1. Introduction
The generalized [
1] or associated Laguerre polynomial (ALP)
is the solution of the differential equation
which is represented by the series
where
is the confluent hypergeometric function, and
is the well-known Pochhammer symbol defined as
The first few terms of the polynomial are given as
ALP has its own significance in various branches of mathematics and physics and has a wide contribution in different aspects in mathematical research. The associated Laguerre polynomials are orthogonal with respect to the gamma distribution
on the interval
. The generalized Laguerre polynomials are widely used in many problems of quantum mechanics, mathematical physics and engineering. In quantum mechanics, the Schrödinger equation for the hydrogen-like atom is exactly solvable by separation of variables in spherical coordinates, and the radial part of the wave function is an ALP [
2]. In mathematical physics, vibronic transitions in the Franck–Condon approximation can also be described by using Laguerre polynomials [
3]. In engineering, the wave equation is solved for the time domain electric field integral equation for arbitrary shaped conducting structures by expressing the transient behaviors in terms of Laguerre polynomials [
4]. The monographs by Szegó [
5], and Andrews, Askey, and Roy [
6] include a wealth of information about ALP and other orthogonal polynomial families.
In this study, we consider
The function
satisfies the normalization condition
and is a solution of the differential equation
The following four functions are also important for this study.
The function
maps
to a leminscate,
shifted
to a disc center at
with radius
,
maps
to the exponential domain, and
maps
to the neuphroid domain as shown in
Figure 1.
Let
denote the class of functions
in the open unit disk
and normalized by the conditions
. If
f and
g are analytic in
, then
f is
subordinate to
g, written
or
if there is an analytic self-map
of
satisfying
and
,
. Especially, if
is univalent in
, then
if and only if
and
. It is worth noting here that
,
,
and
are not subordinate to each other as it is clear from
Figure 1 that the image of
by any one of these functions does not contain the image by others. Differential subordination is an important technique to study geometric functions theory. Details about this technique can be seen in [
7,
8].
Denote by and , respectively the important subclasses of consisting of univalent starlike and convex functions. Geometrically, if the linear segment , lies completely in whenever , while if is a convex domain. Related to these subclasses is the Cárathèodory class consisting of analytic functions p satisfying and in . Analytically, if , while if . It is well-known that the function is starlike and is convex in the unit disk .
A function is lemniscate convex if lies in the region bounded by right half of lemniscate of Bernoulli given by , which is equivalent to the subordination . Similarly, the function f is lemniscate starlike if . On the other hand, the function is lemniscate Carathéodory if . Clearly, a lemniscate Carathéodory function is a Carathéodory function and hence is univalent.
The sufficient conditions of starlikeness associated with lemniscate of Bernoulli are obtained in [
9]. A similar study associated with the exponential domain is conducted in [
10]. One of the motivations of this work is the nephroid curve
Recently, the nephroid curve received attention of researchers in geometric functions theory thanks to the work by Wani and Swaminathan [
11,
12,
13]. This two-cusped kidney-shaped curve was first studied by Huygens and Tschirnhausen in 1697. However, the word nephroid was first used by Richard A. Proctor in 1878 in his book
The Geometry of
Cycloids. For further details related to the nephroid curve, we refer to [
11,
14]. The radius of starlikeness and convexity for functions associated with the nephroid domain is discussed in [
13]. In [
12], the authors discuss the starlike and convex functions associated with the nephroid domain. The Fekete–Szegö kind of inequalities for certain subclasses of analytic functions in association with the nephroid domain is studied in [
15].
Significant findings from the articles [
9,
10] are summarized, respectively, in Lemma 1 and Lemma 2, while Lemma 3 and Lemma 4 highlights the results from the reference [
11]. The special functions, such as Bessel, Struve, Confluent hypergeometric and hypergeometric, are closely associated with the geometric functions theory. The geometric nature of these special functions associated with the leminisciate, the exponential and the nephroid domain are studied in [
9,
10,
13]. The lemniscate convexity of generalized Bessel functions is studied in [
9], while [
10] deals with the exponential starlikeness and convexity of confluent hypergeometric, Lommel and Struve functions.
In this paper, motivated by the aforementioned works, we investigated the inclusion properties of the normalized function
involving ALP that maps the unit disc
into the lemniscate and the exponential domain, respectively, in
Section 2 and
Section 3.
Section 4 deals with the results concerning the shifted disc
, for
. In
Section 5, we derive the conditions under which integration associated with
maps
into the nephroid domain. All the results are interpreted graphically. Several options for the improvement are highlighted.
2. Mapping in the Lemniscate Domain
In this section, we derive the relation between and n for which maps into . To prove the main results related with the lemniscate, the following Lemma 1 is used.
Lemma 1 ([
16])
. Let with and . Let , and satisfy whenever and for , ,If for , then in . In the case of two dimensions, if
satisfy
whenever
and for
,
,
If for , then in .
