Abstract
The purpose of this paper is to find sufficient argument properties, such that the images of some subclasses of functions by the Libera transform have bounded arguments.
1. Introduction
Let denote the class of functions of the form which are analytic in the open unit disk .
A function is said to be in the class of star-like functions of order in , if and only if it satisfies the condition
A function is said to be in the class of convex functions of order in , if and only if it satisfies the condition
It is well-known that , whenever , where and represent, respectively, the class of starlike and convex (normalized) functions.
If satisfies the condition then is said to be star-like of reciprocal order , and we denote this class by .
The above definition was recently discussed by Nunokawa et al. [1] and Ravichandran and Sivaprasad Kumar [2].
Motivated by their works, we define a class of convex functions of reciprocal order as follows.
Definition 1. If satisfies the condition then is said to be convex function of reciprocal order , and we denote this class by .
Definition 2 (see [3, 4]). Suppose that and are two analytic functions in . We say that the function is subordinate to , written , if there exists a Schwarz function , that is, a function analytic in , with and , such that , .
It is well-known that (see, e.g., [3, 4]) if is univalent in , then the following subordination property holds:
Remarks 1. (1) In view of the fact that it follows that a star-like (convex) function of reciprocal order is a star-like (convex) function. Thus, and the equality holds in both cases if and only if .
(2) Let , and suppose that . Then (i)(ii)(3) We note that, for , we have
In the following, we give some examples of functions belonging to the class of star-like functions of reciprocal order and the class of convex functions of reciprocal order.
Example 3. (1) For , define the function by Since and a simple computation shows that we conclude that .
(2) Using the third part of the Remarks 1, we deduce that Since we proved that , and from the above differential equation we obtain it follows that .
Example 4. (1) For , consider the function , defined by This gives that whereSince , it follows that is analytic in . Using the fact that is a convex (not necessary normalized) function in and is symmetric with respect to the real axis, we deduce that for all ; hence .
(2) From the above result, using the same reasons like in the second part of Example 3, we obtain that
We mention that several authors have investigated the strongly star-like functions and the strongly convex functions (see [5–18]). In the present investigation we give some argument properties of analytic functions belonging to , such that the images of these functions by the Libera transform have bounded arguments.
2. Main Results
The following lemma will be used to prove our main results.
Lemma 5 (see [16]). Let be the solution offor a suitable fixed , so that satisfiesand letIf is analytic in , and then
Remark that in the article [19] the author considered some special situations improving many results with a lot of applications. Thus, in [19, Lemma 2], the author proved the next result.
Let be a function defined on satisfying and let be such that . If is analytic in , , then
We emphasize that a special case of this lemma improves the conclusion of Lemma 1 from [19], and in the same article the author derived a number of interesting consequences of it.
Let defined by be the well-known Libera transform [20].
We first prove the following theorem, which is essential for proving our other results.
Theorem 6. For suppose that the Libera transform satisfies the condition (21), withwhere and is defined by (20). If and , then implies
Proof. Let us define the function by Suppose that there exists a number , such that ; since , then . It follows that there exists a unique number and a unique function , analytic in , with , such that From the above relation we deduce that which implies that is a simple pole for the function . Consequently, from (29) we obtain that , and using the fact that satisfies the inequality (21) for all it follows that which contradicts the assumption .
Thus, for all , which implies that the function is analytic in , and .
Now, a simple calculus shows that and our result follows immediately from Lemma 5.
Taking in Theorem 6, for the function defined by (29) we have , and we obtain the following result.
Corollary 7. Suppose that the parameters and satisfy the conditions (20) and (22), for a given . If and ; then implies
Example 8. For and , considering the special cases , and , from the above corollary we obtain respectively the next implications:(1)If , then (2)If , then (3)If , then
Theorem 9. For suppose that the Libera transform , . Suppose that is defined by (20), and for let satisfy the conditionIf and , then implies
Proof. Let be the function defined bySimilarly like in the proof of Theorem 6, we will prove that the function is analytic in . Supposing that there exists a number , such that , since , then . Hence, there exists a unique number and a unique function , analytic in , with , such that From here we deduce that which implies that is a simple pole for the function . Now, using the fact that , we obtain which contradicts the assumption .
Consequently, for all , and this implies that the function is analytic in , with .
From (45) we easily get where the function is given by (29).
The assumption is equivalent to hence there exists a Schwarz function such that By simple calculations, we get that is,Since the circular transform maps the unit disk onto the disk from the subordination property (6) we deduce that the subordination relation (53) is equivalent to With the above notation, the condition (21) becomesWe will prove that if satisfies the assumption (42), that is, , then the subordination (53) implies that the inequality (56) holds, that is, where Since the sets and are symmetric with respect to the real axis, the above inclusion is equivalent towhere The line containing the half-line which is a part of the boundary the set , has the equation Thus, the line intersects the real axis in the point and contains the point (see Figure 1). The smallest set that includes the domain is obtained for the case when the line , containing the point , is tangent to the upper boundary of , which is the half-circle In this case, as it is shown in the figure, the line becomes the tangent line to .
We will determine now the equation of the tangent line . If we consider the family of all the lines containing the point , with nonnegative angular coefficient, that is, we will solve the system It follows that and the line is tangent to the half-circle if and only if the discriminant of the above quadratic form is zero, that is, This last equation has the biggest positive root hence the tangent line to the half-circle , which contains the point , has the equation The tangent intersects the real axis in the point ; hence we deduce that the inclusion (59) holds if and only if that isThus, if the parameter satisfies the assumption (42), then the condition (21) of Lemma 5 is fulfilled, and our result follows from Theorem 6.
Putting in Theorem 9, for , we get the next result.
Corollary 10. Suppose that is defined by (20), and for let satisfy the condition If and , then implies
Example 11. For and , considering the special case , from Corollary 10 we deduce the next implication.
If then
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Dr. Badr S. Alkahtani and Dr. Rubayyi Alqahtani extend their sincere appropriations to the Deanship of Scientific Research at King Saud University for funding this Profile Research Group (PRG-1437-35).