Abstract
In this article, we are interested in finding sufficient conditions on , and which ensure the normalized Coulomb wave function to be Janowski starlike. Sufficient conditions are also obtained for , which readily yield conditions for to be close-to-convex.
1. Introduction
Let be the class of functions which are analytic in the open unit disc and normalized by the conditions An analytic function is subordinate to an analytic function (written as if there exists an analytic function with and for such that In particular, if is univalent in then and Let denote the class of analytic functions such that and
Note that for is the class of analytic functions with satisfying in For , the class defined by is the class of Janowski starlike functions [13]. For is the usual class of starlike functions of order :
These classes have been studied in [6, 8]. A function is said to be close-to-convex of order with respect to a function if In particular case, if and satisfies the condition for all in , then is a close-to-convex of order .
Let denote the Kummer confluent hypergeometric function. The regular Coulomb wave function is defined as where which is the solution of following differential equation:
In this paper, we focus on the following normalized form:
The function satisfies the following homogenous second-order differential equation:
Baricz [9, 10] studied the Turan-type inequalites of regular Coulomb wave functions and zeros of a cross-product of the Coulomb wave and Tricomi hypergeometric functions, respectively. Baricz et al. [11] also investigated the radii of starlikeness and convexity of regular Coulomb wave functions. Recently, Aktas [1] has studied lemniscate and exponential starlikeness of Coulomb wave functions. In some recent papers [2–5, 12], the authors have discussed certain geometric properties of some special functions. The relationships of generalized Bessel function, Bessel-Struve kernal function, and Struve function with the Janowski class have also been studied by various researchers, see [7, 14, 17, 18]. Motivated by the above papers in this subject, in this paper, our aim is to present some geometric results for the normalized regular Coulomb wave function.
The following lemmas are needed in the paper.
Lemma 1 (see [15, 16]). Let , and satisfy whenever , real, . If is analytic in with , and for , then in . In the case , then the condition in Lemma 1 generalized to real, , and .
Lemma 2 (see [19]). Let . If then for
2. Inclusion of Generalized Coulomb Wave Function in the Janowski Class
Our first result is related with Janowski starlikeness of normalized Coulomb wave function.
Theorem 3. Let and Suppose that or, for , If and then .
Proof. Define an analytic function by Then, A rearrangement of (18) gives. Now, define a function by This function is analytic in and Suppose that We know that This function satisfies the following equation: which yields Substituting (17) and (19) in (22), we get or equivalently Now setting Then, for and , we get To get the contradiction, we have to show for We split the proof into two cases. First, consider the case Then, the function becomes that achieve its maximum at and which is nonpositive if and only if Now, consider the case Rewriting in the form where The inequality holds for any real if and or and that holds by hypothesis (14) and (15). Thus, in both cases, the function satisfies the hypothesis of lemma (8) and hence or By definition of subordination, there exist a map in with and which yields Hence,
If take and for in Theorem 3, we obtain following result.
Corollary 4. Let and If then the normalized Coulomb wave function is starlike of order
Theorem 5. Let and satisfy If then
Proof. Define a function by
The function is analytic in and Suppose that . This function satisfies the following equation:
Define the analytic function by
Then, simple computation yields
Thus, using (43)–(45), the differential equation (41) can be rewritten as
Assume and define by
It follows from (47) that To ensure for , from Lemma 1, it is enough to establish in for any real , , and Let . A computation yields
Since Thus,
The proof will be divided into four cases. Consider first . The inequality (49) reduces to
A quadratic function takes nonpositive values for any , if
Last inequality can be rewritten as
that holds, if
which reduces to the assumption. Therefore, the assertion follows.
In second case, we consider According to (49), we have
We note that the function is even with respect to and
that satisfies if
Moreover, , and
with if and only if or
We observe that by the inequality
Additionally,
in view of (60). Hence, and
that holds if
Since,
holds for then the condition (56), (60), and (63) reduce to the assumption (39). Therefore, the assertion follows. Let now By the fact we get
Also, for , we have ; therefore,
for any real Thus,
Since for ,
and the last expression is nonpositive in view of (39); then, the assertion follows. Finally, consider In this case Hence, setting with and using (65), we get from (49)
that is nonpositive due to inequality
that is equivalent to the assumption (39). Evidently, satisfies the hypothesis of Lemma 1, and thus, , that is
Hence, there exists an analytic self-map of with such that
which implies that
If we take and for in Theorem 5, we obtain following result.
Corollary 6. Let and If then that is, is close-to-convex of order
Applying Corollary 6 for and Lemma 2, the following result for close-to-convexity of immediately follows.
Corollary 7. Let If then is close-to-convex (univalent) for
Data Availability
No data were used to support this study.
Ethical Approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.