Hölder and Minkowski type inequalities for pseudo-integral

Applied Mathematics and Computation 217 (2011) 8630–8639

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Hölder and Minkowski type inequalities for pseudo-integral Hamzeh Agahi a, Yao Ouyang b, Radko Mesiar c,d, Endre Pap f,e,⇑, Mirjana Štrboja f a Department of Statistics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424, Hafez Ave., Tehran 15914, Iran b Faculty of Science, Huzhou Teacher’s College, Huzhou, Zhejiang 313000, People’s Republic of China c Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, SK81368 Bratislava, Slovakia d Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Czech Republic e Óbuda University, Becsi út 96/B, H-1034 Budapest, Hungary f Department of Mathematics and Informatics, Faculty of Sciences and Mathematics, University of Novi Sad, Serbia

a r t i c l e

i n f o

Keywords: Hölder’s inequality Minkowski’s inequality Semiring Pseudo-addition Pseudo-multiplication Pseudo-integral

a b s t r a c t There are proven generalizations of the Hölder’s and Minkowski’s inequalities for the pseudo-integral. There are considered two cases of the real semiring with pseudo-operations: one, when pseudo-operations are defined by monotone and continuous function g, the second semiring ([a, b], sup, ), where  is generated and the third semiring where both pseudo-operations are idempotent, i.e.,  = sup and  = inf. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Pseudo-analysis is a generalization of the classical analysis, where instead of the field of real numbers a semiring is defined on a real interval [a, b]  [1, 1] with pseudo-addition  and with pseudo-multiplication , see [19–24,32]. Based on this structure there were developed the concepts of -measure (pseudo-additive measure), pseudo-integral, pseudo-convolution, pseudo-Laplace transform, etc. The advantages of the pseudo-analysis are that there are covered with one theory, and so with unified methods, problems (usually nonlinear and under uncertainty) from many different fields (system theory, optimization, decision making, control theory, differential equations, difference equations, etc.). Pseudo-analysis uses many mathematical tools from different fields as functional equations, variational calculus, measure theory, functional analysis, optimization theory, semiring theory, etc. Similar ideas were developed independently by Maslov and his collaborators in the framework of idempotent analysis and idempotent mathematics, with important applications [8,9,11]. In particular, idempotent analysis is fundamental for the theory of weak solutions to Hamilton–Jacobi equations with non-smooth Hamiltonians, see [8,9,11] and also [22,23,25] (in the framework of pseudo-analysis). In some cases, this theory enables one to obtain exact solutions in the similar form as for the linear equations. Some further developments relate more general pseudo-operations with applications to nonlinear partial differential equations, see [27]. Recently, these applications have become important in the field of image processing [23,25]. On the other side, more general set functions than pseudo-additive measures, as fuzzy measures and corresponding fuzzy integrals had been investigated in [6,14,21,28,31,15], as aggregation functions with important applications, e.g., given in [30,33]. Recently, there were obtained generalizations of the classical integral inequalities for integrals with respect to non-additive measures [1–4,12,16,17,26,33]. The well-known Hölder’s and Minkowski’s inequality is a part of the classical mathematical analysis, see [29]. ⇑ Corresponding author at: Department of Mathematics and Informatics, Faculty of Sciences and Mathematics, University of Novi Sad, Serbia. E-mail addresses: [email protected] (H. Agahi), [email protected] (Y. Ouyang), [email protected] (R. Mesiar), [email protected] (E. Pap), mirjana.strboja@ dmi.uns.ac.rs (M. Štrboja). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.100

H. Agahi et al. / Applied Mathematics and Computation 217 (2011) 8630–8639

8631

Definition 1. If p and g are positive real number such that 1p þ 1q ¼ 1, then we call p and q a pair of conjugate exponents.

