( y ) ∈ [0 , c ] ,then g (sup y ∈ [0 ,k ] h ( y )) = sup y ∈ [0 ,k ] g ( h ( y )) . Proof.
Clearly, sup y ∈ [0 ,k ] g ( h ( y )) g (sup y ∈ [0 ,k ] h ( y )) . On the other hand, let ( y n ) ∞ ∈ [0 , k ] be a sequence such that h ( y n ) ր sup y ∈ [0 ,k ] h ( y ) , where a n ր a means that a non-decreasingsequence ( a n ) ∞ converges to a. By left-continuity and monotonicity of g we have g ( lim n →∞ h ( y n )) = lim n →∞ g ( h ( y n )) sup y ∈ [0 ,k ] g ( h ( y )) . Hence, g (sup y ∈ [0 ,k ] h ( y )) = sup y ∈ [0 ,k ] g ( h ( y )) . Theorem 3.1.
Assume that ¯ y ∈ (0 , ∞ ] , k ∈ (0 , ¯ y ] , m ∈ M ( X, A ) , △ : m ( A ) × m ( A ) → m ( A ) and ⋆, ♦ are non-decreasing fusion functions such that ♦ is left-continuous. Let ϕ i : [0 , ¯ y ] → [0 , ¯ y ] be functions such that ϕ is non-decreasing and ϕ j are increasing and right-continuous, ψ i : [0 , ϕ i (¯ y )] → [0 , ¯ y ] be non-decreasing and ψ j be left-continuous, where i = 1 , , and j = 2 , . Let ◦ i : [0 , ¯ y ] × m ( A ) → [0 , ¯ y ] be non-decreasing such that ϕ i (¯ y ) ◦ i m ( X ) ϕ i (¯ y ) and ¯ y ◦ j with i = 1 , , and j = 2 , . Function F m : [0 , → [0 , defined as F m ( x , x ) = m (( x , ∩ ( x , is called a survival functionassociated with m. a) Assume that ψ ( ϕ ( a ⋆ b ) ◦ ( c △ d )) > ψ ( ϕ ( a ) ◦ c ) ♦ ψ ( ϕ ( b ) ◦ d ) (5) holds for all a, b ∈ [0 , k ] and c, d ∈ m ( A ) . If f | A , g | B ∈ F k ( X, A ) are m -positively dependent withrespect to △ , then ψ (cid:0) I ◦ ,A ∩ B ( m, ϕ ( f ⋆ g )) (cid:1) > ψ (cid:0) I ◦ ,A ( m, ϕ ( f )) (cid:1) ♦ ψ (cid:0) I ◦ ,B ( m, ϕ ( g )) (cid:1) . (6) (b) Suppose that ¯ y ◦ , a △ △ a = 0 for all a ∈ m ( A ) and the condition ( Z ) holds, i.e., for all c, d ∈ m ( A ) there exists sets C, D ∈ A such that c = m ( C ) , d = m ( D ) and m ( C ∩ D ) = m ( C ) △ m ( D ) . If (6) is true for all m -positively dependent functions f | A , g | B ∈F k ( X, A ) with respect to △ , then (5) holds for all a, b ∈ [0 , k ] and c, d ∈ m ( A ) . Proof. (a) Firstly, note that ψ ’s are well-defined in (5) as ϕ j ( a ) ◦ j c ∈ [0 , ϕ j (¯ y )] for all a ∈ [0 , k ] and c ∈ m ( A ) , where j = 2 , , and ϕ ( a ⋆ b ) ◦ ( c △ d ) ∈ [0 , ϕ (¯ y )] for all a, b ∈ [0 , k ] and c, d ∈ m ( A ) , since ϕ i (¯ y ) ◦ i m ( X ) ϕ i (¯ y ) for i = 1 , , . Then m -positive dependency of f | A and g | B implies that m ( A ∩ B ∩ { f > a } ∩ { g > b } ) > m ( A ∩ { f > a } ) △ m ( B ∩ { g > b } ) for all a, b ∈ [0 , k ] . The monotonicity of ⋆ and m yields m ( A ∩ B ∩ { f ⋆ g > a ⋆ b } ) > m ( A ∩ { f > a } ) △ m ( B ∩ { g > b } ) for any a, b ∈ [0 , k ] . As ϕ is non-decreasing and ϕ j are increasing for j = 2 , , we get m (cid:0) A ∩ B ∩ { ϕ ( f ⋆ g ) > ϕ ( a ⋆ b ) } (cid:1) > c a △ d b for all a, b ∈ [0 , k ] with c a := m ( A ∩ { ϕ ( f ) > ϕ ( a ) } ) and d b := m ( B ∩ { ϕ ( g ) > ϕ ( b ) } ) . Fromthe assumption on monotonicity of ◦ and ψ , we obtain ψ (cid:0) ϕ ( a ⋆ b ) ◦ m ( A ∩ B ∩ { ϕ ( f ⋆ g ) > ϕ ( a ⋆ b ) } ) (cid:1) > ψ ( ϕ ( a ⋆ b ) ◦ ( c a △ d b )) for any a, b ∈ [0 , k ] . By the definition of integral I ◦ , function ψ and from (5), we get ψ (cid:0) I ◦ ,A ∩ B ( m, ϕ ( f ⋆ g )) (cid:1) > ψ ( ϕ ( a ) ◦ c a ) ♦ ψ ( ϕ ( b ) ◦ d b ) (7)for any a, b ∈ [0 , k ] . For a fixed y ∈ [0 , ¯ y ] put H ( x ) = x ♦ y for all x ∈ [0 , ¯ y ] . Also, put h ( a ) = ϕ ( a ) ◦ c a for all a ∈ [0 , k ] . The function H ( ψ ) : [0 , ϕ (¯ y )] → [0 , ¯ y ] is non-decreasingand left-continuous as ♦ and ψ are non-decreasing and left-continuous. Then Lemma 3.1 yields sup a ∈ [0 ,k ] H (cid:0) ψ ( h ( a )) (cid:1) = H (cid:0) ψ ( sup a ∈ [0 ,k ] h ( a )) (cid:1) . (8)Moreover, sup a ∈ [0 ,k ] h ( a ) = sup a ∈ [0 , ¯ y ] h ( a ) due to the fact that ϕ (¯ y ) ◦ and f k ¯ y. Putting t = ϕ ( a ) , we get sup a ∈ [0 ,k ] h ( a ) = sup t ∈ ϕ ([0 , ¯ y ]) (cid:8) t ◦ m ( A ∩ { ϕ ( f ) > t } ) (cid:9) , ϕ ([0 , ¯ y ]) is the image of ϕ . Note that sup a ∈ [0 ,k ] h ( a ) = sup t ∈ [0 ,ϕ (0)) { t ◦ m ( A ) } ∨ sup t ∈ ϕ ([0 , ¯ y ]) (cid:8) t ◦ m ( A ∩ { ϕ ( f ) > t } ) (cid:9) ∨ sup t ∈ ( ϕ (¯ y ) , ¯ y ] { t ◦ } since sup t ∈ [0 ,ϕ (0)) { t ◦ m ( A ) } = ϕ (0) ◦ m ( A ) and ¯ y ◦ . From right-continuity of ϕ andmonotonicity of ◦ we have sup a ∈ [0 ,k ] h ( a ) = sup t ∈ [0 , ¯ y ] (cid:8) t ◦ m ( A ∩ { ϕ ( f ) > t } ) (cid:9) . (9)From (7)-(9) we obtain ψ (cid:0) I ◦ ,A ∩ B ( m, ϕ ( f ⋆ g )) (cid:1) > ψ (cid:0) I ◦ ,A ( m, ϕ ( f )) (cid:1) ♦ ψ ( ϕ ( b ) ◦ d b ) for all b ∈ [0 , k ] . Proceeding similarly with the supremum in b ∈ [0 , k ] , we get (6).(b) Fix a, b ∈ [0 , k ] and c, d ∈ m ( A ) . Define f = a X and g = b X , and consider A, B ∈ A satisfying the condition ( Z ) . Then f | A and g | B are m -positively dependent with respect to △ .Indeed, (4) takes the form m ( ∅ ) > m ( C ∩ D ) = m ( C ) △ m ( D ) = 0 for α > a or β > b, where C = { f | A > α } and D = { g | B > β } , as △ x = x △ . If α a and β b, then wehave m ( A ∩ B ) = m ( A ) △ m ( B ) , so the inequality (4) is satisfied. Thus, I ◦ ,A ( m, ϕ ( f )) = ϕ ( a ) ◦ m ( A ) , I ◦ ,B ( m, ϕ ( g )) = ϕ ( b ) ◦ m ( B ) , I ◦ ,A ∩ B ( m, ϕ ( f ⋆ g ))) = ϕ ( a ⋆ b ) ◦ m ( A ∩ B ) = ϕ ( a ⋆ b ) ◦ ( m ( A ) △ m ( B )) , since ¯ y ◦ i for all i. Applying (6) we obtain (5) for all a, b ∈ [0 , k ] and c, d ∈ m ( A ) . Note that Theorem 3.1 generalizes Theorem 2.3 from [13]. Moreover, the inequality (5)has been investigated for some functions in [13]. Now, we mention two examples where theinequality (6) becomes equality.
Example 3.1.
The equality in (6) holds if k = ¯ y = 1 , ◦ i = ∧ , ⋆ = ♦ = · , ψ i ( x ) = x /p i , ϕ i ( x ) = x p i , f = a D and g = b D , where a, b ∈ [0 , , p i > , D ⊂ A ∩ B and ( ab ) p ∨ a p ∨ b p m ( D ) for i = 1 , , . Example 3.2. If k = ¯ y = 1 , ◦ i ∈ S , A = B = X, ⋆ = ♦ = · , f = a X , g = b X , ψ i ( x ) = x q and ϕ i ( x ) = x p , where a, b ∈ [0 , , p, q > and i = 1 , , , then the equality in (6) holds for m ∈ M X, A ) . The condition ( Z ) appears naturally in the context of m -positively dependent functions.In order to avoid it, we have to consider the class of comonotone functions on the whole X asa special case of m -positively dependent functions with respect to △ = ∧ . Firstly, we give thefollowing useful lemma.
Lemma 3.2.
