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## Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(2): 124-128

Published online June 25, 2020

https://doi.org/10.5391/IJFIS.2020.20.2.124

© The Korean Institute of Intelligent Systems

## An Improved Bushell-Okarasinski Type Inequality for Sugeno Integrals

Dug Hun Hong

Department of Mathematics, Myongji University, Yongin, Korea

Correspondence to :
Dug Hun Hong (dhhong@mju.ac.kr)

Received: September 27, 2018; Revised: April 11, 2020; Accepted: April 24, 2020

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Recently, Roman-Flores et al. (2008) proposed a Bushell-Okrasinski type inequality for fuzzy integrals. In this paper, we improve the result of Roman-Flores et al. by findings the optimal constant Hs for which the following Bushell-Okrasinski type inequality for fuzzy integrals Hs((S)01(1-t)s-1g(t)sdμ)((S)01g(t)dμ)s holds, where s ≥ 1, g : [0, 1] → [0, ∞) is a non-increasing function. The case of nondecreasing function is also treated. The result of Roman-Flores et al. is a special case of our result.

Keywords: Fuzzy measure, Sugeno integral, Bushell-Okrasinski type inequality

### 1. Introduction and Preliminaries

A number of studies have examined the Sugeno integral since its introduction in 1974 [1]. Ralescu and Adams [2] generalized a range of fuzzy measures and gave several equivalent definitions of fuzzy integrals. Wang and Klir [3] provided an overview of fuzzy measure theory.

Caballero and Sadarangani [46] proved a Hermite-Hadamarda type inequality, a Cauchy type inequality, and Fritz Carlson’s inequality for fuzzy integrals. Roman-Flores et al. [710] presented several new types of inequalities for Sugeno integrals, including a Prekopa-Leindler type inequality, a Jensen type inequality, a Young type inequality, and some convolution type inequalities. Flores-Franulic et al. [11, 12] presented Chebyshev’s inequality and Stolarsky’s inequality for fuzzy integrals. Ouyang and Fang [13] generalized their main results to prove some optimal upper bounds for the Sugeno integral of the monotone function in [9]. Ouyang et al. [14] generalized a Chebyshev type inequality for the fuzzy integral of monotone functions based on an arbitrary fuzzy measure. Hong [15] extended previous research by presenting a Hardy-type inequality for Sugeno integrals, Hong [16] proposed a Liapunov type inequality for Sugeno integrals, and Hong [17] proposed a Berward and Favard type inequalities for fuzzy integrals. Hong et al. [19] considered Steffensen’s integral inequality for Sugeno integral. Recently, Roman-Flores et al. [10] showed a Bushell-Okrasinski type inequality for fuzzy integrals, but this inequality is not optimal. In this paper, we improve the results of Roman-Flores et al. [10] for s ≥ 1. Specifically, we find an optimal constant for which a Bushell-Okrasinski type inequality for Sugeno integrals holds for non-increasing functions.

### Definition 1

Let Σ be a σ-algebra of subsets of ℝ and let μ : Σ → [0, ∞] be a non-negative, extended real-valued set function. We say that μ is a fuzzy measure if and only if

• μ(∅︀) = 0.

• E, F ∈ Σ and EF imply μ(E) ≤ μ(F) (monotonicity).

• {Ep} ⊆ Σ and E1E2 ⊆ ⋯ imply limpμ(Ep)=μ(p=1Ep) (continuity form below).

• {Ep} ⊆ Σ, E1E2 ⊇ ⋯, and μ(E1) < ∞ imply limpμ(Ep)=μ(p=1Ep) (continuity form above).

If f is a non-negative real-valued function defined on ℝ, then we denote by Fα = {xX|f(x) ≥ α} = {fα} the α-level of f, for α > 0, and F0={xXf(x)>0}¯=supp(f) is the support of f.

We note that

αβ{fβ}{fα}.

If μ is a fuzzy measure on A ⊂ ℝ, then we define the following:

Fμ(A)={f:A[0,)fis μ-measurable}.

