Soft Computing https://doi.org/10.1007/s00500-018-3080-6
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Refinements of Hölder’s and Minkowski’s type inequalities for σ -⊕-measures and pseudo-expectation Hamzeh Agahi1 · Milad Yadollahzadeh2
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract This paper gives some refinements of Hölder’s and Minkowski’s type inequalities and their relations to pseudo-expectation and g-semiring. These inequalities are studied for any σ -⊕-measure with general kernels, including pseudo-expectation and pseudo-convolution integral. In special cases, our results improve and refine the previous results. Keywords σ -⊕-measure · Hölder’s inequality · Minkowski’s inequality · Pseudo-expectation · Probability measure
1 Introduction Hölder’s and Minkowski’s inequalities are important in many applications, such as probability theory and statistics, functional analysis, pseudo-analysis, information sciences and etc. (Agahi et al. 2011; Tian 2014). For example, in information sciences, Tian (2014) obtained a generalized Hölder’s inequality in 2014. In non-additive measure theory, Ouyang et al. studied Minkowski’s inequality for the Sugeno integral on abstract spaces (Ouyang et al. 2010). The theory of nonadditive measures and integrals has been also discussed in nonlinear analysis, uncertain dynamical systems and partial differential equations (Pap 1993). Let us recall the classical Hölder and Minkowski type inequalities in probability. Theorem 1.1 Let (, A, P) be a probability space and X and Y be two random variables.
Communicated by A. Di Nola.
B
Hamzeh Agahi
[email protected] Milad Yadollahzadeh
[email protected]
1
Department of Mathematics, Faculty of Basic Science, Babol Noshirvani University of Technology, Shariati Ave., Babol 47148-71167, Iran
2
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar 47416-1468, Iran
(i) If p > 1 and
1 p
+
1 q
= 1, then Hölder’s inequality
1 1 E |X Y | ≤ E X p p E |Y |q q holds. (ii) If p ≥ 1, then Minkowski’s inequality 1 1 1 E |X + Y | p p ≤ E |X | p p + E |Y | p p , holds. Here E [·] means the mathematical expectation with respect to P. Let (1 , F1 , μ1 ) and (2 , F2 , μ2 ) be two finite measure spaces. Kruli´c et al. (2009) proposed the following generalized fractional operator 2 k, [X ] (ω1 ) id := k (ω1 , ω2 ) X (ω2 ) dμ2 (ω2 ) , ω1 ∈ 1 ,
2
(1.1)
where X is a measurable function on 2 and k : 1 × 2 → R+ is a measurable and non-negative function of two variables which sometimes is called a kernel (for more details see e.g. Himmelreich et al. 2013, Chapter II, p. 15). Example 1.2 Let 1 = 2 = (a, b). If μ1 = μ2 is the Lebesgue measure and k (ω1 , ω2 ) =
(ω1 −ω2 )α−1 (α)
0
a ≤ ω2 < ω1 , ω1 < ω2 ≤ b,
123
H. Agahi, M. Yadollahzadeh
then
(1) If
2 k, id
ω1
[X ] (ω1 ) = a
1 p
+
1 q
= 1, 1 < p < ∞, then by (1.2),
1 1 p q k,D k,H p
k,H q
ϒid X (ω1 ) Y (ω1 ) , ϒid [X Y ] (ω1 ) ≤ ϒid
1 (ω1 − ω2 )α−1 X (ω2 ) dω2 (α)
(1.4)
is the Riemann–Liouville fractional integral of order α > 0 Kilbas et al. (2006) .
for each fixed ω1 ∈ 1 where H = {ω0 ∈ D : X (ω0 )Y (ω0 ) = 0} and
Example 1.3 If k (ω1 , ω2 ) ≡ 1 and μ2 = P is the probability 2 measure, then k, [X ] (ω1 ) = E [X ] is the mathematical id expectation with respect to P.