Theorem 1. Let p and q be conjugate exponents, 1 < p < 1. Let X be a measure space, with measure l. Let f and g be measurable functions on X, with range in [0, 1]. Then (i) (Hölder’s inequality)

Z

fgdl 6

Z

X

1p Z 1q f p dl g q dl X

ð1Þ

X

(ii) (Minkowski’s inequality)

Z

1p Z 1p Z 1p ðf þ gÞp dl 6 f p dl þ g p dl

X

X

ð2Þ

X

In this paper we shall give generalizations of Hölder’s and Minkowski’s inequality for pseudo-integral. Our paper is organized as follows. In Section 2 are given some preliminaries on the pseudo-analysis. In Section 3 special kinds of pseudo-integrals are introduced and the generated case is related to the sup-plus case. We prove generalizations of the Hölder, Minkowski for pseudo-integral in Sections 4 and 5. We note that the third important case  = max and  = min has been studied in [1,2], where the pseudo-integral in such a case yields the Sugeno integral. 2. Pseudo-integral Let [a, b] be a closed (in some cases can be considered semiclosed) subinterval of [1, 1]. The full order on [a, b] will be denoted by . A binary operation  on [a, b] is pseudo-addition if it is commutative, non-decreasing (with respect to ), associative and with a zero (neutral) element denoted by 0. Let [a, b]+ = {x j x 2 [a, b], 0  x}. A binary operation  on [a, b] is pseudo-multiplication if it is commutative, positively non-decreasing, i.e., x  y implies x  z  y  z for all z 2 [a, b]+, associative and with a unit element 1 2 [a, b], i.e., for each x 2 [a, b], 1  x = x. We assume also 0  x = 0 and that  is distributive over , i.e.,

x  ðy  zÞ ¼ ðx  yÞ  ðx  zÞ The structure ([a, b], , ) is a semiring (see [10,21]). In this paper we will consider only semirings with the following continuous operations (continuity of  can be possibly violated in the cases 0  a = a  0 or 0  b = b  0, i.e., in points (0, a) and (a, 0), or (0, b) and (b, 0)), and where the boundary elements of the interval [a, b] are the neutral elements of the pseudooperations: Case I: The pseudo-addition is idempotent operation and the pseudo-multiplication is not. (a) x  y = sup(x, y),  is arbitrary not idempotent pseudo-multiplication on the interval [a, b] cancellative on ]a, b[2. We have 0 = a and the idempotent operation supinduces a full order in the following way: x  y if and only if sup(x, y) = y. Moreover, the pseudo-multiplication  is generated by an increasing bijection g : [a, b] ? [0, 1], x  y = g1(g(x)  g(y)) (this result follows from [7], see also [5]). Observe that 1 = g1(1). (b) x  y = inf(x, y),  is arbitrary not idempotent pseudo-multiplication on the interval [a, b] cancellative on ]a, b[2. We have 0 = b and the idempotent operation infinduces a full order in the following way: x  y if and only if inf(x, y) = y. Moreover, the pseudo-multiplication  is generated by a decreasing bijection g : [a, b] ? [0, 1], x  y = g1(g(x)  g(y)) and 1 = g1(1). Case II: The pseudo-operations are defined by a monotone and continuous function g : [a, b] ? [0, 1], i.e., pseudo-operations are given with

x  y ¼ g 1 ðgðxÞ þ gðyÞÞ and x  y ¼ g 1 ðgðxÞ  gðyÞÞ: If the zero element for the pseudo-addition is a, we will consider increasing generators. Then g(a) = 0 and g(b) = 1. If the zero element for the pseudo-addition is b, we will consider decreasing generators. Then g(b) = 0 and g(a) = 1. If the generator g is increasing (respectively decreasing), the operation induces the usual order (respectively opposite to the usual order) on the interval [a, b] in the following way: x  y if and only if g(x) 6 g(y). Case III: Both operations are idempotent. We have (a) x  y = sup(x, y), x  y = inf(x, y), on the interval [a, b]. We have 0 = a and 1 = b. The idempotent operation supinduces the usual order (x  y if and only if sup(x, y) = y). (b) x  y = inf(x, y), x  y = sup(x, y), on the interval [a, b]. We have 0 = b and 1 = a. The idempotent operation infinduces an order opposite to the usual order (x  y if and only if inf(x, y) = y). ðpÞ For x 2 [a, b]+ and p 2 ]0, 1[, we will introduce the pseudo-power x as follows: if p = n is a natural number then ðnÞ

x ¼ |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} x  x    x: n

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H. Agahi et al. / Applied Mathematics and Computation 217 (2011) 8630–8639 ð1Þ

