1. Introduction
Let
be the class of analytic functions normalized by the condition
in the unit disk
, where
. Now let
be the image of an arc
,
under the function
, and let
be a point not on
. Let
. The arc
is
-spirallike with respect to
if
lies between
and
. Further, an arc
is convex
-spirallike if
lies between
and
. In the form of one variable, equivalently, we say that a function
is
-spirallike of order
if and only if
where
and
. We denote the class of such functions by
. In view of the well-known Alexander’s relation, let
be the class of convex
-spirallike functions of order
, which is defined below:
Spacek [
1] introduced and studied the class
. Each function in
is univalent in
, but they are not necessarily starlike. Further, it is worth mentioning that, for general values of
, a function in
need not be univalent in
. For example,
, but this is not univalent. In this context, see
Figure 1. Indeed,
is univalent if
; see Robertson [
2] and Pfaltzgraff [
3]. Note that, for
, the classes
and
reduce to the classes of starlike and convex functions of order
, given by
which we denote by
and
, respectively.
In the above context and using the idea of uniformly starlike and uniformly convex functions introduced and studied in [
4,
5,
6,
7,
8], Ravichandran et al. [
9] introduced the concept of uniformly
-spiral functions as described below:
Definition 1. ([
9]).
The function f is a uniformly convex γ-spiral function if the image of every circular arc with center at ζ lying in is a convex γ-spiral. We denote the class of such functions by
. These functions have an analytic characterization (see [
9]) as follows:
Indeed, authors in [
9] obtained the one-variable characterization as:
Special functions and their geometric properties frequently appear in univalent function theory. In the recent past, normalized function properties have been explored in terms of radius problems [
10,
11,
12,
13]. In particular, for a normalized special function
f, we define
and, similarly, one may define the
-radius. In this direction, the
-radius of Bessel functions was obtained in [
10], and Ramanujan-type entire functions were dealt with in [
11]. A unified treatment of the radius of Ma–Minda classes [
14] and more for some special functions was studied in [
13,
15,
16].
However, to the best of our knowledge, the -radius has not been studied to date for the above-mentioned special functions. Therefore, we define the -radius here:
Definition 2. Let f in be a special function. Then, the radius of a uniformly convex γ-spirallike is found as: In the present investigation, we consider our special normalized functions to be the derivatives of Bessel functions. Recall that the Bessel function of the first kind of order
is defined by ([
17], p. 217):
We know that all its zeros are real for
. Here we consider the general function
which was studied by Mercer [
18]. Here, as mentioned in [
18], we take
and
. From (
1), we now have the power series representation
where
There are three important reference works dealing with the function
Firstly, in Mercer’s paper [
18], it is proved that the
positive zero of
increases with
in
. Secondly, Ismail and Muldoon [
19], under the conditions
such that
and
or
and
, showed the following behavior of roots:
- (i)
For , the zeros of are either real or purely imaginary.
- (ii)
For , where is the largest real root of the quadratic the the zeros of are real.
- (iii)
If , and , the zeros of are all real, except for a single pair that is conjugate and purely imaginary.
In 2016, Baricz et al. [
20] obtained the sufficient and necessary conditions for the starlikeness of a normalized form of
by using the results of Mercer [
18], Ismail and Muldoon [
19], and Shah and Trimble [
21].
Since the function
does not belong to
, to prove our results, we consider the following normalizations of the function
as given by:
In the rest of this paper, for the quadratic
, we will always assume that
or
Moreover,
is the largest real root of the quadratic
defined according to the above conditions.
Since the functions
and
are entire functions of order zero, then they have infinitely many zeros. According to the Hadamard factorization theorem [
22], we may write
and
where
and
denote the
positive zero of
and
respectively. For recent updates on the geometric properties of Bessel functions, readers are urged to see [
20,
23,
24,
25,
26,
27,
28] and references therein.
In this paper, we find the radius of uniformly convex
-spirallike for the functions
, and
to be defined by (
3)–(
5) when
. The key tools in their proofs are special properties of the zeros of the function
and their derivatives.
2. Zeros of Hyperbolic Polynomials and the Laguerre–Pólya Class of Entire Functions
In this section, we recall some necessary information about polynomials and entire functions with real zeros. An algebraic polynomial is called hyperbolic if all its zeros are real. We formulate the following specific statement that we shall need; see [
25] for more details. By definition, a real entire function
belongs to the Laguerre–Pólya class
if it can be represented in the form
with
, and
Similarly,
is said to be of type
in the Laguerre–Pólya class, written
, if
or
can be represented as
with
, and
The class
is the complement of the space of hyperbolic polynomials in the topology induced by the uniform convergence on the compact sets of the complex plane, while
is the complement of the hyperbolic polynomials whose zeros possess a preassigned constant sign. Given an entire function
with the Maclaurin expansion
its Jensen polynomials are defined by
The next result of Jensen [
29] is a well-known characterization of functions belonging to
.
Lemma 1. The function φ belongs to , respectively) if and only if all the polynomials , are hyperbolic (hyperbolic with zeros of equal sign). Moreover, the sequence converges locally uniformly to .
The following result is a key tool in the proof of the main results.
Lemma 2. ([
30]).
