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Barnes-Godunova-Levin type inequality of the Sugeno integral for an \((\alpha,m)\)-concave function
Journal of Inequalities and Applications volume 2015, Article number: 25 (2015)
Abstract
In this paper, a Barnes-Godunova-Levin type inequality for the Sugeno integral based on an \(( {\alpha,m} )\)-concave function is proved. Some examples are given to illustrate the validity of these inequalities. Finally, several important results, as special cases of an \(( {\alpha,m} )\)-concave function, are also obtained.
1 Introduction
As a tool for modeling non-deterministic problems, the theory of fuzzy measures and fuzzy integrals was introduced by Sugeno in [1]. Many authors generalized the Sugeno integral by using some other operators to replace the special operator(s) ∨ and/or ∧ and introduced Choquet-like integral [2], Shilkret integral [3], ⊥-integral [4], and pseudo-integral [5]. Suárez and Gil [6] presented two families of fuzzy integrals, the so-called seminormed fuzzy integral and semiconormed fuzzy integral. Wang and Klir [7] provided a general overview on fuzzy measurement and fuzzy integration.
Recently, Flores-Franulič et al. [8–21] generalized several classical integral inequalities of the Sugeno integral. Agahi et al. [22] proved a general Barnes-Godunova-Levin type inequality of the Sugeno integral for a concave function. In [23], Mihesan introduced the concept of \(( {\alpha,m} )\)-convex function. For recent results and generalizations concerning m-convex and \(( {\alpha,m} )\)-convex functions, we refer to [24–26]. The purpose of this paper is to prove a Barnes-Godunova-Levin type inequality for the Sugeno integral based on an \(( {\alpha,m} )\)-concave function. Some examples are given to illustrate the results.
After some preliminaries and summarization of previous known results in Section 2, Section 3 deals with a Barnes-Godunova-Levin type inequality for the Sugeno integral, and some examples are given to illustrate the results. Finally, as special cases, some remarks are obtained.
2 Preliminaries
In this section, we recall some basic definitions or properties of a fuzzy integral and an \(( {\alpha,m} )\)-concave function. For details, we refer the reader to Refs. [1, 7, 23].
Suppose that ℘ is a σ-algebra of the subsets of X, and let \(\mu:\wp \to[0,\infty)\) be a non-negative, extended real-valued set function. We say that μ is a fuzzy measure if it satisfies:
-
(1)
\(\mu ( \emptyset ) = 0\);
-
(2)
\(E,F \in\wp\) and \(E \subset F\) imply \(\mu ( E ) \le\mu ( F)\);
-
(3)
\(\{ {{E_{n}}} \} \subset\wp\), \({E_{1}} \subset {E_{2}} \subset \cdots\), imply \({\lim_{n \to\infty}}\mu ( {{E_{n}}} ) = \mu ( {\bigcup_{n = 1}^{\infty}{{E_{n}}} } )\);
-
(4)
\(\{ {{E_{n}}} \} \subset\wp\), \({E_{1}} \supset {E_{2}} \supset \cdots\), \(\mu ( {{E_{1}}} ) < \infty\), imply \({\lim_{n \to\infty}}\mu ( {{E_{n}}} ) = \mu ( {\bigcap_{n = 1}^{\infty}{{E_{n}}} } )\).
Definition 2.1
(Mihesan [23])
The function \(f: [ {0,b} ] \to R\) is said to be \(( {\alpha,m} )\)-concave, where \(( {\alpha,m} ) \in { [ {0,1} ]^{2}}\), if for every \(x,y \in [ {0,b} ]\) and \(t \in [ {0,1} ]\), it satisfies
Note that for \(( {\alpha,m} ) \in \{ { ( {0,0} ), ( {\alpha,0} ), ( {1,0} ), ( {1,m} ), ( {1,1} ), ( {\alpha,1} )} \}\) one obtains the following classes of functions: decreasing, α-starshaped, starshaped, m-concave, concave and α-concave.
If f is a non-negative real-valued function defined on X, we denote the set \(\{ x \in X: f ( x ) \ge\alpha \} = \{ {x \in X:f \ge\alpha} \}\) by \({F_{\alpha }}\) for \(\alpha \ge0\). Note that if \(\alpha \le\beta\) then \({F_{\beta }} \subset {F_{\alpha }}\).