k,A ϒid [X ] (ω1 ) :=
Example 1.4 Let 1 = 2 = (0, + ∞). If μ1 = μ2 is the Lebesgue measure and
In particular, if k (ω1 , ω2 ) ≡ 1 in (1.4), then
k (ω1 , ω2 ) =
k (ω1 − ω2 ) 0
ω1
(1.5)
0 ≤ ω2 < ω1 ; ω1 < ω2 < + ∞,
D
k (ω1 − ω2 ) X (ω2 ) dω2
is convolution integral (Schiff 1999). In the following lemma, we first propose two strong versions of Hölder’s and Minkowski’s inequalities in measure theory. Lemma 1.5 Let (1 , F1 , μ1 ) and (2 , F2 , μ2 ) be two finite measure spaces and D be an arbitrary set in F2 . Let X , Y : 2 → [0, ∞) be two random variables.
1 q
= 1, then a strong version of
p
k,D k,H X id [X Y ] (ω1 ) ≤ id
(ω1 )
1 p
k,H id Y
q
1
p
p
q
q
X dP
Y dP
H
,
(1.6)
H
which is a refinement of the classical Hölder inequality (Petrov 2015). (2) If p ≥ 1, then
0
(I) If 1 < p < ∞, 1p + Hölder’s inequality
1
X Y dP ≤
then 2 k, [X ] (ω1 ) = id
k (ω1 , ω2 ) X (ω2 ) dP, ω1 ∈ 1 , A ∈ F2 . A
(ω1 )
1 q
1 1
p p k,D k,M p
X (ω1 ) ≤ ϒid ϒid (X + Y ) p (ω1 ) 1 p k,M p
Y (ω1 ) , + ϒid
for each fixed ω1 ∈ 1 where M = {ω0 ∈ D : X (ω0 ) + k,A Y (ω0 ) = 0} and ϒid [·] (ω1 ), A ∈ F2 is defined in (1.5). In particular, if k (ω1 , ω2 ) ≡ 1 in (1.7), then
1 (X + Y ) p dP D
,
(1.7)
p
1
≤
X p dP M
p
1
+
Y p dP
p
,
M
(1.8)
(1.2) holds for each fixed ω1 ∈ 1 where H = {ω0 ∈ D : X (ω0 )Y (ω0 ) = 0}. (II) If p ≥ 1, then a strong version of Minkowski’s inequality 1 1
p p k,M p
p k,D + Y (ω X (ω ) ≤ ) (X ) 1 1 id id 1 p
p Y (ω + k,M ) , 1 id
(1.3)
holds for each fixed ω1 ∈ 1 where M = {ω0 ∈ D : X (ω0 ) + Y (ω0 ) = 0}. Here k,A id [·] (ω1 ) for any A ∈ F2 is defined in (1.1) Proof See “Appendix A”.
Remark 1.6 Let μ2 = P, P be the probability measure, in Lemma 1.5.
123
which is a refinement of the classical Minkowski inequality (Petrov 2015). Example 1.7 (Petrov 2015) For any random variable X , we have (E [|X |])2 ≤ P (X = 0) E X 2 . The purposes of this paper are to provide some refinements of Hölder’s and Minkowski’s type inequalities for any σ -⊕-measure with general kernels. In special case, we can obtain some refinements of Hölder’s and Minkowski’s type inequalities for g-convolution integral. The rest of the paper is organized as follows. Some definitions that are necessary in this paper are recalled in Sect. 2. In Sect. 3, we establish some refinements of the Hölder and Minkowski type inequalities for pseudo-integral in general form. Finally, some concluding remarks are given.
Refinements of Hölder’s and Minkowski’s type inequalities for σ -⊕-measures and… (1)
2 Preliminaries
(n)
= sup{v|v u } and for any rational
Moreover, u n
Let [a, b] be a closed subinterval of [− ∞, ∞]. The full order on [a, b] is denoted by . Definition 2.1 A binary operation ⊕ on [a, b] is pseudoaddition if it is continuous, non-decreasing, associative, commutative and with a zero (neutral) element different from b and denoted by 0.