The function has infinitely many zeros, and all of them are positive, if . Denoting by the positive zero of under the same conditions, the Weierstrassian decompositionis valid, and this product is uniformly convergent on compact subsets of the complex plane. Moreover, if we denote by the nth positive zero of where then the positive zeros of are interlaced with those of In other words, the zeros satisfy the chain of inequalities 3. Main Results
Our principal result establishes the radius of
, see
Table 1, and reads as follows:
Theorem 1. Let The following statements hold:
If , , then the radius of uniformly convex γ-spirallikeness of the function is the smallest positive root of the equation If then the radius of uniformly convex γ-spirallikeness of the function is the smallest positive root of the equation If then the radius of uniformly convex γ-spirallikeness of the function is the smallest positive root of the equation
Proof. We first prove part
. From (
3), we have
and by means of (
6) and (
7), we obtain
For
, we get
where
. Moreover, observe that if we use the inequality ([
31], Lemma 2.1)
where
and
such that
, then we see that the inequality (
8) is also valid when
. Here we have seen that the zeros of
and
are interlacing according to Lemma 1. In view of the Definition 2 and the above inequalities (
8), let us define
, which is given by
Clearly, it can be seen that
for all
. Moreover, for
using Lemma 2,
Now let
be the smallest positive root of the equation
Hence, the inequality
holds whenever the following inequality is valid:
The above inequality may be equivalently read as
, which is valid for all
. The sharpness of the radius constant
can be observed, in view of the maximum and minimum modulus principles by taking
such that the following reverse inequality holds:
This completes the proof of part
.
For the other parts, in view of Lemma 2, note that the functions
and
belong to the Laguerre–Pólya class
, which is closed under differentiation, that their derivatives
and
also belong to
, and that the zeros are real. Thus assuming
and
to be the positive zeros of
and
, respectively, we have the following representations:
which yield
Further, reasoning along the same lines as in part
, the result follows at once. □
Table 1.
Radii of uniformly convex spirallike for , and in Theorem 1.
Table 1.
Radii of uniformly convex spirallike for , and in Theorem 1.
| and | and | and |
---|
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
Example 1. For the value , we may express the function in terms of the elementary trigonometric functions as follows:Consequently, we getand An immediate consequence of the proof of Theorem 1 is the following sufficient conditions for functions to be uniformly convex
-spirallike. In fact,
Table 2,
Table 3 and
Table 4 explain the sufficient conditions for uniformly convex
-spirallikeness by giving the minimum value of the
with respect to the given equations in Corollary 1. Also,
Figure 1,
Figure 2 and
Figure 3 represent image domains of the unit disk in view of Corollary 1.
Corollary 1. (Sufficient condition.) Let and . Then, the following statements hold:
Let . The function is uniformly convex γ-spirallike if The function is uniformly convex γ-spirallike if The function is uniformly convex γ-spirallike if
Table 2.
Minimum value of for uniformly convex -spirallike of in Corollary 1.
Table 2.
Minimum value of for uniformly convex -spirallike of in Corollary 1.
| and | and | and |
---|
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
Figure 1.
Image domains of the unit disk under
with
, and
for
, and
with
, and
for
using
Table 2.
Figure 1.
Image domains of the unit disk under
with
, and
for
, and
with
, and
for
using
Table 2.
Remark 1. In light of Spacek [1], here we see that for , and , and for , and , but that these are not univalent in , as shown in Figure 1.
Table 3.
Minimum value of for uniformly convex -spirallike of in Corollary 1.
Table 3.
Minimum value of for uniformly convex -spirallike of in Corollary 1.
| and | and | and |
---|
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
Figure 2.
Image domains of the unit disk under the function
with
, and
for
, and the function
with
, and
for
using
Table 3.
Figure 2.
Image domains of the unit disk under the function
with
, and
for
, and the function
with
, and
for
using
Table 3.
Table 4.
Minimum value of for uniformly convex -spirallike of in Corollary 1.
Table 4.
Minimum value of for uniformly convex -spirallike of in Corollary 1.
| and | and | and |
---|
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
Figure 3.
Image domains of the unit disk under function
with
, and
for
, and function
with
, and
for
using
Table 4.
Figure 3.
Image domains of the unit disk under function
with
, and
for
, and function
with
, and
for
using
Table 4.
Remark 2. In Table 2, Table 3 and Table 4, the entries with the choice generate ample examples of uniformly convex functions via the special function , studied by Mercer [18]. In Theorem 1, letting
, we obtain the radius of uniform convexity for the functions
, and
as defined by (
3), (
4), and (
5), respectively.
Corollary 2. (Radius of uniform convexity). Let . The following statements hold:
If , then the radius of uniform convexity of the function is the smallest positive root of the equation The radius of uniform convexity of the function is the smallest positive root of the equation Then the radius of uniform convexity of the function is the smallest positive root of the equation
Now, from Theorem 1, we deduce that:
Example 2. Let , and , see Figure 4. The following statements are true. The radius of uniformly convex spirallikeness of the function is the smallest positive root of the equation The radius of uniformly convex spirallikeness of the function is the smallest positive root of the equation The radius of uniformly convex spirallikeness of the function is the smallest positive root of the equation
Figure 4.
Image domains of for and for , respectively, with , and for
Figure 4.
Image domains of for and for , respectively, with , and for