Inclusion of the Generalized Bessel Functions in the Janowski Class
Sufficient conditions on , , , , and are determined that will ensure the generalized Bessel function satisfies the subordination . In particular this gives conditions for , , to be close-to-convex. Also, conditions for to be Janowski convex and to be Janowski starlike in the unit disk are obtained.
Let denote the class of analytic functions defined in the open unit disk normalized by the conditions . If and are analytic in , then is subordinate to , written , if there is an analytic self-map of satisfying and . For , let be the class consisting of normalized analytic functions in satisfyingFor instance, if , then is the class of functions satisfying in .
The class of Janowski starlike functions  consists of satisfyingFor , is the usual class of starlike functions of order ; ; and . These classes have been studied, for example, in [2, 3]. A function is said to be close-to-convex of order [4, 5] if for some .
This article studies the generalized Bessel function given by the power serieswhere . The function is analytic in and solution of the differential equationif , , in , such that , and . This normalized and generalized Bessel function of the first kind of order also satisfies the following recurrence relation:which is useful tool to study several geometric properties of . There have been several works [6–11] studying geometric properties of the function , for example, on its close-to-convexity, starlikeness, and convexity, and radius of starlikeness and convexity.
In Section 2 of this paper, sufficient conditions on , , , and are determined that will ensure satisfies the subordination . It is to be understood that a computationally intensive methodology with shrewd manipulations is required to obtain the results in this general framework. The benefits of such general results are that, by judicious choices of the parameters and , they give rise to several interesting applications, which include extending the results of previous works. Using this subordination result, sufficient conditions are obtained for , which next readily gives conditions for to be close-to-convex. Section 3 gives emphasis to the investigation of to be Janowski convex as well as of to be Janowski starlike.
The following lemma is needed in sequel.
In the case , then the condition in Lemma 1 is generalized to is real, , and .
2. Close-To-Convexity of Generalized Bessel Functions
In this section, one main result on the close-to-convexity of the generalized Bessel function with several consequences is discussed in detail.
Theorem 2. Let . Suppose and satisfyFurther let , , , and satisfy either the inequalitywheneveror the inequalitywheneverIf for all , then .
Proof. Define the analytic function by Then, a computation yieldsThus, using identities (14), the Bessel differential equation (4) can be rewritten as Assume , and define byIt follows from (15) that . To ensure for , from Lemma 1, it is enough to establish in for any real , , and .
With in (16), a computation yieldsSince and , Thus where Condition (8) shows that Since for , it is clear that when with . As , the above condition holds whenever that is, whenTo establish inequality (24), consider the polynomial given by where Constraint (10) yields , and thus . Now inequality (9) readily implies that Now consider the case of constraint (12), which is equivalent to . Then the minimum of occurs at , and (11) yields Evidently satisfies the hypothesis of Lemma 1, and thus ; that is,Hence there exists an analytic self-map of with such thatwhich implies that .
Theorem 2 gives rise to simple conditions on and to ensure maps into a half plane.
Corollary 3. Let and . Then
Proof. Choose and in Theorem 2. Then condition (8) is equivalent to and (10) reduces to , and clearly both hold for . The proof will complete if hypothesis (9) holds; that is,Since , it follows that which establishes (31).
Corollary 4. Let . Then if , then .
Next theorem gives the sufficient condition for close-to-convexity when
Theorem 5. Let and satisfySuppose , , , and satisfy either the inequalitywheneveror the inequalitywheneverIf for all , then .
Proof. First, proceed similarly to the proof of Theorem 2 and derive the expression of as given in (17). Now, for , , and , and then, with and , it follows that where Observe that inequality (34) implies that . Thus for all provided ; that is, for , With , it is enough to show, for , which is equivalent towhere If (36) holds, then and , which is nonnegative from (35). On the other hand, if (38) holds, then , , and (37) implies . Either case establishes (44).
Theorem 6. Let . Suppose and with satisfyingFurther let , , , and satisfy either whenever or the inequality when If for all , then .
Theorem 7. Let . Suppose and , such that Suppose , , , and satisfy either whenever or the inequality when If for all , then .
Corollary 8. Let andThen is close-to-convex of order with respect to the identity function.
Corollary 9. Let be a nonzero real number and . Then
3. Janowski Starlikeness of Generalized Bessel Functions
This section contributes to finding conditions to ensure a normalized and generalized Bessel function in the class of Janowski starlike functions. For this purpose, first sufficient conditions for to be Janowski convex are determined, and then an application of relation (5) yields conditions for
Theorem 10. Let be such that for all and . Suppose Further let , , , and satisfyIf and , then
Proof. Define an analytic function byThenA rearrangement of (63) yieldsThus,Now a differentiation of (4) leads to which givesUsing (62) and (65), (67) yields and equivalentlyDefinewhere Thus, (69) yields . Now, with , let For , , Note that condition (58) implies . In this case, has a maximum at . Thus for all real provided Since , it is left to show that . The above inequality is equivalent towhere Since , the left-hand side of inequality (76) satisfies Now it is evident from (59) that which establishes inequality (76).
Thus satisfies the hypothesis of Lemma 1, and hence , or equivalently By definition of subordination, there exists an analytic self-map of with and A simple computation shows that and hence