Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios

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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=M_n(λ, α), whereλ∈[0,1],α∈[0,1)and the functionf (z)=z+a_n+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+a_n+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+a_n+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 nz^γ/n

z 0

h(t)t^γ/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

nz^γ/n z

0h(t)t^γ/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

Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios

Call for Papers

Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.

Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from “Qualitative Theory of Differential Equations,”

allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.

This proposed special edition of the Mathematical Prob- lems in Engineering aims to provide a picture of the impor- tance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.

Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophis- ticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.

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://

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Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

José Roberto Castilho Piqueira,Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, 05508-970 São Paulo, Brazil;

[email protected]

Elbert E. Neher Macau,Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, 12227-010 São Paulo, Brazil ; [email protected] Celso Grebogi,Center for Applied Dynamics Research, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK; [email protected]

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