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Applied Mathematics and Computation 241 (2014) 64–69
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Generalization of the Lyapunov type inequality for pseudo-integrals Dong-Qing Li a, Xiao-Qiu Song a,⇑, Tian Yue a,b, Ya-Zhi Song a a b
College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China School of Science, Hubei University of Automotive Technology, Shiyan, Hubei 442002, China
a r t i c l e
i n f o
Keywords: Lyapunov type inequality Semiring Pseudo-addition Pseudo-multiplication Pseudo-integral
a b s t r a c t We prove two kinds of Lyapunov type inequalities for pseudo-integrals. One discusses pseudo-integrals where pseudo-operations are given by a monotone and continuous function g. The other one focuses on the pseudo-integrals based on a semiring ð½0; 1; sup; Þ, where the pseudo-multiplication is generated. Some examples are given to illustrate the validity of these inequalities. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction As a generalization of classical analysis, Pseudo-analysis [19,24,31] based on a semiring ð½a; b; ; Þ, in which pseudoaddition and pseudo-multiplication are given by a monotone and continuous function generator g, has been investigated. In this structures, many concepts such as -measure (pseudo-additive measure), pseudo- integral, pseudo-convolution, pseudo-Laplace transform, etc. have been proposed. The integral inequalities are good mathematical tools both in theory and application. Different integral inequalities including Chebyshev, Jensen, Hölder and Minkowski inequalities are widely used in various fields of mathematics, such as probability theory, differential equations, decision-making under risk and information sciences. Recently, some classical inequalities have been generalized for fuzzy integrals. Román-Flores et al. [4,13–15,26–30] investigated several kinds of inequalities for Sugeno integral including Geometric inequality, Jensen type inequality, Chebyshev type inequality, Hardy type inequality, Convolution type inequality, Markov type inequality and General Barnes–Godunova–Levin type inequality. Girotto and Holzer [16,17] illustrated a characterization of comonotonicity property by a Chebyshev type inequality for Sugeno integral and Choquet integral. Mesiar and Ouyang [1–3,20–23] proposed Chebyshev, Minkowski and Berwald inequalities for Sugeno integral. Caballero and Sadarangani [8–11] presented Chebyshev, Cauchy–Schwarz, Fritz Carlson and sandor inequalities for Sugeno integral. In addition, some famous inequalities have also been generalized for pseudo-integral. Agahi et al. [5,6] studied Chebyshev, Hölder and Minkowski inequalities for pseudointegrals. Pap and Šrboja [25] discussed generalization of the Jensen type inequalities for pseudo-integrals. Daraby [12] obtained generalization of the Stolarsky type inequality for pseudo-integrals. The purpose of this paper is to prove a generalization of the Lyapunov type inequality for pseudo-integrals. We discuss two kinds of Lyapunov type inequalities for pseudo-integrals. The first discusses pseudo-integrals where pseudo-operations
⇑ Corresponding author. E-mail addresses: [email protected] (D.-Q. Li), [email protected] (X.-Q. Song), [email protected] (T. Yue), [email protected] (Y.-Z. Song). http://dx.doi.org/10.1016/j.amc.2014.05.006 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
D.-Q. Li et al. / Applied Mathematics and Computation 241 (2014) 64–69
65
are given by a monotone and continuous function g. The second focuses on the pseudo-integrals based on a semiring ð½0; 1; sup; Þ with respect to sup-measure. Some examples are given to support the validity of these inequalities. The paper is organized as follows. In Section 2, some necessary preliminaries such as pseudo-operations, pseudo-integrals and r--measure are presented. In Section 3, generalizations of the Lyapunov type inequalities for pseudo-integrals are investigated. In Section 4, some conclusions are given. 2. Preliminaries In this section, we recall some basic definitions of pseudo-operations and pseudo-integrals, see [19,24,31]. Let ½a; b be a closed (in some cases it can be considered semiclosed) subinterval of ½1; 1. The full order on ½a; b is denoted by which can be the usual order of the real line or can be another order. Definition 2.1. A binary operation : ½a; b ½a; b ! ½a; b is called a pseudo-addition, for x; y; z; 0 (zero element) 2 ½a; b it satisfies the following requirements: (i) (ii) (iii) (iv)
x y ¼ y x; ðx yÞ z ¼ x ðy zÞ; x y ) x z y z; 0 x ¼ x.