(m) u n
= u (s) s ∈ (0, ∞) ,
is well defined. If α is not rational, then (s) u (α)
= sup u |s ∈ (0, α) , s is rational . (α)
Evidently, if u v = g −1 (g(u) · g(v)), then u g −1 (g α (u)).
Let [a, b]+ = {u | u ∈ [a, b] , 0 u}. Definition 2.2 A binary operation on [a, b] is called pseudo-multiplication if it is positively non-decreasing, commutative, and associative with a unit element 1 ∈ [a, b]+ . We suppose, further, 0 u = 0 and that is distributive pseudo-multiplication with respect to ⊕, i.e., u (v ⊕ z) = (u v) ⊕ (u z). The structure ([a, b] , ⊕, ) is a semiring (see Kuich 1986). Remark 2.3 Three important cases of semirings with continuous pseudo-operations were suggested in Pap (2002), Pap and Štrboja (2010). Case I The pseudo-addition is idempotent operation and the pseudo-multiplication is not. Case II The pseudo-operations are given by a continuous and monotone function g : [a, b] → [0, ∞], i.e., u ⊕ v = g −1 (g (u) + g (v)) and u v = g −1 (g (u) g (v)) . Case III Both operations are idempotent.
Now, we recall two important cases of pseudo-integrals. Definition 2.6 If pseudo-operations are generated in Case II from Remark 2.3, then the pseudo-integral for a measurable function Y : → [a, b] is defined by
⊕
Y dm = g
−1
(i) v (φ) = 0 (if ⊕ is not idempotent); (ii) v is σ -⊕-(decomposable) measure, i.e. v
∞
Bi
=
∞
i=1
i=1
holds for any sequence {Bi }i∈N of pairwise disjoint sets from A. Definition 2.5 (Agahi et al. 2011) For u ∈ [a, b]+ and α ∈ (α) (0, ∞), the pseudo-power u as follows: if α = n is a natural number then (n)
u = u u · · · u. n-times
(g ◦ Y ) d (g ◦ m) ,
where the integral in the right side is the Lebesgue integral (Pap 1993). In special cases, • When m = g −1 ◦ μ, μ is the Lebesgue measure, then (Mesiar and Pap 1999)
⊕
Y dm = g −1
g (Y (ω)) dμ (ω) .
• When m = g −1 ◦ P, P is the probability measure, then ϒ⊕ [X ]
:= g
−1
(g ◦ Y ) dP = g −1 (E [g (Y )])
is called a pseudo-expectation. Definition 2.7 Let a generator g be the same as in Definition 2.6. Let (1 , F1 ) and (2 , F2 ) be two measurable spaces and Y : 2 → [a, b] be a measurable function. Then, for any σ -⊕-measure μ2 and for each fixed ω1 ∈ 1 , we define an operator k,A ⊕, ,
v(Bi )
Let A be a σ -algebra of subsets of a set . Definition 2.4 (Agahi et al. 2011; Pap 1995) A set function v : A → [a, b]+ (or semiclosed interval) is a ⊕-measure if it satisfies the following conditions:
=
k,A ⊕, [Y ](ω1 ) :=
⊕
(k (ω1 , ω2 ) Y (ω2 )) dμ2 (ω2 )
g (k (ω1 , ω2 ) Y (ω2 )) d (g ◦ μ2 (ω2 )) , = g −1 A
A
(2.1) for any A ∈ F2 by using Definition 2.6, where k : 1 × 2 → [a, b] is a measurable kernel (Agahi et al. 2017). In particular, when μ2 = g −1 ◦ P, P is the probability measure, then we define a generalized pseudo-expectation k,A ϒ⊕, [Y ] (ω1 ) := ϒ⊕A [k (ω1 , ω2 ) Y (ω2 )] .
(2.2)
123
H. Agahi, M. Yadollahzadeh
In Definition 2.7, if [a, b] = [0, ∞], g = id (i.e., g(x) = k,A x for all x), then we define k,A ⊕, [.] = id [.] .
k,A where ϒ⊕, [·] (ω1 ), A ∈ F2 is defined in (2.2).
