- Research Article
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Some Properties of Orthogonal Polynomials for Laguerre-Type Weights
Journal of Inequalities and Applications volume 2011, Article number: 372874 (2011)
Abstract
Let , let
be a continuous, nonnegative, and increasing function, and let
be the orthonormal polynomials with the weight
. For the zeros
of
we estimate
, where
is a positive integer. Moreover, we investigate the various weighted
-norms (
) of
.
1. Introduction and Main Results
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 quasi-increasing 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
.
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)
(11)
(d)the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ2_HTML.gif)
is quasi-increasing in , with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ3_HTML.gif)
(e)there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ4_HTML.gif)
Then, we write . If there also exists a compact subinterval
of
, and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ5_HTML.gif)
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)
(16)
(d)the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ7_HTML.gif)
is quasi-increasing in , with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ8_HTML.gif)
(e)there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ9_HTML.gif)
There exists a compact subinterval of
, and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ10_HTML.gif)
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
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ11_HTML.gif)
Now, we will estimate the higher-order 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ12_HTML.gif)
(c)there exist positive constants ,
and
such that for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ13_HTML.gif)
(d)there exists such that we have one among the following:
(d1) is quasi-increasing on
,
(d2) is nondecreasing on
.
Then, we write .
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ14_HTML.gif)
where is the
th iterated exponential.
(1)When , we consider
, then
.
(2)When ,
is an integer, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ15_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ16_HTML.gif)
Let us denote the zeros of by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ17_HTML.gif)
The Mhaskar-Rahmanov-Saff numbers are defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ18_HTML.gif)
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 higher-order derivatives of
at the zeros of
and to investigate the various weighted
-norms (
) of
. More precisely, we will estimate the higher-order 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 higher-order Hermite-Fejér 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ19_HTML.gif)
Theorem 1.6.
Let and
. Assume that
for
, and if
is bounded, then assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ20_HTML.gif)
where is defined in (1.13). For each
and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ21_HTML.gif)
and in particular if is even, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ22_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ23_HTML.gif)
and as
and
.
Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ24_HTML.gif)
Let us define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ25_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ26_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ27_HTML.gif)
Theorem 1.9.
Let . Let
and
. Then one has for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ28_HTML.gif)
Theorem 1.10.
Let ,
and
. Suppose that
. For
and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ29_HTML.gif)
and for  
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ30_HTML.gif)
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.
2. Preliminaries
Levin and Lubinsky introduced the classes and
as an analogy of the classes
and
which they already defined on
. They defined the following.
Definition 2.1 (see [18]).
We assume that has the following properties:
(a) is continuous in
, with
,
(b) exists and is positive in
,
-
(c)
(21)
(d)the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ32_HTML.gif)
is quasi-increasing in , with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ33_HTML.gif)
(e)there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ34_HTML.gif)
Then, we write . If there also exists a compact subinterval
of
, and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ35_HTML.gif)
then, we write .
Then we see that and
from [1, Lemma  2.2]. In addition, we easily have the following.
Lemma 2.2.
Let . Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ36_HTML.gif)
where .
On , we can consider the corresponding class to
as follows.
Definition 2.3 (cf. [19]).
Let and
be an integer. Let
be a continuous and even function on
. For the exponent
, we assume the following:
(a), for
and
,
(b)there exist positive constants such that for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ37_HTML.gif)
(c)there exist positive constants ,
and
such that for
 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ38_HTML.gif)
(d)there exists such that one has one among the following:
(d1) is quasi-increasing on
,
(d2) is nondecreasing on
.
Then, we write .
Let and
. For
, we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ39_HTML.gif)
Then, we can construct the orthonormal polynomials of degree
with respect to
. That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ40_HTML.gif)
Let us denote the zeros of by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ41_HTML.gif)
Jung and Sakai [5, Theorems  3.3 and  3.6] estimate ,
,
, and we will obtain analogous estimations with respect to
,
,
in Theorems 1.6 and 1.7.
There are many properties of with respect to
,
,
of Definition  2.3 in [4–6, 19–21]. They were obtained by transformations from the results in [1, 2]. In this paper, we consider
and
. In [5] we got the estimations of
,
,
with the weight
. By a transformation of the results with respect to
, we estimate
,
,
. In order to it we will give the transformation theorems in this section. In the following, we will give some applications of them.