Let ½a; bþ ¼ fxjx 2 ½a; b; 0 xg. Definition 2.2. A binary operation : ½a; b ½a; b ! ½a; b is called a pseudo-multiplication, for x; y; z; 1 (unit element) 2 ½a; b it satisfies the following requirements: (i) (ii) (iii) (iv) (v) (vi)
x y ¼ y x; ðx yÞ z ¼ x ðy zÞ; x y ) x z y z; ðx yÞ z ¼ ðx zÞ ðy zÞ; 1 x ¼ x; limn!1 xn and limn!1 yn exist and are finite ) limn!1 ðxn yn Þ ¼ lim xn lim yn . n!1
Definition 2.3. A set function m :
P
! ½a; bþ (or semiclosed interval) is a
n!1
r--measure if there holds:
(i) mð/Þ ¼ 0; S P 1 n (ii) m 1 , where 1 i¼1 Ai ¼ i¼1 mðAi Þ for any sequence fAi g of pairwise disjoint sets from i¼1 xi ¼ limn!1 i¼1 xi . Let m be a -measure, where has a monotone and continuous generator g, then g m is a r-additive measure. In the following, two important cases of pseudo integrals based on semiring ð½a; b; ; Þ are discussed. The first case is when pseudo-operations are generated by a monotone and continuous function g : ½a; b ! ½0; 1, i.e., pseudo-operations are given by: x y ¼ g 1 ðg ðxÞ þ g ðyÞÞ and x y ¼ g 1 ðg ðxÞg ðyÞÞ. If x y ¼ g 1 ðg ðxÞg ðyÞÞ, then
ðpÞ p x ¼ x|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} x x ¼ g 1 ðg ðxÞÞ : p
ðpÞ
Specially, is idempotent, then x ¼ x for any x 2 ½a; b and p 2 ð0; 1Þ. Thus, the pseudo-integral of the function f : X ! ½a; b is defined by
Z
f dm ¼ g 1
Z
X
ðg f Þdðg mÞ ;
X
where the integral applied on the right side is the standard Lebesgue integral. In fact, let m ¼ g 1 l; l is the standard Lebesgue measure on X, then we obtain
Z
f ðxÞdx ¼ g 1
Z
g ðf ðxÞÞdx :
X
X
In the second case, there are x y ¼ max ðx; yÞ and x y ¼ g 1 ðg ðxÞg ðyÞÞ, the pseudo-integral of the function f : R ! ½a; b is given by
Z R
f dm ¼ sup ðf ðxÞ wðxÞÞ:
66
D.-Q. Li et al. / Applied Mathematics and Computation 241 (2014) 64–69
where function w defines sup-measure m. Any sup-measure generated as essential supremum of a continuous density can be obtained as a limit of pseudo-additive measures with respect to generated pseudo-addition [19]. For arbitrary continuous function f : ½0; 1 ! ½0; 1, the integral R f dm can be obtained as a limit of g-integrals, see [19]. We denote l as the usual Lebesgue measure on R, We have
mðAÞ ¼ ess sup ðxjx 2 AÞ ¼ sup fajlðfxjx 2 A; x > agÞ > 0g:
Theorem 2.4 (Mesiar and Pap [19]). Let m be a sup-measure on ð½0; 1; Bð½0; 1ÞÞ, where Bð½0; 1Þ is the Borel r-algebra on ½0; 1; mðAÞ ¼ esssupl ðwðxÞjx 2 AÞ, and w : ½0; 1 ! ½0; 1 is a continuous density. Then, for any pseudo-addition with a generator g there exists a family fmk g of k -measure on ð½0; 1Þ; BÞ, where k is generated by g k (the function g of the power k), k 2 ð0; 1Þ, such that limk!1 mk ¼ m. Theorem 2.5 (Mesiar and Pap [19]). Let ð½0; 1; sup; Þ be a semiring, when is generated with g, i.e., we have x y ¼ g 1 ðg ðxÞg ðyÞÞ for every x; y 2 ð0; 1Þ. Let m be the same as in Theorem 2.4, Then, there exists a family fmk g of k -measure, where k is generated by g k ; k 2 ð0; 1Þ such that for every continuous function f : ½0; 1 ! ½0; 1,
Z
sup
f dm ¼ lim
Z
k
k!1
1 f dmk ¼ lim g k
Z
g k ðf ðxÞÞdx :
k!1
3. Generalization of the Lyapunov type inequality for pseudo-integrals In [7], the classical Lyapunov inequality provides the inequality
Z
1
f ðxÞs dl
rt
Z 6
0
1
rs Z f ðxÞt dl
0
1
f ðxÞr dl
st ;
0
where 0 < t < s < r; f : ½0; 1 ! ½0; 1Þ is an integrable function. In this section, we discuss Lyapunov inequality for pseudo-integrals based on the semiring ð½0; 1; ; Þ. Theorem 3.1. Let 0 < t < s < r; f : ½0; 1 ! ½0; 1 be measurable function and let a generator g : ½0; 1 ! ½0; 1 of the pseudo-addition and the pseudo-multiplication be an increasing function. Then for any d--measure it holds:
Z
!ðrtÞ
ðsÞ f
dm
Z
6
½0;1
!ðrsÞ ðt Þ f
dm
Z
½0;1
!ðstÞ ðr Þ f
dm
½0;1
:
Proof. We apply the classical Lyapunov inequality and then we obtain
Z
1
rt Z s ðg f Þ dðg mÞ 6
0
1
t
ðg f Þ dðg mÞ
rs Z
0
1
st
r
ðg f Þ dðg mÞ
:
0
Since function g is an increasing function, inverse function g 1 is also an increasing function, then there is
g 1
Z
1
rt ! Z s 6 g 1 ðg f Þ dðg mÞ
0
1
t
ðg f Þ dðg mÞ
rs
Z
0
1
r
ðg f Þ dðg mÞ
st ! :
0
Hence,
g 1
Z
1
s g g 1 ðg f Þ dðg mÞ
rt !
6 g 1
0
Z
1
rs Z t g g 1 ðg f Þ dðg mÞ
0
1
r g g 1 ðg f Þ dðg mÞ
st ! :
0
That is
g 1
Z
1
rt ! Z ðsÞ 6 g 1 g f dðg mÞ
0
1
rs Z ðt Þ g f dðg mÞ
0
1
st ! ðr Þ : g f dðg mÞ
0
Hence,
g 1
Z g g 1 0
1
rt ! Z ðsÞ 6 g 1 g f dðg mÞ g g 1
1 0
rs Z ðt Þ g f dðg mÞ g g 1 0
1
st ! ðr Þ : g f dðg mÞ
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D.-Q. Li et al. / Applied Mathematics and Computation 241 (2014) 64–69
i.e.,
g
1
Z
g
!!rt !
ðsÞ f dm
6g
1
g g
1
Z
g
½0;1
Z
ðsÞ
00 0 6 g 1 @@g @
f dm
½0;1
Z
Z
!ðrtÞ
ðsÞ f
ðt Þ f dm
g g
1
g
Z
½0;1
!ðrtÞ
!!rs !!!