ðnÞ

ðmÞ

ðrÞ

Moreover, xn ¼ supfyjy 6 xg. Then xn ¼ x is well defined for any rational r 2 ]0, 1[, independently of representation r ¼ mn ¼ mn11 ; m; n; m1 ; n1 being positive integers (the result follows from the continuity and monotonicity of ). Due to continuity of , if p is not rational, then

n o ðpÞ ðrÞ x ¼ sup x jr 20; p½; r is rational : ðpÞ

ðpÞ

Evidently, if x  y = g1(g(x)  g(y)), then x ¼ g 1 ðg p ðxÞÞ. On the other hand, if  is idempotent, then x ¼ x for any x 2 [a, b] and p 2 ]0, 1[. Let X be a non-empty set. Let A be a r-algebra of subsets of X. Definition 2. A set function m : A ! ½a; b is a r--measure if there hold (i) m(£) = 0 (if  is not idempotent) S 1 (ii) mð 1 i¼1 Ai Þ ¼ ai¼1 mðAi Þ holds for any sequence ðAi Þi2N of pairwise disjoint sets from A. Observe that is the case II, a set function m : A ! ½a; b is a r--measure if and only if g m : A ! ½0; 1 is a classical measure, i.e., a r-additive measure. We suppose that ([a, b], ) and ([a, b], ) are complete lattice ordered semigroups. We suppose that [a, b] is endowed with a metric d compatible with sup and inf, i.e. lim sup xn = x and lim inf xn = x, imply limn?1d(xn, x) = 0, and which satisfies at least one of the following conditions: (a) d(x  y, x0  y0 ) 6 d(x, x0 ) + d(y, y0 ), (b) d(x  y, x0  y0 ) 6 max {d(x, x0 ), d(y, y0 )}. Both conditions (a) and (b) imply: d(xn, yn) ? 0 ) d(xn  z, yn  z) ? 0. Metric d is also monotonic, i.e., x 6 z 6 y ) d(x, y) P sup{d(y, z), d(x, z)}. Let f and h be two functions defined on X and with values in [a, b]. Then for any x 2 X and for any k 2 [a, b] we define (f  g)(x) = f(x)  g(x), (f  g)(x) = f(x)  g(x), and (k  f)(x) = k  f(x). The characteristic function with values in a semiring ([a, b], , ) is defined by

vA ðxÞ ¼



0; x R A; 1; x 2 A: n

A step (measurable) function is a mapping e : X ? [a, b] that has the following representation e ¼ ai¼1 ai  vAi for ai 2 [a, b] and sets Ai 2 A are pairwise disjoint if  is nonidempotent. e e Let e be a positive real number and B  [a, b]. A subset fli gn2N of set B is a e-net on B if for each x 2 B there exists li such e e e e e e that dðli ; xÞ 6 e. If we, also, have li  x, then we call fli g a lower e-net. If li  liþ1 holds, then fli g is monotone, for more details see [13,18,21]. Definition 3. Let m : A ! ½a; b be a r--measure. (i) The pseudo-integral of a step function e : X ? [a, b] m is defined by

Z



n

e  dm ¼ a ai  mðAi Þ:

X

i¼1

(ii) The pseudo-integral of a measurable function f : X ? [a, b], (if  is not idempotent we suppose that for each e > 0 there exists a monotone e-net in f(X)) is defined by

Z X



f  dm ¼ lim

n!1

Z



en ðxÞ  dm;

X

where ðen Þn2N is a sequence of step functions such that d(en(x), f(x)) ? 0 uniformly as n ? 1. For more details see [9,21,23]. 3. Explicit forms of special pseudo-integrals We shall consider the semiring ([a, b], , ) for three (with completely different behavior) cases, namely I(a), II and III(a). Observe that the cases I(b) and III(b) are linked to the cases I(a) and III(a) by duality. First case is when pseudo-operations are generated by a monotone and continuous function g : [a, b] ? [0, 1], case II from Section 2. Then the pseudo-integral for a measurable function f : X ? [a, b] is given by,

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H. Agahi et al. / Applied Mathematics and Computation 217 (2011) 8630–8639

Z



f  dm ¼ g 1

Z

X

 ðg f Þdðg mÞ ;

ð3Þ

X

where the integral applied on the right side is the standard Lebesgue integral. In special case, when X ¼ ½c; d; A ¼ BðX Þ and m = g1 k, k the standard Lebesgue measure on [c, d], then we use notation

Z



f ðxÞdx ¼

Z



f  dm: X

½c;d

By (3),

Z



f ðxÞdx ¼ g 1

Z

!

d

gðf ðxÞÞdx ;

c

½c;d

i.e., we have recovered the g-integral, see [18,19]. Second case is when the semiring is of the form ([a, b], sup, ), cases I(a) and III(a) from Section 2. We will consider complete sup-measure m only and A ¼ 2X , i.e., for any system (Ai)i2I of measurable sets,

!