Now we state and prove the main result for this section.
Theorem 1. For , .
Proof. Let
. Suppose that
. Define
From (
4), it follows
. To prove the result by using Lemma 1, it is enough to show
for
r,
s and
t as stated in (
5). Now
provided
. □
A natural question arises for a fixed : are the values the best possible in Theorem 1? To investigate it, we try to experiment through graphical representation of and . It is worth noting here that when . We present our cases for .
- n = 2
By Theorem 1,
holds for
. However,
Figure 2 indicates that for real
, the subordination property for which
follows for
where the possible value
is any number in the interval
.
- n = 3
As per Theorem 1, in this case
is
.
Figure 3 indicates that for real
, the inclusion
holds for
where the possible value of
is any number in the interval
.
- n = 4
The expected value of
is
, but as per
Figure 4, the value of
can be lower down to a number in
.
4. Mapping in Disc Center at and Radius
The function for maps the unit disc to a disc center at and radius A. In this section, we will derive conditions by which
Theorem 3. For , and , suppose thatThen, . Proof. Consider
A simplification gives
From (
4) it follows that
Let
and define
as
It is clear from (
9) that
. We shall apply Lemma 1 to show
, which implies
.
Now, for
, let
Applying elementary trigonometric identities, we have
Substitute
r,
s and
t in (
5), and a simplification leads to
when
. By Lemma 1, it is proved that
which is equivalent to
for some analytic function
such that
. A simplification of (
10) gives
This completes the proof. □
Graphical representation indicates that there is a provision of improvement for a minimum value of
for fixed
n and
A. For example, set
and
, and suppose that
is real. Then, by Theorem 3,
if
. However,
Figure 10 clearly indicates that the result can hold for
. This claim can also be valid theoretically. The subordination
is equivalent to
which holds for
if
. In particular, if
is a positive real number, and
, then
holds for
Similarly for
, as per Theorem 3, the subordination
holds for
. In particular, for real
and
, the subordination is true when
. However, a direct proof indicates that the subordination holds when
Clearly, the second condition is better than the first condition (derived from Theorem 3). For example, if
is real and
, then
holds for
. The comparison can be seen in
Figure 11.
Based on the above facts, we can conclude that for certain special cases, Theorem 3 has a chance for improvement. Now, we state and prove an improved version of Theorem 3.
Theorem 4. For real and fixed and , suppose that is the largest root ofThen, the subordination holds for all . The result is sharp as is the best lowest value. Proof. The subordination
is equivalent to
Now, for
, it follows
It can be easily verified that for a fixed
n, the function
is decreasing; hence, the inequality (
11) holds for
. Here,
is the largest root of the equation
This completes the proof. □
In the following
Table 1 we have listed value of
for fixed
n and
A.
5. Connection with the Nephroid Domain
In this section, we observe that
or
do not always imply
. For example, consider the case when
and
, the polynomial
as shown in
Figure 12a. Now define the function
In
Figure 12b, we can see that
To state the next result, let us generalize (
12) as follows
We also consider the function
We need the following results in sequence.
Lemma 3 ([
11])
. Let be analytic such that . Then, the following subordination implies - (i)
for
- (ii)
for
Lemma 4 ([
11])
. Let be analytic such that . Then the following subordination imply - (i)
for ,
- (ii)
for
- (iii)
for
- (iv)
for ,
- (v)
for
- (vi)
for
The following subordination holds true.
Theorem 5. For and , the following subordination holds
- (i)
for
- (ii)
for
- (iii)
for
Proof. It is worth noting that for
, Theorem 2 implies that
Now, it follows from (
13) that
Let us denote
; then, a logarithmic differentiation gives
Lastly, a derivative of
in (
14) leads to
The first three cases along with Lemma 4 (part (iv)–(vi)) helps to conclude the result, while the fourth case together with Lemma 3 (part (ii)) implies the result. □
As of a final result, we have the following that can be proved using Theorem 1 and Lemma 4 (part (i)–(iii)) and Lemma 3 (part (i)). We omit the details of the proof.
Theorem 6. For and , the following subordination holds
- (i)
for
- (ii)
for
- (iii)
for
Now, we are going to interpret the result obtained in Theorem 5 graphically. For this, we consider the special case where
. In the case
, we set the smallest value of
. Now, by judicious choice of
n (we chose up to 5000), we can see through
Figure 13 that
holds. This indicates that in the case for all
, the smallest value of
is sharp. However, in case of a fixed
n, there is a possibility to lower the value of
as presented in
Table 2.
Clearly is approaching the value for increasing n.
Similar analysis of results can also be computed for part of Theorem 5 ((ii) & (iii)) and Theorem 6. We avoid such details. However, this fact leads to an open problem as stated below
Problem 1 (Open). Find the exact value of for all n, α such that ; ; and holds for .