SHAPE PRESERVING PROPERTIES AND MONOTONICITY PROPERTIES OF THE SEQUENCES OF CHOQUET TYPE INTEGRAL OPERATORS

. In this paper, for the univariate Bernstein-Kantorovich-Choquet, Sz´asz-Kantorovich-Choquet, Baskakov-Kantorovich-Choquet and Bernstein-Du-rrmeyer-Choquet operators written in terms of the Choquet integrals with respect to monotone and submodular set functions, we study the preservation of the monotonicity and convexity of the approximated functions and the monotonicity of some approximation sequences.


INTRODUCTION
Qualitative results and quantitative uniform, pointwise and L p results in approximation by Bernstein-Durrmeyer-Choquet, Bernstein-Kantorovich-Choquet, Szász-Kantorovich-Choquet and Baskakov-Kantorovich-Choquet operators defined in terms of the Choquet integral with respect to a family of monotone and submodular set functions, were obtained by the author in a series of very recent papers [7]- [11]. As it was pointed out in some of these papers, for large classes of functions, the Choquet type operators approximate better than their classical correspondents.
By analogy with what happens in the case of the classical positive and linear operators, it is a natural question to look for shape preserving properties of these Choquet type operators and for monotonicity of the sequences of approximation.
The aim of the present paper is to give answers to this question.
The plan of the paper goes as follows. Section 2 contains some preliminaries on the Choquet integral. In Section 3 we prove monotonicity and convexity preserving properties for the Bernstein-Kantorovich-Choquet, Szász-Kantorovich-Choquet, Baskakov-Kantorovich-Choquet operators and we discuss these properties for the Bernstein-Durrmeyer-Choquet type operators. Section 4 proves the monotonicity property of the sequences of Baskakov-Kantorovich-Choquet and of Bernstein-Kantorovich-Choquet type operators.

PRELIMINARIES
In this section we present some concepts and results on the Choquet integral which will be used in the main section.
Definition 2.1. Let Ω be a nonempty set and C be a σ-algebra of subsets in Ω.
If f : Ω → R is C-measurable, i.e. for any Borel subset B ⊂ R we have f −1 (B) ∈ C, then for any A ∈ C, the Choquet integral is defined by In what follows, we list some known properties of the Choquet integral.
Remark 2.2. If µ : C → [0, +∞) is a monotone set function, then the following properties hold : that is the Choquet integral is sublinear.

SHAPE PRESERVING PROPERTIES
Firstly, we deal with the Kantorovich-Choquet type operators. Denoting by B I the sigma algebra of all Borel measurable subsets in P(I), everywhere in this section, (Γ n,x ) n∈N,x∈I , will be a collection of families Γ n,x = {µ n,k,x } n k=0 , of monotone, submodular and strictly positive set functions µ n,k,x on B I , with I = [0, 1] in the case of Bernstein-Kantorovich polynomials and I = [0, +∞) in the cases of Szász-Mirakjan-Kantorovich and Baskakov-Kantorovich operators.
We note that in order to be well defined these operators, it is good enough if, for example, we suppose that f : I → R + is a B I -measurable function, bounded on I, where I = [0, 1] for K n,Γn,x (f )(x) and I = [0, +∞) for S n,Γn,x (f )(x) and V n,Γn,x (f )(x).
Since in general, the change of variable does not work for the Choquet integral, we also can introduce the following different Choquet operators, given formally by different from K n,Γn , S n,Γn and V n,Γn , correspondingly.
Remark 3.2. It is known that if all the set functions in the family Γ n one reduce to the Lebesgue measure denoted here by M (which is independent of n too), then K n,M = K n,M , S n,M = S n,M and V n,M = V n,M . But as we will show later, these equalities also hold for some monotone and submodular function different from the Lebesgue measure.
Everywhere in this paper the shape preserving properties will be considered in the case when the set functions in the collection of families Γ n,x are independent of x and and k.
The main result of the paper is the following.
(i) If f is nondecreasing on I, then for all n ∈ N, K n,Γn (f ), S n,Γn (f ) and V n,Γn (f ) are nondecreasing on I; (ii) If f is nondecreasing on I, then for all n ∈ N, K n,Γn (f ), S n,Γn (f ) and V n,Γn (f ) are nondecreasing on I; (iii) Suppose that, in addition, Γ n = {µ n }, n ∈ N, is a family of submodular set functions on B I . If f is nonconcave on I, then for all n ∈ N, K n,Γn (f ), S n,Γn (f ) and V n,Γn (f ) are nonconcave on I.
(iv) If f is nonconcave on I and all µ n (A) = γ n (M (A)), n ∈ N are distorted Lebesgue measures with all the γ n , increasing, concave and continuous on [0, 1], then for all n ∈ N, K n,Γn (f ), S n,Γn (f ) and V n,Γn (f ) are nonconcave on I.

