In this subsection we prove our main results.

#### Proof

Using the mean value theorem, we have

Also it is a valid identity that

Thus, using the above identities, we can write

Also it can be verified that

Combining Equations 3.1–3.3, we get

Applying Remark 2 and Lemma 1 in Equation 3.4, we have

In order to complete the proof of the theorem, it suffices to estimate the terms

*A*
_{
n,r
}(

*f*,

*x*) and

*B*
_{
n,r
}(

*f*,

*x*) as follows:

For estimating the integral

above, we proceed as follows: since

*t*≥2

*x* implies that

*t*≤2(

*t*−

*x*) Schwarz inequality which follows from Lemma 1,

By using the Hölder’s inequality and Remark 2, we get the estimate as follows:

Collecting the estimates from Equations 3.6–3.9, we obtain

On the other hand, to estimate

*B*
_{
n,r
}(

*f*,

*x*) by applying the Lemma 2 with

and integration by parts, we have

where
.

Through combining the Equations 3.4, 3.10, and 3.11, we get the desired results. □

As a consequence of Lemma 3, we can easily prove the following corollary for the derivatives of the operators *G*
_{
n,r,c
}.