Let \(( {X,\wp,\mu} )\) be a fuzzy measure space, we denote by \({M^{+} }\) the set of all non-negative measurable functions with respect to ℘.
Definition 2.2
(Sugeno [1])
Let \(( {X,\wp,\mu} )\) be a fuzzy measure space, \(f \in{M^{+} }\) and \(A \in\wp\). The Sugeno integral (or the fuzzy integral) of f on A, with respect to the fuzzy measure μ, is defined as
when \(A = X\),
where ∨ and ∧ denote the operations sup and inf on \([ {0,\infty} )\), respectively.
The properties of the Sugeno integral are well known and can be found in [7].
Proposition 2.3
Let \(( {X,\wp,\mu} )\) be a fuzzy measure space, \(A,B \in \wp\) and \(f,g \in{M^{+} }\) then:
-
(1)
\({ ( S )\int_{A} {f\,d\mu \le\mu ( A )} }\);
-
(2)
\({ ( S )\int_{A} {k\,d\mu} = k \wedge\mu ( A )}\), k for a non-negative constant;
-
(3)
\({ ( S )\int_{A} {f\,d\mu \le} ( S )\int_{A} {g\,d\mu} }\) for \(f \le g\);
-
(4)
\({ ( S )\int_{A \cup B} {f\,d\mu \ge} ( S )\int_{A} {f\,d\mu} \vee ( S )\int_{B} {f\,d\mu} }\);
-
(5)
\({\mu ( {A \cap \{ {f \ge\alpha} \} } ) \ge\alpha \Rightarrow ( S )\int_{A} {f\,d\mu} \ge \alpha}\);
-
(6)
\({\mu ( {A \cap \{ {f \ge\alpha} \} } ) \le\alpha \Rightarrow ( S )\int_{A} {f\,d\mu} \le \alpha}\);
-
(7)
\({ ( S )\int_{A} {f\,d\mu} > \alpha \Leftrightarrow}\) there exists \(\gamma > \alpha\) such that \(\mu ( {A \cap \{ {f \ge\gamma} \}} ) >\alpha\);
-
(8)
\({ ( S )\int_{A} {f\,d\mu} < \alpha \Leftrightarrow}\) there exists \(\gamma < \alpha\) such that \(\mu ( {A \cap \{ {f \ge\gamma} \}} ) < \alpha\).
Remark 2.4
Consider the distribution function F associated to f on A, that is, \(F ( \alpha ) = \mu ( {A \cap \{ {f \ge \alpha} \}} )\). Then, due to (4) and (5) of Proposition 2.3, we have \(F ( \alpha ) = \alpha \Rightarrow ( S )\int_{A} {f\,d\mu} = \alpha\). Thus, from a numerical point of view, the Sugeno integral can be calculated solving the equation \(F ( \alpha ) = \alpha\).
Definition 2.5
Functions \(f,g:X \to R\) are said to be co-monotone if for all \(x,y \in X\),
and f and g are said to be counter-monotone if for all \(x,y \in X\),
It is clear that if f and g are co-monotone, then for any real numbers s, t either \({F_{s}} \subseteq{G_{t}}\) or \({F_{t}} \subseteq {G_{s}}\).
2.1 Barnes-Godunova-Levin type inequality for the Sugeno integral based on an \(( {\alpha,m} )\)-concave function
The classical Barnes-Godunova-Levin type inequality provides the inequality
where \(p,q > 1\), \(B ( {p,q} ) = \frac{{6{{ ( {b - a} )}^{{\frac{1 }{ p}} + {\frac{1 }{ q}} - 1}}}}{{{{ ( {1 + p} )}^{{\frac{1 }{ p}}}}{{ ( {1 + q} )}^{{\frac{1 }{ q}}}}}}\) and f, g are non-negative concave functions on \([ a,b ]\).
Unfortunately, the following example shows that the Barnes-Godunova-Levin type inequality for the Sugeno integral is not valid.
Example
Consider \(X = [ {0,100} ]\) and \(p = q = 4\). Let m be the Lebesgue measure on X. If we take the functions \(f ( x ) = g ( x ) = \sqrt[4]{x}\), then \(f ( x )\), \(g ( x )\) are two \(( {{\frac{1 }{2}},{\frac{1}{3}}} )\)-concave functions. In fact,
A straightforward calculus shows that
However,
This proves that the Barnes-Godunova-Levin type inequality for the Sugeno integral is not satisfied.
