On the q-Monotonicity Preservation of Durrmeyer-Type Operators

We prove that various Durrmeyer-type operators preserve q-monotonicity in [0, 1] or [0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty )$$\end{document} as the case may be. Recall that a 1-monotone function is nondecreasing, a 2-monotone one is convex, and for q>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q>2$$\end{document}, a q-monotone function possesses a convex (q-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q-2)$$\end{document}nd derivative in the interior of the interval. The operators are the Durrmeyer versions of Bernstein (including genuine Bernstein–Durrmeyer), Szász and Baskakov operators. As a byproduct we have a new type of characterization of continuous q-monotone functions by the behavior of the integrals of the function with respect to measures that are related to the fundamental polynomials of the operators.


Introduction
Let I be an interval on the real line, finite or infinite, open or closed or half open. For a function f , defined on I, denote x ,x+ h ∈ I, 0, otherwise, and for q ≥ 1, ). That is, x ,x+ qh ∈ I, 0, otherwise.

U. Abel et al. MJOM
A function f defined on I, is called q-monotone there, if Δ q h f (x) ≥ 0, x ∈ I, for all h > 0. In particular a 1-monotone function is nondecreasing and a 2-monotone one is convex. It is well-known (it goes back to T. Popoviciu, Mathematica 8 (1934), and R. P. Boas Jr. and D. V. Widder, Duke Math. J. 7 (1940)) that for q > 2, f is q-monotone in I, implies that f possesses a convex (q − 2)nd derivative in the interior of I.
The classical Bernstein polynomials associated with a function f which is defined on [0, 1], are defined by, For the sake of simplicity we also denote p 0,0 (x) ≡ 1 and p n,j (x) ≡ 0, j < 0 and j > n. It is well-known (see, e.g., [11, 1.4(2)]) that the Bernstein polynomials preserve q-monotonicity of all orders q ≥ 1.
Durrmeyer [6] and, independently, Lupaş [12] have modified the Bernstein polynomials and defined what we call the Bernstein-Durrmeyer polynomials. Namely (see [4]), for an integrable f on [0, 1], we set (D n f )(x) := (n + 1) n j=0 p n,j (x)  [4] (see also [5]). We are interested in something else, namely, preservation of q-monotonicity. Adell and de la Cal [2] proved that the Bernstein-Durrmeyer polynomials preserve monotonicity and convexity, applying techniques from probability and stochastic processes. In 2007 Attalienti and Raşa [3] proved that the Bernstein-Durrmeyer polynomials preserve q-monotonicity for all q ≥ 1, using the heavy machinery of Karlin's Total Positivity [10]. In Section 2, we will show by elementary means that the Bernstein-Durrmeyer polynomials preserve q-monotonicity for all q ≥ 1. We will use the proof to prove that the genuine Bernstein-Durrmeyer polynomials [8] (see also [7]) have the same property.
In Section 3, we discuss the Szász-Durrmeyer operators [13] and in Section 4 the Baskakov-Durrmeyer operators [15]. We will prove that both preserve q-monotonicity. For functions differentiable sufficiently many times and all required integrals converge, the q-monotonicity of the Szász-Durrmeyer operators was proved by Pȃltȃnea [14] (see [9] for details and other operators). Finally, in Section 5 we present some Raşa-type inequalities for Durrmeyertype operators.