Let ¯ y ∈ (0 , ∞ ] , k ∈ (0 , ¯ y ] and ⋆, ♦ be fusion functions such that ♦ is non-decreasing. Assume that ϕ i : [0 , ¯ y ] → [0 , ¯ y ] and ψ i : [0 , ϕ i (¯ y )] → [0 , ¯ y ] are non-decreasing for i = 1 , , . Let ◦ i : [0 , ¯ y ] × D → [0 , ¯ y ] be non-decreasing such that ϕ i (¯ y ) ◦ i ¯ d ϕ i (¯ y ) for i = 1 , , , where D ⊂ [0 , ∞ ] and ¯ d = sup D ∈ D. Then the following conditions are equivalent: C ) ψ ( ϕ ( a ⋆ b ) ◦ ( c ∧ d )) > ψ ( ϕ ( a ) ◦ c ) ♦ ψ ( ϕ ( b ) ◦ d ) for all a, b ∈ [0 , k ] and c, d ∈ D ;( C ) ψ ( ϕ ( a ⋆ b ) ◦ c ) > (cid:2) ψ ( ϕ ( a ) ◦ c ) ♦ ψ ( ϕ ( b ) ◦ ¯ d ) (cid:3) ∨ (cid:2) ψ ( ϕ ( a ) ◦ ¯ d ) ♦ ψ ( ϕ ( b ) ◦ c ) (cid:3) forall a, b ∈ [0 , k ] and c ∈ D .Proof. Similar as in the proof of Theorem 3.1, one can show that ψ ’s are well-defined in ( C ) and ( C ) , as ϕ i (¯ y ) ◦ i ¯ d ϕ i (¯ y ) for i = 1 , , . “ ( C ) ⇒ ( C ) ” Putting d = ¯ d in ( C ) and then c = ¯ d in ( C ) we get ( C ) .“ ( C ) ⇒ ( C ) ” By ( C ) and by monotonicity of ◦ , ψ and ♦ , we get ψ ( ϕ ( a ⋆ b ) ◦ c ) > ψ ( ϕ ( a ) ◦ c ) ♦ ψ ( ϕ ( b ) ◦ ¯ d ) > ψ ( ϕ ( a ) ◦ c ) ♦ ψ ( ϕ ( b ) ◦ d ) (10)for all a, b ∈ [0 , k ] and c, d ∈ D. Similarly, we obtain the following inequalities ψ ( ϕ ( a ⋆ b ) ◦ d ) > ψ ( ϕ ( a ) ◦ ¯ d ) ♦ ψ ( ϕ ( b ) ◦ d ) > ψ ( ϕ ( a ) ◦ c ) ♦ ψ ( ϕ ( b ) ◦ d ) (11)for any a, b ∈ [0 , k ] and c, d ∈ D. Combining (10) and (11) with monotonicity of ◦ and ψ weget the condition ( C ) .Now, we can state the second version of Chebyshev type inequality omitting the condition ( Z ) . Theorem 3.2.
Assume that ¯ y ∈ (0 , ∞ ] , k ∈ (0 , ¯ y ] and ⋆, ♦ are non-decreasing fusion functionssuch that ♦ is left-continuous. Let ϕ i : [0 , ¯ y ] → [0 , ¯ y ] be functions such that ϕ is non-decreasingand ϕ j are increasing and right-continuous, ψ i : [0 , ϕ i (¯ y )] → [0 , ¯ y ] be non-decreasing and ψ j beleft-continuous, where i = 1 , , and j = 2 , . Let ◦ i : [0 , ¯ y ] → [0 , ¯ y ] be non-decreasing suchthat ϕ i (¯ y ) ◦ i ¯ y ϕ i (¯ y ) and ¯ y ◦ j , where i = 1 , , and j = 2 , . (a) Suppose that the inequality ψ ( ϕ ( a ⋆ b ) ◦ ( c ∧ d )) > ψ ( ϕ ( a ) ◦ c ) ♦ ψ ( ϕ ( b ) ◦ d ) (12) holds for all a, b ∈ [0 , k ] and c, d ∈ [0 , ¯ y ] . If f, g ∈ F k ( X, A ) are comonotone on X and m ∈ M ( X, A ) such that m ( X ) ¯ y, then ψ (cid:0) I ◦ ( m, ϕ ( f ⋆ g )) (cid:1) > ψ (cid:0) I ◦ ( m, ϕ ( f )) (cid:1) ♦ ψ (cid:0) I ◦ ( m, ϕ ( g )) (cid:1) . (13) (b) Assume that ⋆ k = k ⋆ , ¯ y ◦ and ϕ i (0) ◦ i ¯ y = 0 for i = 1 , , . If (13) is truefor all comonotone functions f, g ∈ F k ( X, A ) on X and any m ∈ M ( X, A ) such that m ( X ) ¯ y, then (12) holds for all a, b ∈ [0 , k ] and c, d ∈ [0 , ¯ y ] . Proof.
Part (a) can be proved in the same way as Theorem 3.1 (a) with △ = ∧ and A = B = X, so we omit it.(b) Define f = a A and g = b X , where a, b ∈ [0 , k ] and A ∈ A . Then I ◦ ( m, ϕ ( f )) = ( ϕ (0) ◦ m ( X )) ∨ ( ϕ ( a ) ◦ m ( A )) ∨ (¯ y ◦
0) = ϕ ( a ) ◦ m ( A ) , ϕ (0) ◦ m ( X ) = 0 and ¯ y ◦ . Similarly, one can check that I ◦ ( m, ϕ ( g )) = ϕ ( b ) ◦ m ( X ) , I ◦ ( m, ϕ ( f ⋆ g )) = ϕ ( a ⋆ b ) ◦ m ( A ) , as ⋆ k = 0 . Applying (13) we obtain ψ ( ϕ ( a ⋆ b ) ◦ m ( A )) > ψ ( ϕ ( a ) ◦ m ( A )) ♦ ψ ( ϕ ( b ) ◦ m ( X )) . Similarly, for f = a X and g = b A we get the inequality ψ ( ϕ ( a ⋆ b ) ◦ m ( A )) > ψ ( ϕ ( a ) ◦ m ( X )) ♦ ψ ( ϕ ( b ) ◦ m ( A )) . In consequence, from (13) for any comonotone functions and any m such that m ( X ) ¯ y, weget ψ ( ϕ ( a ⋆ b ) ◦ c ) > (cid:2) ψ ( ϕ ( a ) ◦ c ) ♦ ψ ( ϕ ( b ) ◦ ¯ y ) (cid:3) ∨ (cid:2) ψ ( ϕ ( a ) ◦ ¯ y ) ♦ ψ ( ϕ ( b ) ◦ c ) (cid:3) for all a, b ∈ [0 , k ] and c ∈ [0 , ¯ y ] . Using Lemma 3.2 with D = [0 , ¯ y ] we obtain (12).The next example demonstrates that in some cases we cannot use Theorem 3.2 (a), butTheorem 3.1 (a) still works. Example 3.3.