The aim of this work is to show a Barnes-Godunova-Levin type inequality for the Sugeno integral with respect to an \(( {\alpha,m} )\)-concave function.
Theorem 2.6
Let \(X= [ {0,1} ]\), \(\alpha,m \in ( {0,1} )\) and f, g be \((\alpha,m )\)-concave functions for all \(x \in X\). If m is a Lebesgue measure on X, then
Case (i). If \(f ( 0 ) \le f ( 1 )\) and \(g ( 0 ) \le g ( 1 )\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case (ii). If \(f ( 0 ) > f ( 1 )\) and \(g ( 0 ) > g ( 1 )\), then
Case (a). If \(\frac{{f ( 1 )}}{{f ( 0 )}} < \frac{{g ( 1 )}}{{g ( 0 )}}\), then
Case 1. If \(m \in ( {0,{\frac{{f ( 1 )} }{{f ( 0 )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 2. If \(m = {\frac{{f ( 1 )} }{{f ( 0 )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 3. If \(m \in ( {{\frac{{f ( 1 )} }{{f ( 0 )}}},{\frac{{g ( 1 )} }{{g ( 0 )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 4. If \(m = {\frac{{g ( 1 )} }{{g ( 0 )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 5. If \(m \in ( {{\frac{{g ( 1 )} }{{g ( 0 )}}},1} )\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case (b). If \(\frac{{f ( 1 )}}{{f ( 0 )}} = \frac{{g ( 1 )}}{{g ( 0 )}}\), then
Case 1. If \(m \in ( {0,{\frac{{f ( 1 )} }{ {f ( 0 )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 2. If \(m = {\frac{{f ( 1 )} }{{f ( 0 )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 3. If \(m \in ( {{\frac{{f ( 1 )} }{{f ( 0 )}}},1} )\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case (c). If \(\frac{{f ( 1 )}}{{f ( 0 )}} > \frac{{g ( 1 )}}{{g ( 0 )}}\), then
Case 1. If \(m \in ( {0,{\frac{{g ( 1 )} }{ {g ( 0 )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 2. If \(m = {\frac{{g ( 1 )} }{{g ( 0 )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 3. If \(m \in ( {{\frac{{g ( 1 )} }{{g ( 0 )}}},{\frac{{f ( 1 )} }{{f ( 0 )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 4. If \(m = {\frac{{f ( 1 )} }{{f ( 0 )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 5. If \(m \in ( {{\frac{{f ( 1 )} }{{f ( 0 )}}},1} )\), then
where \({t_{1}} = { ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Proof
Let \(p,q \in ( {0,\infty} )\), \({ ( { ( S )\int_{0}^{1} {{f^{p}} ( x )\,dx} } )^{{\frac{1 }{ p}}}} = {t_{1}}\) and \({ ( { ( S )\int_{0}^{1} {{g^{q}} ( x )\,dx} } )^{{\frac{1 }{ q}}}} = {t_{2}}\). Since \(f,g: [ {0,1} ] \to [ {0,\infty} )\) are two \(( {\alpha,m} )\)-concave functions for \(x \in [ {0,1} ]\), we have
Case (i). If \(f ( 0 ) \le f ( 1 )\) and \(g ( 0 ) \le g ( 1 )\), then by (3) of Proposition 2.3 and the co-monotonicity of \({h_{1}} ( x )\) and \({h_{2}} ( x )\), we have
Case (ii). If \(f ( 0 ) > f ( 1 )\) and \(g ( 0 ) > g ( 1 )\), then by (3) of Proposition 2.3 and the co-monotonicity of \({h_{1}} ( x )\) and \({h_{2}} ( x )\), we have
Case (a). If \(\frac{{f ( 1 )}}{{f ( 0 )}} < \frac{{g ( 1 )}}{{g ( 0 )}}\), then by (2.24) we obtain
Case 1. If \(m \in ( {0,{\frac{{f ( 1 )} }{ {f ( 0 )}}}} )\), then