q-monotonicity of the ordinary and genuine Bernstein-Durrmeyer polynomials
We first prove the following A crucial lemma is Then, for every n ≥ q and 0 ≤ k ≤ n − q, we have An immediate consequence of Lemma 2.2 and the proof of Theorem 2.1 is a new type of characterization of continuous q-monotone functions in [0, 1], which may be interesting by itself. Proof. If f ∈ C[0, 1] is q-monotone, then (2.1) follows by Lemma 2.2. Conversely, if (2.1) is valid for every n ≥ q and 0 ≤ k ≤ n − q, then by the proof of Theorem 2.1 it follows that D n f , n ≥ 1, is q-monotone. It is well-known (see, e.g., [4,Théorème II.2]) that D n f converges (uniformly) to f in [0, 1], so it follows that f is q-monotone.
Proof of Lemma 2.2. First, we observe that every q-monotone function f ∈ C[0, 1] may be uniformly approximated by q times continuously differentiable q-monotone functions on [0, 1]. For example, one may take the Bernstein polynomials associated with f . Hence, in order to prove (2.1) it suffices to assume that f ∈ C q [0, 1] so that by assumption f (q) (x) ≥ 0, x ∈ [0, 1]. Denote a n,k : and Δ q a n,k := To this end, straightforward differentiation yields that for any m, l ≥ 0, Hence, it readily follows that Δ 1 a n,k = 1 n + 1 Proof of Theorem 2.1. D n f is a polynomial of degree n. Thus, for 0 ≤ n < q it is clearly q-monotone. We write where a n,k is defined in (2.2). Repeating, verbatim, the well-known computations for the Bernstein polynomials (see, e.g., [11, 1.4(2)]), we have, for n ≥ q, Since by Lemma 2.2 Δ q a n,k ≥ 0, for all 0 ≤ k ≤ n − q, n ≥ q, our proof is complete.
The genuine Bernstein-Durrmeyer polynomials associated with a func- Recall that the advantage of the genuine Bernstein-Durrmeyer polynomials is that they preserve linear functions while the ordinary Bernstein-Durrmeyer polynomials only preserve constants.
We have the following.
Proof. We have to separate the proof for q = 1 and for q ≥ 2, and we begin with the latter. Thus assume that q ≥ 2. If 2 ≤ n < q, then there is nothing to prove. Thus, we assume that n ≥ q. Let L be the linear function interpolating f at where a n,k were defined in (2.2). Denote b n,k := a n−2,k−1 , 1 ≤ k ≤ n − 1, and b n,0 = b n,n = 0. Then we rewrite As in (2.6) we have If n = q, q + 1, then this statement is empty. In both cases, we complete the proof if we show that Recall that, as in the proof of Lemma 2.2, we may assume that f ∈ C q [0, 1]. First, assume that n ≥ q + 1. Since f (0) = 0, and by (2.5), Hence, by (2.4), Assuming, by induction, that and noting that (2.3), for q − 1, implies we obtain, by (2.4) and integration by parts, and by (2.5), Hence, Finally, again assuming, by induction, that we obtain as above, that proves the right hand inequality in (2.7). This completes the proof for q ≥ 2 and n ≥ q + 1, so that we proceed to the case n = q ≥ 2.
We begin with n = q = 2, and observe that in this case, which is convex, since f (t) ≤ 0, 0 ≤ t ≤ 1. If n = q > 2, then we have to investigate Δ q b n,0 = Δ q b n,n−q , but both (2.9) and (2.11) are inapplicable. Thus we have to go back to (2.10) and (2.8).
We have This completes the proof for all n ≥ q ≥ 2. As for the case q = 1, we observe that since U n preserves constants, we may assume that f (0) = 0, so that, in turn, (2.12) First let n > 2, and again denote b n,0 = 0, b n,n = 0 and b n,k = a n−2,k−1 , 1 ≤ k ≤ n − 1. Then, To this end, On the other hand, Hence, by virtue of (2.12), U 2 f is nondecreasing in [0, 1]. This completes the proof.

The Szász-Durrmeyer operators
The Favard-Mirakyan-Szász operators associated with a function f defined in [0, ∞), and such that |f (x)| ≤ Ce Ax for some constants C, A > 0, are defined by It is well-known that if, in addition, f ∈ C[0, ∞), then the Favard-Mirakyan-Szász operators associated with f , approximate it uniformly on every compact subinterval of [0, ∞). It is also known that they preserve q-monotonicity on [0, ∞), which readily follows from the equation, We will prove the following.
A weaker result but easier to prove, and which goes along the lines of the proof of Lemma 2.2 is We may characterize q-monotone functions in [0, ∞), with an analog of Corollary 2.3. For this we need the notation a n,k : and only if, Δ q a n,k ≥ 0, n > A and k ≥ 0. Proof. From the proof of Theorem 3.1 we conclude that if f is q-monotone, then (3.3) is valid. Conversely, we observe from the proof of Theorem 3.1 We begin with the proof of Theorem 3.2.
Proof of Theorem 3.2. Similar to (3.1), we obtain, where a n,k are defined in (3.2). Hence, our proof will be complete if we prove that Δ q a n,k ≥ 0, k = 0, 1, 2 . . . , n > A.
Note that due to the integration by parts, we do not have to assume growth conditions on f (q) , for the integral to converge. This completes the proof.
Proof of Theorem 3.1. As above, we need to prove that Δ q a n,k ≥ 0. We observe that Denote s k+q (nt) =: g(t) and let Δ q We will prove that R 1 → 0, as h → 0. To this end, note that for some ξ ∈ (t, t + qh), Hence, Thus, it suffices to prove that for m = 0, 1, 2, . . . , Finally, by (3.5), we have Δ q a n,k = 1 where R := R 1 + 1 (nh) q R 2 = O(h), as h → 0. If f is q-monotone, then we get for all h > 0 that Δ q a n,k ≥ R.
Hence, as the left hand side is independent of h, we conclude that Δ q a n,k ≥ 0, and the proof is complete.

The Baskakov-Durrmeyer operators
The Baskakov operators are defined for a function f of polynomial growth on [0, ∞), by It is well-known that if, in addition, f ∈ C[0, ∞), then the Baskakov operators associated with f , approximate it uniformly on every compact subinterval of [0, ∞). It is also known that they preserve q-monotonicity on [0, ∞), which readily follows from the equation, (4.1) Denoting Remark 4.1. In [15] the definition of the operators is only for integrable functions while one may define the operators for large enough n, for functions of polynomial growth. On the other hand, we prefer to assume that the functions are continuous in [0, ∞), which, for q ≥ 2, amounts to assuming continuity at x = 0. This way we are guaranteed the approximation on compact subintervals of [0, ∞).
Again, a weaker result, which is easier to prove, is Theorem 4.3. Given q ≥ 1, suppose that f ∈ C q [0, ∞) is q-monotone and there exist constants C, A ≥ 0 such that |f (i) Yet another characterization of continuous q-monotone functions in [0, ∞) is the next corollary that is proved exactly as Corollary 3.3, thus we omit the proof. Denote a n,k := ∞ 0 m n,k (t)f (t)dt. We begin with the proof of Theorem 4.3.