Definition 2.8 (Agahi et al. 2011) If the semiring is of the form ([a, b] , sup, ), cases I(a) and III(a) from Remark 2.3, then the pseudo-integral for a function Y : → [a, b] is defined by
replace X , Y and μ2 by (g ◦ u) (g ◦ k) p , (g ◦ υ) (g ◦ k) q and g ◦ μ2 , respectively. Then we obtain
⊕
Proof Denote k (ω1 , ω2 ) by k. We apply Lemma 1.5 (I), and 1
1
(g ◦ k) (g ◦ u)(g ◦ υ)d(g ◦ μ2 ) 1 1 = (g ◦ k) p (g ◦ u) (g ◦ k) q (g ◦ υ)d(g ◦ μ2 ).
D
Y dm = sup (Y (ω) ψ(ω)) ,
(2.3)
ω∈
D
where function ψ is sup-measure m.
≤
Definition 2.9 Let μ2 be a complete sup-measure. Define an 2 operator k, sup, , 2 k, sup, [Y ](ω1 ) :=
1 H
.
1
(g ◦ k)(g ◦ υ) d(g ◦ μ2 ) q
q
.
H sup 2
(k (ω1 , ω2 ) Y (ω2 )) dμ2
If g is increasing, then g −1 is also increasing and we have
= sup ((k (ω1 , ω2 ) Y (ω2 )) ψ(ω2 )), ω2 ∈2
(2.4)
g
−1
(g ◦ k)(g ◦ u)(g ◦ υ)d(g ◦ μ2 ) D
≤g
where k : 1 × 2 → [a, b] is a measurable kernel (Agahi et al. 2017) and Y : 2 → [a, b] is a measurable function.
−1
1
p
(g ◦ k)(g ◦ u) d(g ◦ μ2 ) p
H
1
×
(g ◦ k)(g ◦ υ) d(g ◦ μ2 ) q
q
.
H
3 Main results In this section, we give some refinements of Hölder’s and Minkowski’s type inequalities for any σ -⊕-measure with general kernels. Now, we give the main results which are based on g-principle formulated by Rybárik (1995). Theorem 3.1 Let 1 < p < ∞, 1p + q1 = 1. Let (1 , F1 ) and (2 , F) be two measurable spaces and u, υ : 2 → [a, b] be two measurable functions and let a generator g : [a, b] → [0, ∞] of the pseudo-addition ⊕ and the pseudomultiplication be an increasing function and let D be an arbitrary set in F2 . Then for any σ -⊕-measure μ2 and for each fixed ω1 ∈ 1 , we have
Hence k,D ⊕, [u υ] (ω1 ) =
g
where H = {x ∈ D : g ◦ u(x).g ◦ v(x) = 0}, where k,A ⊕, [·] (ω1 ), A ∈ F2 is defined in (2.1). In particular, if μ2 = g −1 ◦ P, then k,D ϒ⊕, [u υ] (ω1 ) 1 1 p q ( p) (q) k,H k,H ≤ ϒ⊕, u (ω1 )
ϒ⊕, υ (ω1 ) ,
1 p (g ◦ k)(g ◦ u) d(g ◦ μ2 ) p
−1
1 q . (g ◦ k)(g ◦ υ) d(g ◦ μ2 ) q
H
Also, −1
1 p (g ◦ k)(g ◦ u) d(g ◦ μ2 ) p
H
g
−1
1 q (g ◦ k)(g ◦ υ) d(g ◦ μ2 ) q
H
(3.1)
−1
k (u υ) dμ2 H
g
υ] (ω1 ) 1 1 p q ( p) (q) k,H k,H ⊕, u (ω1 )
⊕, υ (ω1 ) ,
⊕ D
≤g
k,D ⊕, [u
123
p
(g ◦ k)(g ◦ u) d(g ◦ μ2 ) p
= g −1
g
−1
1 p g(g −1 ((g ◦ k)(g ◦ u) p ))d(g ◦ μ2 )
H
g(g
−1
1 q ((g ◦ k)(g ◦ υ) ))d(g ◦ μ2 ) q
H
=g
−1
g −1
g k
H
H
( p)
u
1 p d(g ◦ μ2 )
1 q (q) g(k υ )d(g ◦ μ2 )
Refinements of Hölder’s and Minkowski’s type inequalities for σ -⊕-measures and…
=
⊕
k H
=
k,H ⊕,
( p)
u
( p) u
dm
(ω1 )
1 p
1 p
⊕
k H
k,H ⊕,
(q) υ
(q)
υ
(ω1 )
dm
1 q
1 q
k,D ⊕, [u
.
