http://ijmms.hindawi.com
© Hindawi Publishing Corp.
A SUFFICIENT CONDITION FOR STARLIKENESS OF ORDERα
PETRU T. MOCANU and GH. OROS
(Received 28 January 2001 and in revised form 12 June 2001)
Abstract.We obtain a sufficient condition for starlikeness of orderα,|f(z)−λ(f (z)/z)+
λ−1|< M=Mn(λ, α), whereλ∈[0,1],α∈[0,1)and the functionf (z)=z+an+1zn+1+
···is analytic in the unit discU.
2000 Mathematics Subject Classification. 30C45.
1. Introduction and preliminaries. Denote byUthe unit disc of the complex plane U=
z∈C:|z|<1
. (1.1)
LetᏴ[U ]be the space of holomorphic functions inU, and let An=
f∈Ᏼ[U ], f (z)=z+an+1zn+1+···, z∈U
(1.2) withA1=A.
LetᏴ[a, n]denote the class of analytic functions in the unit disc of the form f (z)=a+anzn+an+1zn+1+···, z∈U . (1.3) Let
S∗(α)=
f∈A,Rezf(z)
f (z) > α, z∈U
, 0≤α <1, (1.4) be the class of starlike functions of orderαinU.
Iffandgare analytic inU, then we say thatf is subordinate tog, writtenf≺g orf (z)≺g(z), if there is a functionw analytic inU, withw(0)=0,|w(z)|<1, for anyz∈U, such thatf (z)=g(w(z)), forz∈U.
Ifgis univalent, thenf≺gif and only iff (0)=g(0)andf (U )⊂g(U ).
We use the following subordination result due to Hallenbeck and Ruscheweyh [1, page 71].
Lemma1.1. Lethbe a convex function withh(0)=a, and letγ∈C∗be a complex number withReγ≥0. Ifp∈Ᏼ[a, n]and
p(z)+1
γzp(z)≺h(z), (1.5)
then
p(z)≺q(z), (1.6)
558 P. T. MOCANU AND GH. OROS where
q(z)= γ nzγ/n
z 0
h(t)tγ/n−1dt, q≺h. (1.7) 2. Main results
Theorem2.1. Letλ∈[0,1],α∈[0,1), and
M=Mn(λ, α)= (1−α)(n+1−λ)
|λ−α|+
(1−λ)2+(n+1−λ)2. (2.1) Iff∈Ansatisfies the inequality
f(z)−λf (z)
z +λ−1
< Mn(λ, α), (2.2) withMn(λ, α)given by (2.1), thenf∈S∗(α).
Proof. In the caseλ=1, the proof is given in [3]. We suppose thatλ∈[0,1). If we considerP (z)=f (z)/z, then
f (z)=zP (z), f(z)=P (z)+zP(z), (2.3) and (2.2) can be written in the following form:
P (z)+zP(z) 1−λ −1
< M
1−λ (2.4)
which is equivalent to the differential subordination P (z)+zP(z)
1−λ ≺1+ M
1−λz≡h(z), (2.5)
and by usingLemma 1.1, we obtain P (z)≺q(z)= γ
nzγ/n z
0h(t)tγ/n−1dt=1+ M
1−λ+nz. (2.6) Subordination (2.6) is equivalent to
P (z)−1< M
1−λ+n≡R. (2.7)
After a simple computation, from (2.7) it follows that R < 1−α
|λ−α|. (2.8)
If we put
zf(z)
f (z) =(1−α)p(z)+α, (2.9)
then
f(z)=P (z)
(1−α)p(z)+α (2.10)
and (2.2) can be written as P (z)
(1−α)p(z)+α−λ +λ−1< M=(1−λ+n)R. (2.11) We have to show that (2.11) implies Rep(z) >0 inU. Suppose that this is false.
Sincep(0)=1, there existz0∈Uand a realρ, such thatp(z0)=iρ.
Therefore, in order to show that (2.11) implies Rep(z) >0 inU, it is sufficient to obtain the contradiction from the inequality
P z0
(1−α)p z0
+α−λ +λ−1≥(1−λ+n)R. (2.12) If we letP (z0)=P=u+iv, then
E=P
(1−α)iρ+α−λ +λ−12
= |P|2
(1−α)2ρ2+(α−λ)2 −2(1−λ)Re
P (1−α)iρ+α−λ
+(1−λ)2
=
u2+v2
(1−α)2ρ2+2(1−λ)(1−α)vρ+P (α−λ)−(1−λ)2.