[

m

¼ sup mðAi Þ:

Ai

i2I

i2I

Recall that if X is countable (especially, if X is finite) then any r-sup-measure m is complete and, moreover, m(A) = supx2Aw(x), where w : X ? [a, b] is a density function given by w(x) = m({x}). Then the pseudo-integral for a function f : X ? [a, b] is given by

Z



f  dm ¼ supðf ðxÞ  wðxÞÞ; x2X

X

where function w defines sup-measure m. 4. Hölder’s inequality for pseudo-integral Now we shall give a generalization of the classical Hölder inequality (1). Theorem 2. Let p and q be conjugate exponents, 1 < p < 1. For a given measurable space ðX; AÞ let u, v : X ? [a, b] be two measurable functions and let a generator g : [a, b] ? [0, 1] of the pseudo-addition  and the pseudo-multiplication  be an increasing function. Then for any r--measure m it holds:

Z



ðu  v Þ  dm 6

Z

X



ð1pÞ Z ðpÞ u  dm 

X



ð1qÞ

v ðqÞ   dm

ð4Þ

:

X





Proof. We apply the classical Hölder’s inequality (1) and then we obtain

Z

ðg uÞðg v Þdðg mÞ 6

Z

X

1p Z 1q ðg uÞp dðg mÞ  ðg v Þq dðg mÞ :

X

X

Since function g is increasing function, then g1 is also increasing function and we have

g

1

Z

 Z 1p Z 1q ! p q 1 : ðg uÞðg v Þdðg mÞ 6 g ðg uÞ dðg mÞ  ðg v Þ dðg mÞ

X

X

X

i.e.,

g

1

Z







g g ððg uÞðg v ÞÞ ðg mÞ 6 g 1

1

g g

1

Z

X

1p !!

p

ðg uÞ dðg mÞ

g g

X

X



ðu  v Þ  dm 6 g

1

Z X

p

ðg uÞ dðg mÞ

1p ! g

1

Z

q

ðg v Þ dðg mÞ X

Hence

Z

1

Z

q

ðg v Þ dðg mÞ X

1q ! :

1q !!! :

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H. Agahi et al. / Applied Mathematics and Computation 217 (2011) 8630–8639

For the right side of the inequality holds:

g

Z

1

1p !

p

ðg uÞ dðg mÞ

g

Z

1

X

¼g

1

1q ! ðg v Þ dðg mÞ q

X

Z



p

1

g g ððg uÞ Þðg mÞ



1p !

g

Z

1

X

¼ g 1

Z

q

¼g

1q !

1p ! Z  1q !  ðpÞ ðqÞ g u dðg mÞ g v  dðg mÞ  g 1 X

  Z  1p !   Z  1q ! ðpÞ ðqÞ 1 1 1 g g g u dðg mÞ g g g v  dðg mÞ g X

1



g g ððg v Þ Þ dðg mÞ 1

X

X

¼ g 1



 Z g



X

1p !  Z ðpÞ  g 1 u  dm g

X



1q ! Z ðqÞ ¼ v   dm

X

which completes the proof.

X



ð1pÞ

ðpÞ u ðxÞdm

 

Z



v

ð1qÞ

ðqÞ  ðxÞdm

X

; 

h

Example 1 (i) Let [a, b] = [0, 1] and g(x) = xa for some a 2 [1, 1[. The corresponding pseudo-operations are x  y ¼ x  y = xy. Then (4) reduces on the following inequality

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaffi a a x þ y and

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z Z Z a pa qa a a pa qa uðxÞ v ðxÞ dx 6 uðxÞ dx v ðxÞ dx: ½c;d

½c;d

½c;d x

(ii) Let [a, b] = [1, 1] and g(x) = e . The corresponding pseudo-operations are x  y = ln(ex + ey) and x  y = x + y. Then (4) reduces on the following inequality

ln

Z

euðxÞþv ðxÞ dx 6

½c;d

!

Z

1 ln p

½c;d

!