dm
½0;1
Z
½0;1
ðt Þ f
6
dm
½0;1
:
!ðstÞ 111 ðr Þ AAA: f dm
!ðstÞ
ðr Þ f
dm
:
½0;1
½0;1
Z
!!st !!!! ðr Þ f dm
½0;1
!ðrsÞ 11 0 0 Z ðt Þ AA @g @ f dm
!ðrsÞ
This completes the proof. h Let us give an example to illustrate the Theorem 3.1. Example 3.2. Let g ðxÞ ¼ ln x, then
x y ¼ eln x ln y :
x y ¼ xy;
By Theorem 3.1, the following inequality holds
Z ln
1
eðln f ðxÞÞ
rt
s
Z 6 ln
0
1
eðln f ðxÞÞ
t
rs
Z þ ln
0
1
eðln f ðxÞÞ
r
st :
0
Theorem 3.3. Let 0 < t < s < r; f : ½0; 1 ! ½0; 1 be measurable function and let a generator g : ½0; 1 ! ½0; 1 of the pseudoaddition and the pseudo-multiplication be a decreasing function. Then for any d--measure it holds:
Z
!ðrtÞ
ðsÞ
f dm
½0;1
Z
!ðrsÞ
ðt Þ
f dm
P ½0;1
Z
!ðstÞ ðr Þ
f dm
:
½0;1
Proof. We apply the classical Lyapunov inequality and then we obtain
Z
1
s
ðg f Þ dðg mÞ
rt
Z
1
0
t
ðg f Þ dðg mÞ
6 0
rs Z
1
r
ðg f Þ dðg mÞ
st :
0
Since function g is a decreasing function, inverse function g 1 is also a decreasing function, then there is
g 1
Z
1
s
ðg f Þ dðg mÞ
rt !
P g 1
0
The next proof is similar to Theorem 3.1.
Z
1
rs Z t ðg f Þ dðg mÞ
0
1
r
ðg f Þ dðg mÞ
st ! :
0
h
Example 3.4. Let g ðxÞ ¼ 1 x. The corresponding pseudo-operations are x y ¼ x þ y 1 and x y ¼ x þ y xy. By Theorem 3.3, the following inequality holds rt
ðr tÞðsf ðxÞ f s ðxÞ 1Þ ðsf ðxÞ f s ðxÞ 1Þ
rs P ðr sÞ tf ðxÞ f t ðxÞ 1 tf ðxÞ f t ðxÞ 1 st
þ ðs t Þðrf ðxÞ f r ðxÞ 1Þ ðrf ðxÞ f r ðxÞ 1Þ rs ðr sÞ tf ðxÞ f t ðxÞ 1 tf ðxÞ f t ðxÞ 1 st ðs t Þðrf ðxÞ f r ðxÞ 1Þ ðrf ðxÞ f r ðxÞ 1Þ : In the following, we consider the second case of Lyapunov type inequalities for pseudo integral based on the semiring ð½0; 1; sup; Þ, where is generated by an increasing function g. Theorem 3.5. Let 0 < t < s < r; f : ½0; 1 ! ½0; 1 be measurable function and let be represented by an increasing generator g and m be a complete sup-measure, it holds:
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D.-Q. Li et al. / Applied Mathematics and Computation 241 (2014) 64–69
Z
sup
!ðrtÞ ðsÞ f
Z
dm
!ðrsÞ
sup
ðt Þ f
6
½0;1
dm
½0;1
Z
sup
!ðstÞ ðr Þ f
dm
½0;1
:
1 ¼ g 1 xk .
1
Proof. Since ðg k ðxÞÞ
1 k x y ¼ g 1 ðg ðxÞg ðyÞÞ ¼ g k g ðxÞg k ðyÞ ¼ xk y; in other words, g k is a generator of the . By Theorem 2.5, we obtain
Z
Z
sup
f dm ¼ lim
k!1
½0;1
1
ðsÞ
f dm
; g ; ðg Þ
Z
¼
½0;1
Z
k!1
k 1
k
!ðrtÞ
sup
1 f dmk ¼ lim g k
½0;1
As g is an increasing, so g
Z
k
sup
0
are also increasing function, then
!ðrtÞ Z k 1 k s 1 g g ð f Þ dm ¼ lim g k k!1
½0;1
¼
g k ðf ðxÞÞdx :
1
0
k!1
k!1
rt ! 1 s k g ðf ðxÞÞ dx :
Z
1 ¼ lim g k
ðrtÞ 1 s gk g k ð f ð xÞ Þ dx
ðrtÞ Z 1 s 1 k 1 g k ðf ðxÞÞ dx ¼ gk g lim g k
Z
k!1
gk 0
1 lim g k
1
Z
k!1
rs Z t g ðf ðxÞÞ dx
Z
k!1
st ! g ðf ðxÞÞ dx r
k
0
rs 1 t gk gk g k ð f ð xÞ Þ dx
1 0
Z
1
st ! 1 r k k k g g g ðf ðxÞÞ dx
0
1 ¼ lim g k
Z
k!1
rs Z ðt Þ g k f dx
1 0
1
! st ðr Þ g k f dx
0
1 k k 1 g g ¼ lim g k
Z
k!1
gk
1
k
0
1 ¼ lim g k
1
1
ðt Þ
g k f
rs dx
!!
0
Z
k 1 g
!!! st ðr Þ k g f dx
1
0
1 ¼ lim g k
Z
k!1
1
! Z rs 1 ðt Þ g f dx gk
0
1 ¼ gk
0
Z 1 g k lim g k k!1
Z
gk
sup
1
½0;1
ðr Þ f
st dx
!
! st ðr Þ g f dx k
0
!!rs ! ðt Þ
f dm
1 gk
½0;1 sup
0
k!1
1 ¼ gk
¼
g
k
Z 1 rs ! k 1 ðt Þ k g lim g g k f dx
1 gk
Z
1
k
ðt Þ
Z
sup
!!st ! ðr Þ
f dm
½0;1
!ðrsÞ f dm
gk
Z
sup
½0;1
!ðstÞ ðr Þ
f dm
g k ðf ðxÞÞ
s
dx
0
ð1Þ
0
By the classical Lyapunov inequality, the last equality (1)
1 6 lim g k
1
:
The proof is now completed. h Example 3.6. Let g k ðxÞ ¼ ekx , the corresponding pseudo-operations are
D.-Q. Li et al. / Applied Mathematics and Computation 241 (2014) 64–69
x y ¼ lim
1 ln ekx þ eky ¼ max ðx; yÞ k
x y ¼ lim
1 ln ekx eky ¼ x þ y: k
k!1
69
and
k!1
By Theorem 3.1, the following inequality holds
ðr tÞ sup ðsf ðxÞ þ wðxÞÞ 6 ððr sÞ sup ðtf ðxÞ þ wðxÞÞÞ þ ððs tÞ sup ðrf ðxÞ þ wðxÞÞÞ Remark 3.7. The third important case when ¼ max and ¼ min has been studied in [18] and the pseudo integral in such case yields the Sugeno integral.