Put k = ¯ y = 1 , A = B = X, ϕ i ( x ) = ψ i ( x ) = x for all x ∈ [0 , , ♦ = ⋆ = · , ◦ i = W , m ( A ) = { , } and △ = ∧ in Theorem 3.1 (a), where i = 1 , , . Then the inequality(5) takes the form W( ab, c ∧ d ) > W( a, c ) W( b, d ) for all a, b ∈ [0 , and c, d ∈ { , } . From Theorem 3.1 (a), the Chebyshev type inequality I W ( m, f g ) > I W ( m, f ) I W ( m, g ) holds for any comonotone functions f, g ∈ F X, A ) on X. However, we cannot use Theorem 3.2 (a),since W( ab, c ∧ d ) > W( a, c ) W( b, d ) is not valid for all a, b, c, d ∈ [0 , . Indeed, it is enough to take a = b = 0 . and c = d = 0 . . In this section, we derive some consequences of theorems proven in the previous section forseveral classes of integrals which are known in the literature.8 .1 Q-integral
Recall that a non-decreasing fusion function ⊗ : [0 , → [0 , is said to be a fuzzy conjunction if ⊗ and ⊗ ⊗ ⊗ (see [8]). The most important examples of fuzzyconjunction are: the Gödel conjunction ⊗ G (see Example 2.4) and the contrapositive Gödelconjunction a ⊗ GC b = a { a> − b } ( a, b ) . Dubois et al. [8] introduced and studied the q -integraldefined as Z ⊗ m f = sup t ∈ [0 , (cid:8) m ( { f > t } ) ⊗ t (cid:9) , (14)where ⊗ denotes a fuzzy conjunction and ( m, f ) ∈ M X, A ) ×F X, A ) . This definition is motivatedby alternative ways of using weights of qualitative criteria in min- and max-based aggregations,that make intuitive sense as tolerance thresholds. Note that the research on Chebyshev typeinequality for q-integral has been initiated by Kaluszka et al. [13] even before its formal def-inition by Dubois [8], see for example [13, Theorem 2.1] for Y = [0 , , µ ∈ M X, A ) and a ◦ i b = b ⊗ a. From this point of view, the claim in [20] about starting the research by theauthors is misleading.
Corollary 4.1.
Let ⊗ : [0 , → [0 , be a fuzzy conjunction and ⋆ : [0 , → [0 , be non-decreasing, left-continuous and ⋆ ⋆ . Suppose that ϕ i : [0 , → [0 , is continuousand increasing such that ϕ i (0) = 0 and ⊗ ϕ i (1) ϕ i (1) for i = 1 , , . Then the followingstatements are equivalent:(i) ϕ − ( a ⊗ ϕ ( b ⋆ c )) > (cid:2) ϕ − ( a ⊗ ϕ ( b )) ⋆ ϕ − (1 ⊗ ϕ ( c )) (cid:3) ∨ (cid:2) ϕ − (1 ⊗ ϕ ( b )) ⋆ ϕ − ( a ⊗ ϕ ( c )) (cid:3) for any a, b, c ∈ [0 , (ii) ϕ − (cid:0) R ⊗ m ϕ ( f ⋆ g ) (cid:1) > ϕ − (cid:0) R ⊗ m ϕ ( f ) (cid:1) ⋆ ϕ − (cid:0) R ⊗ m ϕ ( g ) (cid:1) holds for any comonotone functions f, g ∈ F X, A ) on X and any capacity m. Proof.
Put ♦ = ⋆, ¯ y = k = 1 , a ◦ i b = b ⊗ a and ψ i = ϕ − i for i = 1 , , in Theorem 3.2 andLemma 3.2. Functions ψ i are well-defined as ϕ i (0) = 0 . Using Lemma 3.2 with D = [0 , andTheorem 3.2, we get the statement.Note that if we put ϕ i ( x ) = x in Corollary 4.1, we get [20, Theorem 3.5]. Example 4.1.
Put ϕ i = ϕ and ⋆ = · in Corollary 4.1 where ϕ : [0 , → [0 , is continuous andincreasing such that ϕ (0) = 0 and ϕ (1) = 1 . We show that the following inequality ϕ − ( a ⊗ ϕ ( bc )) > (cid:2) ϕ − ( a ⊗ ϕ ( b )) · ϕ − (1 ⊗ ϕ ( c )) (cid:3) ∨ (cid:2) ϕ − (1 ⊗ ϕ ( b )) · ϕ − ( a ⊗ ϕ ( c )) (cid:3) (15)does not hold for any a, b, c ∈ [0 , and ⊗ ∈ {⊗ G , ⊗ GC } . Putting b = 1 in (15), we get ϕ − ( a ⊗ ϕ ( c )) > (cid:2) ϕ − ( a ⊗ · ϕ − (1 ⊗ ϕ ( c )) (cid:3) ∨ ϕ − ( a ⊗ ϕ ( c )) for all a, c ∈ [0 , . Consider a > and ϕ ( c ) > such that a + ϕ ( c ) . Then we have > ϕ − ( a ⊗ · ϕ − (1 ⊗ ϕ ( c )) ⊗ ∈ {⊗ G , ⊗ GC } which leads to the contradiction. In consequence, the Chebyshev typeinequality ϕ − Z ⊗ m ϕ ( f ⋆ g ) ! > ϕ − Z ⊗ m ϕ ( f ) ! ⋆ ϕ − Z ⊗ m ϕ ( g ) ! does not hold for any comonotone functions f, g ∈ F X, A ) on X and any capacity m for ⊗ ∈{⊗ G , ⊗ GC } . Recall that for ◦ = S ∈ S (a semicopula) the generalized Sugeno integral I ◦ coincides withthe seminormed fuzzy integral being the smallest semicopula-based universal integral (see [4]).Observe that the seminormed fuzzy integral is a special case of q-integral with the conjunctionreplaced by the semicopula in the following way a ⊗ b = S( b, a ) . Applying Theorem 3.1 with ¯ y = 1 and ◦ i = S i for i = 1 , , , we get the following Chebyshev type inequality for seminormedfuzzy integral. Corollary 4.2.