I.e., ⊕
υ] (ω1 ) =
k H
◦ λ, where λ is the Lebesgue Example 3.2 Let μ2 = measure. Let D = H = [0, t] for t ∈ (0, ∞) and k(t, x) =
for α > 0 and g(x) = x β for some β ∈ [1, ∞). The corresponding pseudo-operations are x⊕y = β x β + y β and x y = x y. Then β
α u(t)β υ(t)β ≤ I0+
pβ
α u(t) pβ I0+
qβ
( p)
u
⊕
α−1 (t−x) β 1/β (α)
⊕
≥
g −1
k (u υ) dμ2
D
k H
dμ2
(q)
υ
( p) u = k,H ) (ω 1
⊕,
α υ(t)qβ , I0+
dμ2
1 q
1 p
(q) υ
k,H ) (ω 1
⊕,
1 p
1 q
.
α is the Riemann–Liouville fractional integral (Kilwhere I0+ bas et al. 2006).
Theorem 3.3 Let 1 < p < ∞, 1p + q1 = 1. Let (1 , F1 ) and (2 , F2 ) be two measurable spaces and u, v : 2 → [a, b] be two measurable functions and let a generator g : [a, b] → [0, ∞] of the pseudo-addition ⊕ and the pseudomultiplication is a decreasing function and let D be an arbitrary set in F2 . Then for any σ -⊕-measure μ2 and for each fixed ω1 ∈ 1 it holds: 1 p
(q)
k,H (ω1 ) ⊕, υ
1 q
,
(3.2)
where H = {x ∈ D : g ◦ u(x).g ◦ v(x) = 0}, where k,A ⊕, [·] (ω1 ) for any A ∈ F2 is defined in (2.1). In particular if μ2 = g −1 ◦ P, then ( p) k,D k,H ϒ⊕, u (ω1 ) [u υ] (ω1 ) ≥ ϒ⊕,
1 p
(q) k,H υ (ω1 )
ϒ⊕,
1 q
Theorem 3.4 Let p ∈ [1, ∞[. Let (1 , F1 ) and (2 , F2 ) be two measurable spaces and u, υ : 2 → [a, b] be two measurable functions. If an additive generator g : [a, b] → [0, ∞] of the pseudo-addition ⊕ and the pseudomultiplication are increasing and D is an arbitrary set in F2 , then for any σ -⊕-measure μ2 and for each fixed ω1 ∈ 1 , we have 1 1 ( p) ( p) p p k,D k,M ⊕, (u ⊕ υ) (ω1 ) ≤ ⊕, u (ω1 )
1 ( p) p k,M ⊕ ⊕, υ (ω1 ) .
( p) k,H k,D ⊕, [u υ] (ω1 ) ≥ ⊕, u (ω1 )
where M = {x ∈ D : g ◦ u(x) + g ◦ υ(x) = 0}, where k,A ⊕, [·] (ω1 ) for any A ∈ F2 is defined in (2.1). Proof We first denote k (ω1 , ω2 ) by k. Then by Lemma 1.5 (II), we have
,
1 D
k(u + υ) p dμ
p
1
≤
p
ku p dμ
+
M
1 kυ p dμ M
p
.
(3.3) k,A where ϒ⊕, [·] (ω1 ) for any A ∈ F2 is defined in (2.2).