4. Conclusion In this paper, We have proved two classes of the Lyapunov type inequalities for pseudo-integral. For further investigation, we will investigate other integral inequalities for Pseudo-integral. Acknowledgments The work in this paper was supported by ‘‘the National Natural Science Foundation of China (51374199)’’ and ‘‘the Fundamental Research Funds for the Central Universities (2013XK03)’’. We are grateful to the referees for their critical comments and helpful suggestions in improving our paper. References [1] H. Agahi, R. Mesiar, Y. Ouyang, E. Pap, M. Štrboja, Berwald type inequality for Sugeno integral, Appl. Math. Comput. 217 (2010) 4100–4108. [2] H. Agahi, R. Mesiar, Y. Ouyang, General Minkowski type inequalities for Sugeno integrals, Fuzzy Sets Syst. 161 (2010) 708–715. [3] H. Agahi, R. Mesiar, Y. Ouyang, New general extensions of Chebyshev type inequalities for Sugeno integrals, Int. J. Approximate Reasoning 51 (2009) 135–140. [4] H. Agahi, H. Román-Flores, A. Flores-Franulicˇ, General Barnes–Godunova–Levin type inequalities for Sugeno integral, Inf. Sci. 181 (2011) 1072–1079. [5] H. Agahi, R. Mesiar, Y. Ouyang, Chebyshev type inequalities for pseudo-integrals, Nonlinear Anal. 72 (2010) 2737–2743. [6] H. Agahi, Y. Ouyang, R. Mesiar, E. Pap, M. Štrboja, Höder and Minkowski type inequalities for pseudo-integral, Appl. Math. Comput. 217 (2011) 8630– 8639. [7] P.S. Bullen, A Dictionary of Inequalities, Addison Wesley Longman Limited, 1998. [8] J. Caballero, K. Sadarangani, A Cauchy–Schwarz type inequality for fuzzy integrals, Nonlinear Anal. 73 (2010) 3329–3335. [9] J. Caballero, K. Sadarangani, Chebyshev inequality for Sugeno integrals, Fuzzy Sets Syst. 161 (2010) 1480–1487. [10] J. Caballero, K. Sadarangani, Fritz Carlson’s inequality for fuzzy integrals, Comput. Math. Appl. 59 (2010) 2763–2767. [11] J. Caballero, K. Sadarangani, Sandors inequality for Sugeno integrals, Appl. Math. Comput. 218 (2011) 1617–1622. [12] B. Daraby, Generalization of the Stolarsky type inequality for pseudo-integrals, Fuzzy Sets Syst. 194 (2012) 90–96. [13] A. Flores-Franulicˇ, H. Román-Flores, A Chebyshev type inequality for fuzzy integrals, Appl. Math. Comput. 190 (2007) 1178–1184. [14] A. Flores-Franulicˇ, H. Román-Flores, Y. Chalco-Cano, A note on fuzzy integral inequality of Stolarsky type, Appl. Math. Comput. 196 (2008) 55–59. [15] A. Flores-Franulicˇ, H. Román-Flores, Y. Chalco-Cano, Markov type inequalities for fuzzy integrals, Appl. Math. Comput. 207 (2009) 242–247. [16] B. Girotto, S. Holzer, A Chebyshev type inequality for Sugeno integral and comonotonicity, Int. J. Approximate Reasoning 52 (2011) 444–448. [17] B. Girotto, S. Holzer, Chebyshev type inequality for Choquet integral and comonotonicity, Int. J. Approximate Reasoning 52 (2011) 1118–1123. [18] D. Hong, A Liapunov type inequality for Sugeno integrals, Nonlinear Anal. 74 (2011) 7296–7303. [19] R. Mesiar, E. Pap, Idempotent integral as limit of g-integrals, Fuzzy Sets Syst. 102 (2009) 385–392. [20] R. Mesiar, Y. Ouyang, General Chebyshev type inequalities for Sugeno integrals, Fuzzy Sets Syst. 160 (2009) 58–64. [21] Y. Ouyang, R. Mesiar, H. Agahi, An inequality related to Minkowski type for Sugeno integrals, Inf. Sci. 180 (2010) 2793–2801. [22] Y. Ouyang, J. Fang, L. Wang, Fuzzy Chebyshev type inequality, Int. J. Approximate Reasoning 48 (2008) 829–835. [23] Y. Ouyang, J. Fang, Sugeno integral of monotone functions based on Lebesgue measure, Comput. Math. Appl. 56 (2008) 367–374. [24] E. Pap, Generalized real analysis and its applications, Int. J. Approximate Reasoning 47 (2008) 368–386. [25] E. Pap, M. Šrboja, Generalization of the Jensen inequality for pseudo-integral, Inf. Sci. 180 (2010) 543–548. [26] H. Román-Flores, A. Flores-Franulicˇ, Y. Chalco-Cano, A convolution type inequality for fuzzy integrals, Appl. Math. Comput. 195 (2008) 94–99. [27] H. Román-Flores, A. Flores-Franulicˇ, Y. Chalco-Cano, A Hardy-type inequality for fuzzy integrals, Appl. Math. Comput. 204 (2008) 178–183. [28] H. Román-Flores, A. Flores-Franulicˇ, Y. Chalco-Cano, A Jensen type inequality for fuzzy integrals, Inf. Sci. 177 (2007) 3192–3201. [29] H. Román-Flores, Y. Chalco-Cano, Sugeno integral and geometric inequalities, Int. J. Uncertainty 1 (2007) 1–11. [30] H. Román-Flores, A. Flores-Franulicˇ, Y. Chalco-Cano, The fuzzy integral for monotone functions, Appl. Math. Comput. 185 (2007) 492–498. [31] M. Sugeno, T. Murofushi, Pseudo-additive measures and integrals, J. Math. Anal. Appl. 122 (1987) 197–222.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Generalization of the Lyapunov type inequality for pseudo-integrals Dong-Qing Li a, Xiao-Qiu Song a,⇑, Tian Yue a,b, Ya-Zhi Song a a b