Case 2. If \(m = {\frac{{f ( 1 )} }{{f ( 0 )}}}\), then
Case 3. If \(m \in ( {{\frac{{f ( 1 )} }{{f ( 0 )}}},{\frac{{g ( 1 )} }{{g ( 0 )}}}} )\), then
Case 4. If \(m = {\frac{{g ( 1 )} }{{g ( 0 )}}}\), then
Case 5. If \(m \in ( {{\frac{{g ( 1 )} }{{g ( 0 )}}},1} )\), then
Case (b). If \(\frac{{f ( 1 )}}{{f ( 0 )}} = \frac{{g ( 1 )}}{{g ( 0 )}}\), then by (2.24) we obtain
Case 1. If \(m \in ( {0,{\frac{{f ( 1 )} }{ {f ( 0 )}}}} )\), then
Case 2. If \(m = {\frac{{f ( 1 )} }{{f ( 0 )}}}\), then
Case 3. If \(m \in ( {{\frac{{f ( 1 )} }{{f ( 0 )}}},1} )\), then
Case (c). If \(\frac{{f ( 1 )}}{{f ( 0 )}} > \frac{{g ( 1 )}}{{g ( 0 )}}\), then by (2.24) we obtain
Case 1. If \(m \in ( {0,{\frac{{g ( 1 )} }{ {g ( 0 )}}}} )\), then
Case 2. If \(m = {\frac{{g ( 1 )} }{{g ( 0 )}}}\), then
Case 3. If \(m \in ( {{\frac{{g ( 1 )} }{{g ( 0 )}}},{\frac{{f ( 1 )} }{{f ( 0 )}}}} )\), then
Case 4. If \(m = {\frac{{f ( 1 )} }{{f ( 0 )}}}\), then
Case 5. If \(m \in ( {{\frac{{f ( 1 )} }{{f ( 0 )}}},1} )\), then
This completes the proof. □
Example
Consider \(X = [ {0,1} ]\) and \(p = 2\), \(q = 4\). If we take the functions \(f ( x ) =\sqrt[2]{x}\), \(g ( x ) = \sqrt[4]{x}\), then \(f ( x )\), \(g ( x )\) are two \(( {{\frac{2 }{3}},{\frac{1 }{3}}} )\) -concave functions. In fact, \(\sqrt[t]{x} = f ( {x \cdot1 + {\frac{1 }{3}} ( {1 - x} ) \cdot0} ) \ge{x^{{\frac{2 }{3}}}}f ( 1 ) + {\frac{1 }{3}} ( {1 - {x^{{\frac{2 }{ 3}}}}} )f ( 0 ) = {x^{{\frac{2 }{3}}}}\) for \(t \ge{\frac{3 }{2}}\). Let m be the Lebesgue measure on X. A straightforward calculus shows that
By Theorem 2.6, we have
Now, we will prove the general cases of Theorem 2.6.
Theorem 2.7
Let \(X= [ {a,b} ]\), \(\alpha,m \in ( {0,1} )\) and f, g be \(( {\alpha,m} )\)-concave functions for all \(x \in X\). If μ is a Lebesgue measure on X, then
Case (i). If \(f ( a ) \le f ( b )\) and \(g ( a ) \le g ( b )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case (ii). If \(f (a ) > f ( b )\) and \(g ( a ) > g ( b )\), then
Case (a). If \(\frac{{f ( b )}}{{f ( a )}} < \frac{{g ( b )}}{{g ( a )}}\), then
Case 1. If \(m \in ( {0,{\frac{{f ( b )} }{ {f ( a )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 2. If \(m = {\frac{{f ( b )} }{{f ( a )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 3. If \(m \in ( {{\frac{{f ( b )} }{{f ( a )}}},{\frac{{g ( b )} }{{g ( a )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 4. If \(m = {\frac{{g ( b )} }{{g ( a )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 5. If \(m \in ( {{\frac{{g ( b )} }{{g ( a )}}},1} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case (b). If \(\frac{{f ( b )}}{{f ( a )}} = \frac{{g ( b )}}{{g ( a )}}\), then
Case 1. If \(m \in ( {0,{\frac{{f ( b )} }{ {f ( a )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 2. If \(m = {\frac{{f ( b )} }{{f ( a )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 3. If \(m \in ( {{\frac{{f ( b )} }{ {f ( a )}}},1} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case (c). If \(\frac{{f ( b )}}{{f ( a )}} > \frac{{g ( b )}}{{g ( a )}}\), then