### Definition 2

Let μ be a fuzzy measure on (ℝ, Σ). If and A ∈ Σ, then the Sugeno integral (or the fuzzy integral) of f on A, with respect to the fuzzy measure μ, is defined as

(S)Afdμ=supα[0,)[αμ(AFα)].

In particular, if A = X then

(S)fdμ=(S)fdμ=supα[0,)[αμ(Fα)].

The following properties of the Sugeno integral are well known and can be found in [3]:

### Proposition 1 [3]

If μ is a fuzzy measure on ℝ and , then

• (S) ∫A fdμ μ(A);

• (S) ∫A Kdμ = Kμ(A) for any constant K ∈ [0, ∞);

• (S) ∫A fdμ ≤ (S) ∫A gdμ, if fg on A;

• μ (A ∩ {fα}) ≥ α ⇒ (S) ∫A fdμ α;

• μ (A ∩ {fα}) ≤ α ⇒ (S) ∫A fdμ α;

• (S) ∫A fdμ < α ⇔ there exists γ < α such that μ (A ∩ {fγ}) < α;

• (S) ∫A fdμ > α ⇔ there exists γ > α such that μ (A ∩ {fγ}) > α.

### Note 1

Let F(α) = μ (A ∩ {fα}). Then by Proposition 1, (iv), (v),

F(α)=α(S)01f(x)dμ=α.

### Theorem 1 [13]

Let f : [0, ∞) → [0, ∞) be continuous and non-increasing or non-decreasing functions and μ be the Lebesgue measure on ℝ. Let (S)0af(x)dμ=p. If 0 < p < a, then f(p) = p and f(ap) = p, respectively.

### 2. Bushell-Okrasinski Type Inequality

The classical Bushell-Okrasinski inequality provides the following inequality [19]:

0x(x-t)s-1g(t)sdμ(0xg(t)dμ)s,0xb,

where 1 ≤ s, g : [0, 1] → [0, ∞) is a continuous and increasing function. After changing the variable t = sx, Malamud [20] analyzed the Bushell-Okrasinski inequality in the following new form:

s01(1-t)s-1g(t)sdμ(01g(t)dμ)s.

Recently, Roman-Flores et al. [10] showed a Bushell-Okrasinski inequality derived from (1) for Sugeno integrals as follows:

### Theorem 2 (Roman-Flores et al.) [10]

Let g : [0, 1] → [0, ∞) be a continuous decreasing function and let μ be the Lebesgue measure on ℝ. Then

s(S)01(1-t)s-1g(t)sdμ((S)01g(t)dμ)s

holds for all 2 ≤ s.

We now improve the above result.

### Theorem 3 (Fuzzy Bushell-Okrasinski inequality)

Let g : [0, 1] → [0, ∞) be a continuous non-increasing function and let μ be the Lebesgue measure on ℝ. Then

((S)01(1-t)s-1dμ)-1((S)01(1-t)s-1g(t)sdμ)((S)01g(t)dμ)s

holds for all 1 ≤ s.

Proof

Let

Hs=sup{((S)01g(t)dμ)s(S)01(1-t)s-1g(t)sdμ|gG},

where is the set of functions which are non-increasing on [0, 1] and let

Hs(α)=sup{αs(S)01(1-t)s-1g(t)sdμ|gGα},

where Gα={gG|(S)01g(t)dμ=α} for α ∈ [0, 1]. Then

Hs=sup0α1Hs(α).

We consider Hs(α). Let

g0(x)={α,if x[0,α)),0,if x[α,1].

Because μ{g0α} = α, by Note 1

(S)01g0(x)dμ=α.

Then it is easy to check that . Thus, we have

Hs(α)=αs(S)01(1-t)s-1g0(t)sdμ.

Now, let

(S)01(1-t)s-1g0(t)sdμ=t0.

Because g0 is continuous and decreasing on [0, α] and the left limit of g0 at α is less than α, by Theorem 1,

t0=(1-t0)s-1αs.

