Strong Converse Results for Linking Operators and Convex Functions

Lucian Blaga University of Sibiu, Department of Mathematics and Informatics, Str. Dr. I. Ratiu, No. 5-7, RO-550012 Sibiu, Romania School of Mathematics and Natural Sciences, University of Wuppertal, Gaußstraße 20, D-42119 Wuppertal, Germany Technical University of Cluj-Napoca, Faculty of Automation and Computer Science, Department of Mathematics, Str. Memorandumului nr. 28, 400114 Cluj-Napoca, Romania


Introduction
The Baskakov-type operators depending on a real parameter c were introduced by Baskakov in [1]. This class of operators includes the classical Bernstein, Szász-Mirakjan, and Baskakov operators as special cases for c = −1, c = 0, and c = 1, respectively.
Let c ∈ ℝ, n ∈ ℝ, n > c for c ≥ 0, and −n/c ∈ ℕ for c < 0. Furthermore, let I c = ½0,∞Þ for c ≥ 0 and I c = ½0,−1/c for c < 0. Consider f : I c → ℝ given in such a way that the corresponding integrals and series are convergent.
The Baskakov-type operators are defined as follows: where p n,j x ð Þ = , c < 0, and a c, j ≔ Q j−1 l=0 ða + clÞ, a c, 0 ≔ 1: We remark that (2) is well defined also for j ∈ ℝ, j ≥ 0, which will be considered below.
The genuine Baskakov-Durrmeyer-type operators are given by In the last years, a nontrivial link between classical Baskakov-type operators and their genuine Durrmeyer-type modification came into the focus of research. Depending on a parameter ρ ∈ ℝ + , the linking operators are given by where For c ≥ 0, the operators B ½c n,ρ are well defined for functions f ∈ W ρ n having a finite limit f ð0Þ = lim x→0 + f ðxÞ where W ρ n denotes the space of functions f ∈ L 1,loc ½0,∞Þ satisfying certain growth conditions, i. e., there exist constants M > 0, 0 ≤ q < nρ + c, such that a. e. on ½0, ∞Þ.
First, we prove that for fixed n, c and a fixed convex function f , B ½c n,ρ f is decreasing with respect to ρ. We give two proofs, using various probabilistic considerations. Then, we combine this property with some existing direct and strong converse results for classical operators, in order to get such results for the operators B ½c n,ρ applied to convex functions.

The Case c = −1
For the linking Bernstein operator, i.e., c = −1, Rasa and Sta nila [2], (10) proved that for a convex function f ∈ C½0, 1, For the proof, they used that B n,ρ can be written as a combination of the classical Bernstein operator and Beta operator and some corresponding results for the Beta operator from Adell et al. [3], Theorem 1. For the case ρ = 1 and the case ρ = ∞, strong converse results are known [4], Theorem 1.1, [5], p.117 [6], and [7], Theorem 3.2, Theorem 5: where (see [5]) with φ 2 a weight function and Thus leading to i.e., 3. The Case c = 0 Consider the classical Szász-Mirakjan operators and also the operators where r > 0.
Theorem 1 (see [8], Theorem 5 and Remark 6). Let f and x be fixed and f convex, such that G r ðjf j ; xÞ < ∞ for all r > 0. Then, G r ðf ; xÞ is nonincreasing with respect to r. Then, For c = 0, Let f be convex and n and x be fixed, such that G r ðjf j ; xÞ < ∞, for all r > 0.

An Application of Ohlin's Lemma
For more details about the techniques used in this section, the reader is referred to [13] and the references therein.
Lemma 2 (Ohlin's Lemma) (see [14]). Let X and Y be two random variables on the same probability space such that E X = EY. If the distribution functions F X and F Y cross exactly one time, i.e., for some x 0 holds then Ef ðXÞ ≤ Ef ðYÞ, for all convex functions f : ℝ → ℝ.
We have Ð Therefore, μ ½c n,j,ρ is the probability density function of a random variable X ½c n,j,ρ with expectation EX ½c n,j,ρ = j/n and probability distribution function G ½c n,j,ρ ðxÞ = Ð x 0 μ ½c n,j,ρ ðtÞdt. Let ρ < σ. We will apply Ohlin's Lemma to the random variables X ½c n,j,ρ and X ½c n,j,σ . Since their expectation is equal, we have to prove that G ½c n,j,ρ and G ½c n,j,σ cross exactly ones. Let Then, g ′ ðxÞ = μ ½c n,j,ρ ðxÞ − μ ½c n,j,σ ðxÞ. In what follows, we suppose c ≠ 0; the case c = 0 can be treated similarly or we can consider c → 0 in the computations below. For c ≠ 0, we have with positive constants K 1 and K 2 .
First, suppose that j > 0 and, if c < 0, j < −n/c. Then, on int ðI c Þ, the equation g ′ ðxÞ = 0 is equivalent to hðxÞ = x, where is a strictly convex function. The equation hðxÞ = x has at most two roots in int ðI c Þ, and so the derivative g′ has at most two zeroes in int ðI c Þ. If c < 0, gð0Þ = gð−1/cÞ = 0; if c > 0, gð0Þ = lim x→∞ gðxÞ = 0. Therefore, g ′ has at least one zero in int ðI c Þ. Suppose that g ′ has exactly one zero in int ðI c Þ, let it be x 0 . Then, g′ has opposite signs in the two intervals determined by x 0 . But (38) shows that g ′ is positive near the endpoints of I c . This contradiction leads us to the conclusion that g ′ has exactly two zeroes x 1 < x 2 in int ðI c Þ; they are also roots of the equation x = hðxÞ, hðxÞ being a strictly convex function. Moreover, g ′ is positive outside of ðx 1 , x 2 Þ and negative inside it. Therefore, gðxÞ is strictly increasing for x < x 1 and for x > x 2 , and strictly decreasing for x 1 < x < x 2 , with gðx 1 Þ > 0 and gðx 2 Þ < 0.