Case 1. If \(m \in ( {0,{\frac{{g ( b )} }{ {g ( a )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 2. If \(m = {\frac{{g ( b )} }{{g ( a )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 3. If \(m \in ( {{\frac{{g ( b )} }{{g ( a )}}},{\frac{{f ( b )} }{{f ( a )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 4. If \(m = {\frac{{f ( b )} }{{f ( a )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 5. If \(m \in ( {{\frac{{f ( b )} }{{f ( a )}}},1} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Proof
Let \(p,q \in ( {0,\infty} )\), \({ ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{{\frac{1 }{ p}}}} = {t_{1}}\) and \({ ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{{\frac{1 }{ q}}}} = {t_{2}}\). Since \(f,g: [ {a,b} ] \to [ {0,\infty} )\) are two \(( {\alpha,m} )\)-concave functions for \(x \in [ {a,b} ]\), we have
Case (i). If \(f ( a ) \le f ( b )\) and \(g ( a ) \le g ( b )\), then by (3) of Proposition 2.3 and the co-monotonicity of \({h_{1}} ( x )\) and \({h_{2}} ( x )\), we have
Case (ii). If \(f (a ) > f ( b )\) and \(g ( a ) > g ( b )\), then by (3) of Proposition 2.3 and the co-monotonicity of \({h_{1}} ( x )\) and \({h_{2}} ( x )\), we have
Case (a). If \(\frac{{f ( b )}}{{f ( a )}} < \frac{{g ( b )}}{{g ( a )}}\), then by (2.56) we obtain
Case 1. If \(m \in ( {0,{\frac{{f ( b )} }{ {f ( a )}}}} )\), then
Case 2. If \(m = {\frac{{f ( b )} }{{f ( a )}}}\), then
Case 3. If \(m \in ( {{\frac{{f ( b )} }{{f ( a )}}},{\frac{{g ( b )} }{{g ( a )}}}} )\), then
Case 4. If \(m = {\frac{{g ( b )} }{{g ( a )}}}\), then
Case 5. If \(m \in ( {{\frac{{g ( b )} }{{g ( a )}}},1} )\), then
Case (b). If \(\frac{{f ( b )}}{{f ( a )}} = \frac{{g ( b )}}{{g ( a )}}\), then by (2.56) we obtain
Case 1. If \(m \in ( {0,{\frac{{f ( b )} }{ {f ( a )}}}} )\), then
Case 2. If \(m = {\frac{{f ( b )} }{{f ( a )}}}\), then
Case 3. If \(m \in ( {{\frac{{f ( b )} }{{f ( a )}}},1} )\), then
Case (c). If \(\frac{{f ( b )}}{{f ( a )}} > \frac{{g ( b )}}{{g ( a )}}\), then by (2.56) we obtain
Case 1. If \(m \in ( {0,{\frac{{g ( b )} }{ {g ( a )}}}} )\), then
Case 2. If \(m = {\frac{{g ( b )} }{{g ( a )}}}\), then
Case 3. If \(m \in ( {{\frac{{g ( b )} }{{g ( a )}}},{\frac{{f ( b )} }{{f ( a )}}}} )\), then
Case 4. If \(m = {\frac{{f ( b )} }{{f ( a )}}}\), then
Case 5. If \(m \in ( {{\frac{{f ( b )} }{{f ( a )}}},1} )\), then
This completes the proof. □
Example
Consider \(X = [ {2,5} ]\) and \(p = 2\), \(q = 4\). Let m be the Lebesgue measure on X. If we take the function \(f ( x ) = \sqrt[2]{{6 - x}}\), \(g ( x ) = \sqrt[4]{{6 - x}}\), then \(f ( x )\), \(g ( x )\) are two \(( {{\frac{1 }{2}},{\frac{{\sqrt{5} } }{2}}} )\) -concave functions. In fact,
and
A straightforward calculus shows that
By Theorem 2.7, we have
As some special cases of \((\alpha,m )\)-concave functions in Theorem 2.7, we have the following results.