College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China School of Science, Hubei University of Automotive Technology, Shiyan, Hubei 442002, China
a r t i c l e
i n f o
Keywords: Lyapunov type inequality Semiring Pseudo-addition Pseudo-multiplication Pseudo-integral
a b s t r a c t We prove two kinds of Lyapunov type inequalities for pseudo-integrals. One discusses pseudo-integrals where pseudo-operations are given by a monotone and continuous function g. The other one focuses on the pseudo-integrals based on a semiring ð½0; 1; sup; Þ, where the pseudo-multiplication is generated. Some examples are given to illustrate the validity of these inequalities. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction As a generalization of classical analysis, Pseudo-analysis [19,24,31] based on a semiring ð½a; b; ; Þ, in which pseudoaddition and pseudo-multiplication are given by a monotone and continuous function generator g, has been investigated. In this structures, many concepts such as -measure (pseudo-additive measure), pseudo- integral, pseudo-convolution, pseudo-Laplace transform, etc. have been proposed. The integral inequalities are good mathematical tools both in theory and application. Different integral inequalities including Chebyshev, Jensen, Hölder and Minkowski inequalities are widely used in various fields of mathematics, such as probability theory, differential equations, decision-making under risk and information sciences. Recently, some classical inequalities have been generalized for fuzzy integrals. Román-Flores et al. [4,13–15,26–30] investigated several kinds of inequalities for Sugeno integral including Geometric inequality, Jensen type inequality, Chebyshev type inequality, Hardy type inequality, Convolution type inequality, Markov type inequality and General Barnes–Godunova–Levin type inequality. Girotto and Holzer [16,17] illustrated a characterization of comonotonicity property by a Chebyshev type inequality for Sugeno integral and Choquet integral. Mesiar and Ouyang [1–3,20–23] proposed Chebyshev, Minkowski and Berwald inequalities for Sugeno integral. Caballero and Sadarangani [8–11] presented Chebyshev, Cauchy–Schwarz, Fritz Carlson and sandor inequalities for Sugeno integral. In addition, some famous inequalities have also been generalized for pseudo-integral. Agahi et al. [5,6] studied Chebyshev, Hölder and Minkowski inequalities for pseudointegrals. Pap and Šrboja [25] discussed generalization of the Jensen type inequalities for pseudo-integrals. Daraby [12] obtained generalization of the Stolarsky type inequality for pseudo-integrals. The purpose of this paper is to prove a generalization of the Lyapunov type inequality for pseudo-integrals. We discuss two kinds of Lyapunov type inequalities for pseudo-integrals. The first discusses pseudo-integrals where pseudo-operations
⇑ Corresponding author. E-mail addresses: [email protected] (D.-Q. Li), [email protected] (X.-Q. Song), [email protected] (T. Yue), [email protected] (Y.-Z. Song). http://dx.doi.org/10.1016/j.amc.2014.05.006 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
D.-Q. Li et al. / Applied Mathematics and Computation 241 (2014) 64–69
65
are given by a monotone and continuous function g. The second focuses on the pseudo-integrals based on a semiring ð½0; 1; sup; Þ with respect to sup-measure. Some examples are given to support the validity of these inequalities. The paper is organized as follows. In Section 2, some necessary preliminaries such as pseudo-operations, pseudo-integrals and r--measure are presented. In Section 3, generalizations of the Lyapunov type inequalities for pseudo-integrals are investigated. In Section 4, some conclusions are given. 2. Preliminaries In this section, we recall some basic definitions of pseudo-operations and pseudo-integrals, see [19,24,31]. Let ½a; b be a closed (in some cases it can be considered semiclosed) subinterval of ½1; 1. The full order on ½a; b is denoted by which can be the usual order of the real line or can be another order. Definition 2.1. A binary operation : ½a; b ½a; b ! ½a; b is called a pseudo-addition, for x; y; z; 0 (zero element) 2 ½a; b it satisfies the following requirements: (i) (ii) (iii) (iv)
x y ¼ y x; ðx yÞ z ¼ x ðy zÞ; x y ) x z y z; 0 x ¼ x.
