Time-Dependent Billiards

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COEFFICIENT INEQUALITIES FOR CERTAIN ANALYTIC FUNCTIONS

JUNICHI NISHIWAKI and SHIGEYOSHI OWA Received 1 March 2001

For realα (α >1), we introduce subclassesM(α)andN(α)of analytic functionsf (z)with f (0)=0 andf(0)=1 inU. The object of the present paper is to consider the coeﬃcient inequalities for functionsf (z)to be in the classesM(α)andN(α). Further, the bounds ofαfor functionsf (z)to be starlike inUare considered.

2000 Mathematics Subject Classiﬁcation: 30C45.

1. Introduction. LetAdenote the class of functionsf (z)of the form f (z)=z+

∞ n=2

a_nzⁿ (1.1)

which are analytic in the open unit diskU= {z∈C:|z|<1}. LetM(α)be the subclass ofAconsisting of functionsf (z)which satisfy

Re

zf(z) f (z)

< α (z∈U) (1.2)

for someα (α >1). And letN(α)be the subclass ofAconsisting of functionsf (z) which satisfy

Re

1+zf(z) f(z)

< α (z∈U) (1.3)

for someα (α >1). Then, we see thatf (z)∈N(α)if and only ifzf(z)∈M(α). We give examples of functionsf (z)in the classesM(α)andN(α).

Remark 1.1. For 1< α≤4/3, the classes M(α)and N(α) were introduced by Uralegaddi et al. [2].

Example1.2. (i)f (z)=z(1−z)^2(α−¹⁾∈M(α).

(ii) g(z)=(1/(2α−1)){1−(1−z)^2α−1} ∈N(α).

Proof. Sincef (z)∈M(α)if and only if Re

zf(z) f (z)

< α, (1.4)

we can write

α−zf(z)/f (z) α−1 =1+z

1−z, (1.5)

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which is equivalent to

f(z) f (z)−1

z=2(α−1)

1−z . (1.6)

Integrating both sides of the above equality, we have

f (z)=z(1−z)^2(α−¹⁾∈M(α). (1.7) Next, sinceg(z)∈N(α)if and only ifzg(z)∈M(α),

zg(z)=z(1−z)^2(α−¹⁾. (1.8) For functiong(z)∈N(α), it follows that

g(z)= − 1

2α−1(1−z)^2α−¹+ 1

2α−1= 1 2α−1

1−(1−z)^2α−¹

∈N(α). (1.9)

2. Coefficient inequalities for the classesM(α)andN(α). We try to derive suffi- cient conditions forf (z)which are given by using coefficient inequalities.

Theorem2.1. Iff (z)∈Asatisﬁes ∞

n=2

(n−1)+|n−2α+1|an≤2(α−1) (2.1)

for someα (α >1), thenf (z)∈M(α).

Proof. Suppose that ∞ n=2

(n−1)+|n−2α+1|an≤2(α−1) (2.2)

forf (z)∈A.

It suﬃces to show that

zf(z)/f (z)−1 zf(z)/f (z)−(2α−1)

<1 (z∈U). (2.3)

We have

zf(z)/f (z)−1 zf(z)/f (z)−(2α−1)

≤

_∞

n=2(n−1)an|z|ⁿ⁻¹ 2(α−1)−_∞

n=2|n−2α+1|a_n|z|ⁿ⁻¹

<

_∞

n=2(n−1)an 2(α−1)−_∞

n=2|n−2α+1|a_n.

(2.4)

The last expression is bounded above by 1 if ∞

n=2

(n−1)a_n≤2(α−1)− ∞ n=2

|n−2α+1|a_n (2.5) which is equivalent to condition (2.1). This completes the proof of the theorem.

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By usingTheorem 2.1, we have the following corollary.

Corollary2.2. Iff (z)∈Asatisﬁes ∞

n=2

n

(n−1)+|n−2α+1|an≤2(α−1) (2.6) for someα (α >1), thenf (z)∈N(α).