To find Hs we now consider the following optimization problem:

Hs=sup0<α1Hs(α)=Maximizeαst,

where

t=(1-t)s-1αs,0<α1.

We first show that Hs(α)=αst is an increasing function of α. We have

Hs(α)=αs-1t2(st-αdtdα).

From the equation t = (1 − t)s−1αs, we now have

sαs-1=(1-t)s-1+t(s-1)(1-t)s-2(1-t)2s-2dtdα=1+t(s-1)(1-t)-1(1-t)s-1dtdα,

which implies that

αdtdα=11+t(s-1)(1-t)-1stst.

Then Hs(α)0, that is, Hs is nondecreasing and thus

Hs=Hs(1)=1(S)01(1-t)s-1dμ,

which completes the proof.

### Note 2

As shown in the proof of Theorem 3, the continuity assumption of g is not needed.

### Note 3

If g(t) = 1 in the inequality of Theorem 3, then the equality holds. Therefore, we see that the inequality in Theorem 3 is optimal.

### Lemma 1

Let (S)01(1-t)s-1dμ=x*, s ≥ 2. Then

1s1s(2(s-1)s-1(s-1)s-1+ss-2)<x*.
Proof

Suppose that f(t) = (1 − t)s−1t. Then by Theorem 1, f(x*) = 0. We note that

f(t)=-(s-1)(1-t)s-2-1<0,f(0)=-s,

and

f(t)=(s-1)(s-2)(1-t)s-30.

Because f is decreasing and convex, we have

1s=-f(0)f(0)<x*,

and similarly, we have

1s1s-f(1s)f(1s)<x*

and

1s-f(1s)f(1s)=1s-(1-1s)s-1-1s1-(s-1)(1-1s)s-2=1s(2(s-1)s-1(s-1)s-1+ss-2),

which completes the proof.

The following result of Roman-Flores et al. [10] is a special case of our results.

### Corollary 1 (Roman-Flores et al. [?])

Let g : [0, 1] → [0, ∞) be a non-increasing function and let μ be the Lebesgue measure on ℝ. Then

s((S)01(1-t)s-1g(t)sdμ)((S)01g(t)dμ)s

holds for all 2 ≤ s.

### Corollary 2

Let g : [0, 1] → [0, ∞) be a non-increasing function and that μ is the Lebesgue measure on ℝ. Then

s((s-1)s-1+ss-22(s-1)s-1)((S)01(1-t)s-1g(t)sdμ)((S)01g(t)dμ)s

holds for all 2 ≤ s.

### Example 1

We compare our result with that of Roman-Flores et al. [10] for s = 2, 3, 4, 5, 10, 20. That is,

s((S)01(1-t)s-1dμ)-1
22.000
32.618
43.148
53.630
105.692
209.110

Our results indicates much lower than that of Roman-Flores et al. [10].

The case of a non-increasing function is similar.

### Theorem 4

Let g : [0, 1] → [0, ∞) be a non-decreasing function and let μ be the Lebesgue measure on ℝ. Then

((S)01ts-1dμ)-1((S)01ts-1g(t)sdμ)((S)01g(t)dμ)s

holds for all 1 ≥ s.

### Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B027869).