Let ½a; bþ ¼ fxjx 2 ½a; b; 0 xg. Definition 2.2. A binary operation : ½a; b ½a; b ! ½a; b is called a pseudo-multiplication, for x; y; z; 1 (unit element) 2 ½a; b it satisfies the following requirements: (i) (ii) (iii) (iv) (v) (vi)
x y ¼ y x; ðx yÞ z ¼ x ðy zÞ; x y ) x z y z; ðx yÞ z ¼ ðx zÞ ðy zÞ; 1 x ¼ x; limn!1 xn and limn!1 yn exist and are finite ) limn!1 ðxn yn Þ ¼ lim xn lim yn . n!1
Definition 2.3. A set function m :
P
! ½a; bþ (or semiclosed interval) is a
n!1
r--measure if there holds:
(i) mð/Þ ¼ 0; S P 1 n (ii) m 1 , where 1 i¼1 Ai ¼ i¼1 mðAi Þ for any sequence fAi g of pairwise disjoint sets from i¼1 xi ¼ limn!1 i¼1 xi . Let m be a -measure, where has a monotone and continuous generator g, then g m is a r-additive measure. In the following, two important cases of pseudo integrals based on semiring ð½a; b; ; Þ are discussed. The first case is when pseudo-operations are generated by a monotone and continuous function g : ½a; b ! ½0; 1, i.e., pseudo-operations are given by: x y ¼ g 1 ðg ðxÞ þ g ðyÞÞ and x y ¼ g 1 ðg ðxÞg ðyÞÞ. If x y ¼ g 1 ðg ðxÞg ðyÞÞ, then
ðpÞ p x ¼ x|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} x x ¼ g 1 ðg ðxÞÞ : p
ðpÞ
Specially, is idempotent, then x ¼ x for any x 2 ½a; b and p 2 ð0; 1Þ. Thus, the pseudo-integral of the function f : X ! ½a; b is defined by
Z
f dm ¼ g 1
Z
X
ðg f Þdðg mÞ ;
X
where the integral applied on the right side is the standard Lebesgue integral. In fact, let m ¼ g 1 l; l is the standard Lebesgue measure on X, then we obtain
Z
f ðxÞdx ¼ g 1
Z
g ðf ðxÞÞdx :
X
X
In the second case, there are x y ¼ max ðx; yÞ and x y ¼ g 1 ðg ðxÞg ðyÞÞ, the pseudo-integral of the function f : R ! ½a; b is given by
Z R
f dm ¼ sup ðf ðxÞ wðxÞÞ:
66
D.-Q. Li et al. / Applied Mathematics and Computation 241 (2014) 64–69
where function w defines sup-measure m. Any sup-measure generated as essential supremum of a continuous density can be obtained as a limit of pseudo-additive measures with respect to generated pseudo-addition [19]. For arbitrary continuous function f : ½0; 1 ! ½0; 1, the integral R f dm can be obtained as a limit of g-integrals, see [19]. We denote l as the usual Lebesgue measure on R, We have
mðAÞ ¼ ess sup ðxjx 2 AÞ ¼ sup fajlðfxjx 2 A; x > agÞ > 0g:
Theorem 2.4 (Mesiar and Pap [19]). Let m be a sup-measure on ð½0; 1; Bð½0; 1ÞÞ, where Bð½0; 1Þ is the Borel r-algebra on ½0; 1; mðAÞ ¼ esssupl ðwðxÞjx 2 AÞ, and w : ½0; 1 ! ½0; 1 is a continuous density. Then, for any pseudo-addition with a generator g there exists a family fmk g of k -measure on ð½0; 1Þ; BÞ, where k is generated by g k (the function g of the power k), k 2 ð0; 1Þ, such that limk!1 mk ¼ m. Theorem 2.5 (Mesiar and Pap [19]). Let ð½0; 1; sup; Þ be a semiring, when is generated with g, i.e., we have x y ¼ g 1 ðg ðxÞg ðyÞÞ for every x; y 2 ð0; 1Þ. Let m be the same as in Theorem 2.4, Then, there exists a family fmk g of k -measure, where k is generated by g k ; k 2 ð0; 1Þ such that for every continuous function f : ½0; 1 ! ½0; 1,
Z
sup
f dm ¼ lim
Z
k
k!1
1 f dmk ¼ lim g k
Z
g k ðf ðxÞÞdx :
k!1
3. Generalization of the Lyapunov type inequality for pseudo-integrals In [7], the classical Lyapunov inequality provides the inequality
Z
1
f ðxÞs dl
rt
Z 6
0
1
rs Z f ðxÞt dl
0
1
f ðxÞr dl
st ;
0
where 0 < t < s < r; f : ½0; 1 ! ½0; 1Þ is an integrable function. In this section, we discuss Lyapunov inequality for pseudo-integrals based on the semiring ð½0; 1; ; Þ. Theorem 3.1. Let 0 < t < s < r; f : ½0; 1 ! ½0; 1 be measurable function and let a generator g : ½0; 1 ! ½0; 1 of the pseudo-addition and the pseudo-multiplication be an increasing function. Then for any d--measure it holds:
Z
!ðrtÞ
ðsÞ f
dm
Z
6
½0;1
!ðrsÞ ðt Þ f
dm
Z
½0;1
!ðstÞ ðr Þ f
dm
½0;1
:
Proof. We apply the classical Lyapunov inequality and then we obtain
Z
1
rt Z s ðg f Þ dðg mÞ 6
0
1
t
ðg f Þ dðg mÞ
rs Z
0
1
st
r
ðg f Þ dðg mÞ
:
0
Since function g is an increasing function, inverse function g 1 is also an increasing function, then there is
g 1
Z
1
rt ! Z s 6 g 1 ðg f Þ dðg mÞ
0
1
t
ðg f Þ dðg mÞ
rs
Z
0
1
r
ðg f Þ dðg mÞ
st ! :
0
Hence,
g 1
Z
1
s g g 1 ðg f Þ dðg mÞ
rt !
6 g 1
0
Z
1
rs Z t g g 1 ðg f Þ dðg mÞ
0
1
r g g 1 ðg f Þ dðg mÞ
st ! :
0
That is
g 1
Z
1
rt ! Z ðsÞ 6 g 1 g f dðg mÞ
0
1
rs Z ðt Þ g f dðg mÞ
0
1
st ! ðr Þ : g f dðg mÞ
0
Hence,
g 1
Z g g 1 0
1
rt ! Z ðsÞ 6 g 1 g f dðg mÞ g g 1
1 0
rs Z ðt Þ g f dðg mÞ g g 1 0
1
st ! ðr Þ : g f dðg mÞ
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D.-Q. Li et al. / Applied Mathematics and Computation 241 (2014) 64–69
i.e.,
g
1
Z
g
!!rt !
ðsÞ f dm
6g
1
g g
1
Z
g
½0;1
Z
ðsÞ
00 0 6 g 1 @@g @
f dm
½0;1
Z
Z
!ðrtÞ
ðsÞ f
ðt Þ f dm
g g
1
g
Z
½0;1
!ðrtÞ
!!rs !!!
