Relations Between the Bernstein Polynomials and q-monotone Functions

We show that certain inequalities involving differences of the Bernstein basis polynomials and values of a function f∈C[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C[0,1]$$\end{document}, imply that the function is q-monotone. In view of previous results of the authors (see the list of references), the current results provide, among others, a characterization of q-monotone functions in C[0, 1].


Introduction and the Main Results
Given a function f defined on [0, 1], the classical Bernstein polynomials associated with it are defined by [8], The following theorem was proved by J. Mrowiec, T. Rajba and S. Wąsowicz [9]. It has given an affirmative answer to a conjecture by the third author. (1. 2) It follows from Vandermonde's identity So, the first and third authors [3] asked whether one may actually prove a stronger inequality than (1.1), namely, for convex f , prove that and indeed, they proved it in [3].
It is well known (see, e.g., [8, 1.5(2)]) that for q ≥ 1, Hence, the Bernstein polynomials preserve q-monotonicity of all orders q ≥ 1, so that if f ∈ C[0, 1] is q-monotone, then for any pair x, y ∈ [0, 1], The identity analogous to (1.3) for a general q is easily proved, applying the Vandermonde's identity n ν1,...,νq=0 ν1+···+νq=k Namely, (1.7) Hence, one may ask whether, for q > 2, a stronger inequality (analogous to (1.4)) is valid. That is, is it true that for a q-monotone f , In Sect. 2, we give an affirmative answer to this question. Namely, we prove Since, Thus, one may ask whether this statement is true also for the weaker inequality (1.5). In Sect. 3, we answer this question affirmatively. Namely, in Sect. 3, we prove In view of the above, we propose the following open question. Prove or disprove that if f ∈ C[0, 1] and (1.8) holds for all n ≥ 1, then f is q-monotone. We note that the answer is unknown even for q = 2. Proof. To this end, we substitute x = 0, y = 2t, 0 < t ≤ 1 2 , into (1.4), to obtain for n = n k , Hence, By Voronovskaja's theorem for the classical Bernstein polynomials, passing to the limit on {n k }, yields which in turn implies f (t) ≥ 0, 0 < t ≤ 1 2 . Repeating the same with g(s) Hence, we conclude that f is convex in [0, 1].
Final comment, note that, in our proof, we only used (1.4) with x = 0 and x = 1.

Proof of Theorem 1.1
We begin with some preparatory lemmas.
then it is a homogeneous polynomial of total degree q(n − 1), in the variables u and v, with non-negative coefficients.
Proof. Denote s := qn and rewrite where the sum is over 2 q possible w = {w 1 , . . . , w q }, with the variables w 1 , . . . , w q taking the values u and v, and for each w, σ(w) is the number of w i 's that take the value v.
Expanding the s power, applying the multinomial expansion, yields, is a homogeneous polynomial in u and v of total degree m i − 1, the lemma follows.
Thus, we may write P (u, v) =: where all a i ≥ 0, 0 ≤ i ≤ q(n − 1). We will compare the coefficients of P to the coefficients of the homogeneous polynomial of total degree q(n − 1), in the variables u and v, We will prove that To this end, we observe that, by (2.2), P (u, v) is a weighted average of the polynomials Hence, it suffices to prove that, coefficient-bycoefficient, they maximize when m 1 = · · · = m q = n.
Without loss of generality we may assume that v = 1 and u = 1. Proof. For any 1 ≤ m ≤ l, If m < l − 1, then m + 1 ≤ l − 1, so we may replace in the above m by m + 1 and l by l − 1, and obtain Comparing the two equations proves the lemma. Proof. Suppose that for some k, m k = n. Then, there exist two indices such that m i < n < m j . By Lemma 2.2, all coefficients of the polynomial (u−1) q are no smaller than those of P (u). As long as there is an m k = n, we continue. This completes the proof.
is divisible by (u − v) q , and the resulting polynomial is a homogeneous polynomial of total degree q(n − 1) in the variables u and v, with nonnegative coefficients.
Proof. Note that Hence, D is divisible by (u − v) q , and the assertion that the coefficients are nonnegative follows by (2.5).