Z

1 ln q

epuðxÞ dx þ

eqv ðxÞ dx ;

½c;d

i.e.,

Z

e

uðxÞþv ðxÞ

dx 6

!1p

Z

½c;d

puðxÞ

e

dx



!1q

Z

½c;d

qv ðxÞ

e

dx

:

½c;d

Theorem 3. Let p and q be conjugate exponents, 1 < p < 1. For a given measurable space ðX; AÞ let u, v : X ? [a, b] be two measurable functions and let a generator g : [a, b] ? [0, 1] of the pseudo-addition  and the pseudo-multiplication  is a decreasing function. Then for any r--measure m it holds:

Z



ðu  v Þ  dm P

X

Z



ð1pÞ Z ðpÞ u  dm 

X



ð1qÞ

v ðqÞ   dm

X



: 

Proof. In an analogous way as in the proof of Theorem (2) we obtain

g

1

Z



ðg uÞðg v Þdðg mÞ P g

1

X

Z

1p Z 1q ! q : ðg uÞ dðg mÞ  ðg v Þ dðg mÞ p

X

X

i.e.,

Z X



ðu  v Þ  dm P

Z X



ð1pÞ Z ðpÞ u  dm  

X



ð1qÞ

v ðqÞ   dm

:  

Now we consider the second case, when  = sup, and  = g1(g(x)g(y)). Theorem 4. Let  be represented by an increasing generator g and m be a complete sup-measure. Let p and q be conjugate exponents, 1 < p < 1. Then for any functions u, v : X ? [a, b], it holds:

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H. Agahi et al. / Applied Mathematics and Computation 217 (2011) 8630–8639

Z

Z

sup

ðu  v Þ  dm 6

X

sup

ð1pÞ Z ðpÞ u  dm 

X

sup

v ðqÞ   dm

X



ð1qÞ : 

Proof. Recall that

Z

sup

  ðu  v Þ  dm ¼ sup ðuðxÞ  v ðxÞ  wðxÞÞ ¼ g 1 sup ðgðuðxÞÞgðv ðxÞÞgðwðxÞÞÞ ; x2X

X

x2X

where w : X ? [a, b] is a density function related to m. Moreover,

Z

sup

X

0 !1p 1 ! ð1pÞ  1 p 1 @ p 1 A p ¼g u  dm ¼g sup ðg ðuðyÞÞgðwðyÞÞÞ sup gðuðyÞÞg ðwðyÞÞ : y2X



y2X

Similarly,

Z

ð1qÞ

sup

v q  dm

X

   1 ¼ g 1 sup gðv ðzÞÞg q ðwðzÞÞ : z2X



Consequently,

Z

sup

X

1p Z up  dm 

v q  dm

X



1q

sup

!   1 1 ¼ g 1 sup gðuðyÞÞg p ðwðyÞÞ  sup gðv ðzÞÞg q ðwðzÞÞ y2X



P g 1



z2X

  1 1 sup gðuðxÞÞg p ðwðxÞÞgðv ðxÞÞg q ðwðxÞÞ x2X

  Z ¼g sup ðgðuðxÞÞgðv ðxÞÞgðwðxÞÞÞ ¼ 1

x2X

Completing the proof.

sup

ðu  v Þ  dm:

X

h ðpÞ

Remark 1. In the case III(a), i.e., if  = sup and  = inf, for each x 2 [a, b] and p > 0; x ¼ x: The Hölder inequality in this case reduces to the inequality

Z

sup

ðu  v Þ  dm 6

Z

X

sup

 Z u  dm 

X



sup

v  dm

;

X

i.e.,

! sup ðinfðuðxÞ; v ðxÞ; wðxÞÞÞ 6 inf

sup ðinfðuðyÞ; wðyÞÞÞ; sup ðinfðv ðzÞ; wðzÞÞÞ ;

x2X

y2X

z2X

which trivially holds because of distributivity of sup and inf. For a general case I(a) and III(a), i.e., when m is an arbitrary r-sup-measure, suppose that u and ðX; AÞ with values in [a, b]. Then there is a finite partition {E1, . . . , En} of X so that

u ¼ supi2f1;2;...;ng ui  vEi ;

v are step function on

v ¼ supi2f1;2;...;ng v i  vE : i

Define a new measurable space (Y, 2 ) with Y = {y1, . . . , yn}, yi = Ei, i = 1, . . . , n, and define mY : 2Y ! ½a; b; mY ðBÞ ¼ mð Evidently, mY is a complete sup-measure, and Y

Z

sup

u  dm ¼

X

Z

sup

uY  dmY ;

Y

where uY(yi) = ui, i = 1, . . . , n. Similarly,

Z X

sup

v  dm ¼

Z

vY is defined and

sup

v Y  dmY : Y

Now, we are ready to prove the Hölder type theorem for general r-sup-measure as a consequence of Theorem (4).