dm
½0;1
Z
½0;1
ðt Þ f
6
dm
½0;1
:
!ðstÞ 111 ðr Þ AAA: f dm
!ðstÞ
ðr Þ f
dm
:
½0;1
½0;1
Z
!!st !!!! ðr Þ f dm
½0;1
!ðrsÞ 11 0 0 Z ðt Þ AA @g @ f dm
!ðrsÞ
This completes the proof. h Let us give an example to illustrate the Theorem 3.1. Example 3.2. Let g ðxÞ ¼ ln x, then
x y ¼ eln x ln y :
x y ¼ xy;
By Theorem 3.1, the following inequality holds
Z ln
1
eðln f ðxÞÞ
rt
s
Z 6 ln
0
1
eðln f ðxÞÞ
t
rs
Z þ ln
0
1
eðln f ðxÞÞ
r
st :
0
Theorem 3.3. Let 0 < t < s < r; f : ½0; 1 ! ½0; 1 be measurable function and let a generator g : ½0; 1 ! ½0; 1 of the pseudoaddition and the pseudo-multiplication be a decreasing function. Then for any d--measure it holds:
Z
!ðrtÞ
ðsÞ
f dm
½0;1
Z
!ðrsÞ
ðt Þ
f dm
P ½0;1
Z
!ðstÞ ðr Þ
f dm
:
½0;1
Proof. We apply the classical Lyapunov inequality and then we obtain
Z
1
s
ðg f Þ dðg mÞ
rt
Z
1
0
t
ðg f Þ dðg mÞ
6 0
rs Z
1
r
ðg f Þ dðg mÞ
st :
0
Since function g is a decreasing function, inverse function g 1 is also a decreasing function, then there is
g 1
Z
1
s
ðg f Þ dðg mÞ
rt !
P g 1
0
The next proof is similar to Theorem 3.1.
Z
1
rs Z t ðg f Þ dðg mÞ
0
1
r
ðg f Þ dðg mÞ
st ! :
0
h
Example 3.4. Let g ðxÞ ¼ 1 x. The corresponding pseudo-operations are x y ¼ x þ y 1 and x y ¼ x þ y xy. By Theorem 3.3, the following inequality holds rt
ðr tÞðsf ðxÞ f s ðxÞ 1Þ ðsf ðxÞ f s ðxÞ 1Þ
rs P ðr sÞ tf ðxÞ f t ðxÞ 1 tf ðxÞ f t ðxÞ 1 st
þ ðs t Þðrf ðxÞ f r ðxÞ 1Þ ðrf ðxÞ f r ðxÞ 1Þ rs ðr sÞ tf ðxÞ f t ðxÞ 1 tf ðxÞ f t ðxÞ 1 st ðs t Þðrf ðxÞ f r ðxÞ 1Þ ðrf ðxÞ f r ðxÞ 1Þ : In the following, we consider the second case of Lyapunov type inequalities for pseudo integral based on the semiring ð½0; 1; sup; Þ, where is generated by an increasing function g. Theorem 3.5. Let 0 < t < s < r; f : ½0; 1 ! ½0; 1 be measurable function and let be represented by an increasing generator g and m be a complete sup-measure, it holds:
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D.-Q. Li et al. / Applied Mathematics and Computation 241 (2014) 64–69
Z
sup
!ðrtÞ ðsÞ f
Z
dm
!ðrsÞ
sup
ðt Þ f
6
½0;1
dm
½0;1
Z
sup
!ðstÞ ðr Þ f
dm
½0;1
:
1 ¼ g 1 xk .
1
Proof. Since ðg k ðxÞÞ
1 k x y ¼ g 1 ðg ðxÞg ðyÞÞ ¼ g k g ðxÞg k ðyÞ ¼ xk y; in other words, g k is a generator of the . By Theorem 2.5, we obtain
Z
Z
sup
f dm ¼ lim
k!1
½0;1
1
ðsÞ
f dm
; g ; ðg Þ
Z
¼
½0;1
Z
k!1
k 1
k
!ðrtÞ
sup
1 f dmk ¼ lim g k
½0;1
As g is an increasing, so g
Z
k
sup
0
are also increasing function, then
!ðrtÞ Z k 1 k s 1 g g ð f Þ dm ¼ lim g k k!1
½0;1
¼
g k ðf ðxÞÞdx :
1
0
k!1
k!1
rt ! 1 s k g ðf ðxÞÞ dx :
Z
1 ¼ lim g k
ðrtÞ 1 s gk g k ð f ð xÞ Þ dx
ðrtÞ Z 1 s 1 k 1 g k ðf ðxÞÞ dx ¼ gk g lim g k
Z
k!1
gk 0
1 lim g k
1
Z
k!1
rs Z t g ðf ðxÞÞ dx
Z
k!1
st ! g ðf ðxÞÞ dx r
k
0
rs 1 t gk gk g k ð f ð xÞ Þ dx
1 0
Z
1
st ! 1 r k k k g g g ðf ðxÞÞ dx
0
1 ¼ lim g k
Z
k!1
rs Z ðt Þ g k f dx
1 0
1
! st ðr Þ g k f dx
0
1 k k 1 g g ¼ lim g k
Z
k!1
gk
1
k
0
1 ¼ lim g k
1
1
ðt Þ
g k f
rs dx
!!
0
Z
k 1 g
!!! st ðr Þ k g f dx
1
0
1 ¼ lim g k
Z
k!1
1
! Z rs 1 ðt Þ g f dx gk
0
1 ¼ gk
0
Z 1 g k lim g k k!1
Z
gk
sup
1
½0;1
ðr Þ f
st dx
!
! st ðr Þ g f dx k
0
!!rs ! ðt Þ
f dm
1 gk
½0;1 sup
0
k!1
1 ¼ gk
¼
g
k
Z 1 rs ! k 1 ðt Þ k g lim g g k f dx
1 gk
Z
1
k
ðt Þ
Z
sup
!!st ! ðr Þ
f dm
½0;1
!ðrsÞ f dm
gk
Z
sup
½0;1
!ðstÞ ðr Þ
f dm
g k ðf ðxÞÞ
s
dx
0
ð1Þ
0
By the classical Lyapunov inequality, the last equality (1)
1 6 lim g k
1
:
The proof is now completed. h Example 3.6. Let g k ðxÞ ¼ ekx , the corresponding pseudo-operations are
D.-Q. Li et al. / Applied Mathematics and Computation 241 (2014) 64–69
x y ¼ lim
1 ln ekx þ eky ¼ max ðx; yÞ k
x y ¼ lim
1 ln ekx eky ¼ x þ y: k
k!1
69
and
k!1
By Theorem 3.1, the following inequality holds
ðr tÞ sup ðsf ðxÞ þ wðxÞÞ 6 ððr sÞ sup ðtf ðxÞ þ wðxÞÞÞ þ ððs tÞ sup ðrf ðxÞ þ wðxÞÞÞ Remark 3.7. The third important case when ¼ max and ¼ min has been studied in [18] and the pseudo integral in such case yields the Sugeno integral.
