Abstract
We aim at investigating the geometric properties of the solutions of the initial-value problem which involves the following third-order linear differential equations: , , , , where is analytic in the open unit disk .
1. Introduction
Let denote the class of functions normalized by which are analytic in the open unit disk : and .
Also let , , , , and denote the subclasses of consisting of functions which are, respectively, univalent, starlike with respect to the origin, starlike of order in (), convex with respect to the origin, and convex of order in () (see, for details, [1–4]). Furthermore, and denote the subclasses of consisting of functions which are strongly starlike of order and strongly convex of order in , (see, [5, 6]).
For functions with (), we define the Schwarzian derivative of by Note that Nehari [7] had proven the quotient of the linearly independent solution of (1.2) is univalent, while Robertson [8] and Miller [9] proved that the unique solution of the equation: is starlike.
Now, let denote the class of bounded functions analytic in the unit disk for which . If , then by using the Schwarz lemma, the function defined by is also in . Thus, in terms of derivatives, we have In 1999, Saitoh [10] proved that the differential equation where and are analytic in the unit disc , has a solution univalent and starlike in under some conditions. Then in 2004, Owa et al. [11] studied geometric properties of the solutions of initial-value problem (1.6) and later, Saitoh [12] studied geometric properties of the solutions of the following second-order linear differential equation: where is nonconstant polynomial of degree .
In this work, we aim at studying certain geometric properties of the solutions of the following initial-value problem: where is analytic in .
In order to prove our main results, we need the following definitions and theorems.
Definition 1.1 (see [13]). Let be the set of complex functions satisfying the following:(i) is continuous in a domain ;(ii) and ;(iii) when , is real and .
Definition 1.2 (see [13]). Let with corresponding domain . We denote by those functions which are analytic in satisfying(i),(ii).
Theorem 1.3 (see [10]). For any ,
Theorem 1.4 (see [10]). Let and be an analytic function in with . If the differential equation has a solution analytic in , then .
2. Main Results
Theorem 2.1. Let be analytic in with and let denote the solution of the initial value problem (1.8) in . Then
Proof. If we let then is analytic in , such that and (1.8) becomes or, equivalently, where, for convenience, From assumption, we have By using Theorem 1.4, we have which, in view of the relationship (2.3), yields that is,
Letting in Theorem 2.1, we have the following corollary.
Corollary 2.2. Let be analytic in with Let be the solution of the initial-value problem in (1.8) in . Then is convex in .
Example 2.3. Let in Corollary 2.2; the solution of the following initial-value problem: is given by
Theorem 2.4. Let be analytic in with Let be the solution of the initial-value problem in (1.8) in . Then is strongly convex of order , that is, for some () and
Proof. By using the same technique as in the proof of Theorem 2.1, the required result is obtained.
Remark 2.5. Putting in Theorem 2.4, we have Corollary 2.2.
Acknowledgment
The work presented here was partially supported by UKM-ST-FRGS-0244-2010.