Assume that m ∈ M X, A ) , △ : m ( A ) × m ( A ) → m ( A ) , k ∈ (0 , , S i ∈ S for i = 1 , , and ⋆, ♦ : [0 , → [0 , are non-decreasing and ♦ is left-continuous. Let ϕ i : [0 , → [0 , be functions such that ϕ is non-decreasing and ϕ j are increasing and right-continuous, ψ i : [0 , ϕ i (1)] → [0 , be non-decreasing and ψ j be left-continuous, where i = 1 , , and j = 2 , . (a) Suppose that ψ (cid:0) S ( ϕ ( a ⋆ b ) , c △ d ) (cid:1) > ψ (cid:0) S ( ϕ ( a ) , c ) (cid:1) ♦ ψ (cid:0) S ( ϕ ( b ) , d ) (cid:1) (16) holds for all a, b ∈ [0 , k ] and c, d ∈ m ( A ) . If f | A , g | B ∈ F k ( X, A ) are m -positively dependent withrespect to △ , then ψ (cid:0) I S ,A ∩ B ( m, ϕ ( f ⋆ g )) (cid:1) > ψ (cid:0) I S ,A ( m, ϕ ( f )) (cid:1) ♦ ψ (cid:0) I S ,B ( m, ϕ ( g )) (cid:1) . (17) (b) Assume that a △ △ a = 0 for all a ∈ m ( A ) and the condition ( Z ) holds. If (17) is true for all m -positively dependent functions f | A , g | B ∈ F k ( X, A ) with respect to △ , then (16) holds for any a, b ∈ [0 , k ] and c, d ∈ m ( A ) . Next corollary gives a necessary and sufficient condition for Chebyshev type inequality for I S for comonotone functions on X and any monotone measure. Corollary 4.3.
Assume S ∈ S and operation ⋆ : [0 , → [0 , is non-decreasing and left-continuous such that ⋆ ⋆ . Let ϕ i : [0 , → [0 , be functions such that ϕ is non-decreasing and ϕ j are increasing and right-continuous, ψ i : [0 , ϕ i (1)] → [0 , be non-decreasingand ψ j be left-continuous, where i = 1 , , and j = 2 , . Then the Chebyshev type inequality ψ (cid:0) I S (cid:0) m, ϕ ( f ⋆ g ) (cid:1)(cid:1) > ψ (cid:0) I S ( m, ϕ ( f )) (cid:1) ⋆ ψ (cid:0) I S ( m, ϕ ( g )) (cid:1) (18)10 s fulfilled for any comonotone functions f, g ∈ F X, A ) on X and any m ∈ M X, A ) if and onlyif ψ (cid:0) S( ϕ ( a ⋆ b ) , c ) (cid:1) > (cid:2) ψ (cid:0) S( ϕ ( a ) , c ) (cid:1) ⋆ b (cid:3) ∨ (cid:2) a ⋆ ψ (cid:0) S( ϕ ( b ) , c ) (cid:1)(cid:3) (19) holds for all a, b, c ∈ [0 , . Proof.
Use Lemma 3.2 with k = ¯ y = 1 , ♦ = ⋆, D = [0 , and ◦ i = S , and Theorem 3.2.Corollary 4.3 with ψ i = ϕ − i is an improvement of [1, Theorem 4.1 with A = X ], where theintegral inequality (18) has been proved whenever the condition (19) holds. Setting ψ i ( x ) = ϕ i ( x ) = x in Corollary 4.3, we get the following result. Corollary 4.4.
Let S ∈ S , ⋆ : [0 , → [0 , be non-decreasing and left-continuous such that ⋆ ⋆ . The Chebyshev type inequality I S (cid:0) m, f ⋆ g (cid:1) > I S ( m, f ) ⋆ I S ( m, g ) (20) is satisfied for any comonotone functions f, g ∈ F X, A ) on X and any m ∈ M X, A ) if and onlyif S( a ⋆ b, c ) > (S( a, c ) ⋆ b ) ∨ ( a ⋆ S( b, c )) (21) is valid for all a, b, c ∈ [0 , . Corollary 4.4 improves [15, Theorem 3.1 with A = X ]. In [20, Corollary 3.6] authors claim(without any proof) that the inequality (20) is true for any comonotone functions f, g ∈ F X, A ) on X and any capacity m if and only if S( c, a ⋆ b ) > (S( c, a ) ⋆ b ) ∨ ( a ⋆ S( c, b )) (22)for any a, b, c ∈ [0 , . Clearly, condition (21) coincides with (22) only if S ∈ S is commutative.Furthermore, in [6] we can find an incorrect statement about Chebyshev type inequalitywith comonotone functions, since the sufficient condition used therein does not have the form(19). Below, we present a counterexample to [6, Theorem 3.5]. Here we use the same notationof functions as in the original paper [6]. Counterexample 4.1.
