Ohlin-Type Theorem for Convex Set-Valued Maps
- PDF / 334,052 Bytes
- 10 Pages / 439.37 x 666.142 pts Page_size
- 80 Downloads / 184 Views
Results in Mathematics
Ohlin-Type Theorem for Convex Set-Valued Maps Kazimierz Nikodem
and Teresa Rajba
Abstract. A counterpart of the Ohlin theorem for convex set-valued maps is proved. An application of this result to obtain some inclusions related to convex set-valued maps in an alternative unified way is presented. In particular counterparts of the Jensen integral and discrete inequalities, the converse Jensen inequality and the Hermite–Hadamard inequalities are obtained. Mathematics Subject Classification. Primary 26A51; Secondary 39B62, 26D15. Keywords. Ohlin’s lemma, convex set-valued maps, Jensen inequalities, Hermite–Hadamard inequalities.
1. Introduction In Ohlin [12] proved the following interesting and very useful result on convex functions in a probabilistic context (as usual, E[X] denotes the expectation of the random variable X): Lemma 1 [12]. Let X1 , X2 be two real valued random variables such that E[X1 ] = E[X2 ]. If the distribution functions FX1 , FX2 cross one time, i.e. there exists t0 ∈ R such that FX1 (t) ≤ FX2 (t)
if t < t0 and FX1 (t) ≥ FX2 (t)
then E[f (X1 )] ≤ E[f (X2 )] for every convex function f : R → R. 0123456789().: V,-vol
if t > t0 ,
(1)
162
Page 2 of 10
K. Nikodem and T. Rajba
Results Math
For years the above Ohlin lemma was not well-known in the mathematical community. It has been rediscovered by Rajba [14], who found its various applications to the theory of functional inequalities. In [13,15,18], the Ohlin lemma is used, among others, to get a simple proof of the known Hermite– Hadamard inequalities, as well as to obtaining new Hermite–Hadamard type inequalities. In this note we prove counterparts of the Ohlin theorem for convex setvalued maps. We present also applications of these results to obtain some inclusions connected with convex set-valued maps.
2. Preliminaries Let (Y, · ) be a separable Banach space, B be the closed unit ball in Y , (Ω, A, P ) be a probability space with a non-atomic measure P and I ⊂ R be an open interval. Denote by n(Y ) the family of all nonempty subsets of Y and by cl(Y ) the family of all closed nonempty subsets of Y . For a given set-valued map G : Ω → n(Y ) the integral Ω G(ω)dP is understood in the sense of Aumann, i.e. it is the set of integrals of all integrable (in the sense of Bochner) selections of the map G (cf. [1,2]):
G(ω)dP = Ω
Ω
g(ω)dP : g : Ω → Y
is integrable and g(ω) ∈ G(ω), ω ∈ Ω .
A set-valued map G : Ω → n(Y ) is called integrable bounded if there exists a non-negative integrable function k : Ω → R such that G(ω) ⊂ k(ω)B, for all ω ∈ Ω. In this case every measurable selection of G is integrable and, consequently, the Aumann integral of G is nonempty whenever G is measurable. The following properties of the Aumann integral will be needed in our investigations: Lemma 2 [[1], Theorems 8.6.3, 8.6.4 ]. Let G : Ω → cl(Y ) be a measurable set-valued map. a) The closure of the integral of G is convex and G(ω)dP = conv G(ω)dP . Ω
Ω
b) If Y is finite dimensional, then the integral of G is convex.
Data Loading...
