Abstract

We consider a family of operators which is a link between classical Baskakov operators (for ) and their genuine Durrmeyer type modification (for ). First, we prove that for fixed and a fixed convex function , is decreasing with respect to . 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 applied to convex functions.

1. Introduction

The Baskakov-type operators depending on a real parameter were introduced by Baskakov in [1]. This class of operators includes the classical Bernstein, Szász-Mirakjan, and Baskakov operators as special cases for , , and , respectively.

Let , , for , and for . Furthermore, let for and for . Consider given in such a way that the corresponding integrals and series are convergent.

The Baskakov-type operators are defined as follows: where and

We remark that (2) is well defined also for , , 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 the linking operators are given by where with

For , is well defined if with finite limits at the endpoints of the interval , i.e., and .

For , the operators are well defined for functions having a finite limit where denotes the space of functions satisfying certain growth conditions, i. e., there exist constants , , such that a. e. on .

First, we prove that for fixed and a fixed convex function , is decreasing with respect to . 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 applied to convex functions.

2. The Case

For the linking Bernstein operator, i.e., , Raa and Stnil [2], (10) proved that for a convex function ,

For the proof, they used that 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 and the case , strong converse results are known [4], Theorem 1.1, [5], p.117 [6], and [7], Theorem 3.2, Theorem 5: where (see [5]) with a weight function and .

Thus, for convex, leading to

i.e.,

3. The Case

Consider the classical Szász-Mirakjan operators and also the operators where .

Moreover, for , let (see [8]).

Theorem 1 (see [8], Theorem 5 and Remark 6). Let and be fixed and convex, such that for all . Then, is nonincreasing with respect to .
Then, For , Let be convex and and be fixed, such that , for all .
Let . Then, by (16) and Theorem 1, Now by (17), Thus, Strong converse results are known also in this case (see [6], Theorem 1.2 and [9], Theorem 5.1, Theorem 5.2): Thus, for convex,

4. The Case

To treat this case, we need some preliminaries.

If and are independent random variables with densities , , and , then has density

Let and be two independent gamma processes (see [8], p.129), i. e., has density , . Let and . Then, has density

Substitute . Then,

Let .

Consequently, compare with [8], (9).

As in [10], Proof of Lemma 2, let , . Since the random vectors and are independent, we have with [10], Lemma 1. Moreover, as in [10], (19), we get

Thus,

As at the end of [10], Proof of Lemma 2, taking here the conditional expectation w.r.t. , we get

Now (30) is exactly the assumption of [8], Theorem 5 (a). Accordingly, [8], Theorem 5 (a) and Remark 6 show that

This implies where are the classical Baskakov operators.

Since, , we infer that

The direct and strong converse results are known also in this case (see [11], Theorem 1.2, Theorem 1.3 and [12], Theorem 1.1):

Thus, for convex,

5. 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 and be two random variables on the same probability space such that . If the distribution functions and cross exactly one time, i.e., for some holds then , for all convex functions .

We have and .

Therefore, is the probability density function of a random variable with expectation and probability distribution function .

Let . We will apply Ohlin’s Lemma to the random variables and . Since their expectation is equal, we have to prove that and cross exactly ones.

Let

Then, .

In what follows, we suppose ; the case can be treated similarly or we can consider in the computations below. For , we have with positive constants and .

On , the equation is equivalent to .

First, suppose that and, if , . Then, on , the equation is equivalent to , where is a strictly convex function. The equation has at most two roots in , and so the derivative has at most two zeroes in . If , ; if , . Therefore, has at least one zero in .

Suppose that has exactly one zero in , let it be . Then, has opposite signs in the two intervals determined by . But (38) shows that is positive near the endpoints of . This contradiction leads us to the conclusion that has exactly two zeroes in ; they are also roots of the equation , being a strictly convex function. Moreover, is positive outside of and negative inside it. Therefore, is strictly increasing for and for , and strictly decreasing for , with and .