### Conflict of Interest

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2. D. Ralescu, G. Adams, "The fuzzy integral," Journal of Mathematical Analysis and Applications, vol. 75, no. 2, pp. 562-570, 1980. https://doi.org/10.1016/0022-247X(80)90101-8
3. Z. Wang, G. J. Klir, Fuzzy Measure Theory. New York, NY: Plenum, 1992.
4. J. Caballero, K. Sadarangani, "Hermite–Hadamard inequality for fuzzy integrals," Applied Mathematics and Computation, vol. 215, no. 6, pp. 2134-2138, 2009. https://doi.org/10.1016/j.amc.2009.08.006
5. J. Caballero, K. Sadarangani, "Fritz Carlson’s inequality for fuzzy integrals," Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2763-2767, 2010. https://doi.org/10.1016/j.camwa.2010.01.045
6. J. Caballero, K. Sadarangani, "A Cauchy–Schwarz type inequality for fuzzy integrals," Nonlinear Analysis: Theory, Methods and Applications, vol. 73, no. 10, pp. 3329-3335, 2010. https://doi.org/10.1016/j.na.2010.07.013
7. H. Roman-Flores, A. Flores-Franulic, Y. Chalco-Cano, "A Hardy-type inequality for fuzzy integrals," Applied Mathematics and Computation, vol. 204, no. 1, pp. 178-183, 2008. https://doi.org/10.1016/j.amc.2008.06.027
8. H. Roman-Flores, A. Flores-Franulic, Y. Chalco-Cano, "A Jensen type inequality for fuzzy integrals," Information Sciences, vol. 177, no. 15, pp. 3192-3201, 2007. https://doi.org/10.1016/j.ins.2007.02.006
9. H. Roman-Flores, A. Flores-Franulic, Y. Chalco-Cano, "The fuzzy integral for monotone functions," Applied Mathematics and Computation, vol. 185, no. 1, pp. 492-498, 2007. https://doi.org/10.1016/j.amc.2006.07.066
10. H. Roman-Flores, A. Flores-Franulic, Y. Chalco-Cano, "A convolution type inequality for fuzzy integrals," Applied Mathematics and Computation, vol. 195, no. 1, pp. 94-99, 2008. https://doi.org/10.1016/j.amc.2007.04.072
11. A. Flores-Franulic, H. Roman-Flores, "A Chebyshev type inequality for fuzzy integrals," Applied Mathematics and Computation, vol. 190, no. 2, pp. 1178-1184, 2007. https://doi.org/10.1016/j.amc.2007.02.143
12. A. Flores-Franulic, H. Roman-Flores, Y. Chalco-Cano, "A note on fuzzy integral inequality of Stolarsky type," Applied Mathematics and Computation, vol. 196, no. 1, pp. 55-59, 2008. https://doi.org/10.1016/j.amc.2007.05.032
13. Y. Ouyang, J. Fang, "Sugeno integral of monotone functions based on Lebesgue measure," Computers & Mathematics with Applications, vol. 56, no. 2, pp. 367-374, 2008. https://doi.org/10.1016/j.camwa.2007.11.044
14. Y. Ouyang, J. Fang, L. Wang, "Fuzzy Chebyshev type inequality," International Journal of Approximate Reasoning, vol. 48, no. 3, pp. 829-835, 2008. https://doi.org/10.1016/j.ijar.2008.01.004
15. D. H. Hong, "A sharp Hardy-type inequality of Sugeno integrals," Applied Mathematics and Computation, vol. 217, no. 1, pp. 437-440, 2010. https://doi.org/10.1016/j.amc.2010.05.071
16. D. H. Hong, "A Liapunov type inequality for Sugeno integrals," Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 18, pp. 7296-7303, 2011. https://doi.org/10.1016/j.na.2011.07.046
17. D. H. Hong, "Berwald and Favard type inequalities for fuzzy integrals," International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 24, no. 1, pp. 47-58, 2016. https://doi.org/10.1142/S0218488516500033
18. D. H. Hong, E. L. Moon, J. D. Kim, "Steffensen’s integral inequality for the Sugeno integral," International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 22, no. 2, pp. 235-241, 2014. https://doi.org/10.1142/S0218488514500111
19. P. J. Bushell, W. Okrasinski, "Nonlinear Volterra integral equations with convolution kernel," Journal of the London Mathematical Society, vol. s2–41, no. 3, pp. 503-510, 1990. https://doi.org/10.1112/jlms/s2-41.3.503
20. S. M. Malamud, "Some complements to the Jensen and Chebyshev inequalities and a problem of W. Walter," Proceedings of the American Mathematical Society, vol. 129, pp. 2671-2678, 2001. https://doi.org/10.1090/S0002-9939-01-05849-X

Dug Hun Hong received the B.S. and M.S. degrees in Mathematics from Kyungpook National University, Taegu Korea in 1981 and 1983, respectively. He received the M.S. and Ph.D. degrees from the University of Minnesota in 1988 and 1990, respectively. His research interests include probability theory and fuzzy theory with applications.