Proof By a similar way as in the proof of Theorem 3.1, we obtain
(g ◦ k)(g ◦ u)(g ◦ υ)d(g ◦ μ2 ) g −1 D
≥g
−1
1 (g ◦ k)(g ◦ u) d(g ◦ μ2 ) p
H
.
(g ◦ k)(g ◦ υ)q d(g ◦ μ2 ) H
1 q
.
We apply (3.3) and replace u, υ, k and μ by g ◦ u, g ◦ υ, g ◦ k and g ◦ μ2 , respectively.
1
p
(g ◦ k)(g ◦ u + g ◦ υ) d(g ◦ μ2 ) p
D
p
1
≤
(g ◦ k)(g ◦ u) d(g ◦ μ2 ) p
M
+
p
1 (g ◦ k)(g ◦ υ) d(g ◦ μ2 ) p
p
.
M
123
H. Agahi, M. Yadollahzadeh
If g is increasing, then g −1 is also increasing and we have
g −1
1 (g ◦ k)(g ◦ u + g ◦ υ) p d(g ◦ μ2 )
≤ g −1
M
+
p
(g ◦ k)(g ◦ υ) d(g ◦ μ2 ) p
g
1 p (g ◦ k)(g ◦ u + g ◦ υ) d(g ◦ μ2 ) D
= g −1 =g
−1
1 p (g ◦ k)g(g −1 (g(u ⊕ υ)) p )d(g ◦ μ2 ) D
=g
−1
2 k, sup, [u
1 p ( p) −1 g g g k (u ⊕ υ) d(g ◦ μ2 ) ⊕
( 1p )
( p)
k (u ⊕ υ) dμ2
D
( p) = k,D (ω1 ) ⊕, (u ⊕ υ)
2 k, sup, [u υ] (ω1 ) =
g
k,M ⊕,
( p) υ (ω1 )
1 p
.
2 k, sup,
( p) u
2 k, sup,
(ω1 )
(q) υ
1 p
(ω1 )
1 q
, (3.4)
k (ω1 , y) (u(y) υ(y)) dμ2
= sup (k (ω1 , y) u(y) υ(y) ψ(y)) y∈2
= g
1 (g ◦ k)(g ◦ u) d(g ◦ μ2 )
+ M
= g −1 ⊕ g −1
M
⎛ −1 ⎝ g ⊕g
⊕ M
k
123
( p)
k u
⊕
k M
M
d(g ◦ μ2 )
g(g −1 ((g ◦ k)g(υ )))d(g ◦ μ2 )
⎛ −1 ⎝ g =g
⊕
( p)
g g −1 (g ◦ k)g u
1
( p)
M
( p)
u
( p)
υ
dm
dm
dm
1 p
sup (g (k (ω1 , y)) g(u(y))g(υ(y))g(ψ(y)) ,
for each fixed ω1 ∈ 1 where ψ : 2 → [a, b] is a density function related to m. Moreover,
p
(g ◦ k)(g ◦ υ) p d(g ◦ μ2 )
y∈2
1
−1
p
p
M
=
sup 2
Also, −1
⊕
Proof
.
1 p
2 for each fixed ω1 ∈ 1 , where k, sup, [·] (ω1 ) is defined in (2.4).
1 p
D
=
υ] (ω1 ) ≤
1 p ( p) (g ◦ k)g((u ⊕ υ) )d(g ◦ μ2 )
D
Theorem 3.5 Let 1 < p < ∞, 1p + q1 = 1, μ2 be a complete sup-measure and be represented by an increasing generator g. If u, v : 2 → [a, b] are two measurable functions, then
p
dm
1 p
Now we consider the second case, when ⊕ = sup, and
= g −1 (g(x)g(y)).
.