Put
T = ⋆ = W , f = 0 . A , g = 0 . A , µ ( A ) = 0 . and s = 2 in [6,Theorem 3.5]. Clearly, functions f and g are comonotone. Then the sufficient condition (3.4)from [6] in the form (cid:0) ( a + b − + + c − (cid:1) + > (cid:0) ( a + c − + + b − (cid:1) + ∨ (cid:0) a + ( b + c − + − (cid:1) + is satisfied for all a, b, c ∈ [0 , . After a simple calculation we get q I W ,A ( µ, W ( f, g )) = p W(0 . , .
9) = 0 , W (cid:16)q I W ,A ( µ, f ) , q I W ,A ( µ, g ) (cid:17) = (cid:16)p W(0 . , .
9) + p W(0 . , . − (cid:17) + ≈ . , which contradicts the Chebyshev type inequality (3.5) from Theorem 3.5 in [6].11 .3 Sugeno integral Chebyshev type inequalities for Sugeno integral I M in case of functions from F X, A ) can beobtained from the corresponding results for I S (with the Sugeno integral regarding as a specialcase of seminormed integral), see e.g. Corollary 4.3. On the other hand, Chebyshev typeinequalities for Sugeno integral in case of functions from F ¯ y ( X, A ) , ¯ y > , can be obtained usingTheorem 3.1 or Theorem 3.2 in a similar manner as in the previous subsection. However,this may lead to some functional inequalities that need not be easy to verify. In the case ofSugeno integral and comonotone functions, the functional inequality may be replaced by easiercondition ⋆ ∧ . We start with the helpful lemma.
Lemma 4.1.
Let D ⊂ [0 , ∞ ] and ⋆ be a non-decreasing fusion function. Assume that ϕ i : [0 , ¯ y ] → [0 , ¯ y ] and ψ i : [0 , ϕ i (¯ y )] → [0 , ¯ y ] are non-decreasing for i = 1 , , such that ϕ (¯ y ) = ϕ j (¯ y ) ,ψ > ψ j and ψ j ( ϕ j ( x )) x ψ ( ϕ ( x )) for any x, where j = 2 , . If ⋆ ∧ , i.e., a ⋆ b a ∧ b for any a, b ∈ [0 , ¯ y ] , then ψ ( ϕ ( a ⋆ b ) ∧ c ∧ d ) > ψ ( ϕ ( a ) ∧ c ) ⋆ ψ ( ϕ ( b ) ∧ d ) (23) for all a, b ∈ [0 , ¯ y ] and c, d ∈ D. Proof.
In a similar manner as in the proof of Theorem 3.1 we can show that ψ ’s are well-definedin (23). To shorten the notation, we put P ( a, b, c, d ) := ψ ( ϕ ( a ) ∧ c ) ⋆ ψ ( ϕ ( b ) ∧ d ) . From the obvious inequalities and assumptions on ψ i and ϕ i , we have P ( a, b, c, d ) ψ ( ϕ ( a )) ⋆ ψ ( ϕ ( b )) a ⋆ b ψ ( ϕ ( a ⋆ b )) (24)for all a, b ∈ [0 , ¯ y ] and c, d ∈ D. Let c, d ∈ D. Consider four cases:(i) Firstly, suppose c ϕ (¯ y ) and d ϕ (¯ y ) . Applying ψ j ψ and ⋆ ∧ , we get P ( a, b, c, d ) ψ ( c ) ⋆ ψ ( d ) ψ ( c ∧ d ) for any a, b ∈ [0 , ¯ y ] . By (24) and monotonicity of ψ we obtain (23).(ii) Let c > ϕ (¯ y ) and d > ϕ (¯ y ) . From (24) we conclude that P ( a, b, c, d ) ψ ( ϕ ( a ⋆ b )) = ψ ( ϕ ( a ⋆ b ) ∧ ϕ (¯ y )) for any a, b ∈ [0 , ¯ y ] . By ϕ (¯ y ) = ϕ j (¯ y ) for j = 2 , ,P ( a, b, c, d ) ψ ( ϕ ( a ⋆ b ) ∧ ϕ (¯ y ) ∧ ϕ (¯ y )) ψ ( ϕ ( a ⋆ b ) ∧ c ∧ d ) for all a, b ∈ [0 , ¯ y ] . c > ϕ (¯ y ) and d ϕ (¯ y ) , then P ( a, b, c, d ) ψ ( ϕ ( a )) ∧ ψ ( d ) a ∧ ψ ( d ) ψ ( ϕ ( a ) ∧ d ) (25)for any a, b ∈ [0 , ¯ y ] . Combining (24) and (25), we get P ( a, b, c, d ) ψ ( ϕ ( a ⋆ b ) ∧ ϕ ( a ) ∧ d ) ψ ( ϕ ( a ⋆ b ) ∧ ϕ (¯ y ) ∧ d ) ψ ( ϕ ( a ⋆ b ) ∧ c ∧ d ) for all a, b ∈ [0 , ¯ y ] . (iv) The case c ϕ (¯ y ) and d > ϕ (¯ y ) is similar to (iii).By (i)–(iv) we get (23) for any a, b ∈ [0 , ¯ y ] and c, d ∈ D. Theorem 4.1.