(2.13)
By using (2.7) and the well-known triangle inequality, one obtains P (α−λ)−(1−λ)=P (α−λ)+α−λ−α+λ−1+λ
=(α−λ)(P−1)−(1−α)
≥1−α−|λ−α|R
(2.14)
and we deduce E≥
u2+v2
(1−α)2ρ2+2(1−λ)(1−α)vρ+
(1−α)−(λ−α)R 2. (2.15) If we let
F (ρ)=E−M2
≥
u2+v2
(1−α)2ρ2+2(1−λ)(1−α)vρ +
(1−α)−|λ−α|R 2−(1−λ+n)2R2,
(2.16)
then (2.12) holds ifF (ρ)≥0, for any real numberρ.
Because(u2+v2)(1−α)2>0, the inequalityF (ρ)≥0 holds if the discriminant∆ is negative, that is,
∆=(1−α)2
(1−λ)2v2−
u2+v2
1−α−|λ−α|R2
−(1−λ+n)2R2 ≤0. (2.17) The last inequality is equivalent to
v2
(1−λ)2−
1−α−|λ−α|R2
+(1−λ+n)2R2]
≤u2
1−α−|λ−α|R2
−(1−λ+n)2R2 .
(2.18)
After an easy computation, by using (2.7) we obtain the inequality v2
u2≤ R2 1−R2≤
1−α−|λ−α|R2
−(1−λ+n)2R2 (1−λ)2−
1−α−|λ−α|R2
+(1−λ+n)2R2, (2.19) which is equivalent to∆≤0. ThereforeF (ρ) 0, a contradiction of (2.11). It follows
560 P. T. MOCANU AND GH. OROS that Rep(z) >0, and
Rezf(z)
f (z) =Re(1−α)p(z)+α=(1−α)Rep(z)+α≥α (2.20) hencef∈S∗(α).
Ifλ=0 then
Mn(0, α)= (1−α)(n+1) α+
(n+1)2+1 (2.21)
and we obtain the following corollary.
Corollary2.2. Iff∈Anand
f(z)−1< (1−α)(n+1) α+
(n+1)2+1, (2.22)
thenf∈S∗(α).
Forα=0 this result was obtained in [2].
Ifλ=1,
Mn(1, α)= n(1−α)
n+1−α, (2.23)
and we obtain the following corollary.
Corollary2.3(see [3]). Iff∈Anand f(z)−f (z)
z
<n(1−α)
n+1−α, (2.24)
thenf∈S∗(α).
Ifλ=α,
Mn(α, α)= (1−α)(n+1−α)
(1−α)2+(1−α+n)2. (2.25) Corollary2.4. Iff∈Anand
f(z)−αf (z)
z +α−1
< (1−α)(n+1−α)
(1−α)2+(1−α+n)2, (2.26) thenf∈S∗(α).
References
[1] S. S. Miller and P. T. Mocanu,Differential Subordinations: Theory and Applications, Mono- graphs and Textbooks in Pure and Applied Mathematics, vol. 225, Marcel Dekker, New York, 2000.MR 2001e:30036. Zbl 0954.34003.
[2] P. T. Mocanu,Some simple criteria for starlikeness and convexity, Libertas Math.13(1993), 27–40.MR 94k:30027. Zbl 0793.30008.
[3] G. Oros,On a condition for starlikeness, The Second International Conference on Basic Sciences and Advanced Technology (Assiut, Egypt, November 5–8), 2000, pp. 89–94.
Petru T. Mocanu: Department of Mathematics, Babes-Bolyai University,3400Cluj- Napoca, Romania
E-mail address:[email protected]
Gh. Oros: Department of Mathematics, University of Oradea,3700Oradea, Romania
Special Issue on
Time-Dependent Billiards
Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi accelera- tion (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.
This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://
mts.hindawi.com/according to the following timetable:
Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009 Publication Date September 1, 2009
Guest Editors
Edson Denis Leonel,Department of Statistics, Applied Mathematics and Computing, Institute of Geosciences and Exact Sciences, State University of São Paulo at Rio Claro, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil; [email protected]
Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;
Hindawi Publishing Corporation http://www.hindawi.com