S

yi 2B Ei Þ:

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H. Agahi et al. / Applied Mathematics and Computation 217 (2011) 8630–8639

Corollary 1. Let  be represented by an increasing generator g and m be r-sup-measure. Let p and q be conjugate exponents with 1 < p < 1. Then for any measurable functions u, v : X ? [a, b], it holds:

Z

sup

ðu  v Þ  dm 6

Z

X

sup

ð1pÞ Z ðpÞ u  dm 

X

sup

v ðqÞ  dm

ð1qÞ

X



ð5Þ

: 

Proof. Consider the sequences ðen Þn2N and ðfn Þn2N of step function from Definition (3)(ii) such that d(en(x), u(x)) ? 0 and  ðpÞ ðpÞ d(fn(x), v(x)) ? 0 uniformly as n ? 1. Then also d((en  fn)(x), (u  v)(x)) ? 0, as well as d ðen Þ ðxÞ; u ðxÞ ! 0 and  ðqÞ d ðfn ÞðqÞ  ðxÞ; v  ðxÞ ! 0 uniformly as n ? 1. Due to Theorem (4), inequality (5) holds for each pair (en, fn) and thus, due to Definition (3)(ii), also for desired pair (u, v).

h

Example 2. For [a, b] = [1, 1], let g generating  be given by g(x) = ex. Then

x  y ¼ x þ y; and Hölder type inequality from Theorem (4) reduces on

supðuðxÞ þ v ðxÞ þ wðxÞÞ 6 x2X

1 1 supðp  uðxÞ þ wðxÞÞ þ supðq  v ðxÞ þ wðxÞÞ; p x2X q x2X

where u, v, w are arbitrary real function on X. 5. Minkowski’s inequality for pseudo-integral Theorem 5. Let u, v : X ? [a, b] be two measurable functions and p 2 [1, 1[. If an additive generator g : [a, b] ? [0, 1] of the pseudo-addition  and the pseudo-multiplication  are increasing. Then for any r--measure m it holds:

Z X



ðu  v ÞðpÞ   dm

ð1pÞ

Z



6 

ð1pÞ Z ðpÞ u  dm 

X



ð1pÞ

v ðpÞ   dm

X



: 

Proof. We apply the classical Minkowski’s inequality (1) on compositions g u and g v and then we obtain

Z

1p Z 1p Z 1p ðg u þ g uÞp dðg mÞ 6 ðg uÞp dðg mÞ þ ðg v Þp dðg mÞ :

X

X

X

Since function g is increasing function, then g1 is also increasing function and we have

g

1

Z

1p ! Z 1p Z 1p ! p p 1 ðg u þ g uÞ dðg mÞ ðg uÞ dðg mÞ þ ðg v Þ dðg mÞ 6g : p

X

X

X

Hence

g

1

Z

p

ðg u þ g v Þ dðg mÞ

1p !

¼g

1

X

Z

 1 p gðg ðg u þ g v ÞÞ dðg mÞ

1p !

X

¼ g 1

Z

1p   p dðg mÞ g g 1 gðg 1 ðg u þ g v ÞÞ

!

X

¼g

1

Z

1p !  1  p  g g ðgðu  v ÞÞ dðg mÞ

X

¼ g 1

Z X

¼ g 1

1p  g ðu  v ÞðpÞ dðg mÞ

!

  Z  1p ! Z ðpÞ 1 g g g ðu  v Þ dðg mÞ ¼ X

X



ð1pÞ ðu  v ÞðpÞ  dm :  

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H. Agahi et al. / Applied Mathematics and Computation 217 (2011) 8630–8639

On the other side, we have

g

Z

1

1p

p

ðg uÞ dðg mÞ

þ

Z

X

p

ðg v Þ dðg mÞ

1p ! ¼g

1

g g

Z

1

X



 g g ððg uÞ Þdðg mÞ p

1

1p !!

X

þg g

1p !!! g g ððg v Þ Þ dðg mÞ

Z

1



p

1



X

Z

¼ g 1

1p ðpÞ g u dðg mÞ 

!

X

g

Z

1

g



v

ðpÞ  ðxÞ

1p ! dðg mÞ

X

 Z g

¼ g 1



1p !  Z ðpÞ u  dm g  g 1

X

¼

Z

which completes the proof.

v ðpÞ   dm

1p !