Theorem 2.4 (see [21, Theorem  2.1]).
Let with
. Then, the orthonormal polynomials
on
can be entirely reduced to the orthonormal polynomials
in
as follows: for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ42_HTML.gif)
In this paper, we will use the fact that is transformed into
as meaning that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ43_HTML.gif)
Theorem 2.5.
Let . Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ44_HTML.gif)
In particular, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ45_HTML.gif)
where is defined in (1.13).
Proof.
Let . Then, from Lemma 2.2, one has
. Let
denote the maximum integer as
(Gaussian symbol). For
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ46_HTML.gif)
Therefore, we easily see that (a) of Definition 2.3 holds. Let . Since
is increasing for
and
, there exists
with
such that for
, 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ47_HTML.gif)
Then, since for , 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ48_HTML.gif)
one has by (b) of Definition 1.3 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ49_HTML.gif)
Similarly, one has by (2.16), (d) of Definition 1.2, and (b) of Definition 1.3 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ50_HTML.gif)
Consequently, one has (b) in Definition 2.3. We know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ51_HTML.gif)
and since on
, one has from (1.13) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ52_HTML.gif)
Therefore, one has by (2.16)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ53_HTML.gif)
where is defined in (1.13). The inequalities (d1) and (d2) of Definition 2.3 follow easily from (d1) and (d2) of Definition 1.3. Therefore, one has (2.14).
3. Proofs of Theorems 1.5, 1.6, and 1.7
For convenience, in the rest of this paper, we put as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ54_HTML.gif)
and . Then, we know that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ55_HTML.gif)
In the following, we introduce some useful notations.
(a)The Mhaskar-Rahmanov-Saff numbers and
are defined as the positive roots of the following equations:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ56_HTML.gif)
(b)Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ57_HTML.gif)
Then, one has the following.
Lemma 3.1 (see [1, (2.5), (2.7), (2.9)]).
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ58_HTML.gif)
To prove Theorem 1.6, we need some lemmas as follows.
Lemma 3.2 (see [21, Theorem  2.2, Lemma  3.7]).
For the minimum positive zero (
is the largest integer
), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ59_HTML.gif)
and for the maximum zero , one has for large enough
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ60_HTML.gif)
Moreover, for some constant , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ61_HTML.gif)
Lemma 3.3 (see [6, Theorem  2.5]).
Let and
. Then, uniformly for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ62_HTML.gif)
Lemma 3.4 (see [5, Theorem  3.6 and Lemma  3.7 (3.20)]).
Let and
,
. Assume that
for
and if
is bounded, then assume
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ63_HTML.gif)
where is defined in (2.8). If
, then one has for
 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ64_HTML.gif)
and in particular, if is even, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ65_HTML.gif)
Remark 3.5.
Let . Then, from [19, Theorem  1.6] we know that when
is unbounded, for any
, there exists
such that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ66_HTML.gif)
In addition, since , we know that
(i) is bounded
is bounded,
(ii) is unbounded
for any
,
(iii) for some constant
.
Lemma 3.6.
For , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ67_HTML.gif)
where satisfy that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ68_HTML.gif)
and for  
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ69_HTML.gif)
Proof.
It is easily proved, using the mathematical induction on .
Proof of Theorem 1.5.
By Lemmas 3.3, 3.6 and (3.2), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ70_HTML.gif)
Since by Lemma 3.2
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ71_HTML.gif)
one has from Lemma 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ72_HTML.gif)
Proof of Theorem 1.6.
Since , we know that
and we know that
by Theorem 2.5 and from (3.1), (3.2), and Lemma 3.1 that
(i),
(ii) for
for
,
(iii).
Then, using Remark 3.5, we can apply Lemma 3.4 to ,
. In a similar way to the proof of Theorem 1.5, one has from Lemma 3.4 and Lemma 3.1
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ73_HTML.gif)
Let be even. Then, one has from Lemma 3.4 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ74_HTML.gif)
Since by Lemma 3.2 and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ75_HTML.gif)
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ76_HTML.gif)
Therefore, when is even, one has by Lemma 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ77_HTML.gif)
Next, we will prove Theorem 1.7. To prove it, we need two lemmas as follows.
