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^{2}

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Starting from the study of the

Starting from the study of the

The first aim of this note is to obtain the same order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of

Section

For the proof of the main results, we need some general considerations on the so-called nonlinear operators of max-prod kind. Over the set of positive reals,

Let

In this section we present some general results on these kinds of operators which will be useful later in the study of the Bernstein max-product-type operator considered in Section

Let

if

Then for all

Since it is very simple, we reproduce here the proof in [

Writing now

(

(

Let

The proof is identical with that for positive linear operators and because of its simplicity, we reproduce it what follows. Indeed, from the identity

An immediate consequence of Corollary

Suppose that in addition to the conditions in Corollary

Since it is easy to check that

For each

Also, for each

Let

for all

for all

(i) The inequality

On the other hand,

(ii) The inequality

On the other hand,

For all

We have two cases: 1)

Since clearly the function

We get

Let

If

If

(i) We observe that

(ii) We observe that

Also, a key result in the proof of the main result is the following.

One has

First we show that for fixed

If

If

It is easy to check that the max-product Bernstein operators fulfill the conditions in Corollary

Let

In order to prove (

If

If

If

Suppose first that

Suppose now that

Therefore, in both subcases, by Lemma

Suppose first that

Suppose now that

In both subcases, by Lemma

In conclusion, collecting all the estimates in the above cases and subcases we easily get the relationship (

Since

Because

(

In what follows, we will prove that for large subclasses of functions

For this purpose, for any

Let

We distinguish the two following cases.

Let

Let

Let

We distinguish the two following cases:

Let

Let

Let

the function

the function

(i) Let

(ii) Let

Let

Let

Now let

(i) If

(ii) If

(i) Since

(ii) Since

By simple reasonings, it follows that if

An example of function satisfying the above conditions is

Analogously, if

In this section, we will present some shape preserving properties, by proving that the max-product Bernstein operator preserves the monotonicity and the quasiconvexity. First, we have the following simple result.

For any arbitrary function

Since

Note that because of the continuity of

As in Section

If

Because

If

Because

If

Because

So let

If

Because

So let

In what follows, let us consider the following concept generalizing the monotonicity and convexity.

Let

By [

If

If

Suppose now that there exists

It is clear that

However, it is easy to show (see also Remark

We have

In this case, it is clear that there exists a point

The preservation of the quasiconvexity by the linear Bernstein operators was proved in [

It is of interest to exactly calculate

For all

The formula

To find the formula for

Indeed, since

Despite of the absence of the preservation of the convexity, we can prove the interesting property that for any arbitrary function

For any function

For any

We will prove that for any fixed

Since

For

For

For

If

If

Since all the functions

At the end of this section, let us note that although

Indeed, in this sense, for example, we present the following.

For

Indeed, since

Also, writing

In this section, we compare the max-product Bernstein operator

Now, if

On the other hand, in other cases (e.g., for differentiable functions), the linear Bernstein operator has better approximation properties than the max-product Bernstein operator, as can be seen from the formula for

Concerning now the shape-preserving properties, it is clear from Section