E-mail: dhhong@mju.ac.kr

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(2): 124-128

Published online June 25, 2020 https://doi.org/10.5391/IJFIS.2020.20.2.124

## An Improved Bushell-Okarasinski Type Inequality for Sugeno Integrals

Dug Hun Hong

Department of Mathematics, Myongji University, Yongin, Korea

Correspondence to:Dug Hun Hong (dhhong@mju.ac.kr)

Received: September 27, 2018; Revised: April 11, 2020; Accepted: April 24, 2020

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Recently, Roman-Flores et al. (2008) proposed a Bushell-Okrasinski type inequality for fuzzy integrals. In this paper, we improve the result of Roman-Flores et al. by findings the optimal constant Hs for which the following Bushell-Okrasinski type inequality for fuzzy integrals Hs((S)01(1-t)s-1g(t)sdμ)((S)01g(t)dμ)s holds, where s ≥ 1, g : [0, 1] → [0, ∞) is a non-increasing function. The case of nondecreasing function is also treated. The result of Roman-Flores et al. is a special case of our result.

Keywords: Fuzzy measure, Sugeno integral, Bushell-Okrasinski type inequality

### 1. Introduction and Preliminaries

A number of studies have examined the Sugeno integral since its introduction in 1974 [1]. Ralescu and Adams [2] generalized a range of fuzzy measures and gave several equivalent definitions of fuzzy integrals. Wang and Klir [3] provided an overview of fuzzy measure theory.

Caballero and Sadarangani [46] proved a Hermite-Hadamarda type inequality, a Cauchy type inequality, and Fritz Carlson’s inequality for fuzzy integrals. Roman-Flores et al. [710] presented several new types of inequalities for Sugeno integrals, including a Prekopa-Leindler type inequality, a Jensen type inequality, a Young type inequality, and some convolution type inequalities. Flores-Franulic et al. [11, 12] presented Chebyshev’s inequality and Stolarsky’s inequality for fuzzy integrals. Ouyang and Fang [13] generalized their main results to prove some optimal upper bounds for the Sugeno integral of the monotone function in [9]. Ouyang et al. [14] generalized a Chebyshev type inequality for the fuzzy integral of monotone functions based on an arbitrary fuzzy measure. Hong [15] extended previous research by presenting a Hardy-type inequality for Sugeno integrals, Hong [16] proposed a Liapunov type inequality for Sugeno integrals, and Hong [17] proposed a Berward and Favard type inequalities for fuzzy integrals. Hong et al. [19] considered Steffensen’s integral inequality for Sugeno integral. Recently, Roman-Flores et al. [10] showed a Bushell-Okrasinski type inequality for fuzzy integrals, but this inequality is not optimal. In this paper, we improve the results of Roman-Flores et al. [10] for s ≥ 1. Specifically, we find an optimal constant for which a Bushell-Okrasinski type inequality for Sugeno integrals holds for non-increasing functions.

### Definition 1

Let Σ be a σ-algebra of subsets of ℝ and let μ : Σ → [0, ∞] be a non-negative, extended real-valued set function. We say that μ is a fuzzy measure if and only if

• μ(∅︀) = 0.

• E, F ∈ Σ and EF imply μ(E) ≤ μ(F) (monotonicity).

• {Ep} ⊆ Σ and E1E2 ⊆ ⋯ imply $limp→∞ μ(Ep)=μ (∪p=1∞Ep)$ (continuity form below).

• {Ep} ⊆ Σ, E1E2 ⊇ ⋯, and μ(E1) < ∞ imply $limp→∞ μ(Ep)=μ (∩p=1∞Ep)$ (continuity form above).

If f is a non-negative real-valued function defined on ℝ, then we denote by Fα = {xX|f(x) ≥ α} = {fα} the α-level of f, for α > 0, and $F0={x∈X∣f(x)>0}¯=supp(f)$ is the support of f.