Hence
M
−1
k
p
1
⊕
( p)
υ
( p) = k,M ⊕, u (ω1 )
1 (g ◦ k)(g ◦ u) p d(g ◦ μ2 )
⊕ M
p
D
1p
1p
1 p
sup 2
⎞ ⎠
dμ2
⎛
1 p
1 ⎞ p = g −1 ⎝ sup (g (k (ω1 , z)) g p (u(z))g(ψ(z))) ⎠
⎞ ⎠
k
p
( p)
u
=g
−1
z∈2
1 p
1 p
sup g (k (ω1 , z)) g(u(z))g (ψ(z))
,
z∈2
for each fixed ω1 ∈ 1 . Similarly,
sup
k
2
=g
(q)
υ
−1
sup
w∈2
dμ2 1 q
1 q
1 q
g (k (ω1 , w)) g(υ(w))g (ψ(w))
,
Refinements of Hölder’s and Minkowski’s type inequalities for σ -⊕-measures and…
for each fixed ω1 ∈ 1 . Consequently,
k,H sup, =
( p) u (ω1 )
sup
k
2
1 p
sup
k
2
= g −1
( p)
u (q)
υ
Acknowledgements The authors are very grateful to the anonymous reviewers for their suggestions which have led to an improved version of this paper.
( 1 ) q (q) υ
k,H (ω ) 1
sup,
dμ2
dμ2
Compliance with ethical standards
1 p
Conflict of interest The authors declare that they have no conflict of interest.
1 q
1
1
sup g p (k (ω1 , z)) g(u(z))g p (ψ(z))
5 Appendix A
z∈2
. sup
w∈2
≥g
g (k (ω1 , w)) g(υ(w))g (ψ(w)) 1 q
−1
1 p
1 p
sup g (k (ω1 , y)) g(u(y))g (ψ(y))g y∈2 1 q
(k (ω1 , y))g(υ(y))g (ψ(y)) = g −1 =
Proof of Lemma 1.5 Part (I). We apply the inequality |r s| ≤ |r | p |s|q p + q for any r and s. Let
1 q
r =
1 q
2
sup (g (k (ω1 , y)) g(u(y))g(υ(y))g(ψ(y)))
− 1
p
1
k p (ω1 , ω2 ) X (ω2 ) ,
− 1 q 1 k (ω1 , ω2 ) Y q (ω2 ) dμ2 (ω2 ) k q (ω1 , ω2 ) Y (ω2 ) ,
H
for each fixed ω1 ∈ 1 . Then integrating both parts of the inequality over H implies that
2 k (u υ) dμ2 = k, sup, [u υ] (ω1 ) ,
for each fixed ω1 ∈ 1 .
H
s=
y∈2 sup
k (ω1 , ω2 ) X p (ω2 ) dμ2 (ω2 )
k (ω1 , ω2 ) X (ω2 ) Y (ω2 ) dμ2 (ω2 ) = k (ω1 , ω2 ) X (ω2 ) Y (ω2 ) dμ2 (ω2 )
H
D
1
p 1 1 p + k (ω1 , ω2 ) X (ω2 ) dμ2 (ω2 ) ≤ p q H
1 q q k (ω1 , ω2 ) Y (ω2 ) dμ2 (ω2 )
4 Concluding remarks We have established some refinements of Hölder’s and Minkowski’s type inequalities for pseudo-integral with general kernels on a semiring ([a, b] , ⊕, ) . Our results significantly generalize the previous results in this fields. As we have seen, • For k (ω1 , ω2 ) ≡ 1 and μ2 = g −1 ◦ P, in Theorems 3.1 and 3.4, we get some refinements of Hölder’s and Minkowski’s type inequalities. In particular, taking [a, b] = [0, ∞], g = id, 2 ∈ F2 and D = 2 , we get the classical H ölder’s and Minkowski’s type inequalities. • For k (ω1 , ω2 ) ≡ 1 in Theorems 3.1 and 3.4, we have some refinements of Hölder’s and Minkowski’s type inequalities for pseudo-integral obtained by Agahi et al. (2011). • For k (ω1 , ω2 ) = k (ω1 − ω2 ) and μ2 = g −1 ◦ μ, μ is the Lebesgue measure, 2 = [0, ω1 ] for each fixed ω1 , Theorems 3.1 and 3.4, we get the some refinements of Hölder’s and Minkowski’s type inequalities for gconvolution integral.