Proof. (i) Denoting
Since f is nondecreasing on I, by applying the properties in Remark 2.2, (iii), we get The proofs in the cases of B n,k and the corresponding S n,Γn (f ) and V n,Γn (f ) operators are similar.
(ii) Since f is nondecreasing on I (therefore is B I -measurable) and for any k, n, f k+t n+1 remains nondecreasing on I, it follows that f k+t n+1 is B Imeasurable as function of t. Now, by simple calculations we immediately obtain Since f is nondecreasing on I, it follows that (for all t ∈ [0, 1], n, k) Applying the property in Remark 2.2, (iii), we get This implies that the first derivatives of these operators are positive, that is the operators K n,Γn (f ), S n,Γn (f ) and V n,Γn (f ) also are nondecreasing for any n ∈ N.
(iii) Since f is nonconcave on I (therefore is B I -measurable) and for any k, n, f k+t n+1 remains nonconcave on I, it follows that f k+t n+1 is B I -measurable as function of t. Now, by the calculations for the classical Kantorovich variants of the operators (see, e.g. again [1, pp. Since f is nonconcave on I, it follows that (for all t ∈ [0, 1], n, k ∈ N, 0 ≤ k ≤ n) we have Since every µ n is submodular, applying consecutively the properties in Remark 2.2, (iii) and (ii), we obtain This implies that the second derivatives K n,Γn (f )(x) is ≥ 0 on I, that is the operator K n,Γn (f )(x) is nonconcave on I. The proof in the cases of the other two operators is similar.
(iv) We use the notations for A n,k and B n,k from the point (i) and the ideas of calculation from the point (iii). We present here only the proof in the case of K n,Γn (f ), because the proofs in the case of B n,k and of the operators S n,Γn (f ) and V n,Γn (f ) are similar. Thus, we have Since the Lebesgue measure M is invariant at translations, it immediately follows that with a n,k = (C) k+1 n+1 k n+1 f (t)dµ n (t). But we can write a n,k+1 = (C) k+2 n+1 k+1 n+1 Since in the similar way we get a n,k+2 = (C) k+1 n+1 k n+1 f (t + 2 n+1 )dµ n (t), it follows that a n,k+2 − 2a n,k+1 + a n,k =(C) k+1 n+1 k n+1 But since f is nonconcave on I = [0, 1], we have ]. Since every µ n is submodular, applying the Choquet integral on [ k n+1 , k+1 n+1 ] to the previous inequality and applying consecutively the properties in Remark 2.2, (iii) and (ii), reasoning exactly as at the above point (iii), we arrive at a n,k+2 − 2a n,k+1 + a n,k ≥ 0, which leads to the desired conclusion.
Remark 3.4. From the proofs, it easily follows that Theorem 3.3, (i), (ii) hold if we replace in their statements the word nondecreasing with the word nonincreasing. But if we replace in Theorem 3.3, (iii) and (iv) the word nonconcave with the word nonconvex, it is easy to see that their proofs do not work since the subaditivity of the Choquet integral is not helpful. However, under some additional hypothesis, we can prove the shape preserving properties concerning the nonconvexity, as follows.
Since as functions of t, f k+2+t n+1 and f k+t n+1 are of the same monotonicity, they are comonotonic and applying to the previous inequality the Choquet integral and the property in Remark 2.2, (ii), we obtain Using the relationship for K n,Γn (f )(x) in the proof of Theorem 3.3, (iii), this immediately implies that the second derivatives K n,Γn (f )(x) is ≤ 0 on I, that is the operator K n,Γn (f )(x) is nonconvex on I. The proof in the cases of the other two operators is similar. Indeed, this is immediate from the fact that if f is nondecreasing (nonconcave), then f − m remains nondecreasing (nonconcave, respectively).
In continuation to the comments in Remark 3.2, we can prove the following result. Proof. For any fixed α ≥ 0, let us make the notations It is clear that B n,k (α) is obtained by applying to A n,k (α) the linear transform w(t) = t n + k n = 1 n (t + k), t ∈ [0, 1]. By the well-known properties of the Lebesgue measure, we get M (B n,k (α)) = 1 n M (A n,k (α)), which evidently implies µ(B n,k (α)) = 1 √ n µ(A n,k (α)). Therefore, we get , which proves our assertion. Evidently that the above relationship remains valid by replacing n with n + 1. Remark 3.9. In the papers [9]- [11], were introduced and studied the qualitative and quantitative approximation properties of the multivariate Bernstein-Durrmeyer-Choquet polynomials, which in the univariate are given by the where {µ n,k,x }, n ∈ N, k ∈ {0, 1, ..., n}, x ∈ [0, 1], is a family of monotone, submodular and strictly positive set functions on It is well-known that the proof of the shape preserving properties for the classical Bernstein-Durrmeyer operators is based on the integration by parts, rule which does not hold for the general Choquet integral. This fact induces much difficulty in any attempt to prove these properties for Bernstein-Durrmeyer-Choquet polynomials and for this reason, it remains as an open question under which conditions still they hold.
However, we can show that, in general, the shape preserving properties for these polynomials do not hold. Indeed, for example, let us consider the Bernstein-Durrmeyer-Choquet polynomials introduced by [10, Example 5.2], given by for all sufficiently large n ∈ N.
From the continuity of the polynomial D n,Γn (f )(x), the last inequality implies that for any sufficiently large n ∈ N, there exists a small neighborhood of 1, such that D n,Γn (f )(x) < 0 for all x in that neighborhood, contradicting a possible preservation of the nondecreasing monotonicity of f .