Suppose that ¯ y ∈ (0 , ∞ ] , m ∈ M ( X, A ) and ⋆ is a non-decreasing and left-continuous fusion function such that ⋆ ∧ . Let ϕ i : [0 , ¯ y ] → [0 , ¯ y ] be functions such that ϕ (¯ y ) = ϕ j (¯ y ) , ϕ is non-decreasing and ϕ j are increasing and right-continuous, ψ i : [0 , ϕ i (¯ y )] → [0 , ¯ y ] be non-decreasing and ψ j be left-continuous such that ψ > ψ j and ψ j ( ϕ j ( x )) x ψ ( ϕ ( x )) for any x, where i = 1 , , and j = 2 , . Then the Chebyshev type inequality ψ (cid:0) I M ,A ( m, ϕ ( f ⋆ g )) (cid:1) > ψ (cid:0) I M ,A ( m, ϕ ( f )) (cid:1) ⋆ ψ (cid:0) I M ,A ( m, ϕ ( g )) (cid:1) is valid for all comonotone functions f, g ∈ F ¯ y ( X, A ) on A ∈ A . Proof.
Put k = ¯ y, ♦ = ⋆, B = A, △ = ∧ , and ◦ i = ∧ for i = 1 , , in Theorem 3.1 (a). FromLemma 4.1 with D = m ( A ) the inequality (23) is valid for any a, b ∈ [0 , ¯ y ] and c, d ∈ m ( A ) . From Theorem 3.1 (a) we get the statement.As the special case with ψ i = ϕ − i and ϕ i = ϕ we obtain the following result. Corollary 4.5.
Let ¯ y ∈ (0 , ∞ ] , m ∈ M ( X, A ) and ⋆ be a non-decreasing and left-continuousfusion function such that ⋆ ∧ . Assume that ϕ : [0 , ¯ y ] → [0 , ¯ y ] is continuous and increasingfunction such that ϕ (0) = 0 , where i = 1 , , and j = 2 , . Then the Chebyshev type inequality ϕ − (cid:0) I M ,A ( m, ϕ ( f ⋆ g )) (cid:1) > ϕ − (cid:0) I M ,A ( m, ϕ ( f )) (cid:1) ⋆ ϕ − (cid:0) I M ,A ( m, ϕ ( g )) (cid:1) (26) is valid for all comonotone functions f, g ∈ F ¯ y ( X, A ) on A ∈ A . The following example shows that the assumption ϕ (0) = 0 cannot be abandoned fromCorollary 4.5. Example 4.2.
Let ϕ ( x ) = 0 . x + 1) [0 , ( x ) , f = g = 0 . X and consider a capacity m suchthat m ( A ) = 0 . , where A ∈ A . Clearly, all assumptions of Corollary 4.5 with ¯ y = 1 , ϕ i = ϕ and ⋆ = · are satisfied except ϕ (0) = 0 . Then ϕ − (cid:0) I M ,A ( m, ϕ ( f g )) (cid:1) > ϕ − (cid:0) I M ,A ( m, ϕ ( f )) (cid:1) · ϕ − (cid:0) I M ,A ( m, ϕ ( g )) (cid:1) ,ϕ − (0 . ∧ . > ϕ − (0 . ∧ . · ϕ − (0 . ∧ . . However, the value ϕ − (0 . is not defined, so the Chebyshev type inequality cannot hold.13 emark 4.1. In [2, Theorem 3.1] the Chebyshev type inequality of the form (26) for a fixed A ∈ A has been obtained without the assumption on ϕ (0) = 0 . As a consequence of the Chebyshev type inequality, we get the following Liapunov typeinequality.
Corollary 4.6.
Let ¯ y ∈ (0 , ∞ ] , m ∈ M ( X, A ) and ϕ i : [0 , ¯ y ] → [0 , ¯ y ] be increasing and right-continuous, ψ i : [0 , ϕ i (¯ y )] → [0 , ¯ y ] , i = 1 , , such that ϕ (¯ y ) = ϕ (¯ y ) , ψ > ψ and ψ ( ϕ ( x )) x ψ ( ϕ ( x )) for any x. Then the Liapunov type inequality ψ (cid:0) I M ,A ( m, ϕ ( f )) (cid:1) > ψ (cid:0) I M ,A ( m, ϕ ( f )) (cid:1) holds for each f ∈ F ¯ y ( X, A ) and A ∈ A . Proof.
Put ⋆ = ∧ , g = f, ϕ = ϕ , and ψ = ψ in Theorem 4.1. In this section, we show that, due to the definition of m -positively dependent functions, one canderive the Chebyshev type inequality for any functions strengthening assumptions on monotonemeasure. Theorem 5.1.
Assume that ¯ y ∈ (0 , ∞ ] , k ∈ (0 , ¯ y ] and m ∈ M ( X, A ) such that m ( C ∩ D ) > m ( C ) △ m ( D ) for all C, D ∈ A with △ : m ( A ) × m ( A ) → m ( A ) . Let ϕ i : [0 , ¯ y ] → [0 , ¯ y ] be in-creasing and right-continuous, and ψ i : [0 , ϕ i (¯ y )] → [0 , ¯ y ] be non-decreasing and left-continuousfor i = 1 , , . Suppose that ⋆ is a non-decreasing and left-continuous fusion function, ◦ i : [0 , ¯ y ] × m ( A ) → [0 , ¯ y ] are non-decreasing such that ϕ i (¯ y ) ◦ i m ( X ) ϕ i (¯ y ) and ¯ y ◦ j for i = 1 , , and j = 2 , . If (5) with ♦ = ⋆ holds for any a, b ∈ [0 , k ] and c, d ∈ m ( A ) , then the Chebyshevtype inequality (6) is satisfied for all functions f | A , g | B ∈ F k ( X, A ) .Proof. Observe that all functions f | A and g | B are m -positively dependent with respect to △ , since m ( C ∩ D ) > m ( C ) △ m ( D ) for all C, D ∈ A . Applying Theorem 3.1 (a) with ♦ = ⋆ weget the statement.Using Theorem 5.1, we can obtain many versions of Chebyshev type inequality for anyfunctions. Below, we present one proposition. Corollary 5.1.
