Geometric Properties Solutions of a Class of Third-Order Linear Differential Equations

which are analytic in the open unit diskU {z: z ∈ C and |z| < 1}. Also let S, S∗, S∗ α , C, and C α denote the subclasses of A consisting of functions which are, respectively, univalent, starlike with respect to the origin, starlike of order α in U 0 ≤ α < 1 , convex with respect to the origin, and convex of order α in U 0 ≤ α < 1 see, for details, 1–4 . Furthermore, SS∗ β and C∗C β denote the subclasses of A consisting of functions which are strongly starlike of order β and strongly convex of order β inU, 0 < β ≤ 1 see, 5, 6 . For functions f ∈ A with f ′ z / 0 z ∈ U , we define the Schwarzian derivative of f by


Introduction
Let A denote the class of functions f normalized by a n z n , 1.1 which are analytic in the open unit disk U {z: z ∈ C and |z| < 1}.Also let S, S * , S * α , C, and C α denote the subclasses of A consisting of functions which are, respectively, univalent, starlike with respect to the origin, starlike of order α in U 0 ≤ α < 1 , convex with respect to the origin, and convex of order α in U 0 ≤ α < 1 see, for details, 1-4 .Furthermore, SS * β and C * C β denote the subclasses of A consisting of functions which are strongly starlike of order β and strongly convex of order β in U, 0 < β ≤ 1 see, 5, 6 .
For functions f ∈ A with f z / 0 z ∈ U , we define the Schwarzian derivative of f by

ISRN Applied Mathematics
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: , then by using the Schwarz lemma, the function ω z defined by is also in B J .Thus, in terms of derivatives, we have In 1999, Saitoh 10 proved that the differential equation where a z and b z are analytic in the unit disc U, has a solution ω z univalent and starlike in U 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 P n z is nonconstant polynomial of degree n ≥ 1.
In this work, we aim at studying certain geometric properties of the solutions of the following initial-value problem: In order to prove our main results, we need the following definitions and theorems.Definition 1.1 see 13 .Let H J be the set of complex functions h u, v satisfying the following: ii 0, 0 ∈ D and |h 0, 0 | < J; iii |h Je iθ , Ke iθ | ≥ J when Je iθ , Ke iθ ∈ D, θ is real and K ≥ J.

Main Results
Theorem 2.1.
and let ω z denote the solution of the initial value problem 1.8 in U. Then Proof.If we let then u z is analytic in U, such that u 0 0 and 1.8 becomes or, equivalently, where, for convenience, h ξ, η ξ 2 − ξ η.

ISRN Applied Mathematics
From assumption, we have By using Theorem 1.4, we have which, in view of the relationship 2.3 , yields zω z ω z < J, 2.9 that is, Letting J 1 in Theorem 2.1, we have the following corollary.

2.11
Let ω z be the solution of the initial-value problem in 1.8 in U. Then ω z is convex in U.
Example 2.3.Let Q z 1 in Corollary 2.2; the solution of the following initial-value problem:

2.14
Let ω z be the solution of the initial-value problem in 1.8 in U. Then ω z is strongly convex of order α, that is, for some α (0 < α ≤ 1) and α 2 π sin −1 J 0 < J ≤ 1 .

2.16
Proof.By using the same technique as in the proof of Theorem 2.1, the required result is obtained.

Mathematics 3
Definition 1.2 see 13 .Let h ∈ H J with corresponding domain D. We denote by B J h those functions ω z ω 1 z ω 2 z 2 • • • which are analytic in U satisfying Theorem 1.4 see 10 .Let h ∈ H J and b z be an analytic function in U with |b z | < J.