MONOTONICITY OF THE APPROXIMATION SEQUENCES
In this section we present two samples concerning the monotonicity of the sequences of Choquet type operators, the rest of the cases being leaved as open questions to the readers.
In this sense, we can state the following.

Proof. (i) For the classical Baskakov operators
it is known the formula (see, [14], or also [1, pp. 176-177]), where Denoting F ( k n ) = (C) 1 0 f k+t n dµ(t), by the above calculations we get If we prove that for all t ∈ [0, 1], n, k ∈ N we have then applying here the Choquet integral and taking into account its properties in Remark 2.2, (iii) and (ii), it follows E(F ) ≥ 0 and the required conclusion.
In this sense, let us observe that from the above considerations, G(0) ≥ 0 means exactly the nonconcavity of f . Let n and k arbitrary fixed. It follows that if we prove that G (t) ≥ 0 for all t ≥ 0, then we arrive at the desired conclusion. Indeed, we have n+1 > k+t n+1 and by hypothesis f is nondecreasing, firstly it follows f ( k+1+t n ) − f ( k+1+t n+1 ) ≥ 0. Then, since f is nonconvex, by e.g. [13, pp. 44], it follows that f (x) x is nonincreasing on (0, +∞), which will imply that k+1 n+1 f ( k+t n+1 )− k n+1 f ( k+1+t n+1 ) ≥ 0, finishing the proof. Indeed, we get which leads to the desired conclusion.