Let m ∈ M ( X, A ) be minitive and ⋆ be a non-decreasing left-continuous fu-sion function such that ⋆ ∧ . Let ϕ i : [0 , ¯ y ] → [0 , ¯ y ] be increasing and right-continuous and ψ i : [0 , ϕ i (¯ y )] → [0 , ¯ y ] be non-decreasing and left-continuous, where i = 1 , , . Assume that ϕ (¯ y ) = ϕ j (¯ y ) , ψ > ψ j and ψ j ( ϕ j ( x )) x ψ ( ϕ ( x )) for all x and j = 2 , . Then theChebyshev type inequality ψ (cid:0) I M ,A ∩ B ( m, ϕ ( f ⋆ g )) (cid:1) > ψ (cid:0) I M ,A ( m, ϕ ( f )) (cid:1) ⋆ ψ (cid:0) I M ,B ( m, ϕ ( g )) (cid:1) is fulfilled for all functions f | A , g | B ∈ F ¯ y ( X, A ) . roof. Put ◦ i = △ = ∧ and k = ¯ y in Theorem 5.1. By assumptions we obtain the inequality(5) for all a, b ∈ [0 , ¯ y ] and c, d ∈ m ( A ) from Lemma 4.1 with D = m ( A ) and ◦ i = ∧ . ThenTheorem 5.1 provides the statement.The following examples illustrate the above results.
Example 5.1.
Let X = [0 , and consider on ( X, A ) the minitive capacity m ( A ) = 1 − sup x ∈ A c x. Assume that ⋆ = · , ψ i ( x ) = √ x , ϕ j ( x ) = x , ϕ ( x ) = x, f ( x ) = [0 , . ( x ) +0 . (0 . , ( x ) and g ( x ) = x , where i = 1 , , and j = 2 , . Fix A = B = X . Then allassumptions in Corollary 5.1 are valid with ¯ y = 1 . By Corollary 5.1 the Chebyshev typeinequality has the form I M ( m, x · [0 , . ( x ) + 0 . x · (0 . , ( x )) > I M ( m, [0 , . ( x ) + 0 . · (0 . , ( x )) · I M ( m, x ) . By simple calculation, we get I M ( m, x [0 , . ( x ) + 0 . x · (0 . , ( x )) = sup t ∈ [0 , . { t ∧ m ([ t, } ∨ sup t ∈ (0 . , . { t ∧ m ([ t, . ∪ [2 t, } = sup t ∈ [0 , . { t ∧ (1 − t ) } ∨ sup t ∈ (0 . , . { t ∧ (1 − t ) } = 1 / . In a similar manner we obtain I M ( m, [0 , . ( x ) + 0 . · (0 . , ( x )) = sup t ∈ [0 , . { t ∧ m ( X ) } ∨ sup t ∈ (0 . , { t ∧ } = 0 . , I M ( m, x ) = sup t ∈ [0 , { t ∧ (1 − √ t ) } = (3 − √ / . Let A , B : [0 , → [0 , be two fusion functions. We say that B is dominated by A , if theinequality A(B( a, b ) , B( c, d )) > B(A( a, c ) , A( b, d )) holds for all a, b, c, d ∈ [0 , . Example 5.2.
Let m is the Lebesgue measure on X = [0 , . Consider ⋆ = △ = W , k = ¯ y = 1 , ◦ i = ∧ , ϕ i ( x ) = x, ψ ( x ) = √ x and ψ j ( x ) = x for i = 1 , , and j = 2 , in Theorem 5.1. Notethat m ( C ∩ D ) > W( m ( C ) , m ( D )) for any C, D ∈ A since m is supermodular, see Example 2.3.The inequality (5) with ♦ = ⋆ holds by the reason of W being dominated by M . By Theorem 5.1with A = B = X the Chebyshev type inequality p I M ( m, W( f, g )) > W (cid:0)(cid:0) I M ( m, f ) (cid:1) , (cid:0) I M ( m, g ) (cid:1) (cid:1) (27)is true for any f, g ∈ F X, A ) . For f ( x ) = 2 x − x + 1 and g ( x ) = − x + 2 x we get I M ( m, W( f, g )) = 0 and I M ( m, f ) = 0 . ∨ sup t ∈ (0 . , { t ∧ (1 − p . t − . } = 2 − √ , I M ( m, g ) = sup t ∈ (0 , . { t ∧ p . . − t ) } = √ − , so the inequality in (27) becomes equality.Example 5.2 illustrates that the equality in the Chebyshev type inequality may be achieved bynon-constant functions f and g (cf. Examples 3.1–3.2).15 onclusion In the present paper, we have focused on Chebyshev type inequalities for generalized (upper)Sugeno integral for m -positively dependent functions, and for any comonotone functions andmonotone measure. We have presented a technique how to obtain some known results fromthe literature. Moreover, we have provided a few unknown results in the literature so far. Forinstance, when restricting the considered fusion functions to fuzzy conjunctions, we can stategeneral Chebyshev type inequalities for q-integral recently introduced and studied by Duboiset al. in [8]. In the point of view that the considered integrals are aggregation functions, weexpect applications of our results everywhere where some bounds on the aggregation process isneeded, such as information aggregation, or decision making. Acknowledgement
This work was supported by the Slovak Research and Development Agency under the con-tract No. APVV-16-0337. The work is also cofinanced by bilateral call Slovak-Poland grantscheme No. SK-PL-18-0032 together with the Polish National Agency for Academic ExchangePPN/BIL/2018/1/00049/U/00001.