X

ð1pÞ Z  ðpÞ u  dm  X



ð1qÞ  v ðqÞ  dm ;

X





h

In the case of idempotent pseudo-addition  = sup also a version of Minkowski inequality holds. Observe that if  = inf, ðpÞ then the corresponding inequality means (recall that x ¼ x for each x 2 [a, b], p > 0)

Z



ðu  v Þ  dm 6 sup

Z

X



u  dm;

X

Z





v  dm

;

X

i.e.,

! sup infðsupðuðxÞ; v ðxÞÞ; wðxÞÞ 6 sup supðinfðuðyÞ; wðyÞÞÞ; sup ðinfðv ðzÞ; wðzÞÞÞ ; x2X

y2X

z2X

which holds due to the distributivity of sup and inf. Finally, we turn our attention to the case I(a). Theorem 6. Let  be represented by an increasing generator g, m be a complete sup-measure and p 2 ]0, 1[. Then for any functions u, v : X ? [a, b], it holds:

Z

sup

X

ð1pÞ Z ðsupðu; v ÞÞðpÞ  dm ¼ sup 

sup

ð1pÞ Z ðpÞ u  dm ;

X



sup

X



ð1pÞ !

v ðpÞ   dm

: 

Proof. Let w be the density function related to m. As already was shown in proof of Theorem (4),

Z

sup

! ð1pÞ  1 up  dm ¼ g 1 sup gðuðyÞÞg p ðwðyÞÞ ; y2X

X

and

Z X

ð1pÞ

sup

v p  dm

   1 ¼ g 1 sup gðv ðzÞÞg p ðwðzÞÞ : z2X

Therefore

sup

Z X

sup

ð1pÞ Z ðpÞ u  dm ; 

sup

ð1pÞ !

v ðpÞ   dm

X



!!   1 1 ¼ g 1 sup sup gðuðyÞÞg p ðwðyÞÞ ; sup gðv ðzÞÞg p ðwðzÞÞ y2X

z2X

   1 ¼ g 1 sup g ðsupðuðxÞ; v ðxÞÞÞ  g p ðwðxÞÞ x2X

¼

Z X

what is the result we have to prove.

h



ð1pÞ ðsupðu; v ÞÞðpÞ  dm ;  

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H. Agahi et al. / Applied Mathematics and Computation 217 (2011) 8630–8639

6. Conclusion We have proved the Hölder and Minkowski integral type inequality for the pseudo-integral for its characteristic cases. There are several classical inequalities related to the Lebesgue integral which can be generated for non-linear integrals, including the pseudo-integrals. Up to the published results mentioned in the introduction, recall, for example, the Markov inequality

mðf P cÞ 6

Z

1 c

fdm;

X

valid for any integrable non-negative function f and positive constant c. Rewriting the Markov inequality into its equivalent form

c  mðf P cÞ 6

Z

fdm;

X

we have a straightforward generalization for pseudo-integrals, namely

c  mðf P cÞ 6

Z



f  dm;

ð6Þ

X

whenever c 2 [a, b]+n{0} and f : X ? [a, b]+. This inequality is trivial if induced the standard ordering on [a, b], i.e., in cases I(a), III(a) and II with increasing generator g, due to the non-decreasingness of the corresponding pseudo-integrals, see [18,21], and the fact that the function fc : X ? [a, b] given by

fc ðxÞ ¼



c;

if f ðxÞ P c;

0; else

satisfies fc 6 f, and

Z



f  dm ¼ c  mðf P cÞ:

X

Moreover, if  is cancellative on ]a, b[ (see cases I(a) and II with g increasing), then (6) generalized into ðpÞ

c  mðf P cÞ 6

Z



ðpÞ

f  dm;

X

where p is an arbitrary positive constant (recall the case p = 2 for the classical Lebesgue integral, which is a version of Chebyshev inequality in statistics). Acknowledgments The work on this paper was supported by grants APVV-0012-07, GACR 402/08/0618, and bilateral project SK-CN-019-07 between Slovakia and China, as well as by the project of Zhejiang Education Committee Y200700477. The fourth and fifth authors were supported in part by the Project MNZZˇSS 174009, and by the project ‘‘Mathematical Models of Intelligent Systems and their Applications’’ of Academy of Sciences and Arts of Vojvodina supported by Provincial Secretariat for Science and Technological Development of Vojvodina. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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