1. Introduction
Special functions have an indispensable role in many branches of mathematics and applied mathematics. Therefore, it is important to examine its properties from many aspects. In recent years, there has been intense interest in some special functions in terms of geometric function theory, especially q-special functions, which have an important place in the fields of mathematical physics and engineering.
The q-analogs are a technique employed in mathematical physics and mathematical analysis, crucial for accurately defining various physical and mathematical systems. Also, q-Bessel and q-Bessel-Struve functions hold significant importance within the geometric function theory. These functions exhibit similarities to classical Bessel and Struve functions but have been modified using q-analog techniques.
There is an extensive literature dealing with geometric properties of different types of special functions. For instance, in 2014, Baricz et al. [
1], by considering a much simpler approach, succeeded in determining the radius of starlikeness of the normalized Bessel functions. In the same year, Baricz and Szász [
2] obtained the radius of convexity of the normalized Bessel functions. Finding radii of starlikeness and convexity using the zeros of Bessel functions was later extended to other special functions. Their geometric properties such as starlikeness, convexity, close-to-convexity, uniformly convex, lemniscate starlikeness and convexity, Janowski starlikeness and convexity,
-spirallike and convex
-spirallike have been investigated by many mathematicians. For example, Bessel functions [
1,
2,
3,
4,
5,
6,
7,
8], Lommel and Struve functions [
9,
10,
11,
12], Wright functions [
12,
13,
14], Mittag-Leffler functions [
12,
13,
15,
16,
17,
18], Ramanujan functions [
12,
17,
19], Legendre polynomials [
12,
20], the function
[
21,
22,
23,
24,
25] and
q-special functions [
6,
7,
26,
27,
28,
29,
30,
31,
32].
Building on the motivation from the literature discussing geometric properties of special functions, particularly in relation to q-Bessel and q-Bessel-Struve functions, I have explored the radii of -spirallike and convex -spirallike properties of order These properties are crucial in geometric function theory as they define the regions in which these functions map the unit disk.
In conclusion, the investigation of these geometric properties for q-Bessel and q-Bessel-Struve functions builds upon the rich theoretical foundation laid out in the literature, offering new perspectives and avenues for future research in mathematical analysis and its applications.
Under the normalization
, let
show the analytic functions class in the unit disk
, where
. We refer to that function
is
-spirallike of order
if and only if
where
and
. We denote the class of functions like this
. We also denote its convex analog, that is the class
of convex
-spirallike functions of order
, which is defined below
The class
was presented by Spacek [
33]. Every function in
is univalent in
, although they need not be starlike. In addition, it is worth noting that for overall values of
, a function in
need not be univalent in
. For instance:
, but not univalent. Actually,
is univalent if
, see Robertson [
34] and Pfaltzgraff [
35]. Note, that for
, the classes
and
reduce to the classes convex and starlike functions with order
, given by
which we denote this with
and
, respectively.
Recently, linkages between special functions and their geometrical features have been demonstrated using radius problems [
1,
3,
4,
9,
10,
11,
14,
15,
17,
20,
23,
25]. In this manner, the behavior of the positive roots of a specific function, as well as the Laguerre-Pólya class, are clearly important. A self-mapping real entire function
L of the real line is in the Laguerre-Pólya class
, if for
,
,
we have
with
, see [
9], ([
36] p. 703), [
37] and the references therein. The class
is of entire functions approximated uniformly on the complex plane’s compact sets by polynomials with only real zeros. This class is closed under differentiation.
We will also need to recall the following results for further development:
Lemma 1 (see [
38,
39])
. Take a look at the power series and where and for all Assume that both series converge on for some If the sequence is decreasing (increasing), then the function is decreasing (increasing) too on This is true for the power series The
-radius, which is given below
similarly, the
-radius has recently been obtained for some normalized forms of Bessel functions in [
1,
3,
4] (Watson’s treatise [
40] is an excellent resource on Bessel function), Struve functions studied in [
9,
11], Wright functions in [
14], Lommel functions radii properties studied in [
9,
11], Legendre polynomials of odd degree radii results deal in [
20] and recently, Ramanujan type entire functions were studied in [
19]. These function’s radii problems collectively for unified subclasses of starlike and convex functions have been studied in [
17].
According to the literature review,
-radius and
-radius for special functions have not been discussed so far. So, in this study, we now aim to derive the radius of
-spirallike of order
, which is provided below
and also the radius of convex
-spirallike of order
, which is
for the function
ℏ in
to be a special function.
