Abstract
Bessel functions are related with the known Bessel differential equation. In this paper, we determine the radius of starlikeness for starlike functions with symmetric points involving Bessel functions of the first kind for some kinds of normalized conditions. Our prime tool in these investigations is the Mittag-Leffler representation of Bessel functions of the first kind.
1. Introduction and Definitions
Let and denote the interior of the unit circle with center at origin. Suppose that represent functions in
Obviously, along with . The subclass A only contains univalent (one-to-one) functions and represent the set of starlike functions. A function for which is star-shaped is starlike if . Also, ifand if
Letbe the radii of the classes defined above. We note that is the maximum value of the radius such that and is the maximum value of the radius such that with symmetric points. Consider the following representation of the function as in [1], which satisfies the well-known Bessel differential equation:where such that . Observe that A. Thus, we consider the following normalizations:
Clearly, the function . We see that
The geometric behavior and properties of the functions , , and were studied by Brown, Kreyszig, Robertson, and many others (for detail, see [2–4] and also the references therein). The related problems were also studied in [3, 5–8] with references therein. We study the radius problems for the functions , , and starlike with symmetric points. Mittag-Leffler expansion for Bessel functions is used as a prime tool along with the conclusion that the specific positive roots of the Dini functions are always smaller than the related zeros , for reference, see [9].
2. Preliminaries
Lemma 1. Let be a transcendental function having the following expansion:where have the same argument. For a univalent function in , we have
This result holds if and only if , and each of its derivatives is close to convex in the open unit disk . Furthermore, for , the zeroes of the derivative of , , and are univalent in and for is a convex-shaped if and only if
Lemma 2. The functionand each of its derivative is close to convex in if and only if , where is a unique zero of on
The proof of Lemma 1 and Lemma 2 is found in [10].
Lemma 3. The functionin if and only if , where is the unique solution ofIn particular, in if and only if , where is a unique zero of
Lemma 4. The functionin if and only if , where is the unique zero oflies in , where is the unique root of and is the first positive zero of In particular, in if and only if , where is a unique zero of
The proof of Lemma 3 and Lemma 4 can be seen in [11].
Lemma 5. If and , then
For the detail of the above Lemma 5, we refer to [3]
3. Main Results
Theorem 6. Let , and Then, , is a unique positive zero ofwhere . Moreover, if , then is the least positive zero of
Proof. Using Lemma 3, we see that the function in with respect to iff , where is a unique zero oflies in Suppose is the th positive root of . By using infinite product representation,Also, as given in [12], we see that has the following form:From Lemma 3, we see that for the unique value of the root of or , we have , and . The above result is immediate, if is increasing on Using (24) and (25), we can writeAlso, from (6), we havewhich in the context of is equivalent to the Mittag-Leffler representation:Consequently,In view of (6), (25), and (27), we can writeUsing Lemma 1, we find that for , and the following inequalityimplies thatWhen we observe thatAs in [12], we see that on for a fixed Thus, is increasing on and is decreasing on . Also,where is the unique zero ofWe also note thatwhen For , the Dini function has real roots except a pair of complex conjugate roots (for detail, see [1]). Thus,in if and only if Considering (5) (30), (33), and (36), we haveAlso, from (38), it is obvious thatAs in [1], for and , is the unique value of positive zero of Moreover, if , then we have which is the least positive zero of
Theorem 7. If , then is the smallest zero ofwhere
Proof. By Lemma 4, the functionfor in the open unit disk if and only if , where is the unique value of the zero of the following equation:Suppose that is the th positive zero of given by (24) and (25). Consider the normalization (7) such thatWe writeFor and , we see thatFor detail, we refer to [12]. Since the function on for fixed , thus is increasing on , and is decreasing on and if and only if , where is the unique value of the root ofThus,and equality holds for The above inequality implies that the function , in if and only if . Considering normalization in (7), we can writeFrom (48), it is known thatFor and , we see that is the least positive zero of the following equation:where
Theorem 8. If and , then is the least positive zero of the differential equation , where
Proof. Assume that is the th positive zero of given by (24) and (25). Considering the normalization given in (7), we writeSince so we can writeThis result shows thatorThe equality holds for The principle of minimum value for harmonic functions along with (7) shows thatis valid if and only if , and is the minimum positive root of the equationThus, we haveorsince on for a fixed Thus, by using (58) and (60), we see that satisfies (7) and by applying Lemma 1 and Lemma 2, we obtain that and decreasing on and by considering (8), we can writeWe also writeWe can writeSinceso we haveFrom (63) along with (65), we see thatAs in [11], we observe thatFor and , is the least positive zero ofwhere .
4. Conclusion
The class of Bessel functions is originated as a solution of the well-known Bessel differential equation. We studied the radius problems of starlike functions with symmetric points involving Bessel functions under some kind of normalized conditions. We used the Mittag-Leffler representation of Bessel functions and derived our main results.
Data Availability
There is no data available.
Conflicts of Interest
The authors declare that they have no conflicts of interest.