Let and . Let be a continuous, nonnegative, and increasing function. Consider the exponential weights , , and then we construct the orthonormal polynomials with the weight . In this paper, for the zeros of we estimate , where is a positive integer. Moreover, we investigate the various weighted norms () of .
We say that is quasiincreasing if there exists such that for . The notation means that there are positive constants such that for the relevant range of , . The similar notation is used for sequences and sequences of functions.
Throughout, denote positive constants independent of . The same symbol does not necessarily denote the same constant in different occurrences. We denote the class of polynomials with degree by .
First, we introduce some classes of weights.
Levin and Lubinsky [1, 2] introduced the class of weights on as follows. Let , where .
Definition 1.1 (see [1, 2]).
We assume that has the following properties. Let ,
(a) is continuous in , with limit 0 at 0 and ,
(b) exists in , while is positive in ,

(c)
(d)the function
is quasiincreasing in , with
(e)there exists such that
Then, we write . If there also exists a compact subinterval of , and such that
then we write .
We consider the case , that is, the space , and we strengthen Definition 1.1 slightly.
Definition 1.2.
We assume that has the following properties:
(a) are continuous, positive in , with , ,
(b) exists in ,

(c)
(d)the function
is quasiincreasing in , with
(e)there exists such that
There exists a compact subinterval of , and such that
then we write .
Let us consider the weight in Definition 1.2. Levin and Lubinsky [2, Theorem 1.3] have given the following theorem.
Theorem A (see [3, Theorem 1.3]).
Let and . There exists such that uniformly for , ,
Now, we will estimate the higherorder derivatives of the orthonormal polynomials . However, we need to focus on a smaller class of weights.
Definition 1.3.
Let and be an integer. For the exponent , we assume the following:
(a), for and , and , .
(b)there exist positive constants such that for
(c)there exist positive constants , and such that for
(d)there exists such that we have one among the following:
(d1) is quasiincreasing on ,
(d2) is nondecreasing on .
Then, we write .
Example 1.4 (see [1, 4]).
Let be a fixed integer. There are some typical examples satisfying all conditions of Definition 1.3 constructed as follows: let , , where is an integer. Then, we define
where is the th iterated exponential.
(1)When , we consider , then .
(2)When , is an integer, we define
Then, .
In the rest of this paper, we consider the classes and ; let or (). For , we set . Then, we can construct the orthonormal polynomials of degree with respect to . That is,
Let us denote the zeros of by
The MhaskarRahmanovSaff numbers are defined as follows:
In this paper, we will consider the orthonormal polynomials with respect to the weight class . Our main themes in this paper are to estimate the higherorder derivatives of at the zeros of and to investigate the various weighted norms () of . More precisely, we will estimate the higherorder derivatives of at all zeros of for two cases of an odd order and of an even order. In addition, we will give asymptotic relation of the odd order derivatives of at the zeros of in a certain finite interval. These estimations will play an important role in investigating convergence or divergence of higherorder HermiteFejér interpolation polynomials (see [3, 5–17]).
Then, our main purpose is to obtain estimations with respect to , , as follows.
Theorem 1.5.
Let and . For each and one has
Theorem 1.6.
Let and . Assume that for , and if is bounded, then assume that
where is defined in (1.13). For each and , one has
and in particular if is even, then
Theorem 1.7.
Let and . Let , , and , . Then, under the same conditions as the assumptions of Theorem 1.6, there exist , for absolute constants , such that the following equality holds:
and as and .
Define
Let us define
We consider the class of weights, , which is defined in Definition 2.1 below. Levin and Lubinsky have obtained the following theorem.
Theorem B (see [18, Theorem 13.6]).
Assume that . Let . Then uniformly for ,
We remark that Levin and Lubinsky have shown Theorem B for more wider class . In the following, we investigate the various weighted norms () of .
Theorem 1.8.
Let . Let and . Then one has for ,
Theorem 1.9.
Let . Let and . Then one has for ,
Theorem 1.10.
Let , and . Suppose that . For and , one has
and for
This paper is organized as follows. In Section 2, we will introduce the weight class as an analogy of the class and the known results of orthonormal polynomials with respect to in order to prove the main results. In Section 3, we will prove Theorems 1.5, 1.6, and 1.7. Finally, we will prove the results for the various weighted norms () of , that is, Theorems 1.8, 1.9, and 1.10, in Section 4.