Lemma 3.7 ([5, Theorem  3.3]).
Let ,
. Let
,
, and
. Then, under the same conditions as the assumptions of Lemma 3.4, there exist
,
for absolute constants
,
such that the following equality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ78_HTML.gif)
and as
and
.
From Lemma 3.3, we easily have the following.
Lemma 3.8.
Let and
. Then, uniformly for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ79_HTML.gif)
Proof of Theorem 1.7..
By Lemmas 3.4, 3.6 and Theorem 2.4, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ80_HTML.gif)
Since we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ81_HTML.gif)
by the same reason as the proof of Theorem 1.6, we can apply Lemma 3.7 to . Then, using Lemmas 3.7 and 3.6, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ82_HTML.gif)
Here, ,
. Since from (3.28) we see that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ83_HTML.gif)
one has that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ84_HTML.gif)
as and
. If we let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ85_HTML.gif)
then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ86_HTML.gif)
and as
and
. On the other hand, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ87_HTML.gif)
Here, one has from Lemma 3.8 and (3.28) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ88_HTML.gif)
Finally, if we let , then the result is proved.
4. Proofs of Theorems 1.8, 1.9, and 1.10
Lemma 4.1.
Let , and let
and
. Then, one has for
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ89_HTML.gif)
Proof.
In [21, theorem  2.6] we showed that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ90_HTML.gif)
But, seeing our proof of [21, Theorem  2.6] carefully, we can easily prove the first equivalence.
Lemma 4.2 (see [21, Theorem  2.4]).
Let ,
and
. Let
. Then, given
, there exists a positive constant
such that one has for any polynomial
 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ91_HTML.gif)
Proof of Theorem 1.8.
From Theorem 2.4 and Lemmas 4.2 and 4.1, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ92_HTML.gif)
On the other hand, one has by Theorem 2.4 and Lemma 4.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ93_HTML.gif)
Consequently, using Lemma 3.1, one has the result.
Lemma 4.3.
Let , and let
. Then, uniformly for
and
, one has the following:
-
(a)
(46)
(b)for and
, 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ95_HTML.gif)
(c)for , 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ96_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ97_HTML.gif)
Proof.
-
(a)
It is from [1, Theorem  1.2]. (b) It is from [2, Theorem  1.3]. (c) It is from [2, Theorem  1.4].
Proof of Theorem 1.9.
By Theorem 1.8, one has for , 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ98_HTML.gif)
For , we know by (4.7) and (4.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ99_HTML.gif)
Then, for  
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ100_HTML.gif)
Therefore, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ101_HTML.gif)
On the other hand, one has from Theorem 1.8 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ102_HTML.gif)
and by (4.6) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ103_HTML.gif)
Therefore, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ104_HTML.gif)
From Remark 3.5(iii), we see that . So, consequently, one has the result.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ105_HTML.gif)
Then, we obtain by Lemma 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ106_HTML.gif)
Lemma 4.4 (see [21, Theorem  2.7]).
Let and
. For
and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ107_HTML.gif)
Proof of Theorem 1.10.
By Theorem 2.4, we can transform on
to
on
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ108_HTML.gif)
Using Lemma 4.4 and noting (3.1), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ109_HTML.gif)
On the other hand, by Lemma 4.2, we see
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ110_HTML.gif)
where is a constant. Therefore, using Lemma 4.4 and noting (3.1) and the definition of
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ111_HTML.gif)
Therefore, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372874/MediaObjects/13660_2010_Article_2337_Equ112_HTML.gif)
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Acknowledgments
The authors thank the referees for many kind suggestions and comments. H. S. Jung was supported by SEOK CHUN Research Fund, Sungkyunkwan University, 2010.
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Jung, H., Sakai, R. Some Properties of Orthogonal Polynomials for Laguerre-Type Weights. J Inequal Appl 2011, 372874 (2011). https://doi.org/10.1155/2011/372874
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DOI: https://doi.org/10.1155/2011/372874