2. -Bessel Functions
The infinite series representation of the first kind of Bessel function [
40] is defined as follows:
where
and
such that
Lommel’s well-known conclusion represents that if
then the zeros of the Bessel function
are all real. Thus, if
and
denote the
k-th positive zeros of
and
respectively, then the Bessel function and its derivative admit the Weierstrassian decomposition of the forms [
40]
and
The convergence is uniform on each compact subset of
for the above infinite products. The Jackson and Hahn-Exton
q-Bessel functions are clearly stated by
and
where
and
Particularly, these analytic functions are
q-extensions of
Namely, for fixed
we have
and
as
Watson’s treatise [
40] is an excellent resource for Bessel functions. Recent developments on Bessel and its extension can be found in [
41,
42,
43,
44] and the references therein.
In this section, we investigate the geometric properties of the normalized Jackson and Hahn-Exton
q-Bessel functions. Combining the methods deployed in [
1,
2,
9] we determine precisely the radii of
-spirallike of order
and convex
-spirallike of order
for each of the six functions. In between the proofs, we exercise the simple properties of the Laguerre-Pólya class, as well as the interlacing property of the zeros of the Jackson and Hahn-Exton
q-Bessel functions.
The current study shows that there is no important difference between Jackson and Hahn-Exton q-Bessel functions when treating the problem about the radii of -spirallike of order and convex -spirallike of order . As a result, one may predict that the other geometric features of these two q-extensions of Bessel’s function are comparable.
Observe that neither
, nor
belongs to
, and therefore,
we define three normalized functions that stem from
:
where
Similarly,
Clearly, the functions
belong to the class
The primary rationale for considering these six functions stems from their connections to the limiting cases found in the literature concerning Bessel functions, as discussed in [
16] and the associated references.
Lemma 2 ([
28])
. If then and are entire functions of order . Consequently, their Hadamard factorization for are of the formwhere and are the k-th positive zeros of the functions and Lemma 3 ([
28])
. If then and are entire functions of order . Consequently, their Hadamard factorization for are of the formandwhere and are the k-th positive zeros of and Lemma 4 ([
28])
. Let and Then the functions and can be expressed in the following waywhere and are entire functions from the Laguerre-Pólya class Furthermore, the lowest positive zero of does not exceed the first positive zero and the lowest positive zero of is smaller than Lemma 5 ([
28])
. Between any two consecutive roots of the function has precisely one zero when and The first principal result gives radii of -spirallike of order .
Theorem 1. Let and . The following statements are true:
- (i)
If then the radius is the lowest positive roots of the equation Moreover, if then the radius is the lowest positive roots of the equation - (ii)
If then the radius is the lowest positive roots of the equation - (iii)
If then the radius is the lowest positive roots of the equation
Proof. The proofs for the cases and are nearly identical; the only difference is that the zeros and in the proofs are different. As a result, we will only present the proof for the case , and for clarity, we will use the notations: and
First we prove part
for
and parts
and
for
We need to show that for
and
the inequalities
and
are valid for
and
accordingly, and none of the disparities listed above apply in larger disks.
Using (
5), we obtain after the logarithmic differentiation:
It is known [
1] that if
and
are such that
then
Then, the inequality
holds for every
which in turn implies that
and
with equality when
Because of the latter inequalities and the minimum principle for harmonic functions, the corresponding inequalities in (
8) and (
9) hold if and only if
and
respectively, where
and
are the lowest positive roots of the equations
and
Because their solutions correspond to the zeros of the functions
and
the required result is obtained by substituting
and
for
a in Lemma 4, respectively. In another saying, Lemma 4 demonstrates that all of the above three functions have real zeros and their first positive zeros do not exceed the first positive zeros
, respectively. This assures that the aforementioned inequalities hold, so completing the proof of part
when
, and parts
and
when
To demonstrate the statement for component
when
first observe that the counterpart of (
11), that is,
is valid for all
and
such that
By using (
12), for all
and
the inequality
which holds for every
and it implies that
In this situation, equality occurs if
Furthermore, the last inequality explains that
if and only if
where
denotes the equation’s lowest positive root
As a result of Lemma 4 the first positive zero of
does not exceed
ensuring that the preceding inequalities are satisfied. We just need to demonstrate that the function mentioned above only has one zero in
Take note that, according to Lemma 1, the function
is increasing on
as a quotient of two power series whose positive coefficients form the increasing “quotient sequence”
When
the above function tends to
, so its graph can only intersect the horizontal line
once. This completes the proof of part
of the theorem for
□
Remark 1. Taking in Theorem 1 yields ([28] Theorem 1). Our second result in this section concerns the radii of convex -spirallike of order
Theorem 2. Let and .