H
1
=
p
k (ω1 , ω2 ) X (ω2 ) dμ2 (ω2 ) p
H
1 k (ω1 , ω2 ) Y (ω2 ) dμ2 (ω2 ) q
q
.
(5.1)
H
Therefore, 1 1 q
p q k,H p
k,H X (ω1 ) Y (ω1 ) , k,D id [X Y ] (ω1 ) ≤ id id
which completes the proof.
Part (II) Let H1 = {ω0 ∈ D : X (ω0 )(X (ω0 )+Y (ω0 )) p−1 = 0}, H2 = {ω0 ∈ D : Y (ω0 )(X (ω0 ) + Y (ω0 )) p−1 = 0} and M = {ω0 ∈ D : X (ω0 ) + Y (ω0 ) = 0}. Applying inequality (5.1 ), we have
123
H. Agahi, M. Yadollahzadeh
k(ω1 , ω2 )(X (ω2 ) + Y (ω2 )) p dμ2 (ω2 ) D = k(ω1 , ω2 )X (ω2 ) (X (ω2 ) + Y (ω2 )) p−1 dμ2 (ω2 ) D + k(ω1 , ω2 )Y (ω2 ) (X (ω2 ) + Y (ω2 )) p−1 dμ2 (ω2 )
D
1
≤
k(ω1 , ω2 ) (X (ω2 ) + Y (ω2 ))( p−1)q dμ2 (ω2 )
1 p p ≤ k(ω1 , ω2 )X (ω2 )dμ2 (ω2 )
q1
M
M
1 k(ω1 , ω2 )Y (ω2 )dμ2 (ω2 ) p
p
.
Consequently, for each fixed ω1 ∈ 1 , we have k(ω1 , ω2 ) (X (ω2 )
H1
+ Y (ω2 ))( p−1)q dμ2 (ω2 )
1 1
p p k,M p
p + Y (ω X (ω k,D ) ≤ ) (X ) 1 1 id id
1 q
1 p
p Y (ω1 ) . + k,M id
1
+
p
k(ω1 , ω2 )Y p (ω2 )dμ2 (ω2 ) H2
k(ω1 , ω2 ) (X (ω2 ) H2
+Y (ω2 ))( p−1)q dμ2 (ω2 )
References 1 q
≤
1 k(ω1 , ω2 )X (ω2 )dμ2 (ω2 ) p
p
M
k(ω1 , ω2 ) (X (ω2 ) M
+ Y (ω2 ))( p−1)q dμ2 (ω2 )
1 q
1
+
k(ω1 , ω2 )Y (ω2 )dμ2 (ω2 ) p
p
M
k(ω1 , ω2 ) (X (ω2 ) M
+ Y (ω2 ))( p−1)q dμ2 (ω2 )
1 q
,
for each fixed ω1 ∈ 1 . Then
k(ω1 , ω2 )(X (ω2 ) + Y (ω2 )) p dμ2 (ω2 )
M
H1
D
+
p
k(ω1 , ω2 )X p (ω2 )dμ2 (ω2 )
≤
D
k(ω1 , ω2 )(X (ω2 ) + Y (ω2 )) p dμ2 (ω2 )
k(ω1 , ω2 ) (X (ω2 ) + Y (ω2 ))( p−1)q dμ2 (ω2 )
1 p ≤ k(ω1 , ω2 )X p (ω2 )dμ2 (ω2 )
q1
M
M
1
+
k(ω1 , ω2 )Y p (ω2 )dμ2 (ω2 )
p
.
M
So,
1
p
k(ω1 , ω2 )(X (ω2 ) + Y (ω2 )) dμ2 (ω2 ) D p D k(ω1 , ω2 )(X (ω2 ) + Y (ω2 )) dμ2 (ω2 ) = 1 ( p−1)q dμ2 (ω2 ) q D k(ω1 , ω2 ) (X (ω2 ) + Y (ω2 )) p
123
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