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

Authors: Afaf Ali Abubaker; Maslina Darus
Journal: ISRN Applied Mathematics
 (2011)

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## 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:ω′′′(z)+Q(z)ω′(z)=0, ω(0)=0, ω′(0)=1, ω′′(0)=0, where Q(z) is analytic in the open unit disk U.


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## Body

## 1. Introduction

LetA denote the class of functions f normalized by
(1.1)f(z)=z+∑n=2∞anzn,
which are analytic in the open unit disk U={z: z∈ℂ and |z|<1}.Also letS, 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 functionsf∈A with f′(z)≠0 (z∈U), we define the Schwarzian derivative of f by
(1.2)S(f,z):=(f′′(z)f′(z))′-12(f′′(z)f′(z))2,(f∈A;f′(z)≠0,z∈U).
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:(1.3)W′′(z)+a(z)W(z)=0,W(0)=0,W′(0)=1
is starlike.Now, letBJ denote the class of bounded functions ω(z)=ω1z+ω2z2+⋯ analytic in the unit disk U for which |ω(z)|<J. If g(z)∈BJ, then by using the Schwarz lemma, the function ω(z) defined by
(1.4)ω(z)=z-1/2∫0zg(t)t-1/2dt
is also in BJ. Thus, in terms of derivatives, we have
(1.5)|12ω(z)+zω′(z)|<J,(z∈U),⟹|ω(z)|<J,(z∈U).
In 1999, Saitoh [10] proved that the differential equation(1.6)ω′′(z)+a(z)ω′(z)+b(z)ω(z)=0,
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:
(1.7)ω′′(z)+Pn(z)ω(z)=0,
where Pn(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:(1.8)ω′′′(z)+Q(z)ω′(z)=0,ω(0)=0,ω′(0)=1,ω′′(0)=0,
where Q(z)=∑n=0∞bnzn is analytic in U.In order to prove our main results, we need the following definitions and theorems.Definition 1.1 (see [13]).
LetHJbe the set of complex functionsh(u,v)satisfying the following:(i)
h(u,v) is continuous in a domain D⊂ℂ×ℂ;(ii)
(0,0)∈D and |h(0,0)|<J;(iii)
|h(Jeiθ,Keiθ)|≥J when (Jeiθ,Keiθ)∈D, θ is real and K≥J.Definition 1.2 (see [13]).
Leth∈HJwith corresponding domainD. We denote byBJ(h)those functionsω(z)=ω1z+ω2z2+⋯which are analytic inUsatisfying(i)
(ω(z),zω′(z))∈D,(ii)
|h(ω(z),zω′(z))|<J(z∈U).Theorem 1.3 (see [10]).
For anyh∈HJ,
(1.9)BJ(h)⊂BJ,(h∈HJ;J>0).Theorem 1.4 (see [10]).
Leth∈HJ and b(z) be an analytic function in U with |b(z)|<J. If the differential equation
(1.10)h(ω(z),zω′(z))=b(z),ω(0)=0,ω(0)=1
has a solution ω(z) analytic in U, then |ω(z)|<J.


## 2. Main Results

Theorem 2.1.
LetQ(z)=∑n=0∞bnzn be analytic in U with
(2.1)∑n=0∞|bn|<J(z∈U,J>0),
and let ω(z) denote the solution of the initial value problem (1.8) in U. Then
(2.2)1-J<R{1+zω′′(z)ω′(z)}>1+J(z∈U,J>0).Proof.
If we let(2.3)u(z)=zω′′(z)ω′(z),
then u(z) is analytic in U, such that u(0)=0 and (1.8) becomes
(2.4)[u(z)]2-u(z)+zu′(z)=-z2∑n=0∞bnzn
or, equivalently,
(2.5)h(u(z),zu′(z))=-z2∑n=0∞bnzn,
where, for convenience,
(2.6)h(ξ,η)=ξ2-ξ+η.
From assumption, we have
(2.7)∑n=0∞|bn|<J(z∈U,J>0).
By using Theorem 1.4, we have
(2.8)|u(z)|<J
which, in view of the relationship (2.3), yields
(2.9)|zω′′(z)ω′(z)|<J,
that is,
(2.10)1-J<R{1+zω′′(z)ω′(z)}>1+J(z∈U,J>0).LettingJ=1 in Theorem 2.1, we have the following corollary.Corollary 2.2.
LetQ(z)=∑n=0∞bnzn be analytic in  U with
(2.11)∑n=0∞|bn|<1.
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.
LetQ(z)=1 in Corollary 2.2; the solution of the following initial-value problem:
(2.12)ω′′′(z)+ω′(z)=0,ω(0)=0,ω′(0)=1,ω′′(0)=0
is given by
(2.13)ω(z)=sin⁡⁡z∈C.Theorem 2.4.
LetQ(z)=∑n=0∞bnzn be analytic in U with
(2.14)∑n=0∞|bn|<J(z∈U,0<J≤1).
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,
(2.15)|arg⁡{1+zω′′(z)ω′(z)}|<π2α
for some α (0<α≤1) and
(2.16)α=2πsin⁡-1J(0<J≤1).Proof.
By using the same technique as in the proof of Theorem2.1, the required result is obtained.Remark 2.5.
Puttingα=1 in Theorem 2.4, we have Corollary 2.2.



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