- (i)
If then the radius is the lowest positive roots of the equation - (ii)
If then the radius is the lowest positive roots of the equation - (iii)
If then the radius is the lowest positive roots of the equation
Proof. The proofs for the conditions and are nearly identical; the only variation is that the zeros in the proofs are different. As in the last argument, we will explain the proof just for the case , and for clarity, we will employ the following notations: and
and by means of (
6) and (
7) we have
For
by using the inequality (
11), for all
we obtain the inequality
where
. Furthermore, notice that if we use the inequality ([
2] Lemma 2.1)
where
and
such that
, then we obtain that the inequality (
13) is also correct for
. Here, we used the zeros
and
interlace according to Lemma 5. The inequality (
13) implies for
Now, we define the function
which is strictly decreasing since
for
and
Observe also that
and
which means that for
we have
if and only if
is the unique root of
situated in
For the other parts, note that the functions
and
for
belong to the Laguerre-Pólya class
, which is closed under differentiation, their derivatives
and
also belong to
and the zeros are real for
Thus, assuming
and
are the
k-th positive zeros of
and
, while
and
are the
k-th positive zeros of
and
, respectively, we have the following representations:
and
Similarly, we can prove parts
and
□
Remark 2. Taking in Theorem 2 yields ([28] Theorem 2). 3. -Bessel-Struve Functions
The Struve functions applications in other active research areas can be found in [
45,
46]. Recently, Oraby and Mansour [
30] introduced the
q-Struve-Bessel functions
defined by
and
where
we follow [
47] for the definitions of
q-shifted factorial,
q-gamma function, the
q-binomial coefficients, Jackson
q-difference and
q-integral operators and
q-numbers.
The functions
are
q-form of the Struve function ([
40] p. 328)
That is,
One can see that the functions
are entire functions of order zero and have infinitely many zeros. So Hadamard factorization theorem [
37] says that
and
where
and
denote the
n-th positive zeros of the functions
and
respectively. Oraby and Mansour [
30] proved that for
the functions
have only real simple zeros. We also refer to [
28,
31,
32].
Since
, we consider some normalizations as in [
10,
28]. For
we associate
with the normalized functions:
In a similar manner, we connect with
the functions:
Clearly, the functions
and
belong to the class
.
The following lemma is essential for the coming results.
Lemma 6 ([
31])
. Between any two consecutive the functions roots the function has precisely one zero when Now we can determine the radii of -spirallike of order of the above mentioned normalized functions.
Theorem 3. Let and . Then, the following statements are true:
- (i)
is the lowest positive roots of the equation - (ii)
is the lowest positive roots of the equation - (iii)
is the lowest positive roots of the equation
Proof. The only difference between the proofs for the cases and is that the proofs use the different zeros and . As a result, we will only provide the proof for the case in the following, and we will use the following notations for simplicity: and
From (
17), we have
where
stands to the
n-th positive zero
It follows that
On the other hand, from the inequality (
11)
holds for every
and
which in turn, implies that for
and
Then the minimum principle for harmonic functions [
37] implies that
where
is the lowest positive root of
Similarly,
where
is the lowest positive root of
and
where
is the lowest positive root of
□
Remark 3. Taking in Theorem 3 yields ([31] Theorems 3 and 4). Our second result in this section concerns the radii of convex -spirallike of order
Theorem 4. Let and . Then, the following statements hold.
- (i)
The radius of convex γ-spirallike of order σ of is the lowest positive roots of the equation - (ii)
The radius of convex γ-spirallike of order σ of is the lowest positive roots of the equation - (iii)
The radius of convex γ-spirallike of order σ of is the lowest positive roots of the equation
Proof. We only present the proof for the case s = 2, and the rest follows in parallel lines. For the sake of clarity, we will use the following notations in what follows: , and
Let
and
be the positive roots of
and
respectively. In ([
31] p. 76) the following equality was shown:
Now, suppose that
By using the inequality (
11), for all
we obtain
where
Moreover, if we use the inequality (
14), then we obtain inequality (
20) for
Here, we used that the zeros
and
interlace according to Lemma 6. The inequality (
20) implies for
Opposed to that, we define the function
is strictly decreasing for
and
Since
and
then for
we have
if and only if
is the equation’s lowest positive root
situated in
Likewise, we can show parts
and
□
Remark 4. Taking in Theorem 4 yields ([31] Theorem 8 and 9).