We note that

$α≤β⇒{f≥β}⊆{f≥α}.$

If μ is a fuzzy measure on A ⊂ ℝ, then we define the following:

$Fμ(A)={f:A→[0,∞)∣f is μ-measurable}.$

### Definition 2

Let μ be a fuzzy measure on (ℝ, Σ). If and A ∈ Σ, then the Sugeno integral (or the fuzzy integral) of f on A, with respect to the fuzzy measure μ, is defined as

$(S)∫Afdμ=supα∈[0,∞)[α∧μ(A∩Fα)].$

In particular, if A = X then

$(S)∫ℝfdμ=(S)∫fdμ=supα∈[0,∞)[α∧μ(Fα)].$

The following properties of the Sugeno integral are well known and can be found in [3]:

### Proposition 1 [3]

If μ is a fuzzy measure on ℝ and , then

• (S) ∫A fdμ μ(A);

• (S) ∫A Kdμ = Kμ(A) for any constant K ∈ [0, ∞);

• (S) ∫A fdμ ≤ (S) ∫A gdμ, if fg on A;

• μ (A ∩ {fα}) ≥ α ⇒ (S) ∫A fdμ α;

• μ (A ∩ {fα}) ≤ α ⇒ (S) ∫A fdμ α;

• (S) ∫A fdμ < α ⇔ there exists γ < α such that μ (A ∩ {fγ}) < α;

• (S) ∫A fdμ > α ⇔ there exists γ > α such that μ (A ∩ {fγ}) > α.

### Note 1

Let F(α) = μ (A ∩ {fα}). Then by Proposition 1, (iv), (v),

$F(α)=α⇒(S)∫01f(x)dμ=α.$

### Theorem 1 [13]

Let f : [0, ∞) → [0, ∞) be continuous and non-increasing or non-decreasing functions and μ be the Lebesgue measure on ℝ. Let $(S)∫0af(x)dμ=p$. If 0 < p < a, then f(p) = p and f(ap) = p, respectively.

### 2. Bushell-Okrasinski Type Inequality

The classical Bushell-Okrasinski inequality provides the following inequality [19]:

$∫0x(x-t)s-1g(t)sdμ≤(∫0xg(t)dμ)s, 0≤x≤b,$

where 1 ≤ s, g : [0, 1] → [0, ∞) is a continuous and increasing function. After changing the variable t = sx, Malamud [20] analyzed the Bushell-Okrasinski inequality in the following new form:

$s∫01(1-t)s-1g(t)sdμ≤(∫01g(t)dμ)s.$

Recently, Roman-Flores et al. [10] showed a Bushell-Okrasinski inequality derived from (1) for Sugeno integrals as follows:

### Theorem 2 (Roman-Flores et al.) [10]

Let g : [0, 1] → [0, ∞) be a continuous decreasing function and let μ be the Lebesgue measure on ℝ. Then

$s(S)∫01(1-t)s-1g(t)sdμ≤((S)∫01g(t)dμ)s$

holds for all 2 ≤ s.

We now improve the above result.

### Theorem 3 (Fuzzy Bushell-Okrasinski inequality)

Let g : [0, 1] → [0, ∞) be a continuous non-increasing function and let μ be the Lebesgue measure on ℝ. Then

$((S)∫01(1-t)s-1dμ)-1((S)∫01(1-t)s-1g(t)sdμ)≥((S)∫01g(t)dμ)s$

holds for all 1 ≤ s.