4. Conclusion In this paper, We have proved two classes of the Lyapunov type inequalities for pseudo-integral. For further investigation, we will investigate other integral inequalities for Pseudo-integral. Acknowledgments The work in this paper was supported by ‘‘the National Natural Science Foundation of China (51374199)’’ and ‘‘the Fundamental Research Funds for the Central Universities (2013XK03)’’. We are grateful to the referees for their critical comments and helpful suggestions in improving our paper. References [1] H. Agahi, R. Mesiar, Y. Ouyang, E. Pap, M. Štrboja, Berwald type inequality for Sugeno integral, Appl. Math. Comput. 217 (2010) 4100–4108. [2] H. Agahi, R. Mesiar, Y. Ouyang, General Minkowski type inequalities for Sugeno integrals, Fuzzy Sets Syst. 161 (2010) 708–715. [3] H. Agahi, R. Mesiar, Y. Ouyang, New general extensions of Chebyshev type inequalities for Sugeno integrals, Int. J. Approximate Reasoning 51 (2009) 135–140. [4] H. Agahi, H. Román-Flores, A. Flores-Franulicˇ, General Barnes–Godunova–Levin type inequalities for Sugeno integral, Inf. Sci. 181 (2011) 1072–1079. [5] H. Agahi, R. Mesiar, Y. Ouyang, Chebyshev type inequalities for pseudo-integrals, Nonlinear Anal. 72 (2010) 2737–2743. [6] H. Agahi, Y. Ouyang, R. Mesiar, E. Pap, M. Štrboja, Höder and Minkowski type inequalities for pseudo-integral, Appl. Math. Comput. 217 (2011) 8630– 8639. [7] P.S. Bullen, A Dictionary of Inequalities, Addison Wesley Longman Limited, 1998. [8] J. Caballero, K. Sadarangani, A Cauchy–Schwarz type inequality for fuzzy integrals, Nonlinear Anal. 73 (2010) 3329–3335. [9] J. Caballero, K. Sadarangani, Chebyshev inequality for Sugeno integrals, Fuzzy Sets Syst. 161 (2010) 1480–1487. [10] J. Caballero, K. Sadarangani, Fritz Carlson’s inequality for fuzzy integrals, Comput. Math. Appl. 59 (2010) 2763–2767. [11] J. Caballero, K. Sadarangani, Sandors inequality for Sugeno integrals, Appl. Math. Comput. 218 (2011) 1617–1622. [12] B. Daraby, Generalization of the Stolarsky type inequality for pseudo-integrals, Fuzzy Sets Syst. 194 (2012) 90–96. [13] A. Flores-Franulicˇ, H. Román-Flores, A Chebyshev type inequality for fuzzy integrals, Appl. Math. Comput. 190 (2007) 1178–1184. [14] A. Flores-Franulicˇ, H. Román-Flores, Y. Chalco-Cano, A note on fuzzy integral inequality of Stolarsky type, Appl. Math. Comput. 196 (2008) 55–59. [15] A. Flores-Franulicˇ, H. Román-Flores, Y. Chalco-Cano, Markov type inequalities for fuzzy integrals, Appl. Math. Comput. 207 (2009) 242–247. [16] B. Girotto, S. Holzer, A Chebyshev type inequality for Sugeno integral and comonotonicity, Int. J. Approximate Reasoning 52 (2011) 444–448. [17] B. Girotto, S. Holzer, Chebyshev type inequality for Choquet integral and comonotonicity, Int. J. Approximate Reasoning 52 (2011) 1118–1123. [18] D. Hong, A Liapunov type inequality for Sugeno integrals, Nonlinear Anal. 74 (2011) 7296–7303. [19] R. Mesiar, E. Pap, Idempotent integral as limit of g-integrals, Fuzzy Sets Syst. 102 (2009) 385–392. [20] R. Mesiar, Y. Ouyang, General Chebyshev type inequalities for Sugeno integrals, Fuzzy Sets Syst. 160 (2009) 58–64. [21] Y. Ouyang, R. Mesiar, H. Agahi, An inequality related to Minkowski type for Sugeno integrals, Inf. Sci. 180 (2010) 2793–2801. [22] Y. Ouyang, J. Fang, L. Wang, Fuzzy Chebyshev type inequality, Int. J. Approximate Reasoning 48 (2008) 829–835. [23] Y. Ouyang, J. Fang, Sugeno integral of monotone functions based on Lebesgue measure, Comput. Math. Appl. 56 (2008) 367–374. [24] E. Pap, Generalized real analysis and its applications, Int. J. Approximate Reasoning 47 (2008) 368–386. [25] E. Pap, M. Šrboja, Generalization of the Jensen inequality for pseudo-integral, Inf. Sci. 180 (2010) 543–548. [26] H. Román-Flores, A. Flores-Franulicˇ, Y. Chalco-Cano, A convolution type inequality for fuzzy integrals, Appl. Math. Comput. 195 (2008) 94–99. [27] H. Román-Flores, A. Flores-Franulicˇ, Y. Chalco-Cano, A Hardy-type inequality for fuzzy integrals, Appl. Math. Comput. 204 (2008) 178–183. [28] H. Román-Flores, A. Flores-Franulicˇ, Y. Chalco-Cano, A Jensen type inequality for fuzzy integrals, Inf. Sci. 177 (2007) 3192–3201. [29] H. Román-Flores, Y. Chalco-Cano, Sugeno integral and geometric inequalities, Int. J. Uncertainty 1 (2007) 1–11. [30] H. Román-Flores, A. Flores-Franulicˇ, Y. Chalco-Cano, The fuzzy integral for monotone functions, Appl. Math. Comput. 185 (2007) 492–498. [31] M. Sugeno, T. Murofushi, Pseudo-additive measures and integrals, J. Math. Anal. Appl. 122 (1987) 197–222.