Remark 2.8
Let \(X= [ {a,b} ]\), \(\alpha= m= 0 \) and f, g be two decreasing functions for all \(x \in X\). If μ is a Lebesgue measure on X, then
Remark 2.9
Let \(X= [ {a,b} ]\), \(\alpha=1\), \(m= 0 \) and f, g be two starshaped functions for all \(x \in X\). If μ is a Lebesgue measure on X, then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Remark 2.10
Let \(X= [ {a,b} ]\), \(\alpha=1\), \(m\in ({0,1} ) \) and f, g be two m-concave functions for all \(x \in X\). If μ is a Lebesgue measure on X, then
Case (i). If \(f ( a ) \le f ( b )\) and \(g ( a ) \le g ( b )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case (ii). If \(f (a ) > f ( b )\) and \(g ( a ) > g ( b )\), then
Case (a). If \(\frac{{f ( b )}}{{f ( a )}} < \frac{{g ( b )}}{{g ( a )}}\), then
Case 1. If \(m \in ( {0,{\frac{{f ( b )} }{ {f ( a )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 2. If \(m = {\frac{{f ( b )} }{{f ( a )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 3. If \(m \in ( {{\frac{{f ( b )} }{{f ( a )}}},{\frac{{g ( b )} }{{g ( a )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 4. If \(m = {\frac{{g ( b )} }{{g ( a )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 5. If \(m \in ( {{\frac{{g ( b )} }{{g ( a )}}},1} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case (b). If \(\frac{{f ( b )}}{{f ( a )}} = \frac{{g ( b )}}{{g ( a )}}\), then
Case 1. If \(m \in ( {0,{\frac{{f ( b )} }{ {f ( a )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 2. If \(m = {\frac{{f ( b )} }{{f ( a )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 3. If \(m \in ( {{\frac{{f ( b )} }{ {f ( a )}}},1} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case (c). If \(\frac{{f ( b )}}{{f ( a )}} > \frac{{g ( b )}}{{g ( a )}}\), then
Case 1. If \(m \in ( {0,{\frac{{g ( b )} }{ {g ( a )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 2. If \(m = {\frac{{g ( b )} }{{g ( a )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 3. If \(m \in ( {{\frac{{g ( b )} }{{g ( a )}}},{\frac{{f ( b )} }{{f ( a )}}}} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 4. If \(m = {\frac{{f ( b )} }{{f ( a )}}}\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case 5. If \(m \in ( {{\frac{{f ( b )} }{{f ( a )}}},1} )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Remark 2.11
[22]
Let \(X= [ {a,b} ]\), \(\alpha=1\), \(m=1 \) and f, g be two concave functions for all \(x \in X\). If μ is a Lebesgue measure on X, then
Case (i). If \(f (a ) < f ( b )\) and \(g ( a ) < g ( b )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case (ii). If \(f (a ) = f ( b )\) and \(g ( a ) = g ( b )\), then
Case (iii). If \(f (a ) > f ( b )\) and \(g ( a ) > g ( b )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Remark 2.12
Let \(X= [ {a,b} ]\), \(\alpha\in ({0,1} )\), \(m=0 \) and f, g be two α-starshaped functions for all \(x \in X\). If μ is a Lebesgue measure on X, then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Remark 2.13
Let \(X= [ {a,b} ]\), \(\alpha\in ({0,1} )\), \(m=1 \) and f, g be two α-concave functions for all \(x \in X\). If μ is a Lebesgue measure on X, then
Case (i). If \(f (a ) < f ( b )\) and \(g ( a ) < g ( b )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
Case (ii). If \(f (a ) = f ( b )\) and \(g ( a ) = g ( b )\), then
Case (iii). If \(f (a ) > f ( b )\) and \(g ( a ) > g ( b )\), then
where \({t_{1}} = { ( { ( S )\int_{a}^{b} {{f^{p}} ( x )\,dx} } )^{\frac{1}{p}}}\), \({t_{2}} = { ( { ( S )\int_{a}^{b} {{g^{q}} ( x )\,dx} } )^{\frac{1}{q}}}\).
3 Conclusion
In this paper, we have investigated the Barnes-Godunova-Levin type inequality of the Sugeno integral with respect to an \(( {\alpha ,m} )\)-concave function. For further investigations, we will continue to explore other integral inequalities for the Sugeno integral related to \(( {\alpha,m} )\)-concavity.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (61273143, 61472424) and Fundamental Research Funds for the Central Universities (2013RC10, 2013RC12 and 2014YC07).
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Li, DQ., Cheng, YH., Wang, XS. et al. Barnes-Godunova-Levin type inequality of the Sugeno integral for an \((\alpha,m)\)-concave function. J Inequal Appl 2015, 25 (2015). https://doi.org/10.1186/s13660-015-0556-0
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DOI: https://doi.org/10.1186/s13660-015-0556-0
MSC
- 03E72
- 28B15
- 28E10
- 26D10
Keywords
- Barnes-Godunova-Levin type inequality
- Sugeno integral
- \((\alpha, m )\)-concave function