Proof

Let

$Hs=sup {((S)∫01g(t)dμ)s(S)∫01(1-t)s-1g(t)sdμ|g∈G},$

where is the set of functions which are non-increasing on [0, 1] and let

$Hs(α)=sup {αs(S)∫01(1-t)s-1g(t)sdμ|g∈Gα},$

where $Gα={g∈G|(S)∫01g(t)dμ=α}$ for α ∈ [0, 1]. Then

$Hs=sup0≤α≤1Hs(α).$

We consider Hs(α). Let

$g0(x)={α,if x∈[0,α)),0,if x∈[α,1].$

Because μ{g0α} = α, by Note 1

$(S)∫01g0(x)dμ=α.$

Then it is easy to check that . Thus, we have

$Hs(α)=αs(S)∫01(1-t)s-1g0(t)sdμ.$

Now, let

$(S)∫01(1-t)s-1g0(t)sdμ=t0.$

Because g0 is continuous and decreasing on [0, α] and the left limit of g0 at α is less than α, by Theorem 1,

$t0=(1-t0)s-1αs.$

To find Hs we now consider the following optimization problem:

$Hs=sup0<α≤1 Hs(α)=Maximize αst,$

where

$t=(1-t)s-1αs, 0<α≤1.$

We first show that $Hs(α)=αst$ is an increasing function of α. We have

$Hs′(α)=αs-1t2 (st-αdtdα).$

From the equation t = (1 − t)s−1αs, we now have

$sαs-1=(1-t)s-1+t(s-1)(1-t)s-2(1-t)2s-2dtdα=1+t(s-1)(1-t)-1(1-t)s-1dtdα,$

which implies that

$αdtdα=11+t(s-1)(1-t)-1st≤st.$

Then $Hs′(α)≥0$, that is, Hs is nondecreasing and thus

$Hs=Hs(1)=1(S)∫01(1-t)s-1dμ,$

which completes the proof.

### Note 2

As shown in the proof of Theorem 3, the continuity assumption of g is not needed.

### Note 3

If g(t) = 1 in the inequality of Theorem 3, then the equality holds. Therefore, we see that the inequality in Theorem 3 is optimal.

### Lemma 1

Let $(S)∫01(1-t)s-1dμ=x*$, s ≥ 2. Then

$1s≤1s (2(s-1)s-1(s-1)s-1+ss-2)
Proof

Suppose that f(t) = (1 − t)s−1t. Then by Theorem 1, f(x*) = 0. We note that

$f′(t)=-(s-1)(1-t)s-2-1<0, f′(0)=-s,$

and

$f″(t)=(s-1)(s-2)(1-t)s-3≥0.$

Because f is decreasing and convex, we have

$1s=-f(0)f′(0)

and similarly, we have

$1s≤1s-f(1s)f′(1s)

and

$1s-f(1s)f′(1s)=1s-(1-1s)s-1-1s1-(s-1)(1-1s)s-2=1s (2(s-1)s-1(s-1)s-1+ss-2),$

which completes the proof.

The following result of Roman-Flores et al. [10] is a special case of our results.

### Corollary 1 (Roman-Flores et al. [?])

Let g : [0, 1] → [0, ∞) be a non-increasing function and let μ be the Lebesgue measure on ℝ. Then

$s ((S)∫01(1-t)s-1g(t)sdμ)≥((S)∫01g(t)dμ)s$

holds for all 2 ≤ s.

### Corollary 2

Let g : [0, 1] → [0, ∞) be a non-increasing function and that μ is the Lebesgue measure on ℝ. Then

$s ((s-1)s-1+ss-22(s-1)s-1)((S)∫01(1-t)s-1g(t)sdμ)≥((S)∫01g(t)dμ)s$

holds for all 2 ≤ s.

### Example 1

We compare our result with that of Roman-Flores et al. [10] for s = 2, 3, 4, 5, 10, 20. That is,

s((S)01(1-t)s-1dμ)-1
22.000
32.618
43.148
53.630
105.692
209.110

Our results indicates much lower than that of Roman-Flores et al. [10].

The case of a non-increasing function is similar.

### Theorem 4

Let g : [0, 1] → [0, ∞) be a non-decreasing function and let μ be the Lebesgue measure on ℝ. Then

$((S)∫01ts-1dμ)-1 ((S)∫01ts-1g(t)sdμ)≥((S)∫01g(t)dμ)s$

holds for all 1 ≥ s.

s((S)01(1-t)s-1dμ)-1
22.000
32.618
43.148
53.630
105.692
209.110

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