Proof. Fromf (z)∈N(α)if and only ifzf(z)∈M(α), replacinga_nbyna_nin Theorem 2.1we have the corollary.

In view ofTheorem 2.1andCorollary 2.2, if 1< α≤3/2, thenn−2α+1≥0 for all n≥2. Thus we have the following corollary.

Corollary2.3. (i) Iff (z)∈Asatisﬁes ∞

n=2

(n−α)an≤α−1 (2.7)

for someα (1< α≤3/2), thenf (z)∈M(α).

(ii) Iff (z)∈Asatisﬁes ∞ n=2

n(n−α)a_n≤α−1 (2.8)

for someα (1< α≤3/2), thenf (z)∈N(α).

3. Starlikeness for functions inM(α)andN(α). By Silverman [1], we know that if f (z)∈Asatisﬁes

∞ n=2

nan≤1, (3.1)

thenf (z)∈S^∗, whereS^∗ denotes the subclass ofAconsisting of all univalent and starlike functionsf (z)inU. Thus we have the following theorem.

Theorem3.1. Iff (z)∈Asatisﬁes ∞ n=2

(n−α)an≤α−1 (3.2)

for some α (1< α≤4/3), then f (z)∈S^∗∩M(α), therefore, f (z)is starlike inU.

Further, iff (z)∈Asatisﬁes ∞ n=2

n(n−α)an≤α−1 (3.3)

for someα (1< α≤3/2), thenf (z)∈S^∗∩N(α), therefore,f (z)is starlike inU.

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Proof. Considerαsuch that ∞ n=2

nan≤ ∞ n=2

n−α

α−1an≤1. (3.4)

Then we havef (z)∈S^∗∩M(α)by means ofTheorem 2.1. This inequality holds true if n≤n−α

α−1 (n=2,3,4,...). (3.5)

Therefore, we have

1< α≤2− 2

n+1 (n=2,3,4,...), (3.6) which shows that 1< α≤4/3. Next, consideringαsuch that

∞ n=2

na_n≤ ∞ n=2

n(n−α)

α−1 a_n≤1, (3.7)

we have

n≤n(n−α)

α−1 (n=2,3,4,...), (3.8)

which is equivalent to

1< α≤n+1

2 (n=2,3,4,...). (3.9)

This implies that 1< α≤3/2.

Finally, by virtue of the result for convex functions by Silverman [1], we have, if f (z)∈Asatisﬁes

∞ n=2

n²an≤1, (3.10)

then f (z)∈K, where K denotes the subclass ofA consisting of all univalent and convex functionsf (z)inU. Using the same method as in the proof ofTheorem 3.1, we derive the following theorem.

Theorem3.2. Iff (z)∈Asatisﬁes ∞ n=2

n(n−α)an≤α−1 (3.11)

for someα (1< α≤4/3), thenf (z)∈K∩N(α), therefore,f (z)is convex inU.

4. Bounds ofαfor starlikeness. Note that the suﬃcient condition forf (z)to be in the classM(α)is given by

∞ n=2

(n−1)+|n−2α+1|a_n≤2(α−1). (4.1)

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Since, iff (z)∈Asatisﬁes

∞ n=2

nan≤1, (4.2)

thenf (z)∈S^∗(cf. [1]). It is interesting to ﬁnd the bounds ofαfor starlikeness of f (z)∈M(α). To do this, we have to consider the following inequality:

∞ n=2

nan≤ 1 2(α−1)

∞ n=2

(n−1)+|n−2α+1|an≤1 (4.3)

which is equivalent to ∞ n=2

|n−2α+1|+(3−2α)na_n≥0. (4.4)

We deﬁne

F(n)= |n−2α+1|+(3−2α)n (n≥2). (4.5) Then, ifF(n)satisﬁes

∞ n=2

F(n)an≥0, (4.6)

thenf (z)belongs toS^∗.

Theorem4.1. Letf (z)∈Asatisfy ∞

n=2

(n−1)+|n−2α+1|an≤2(α−1) (4.7)

for someα >1. Further, letδ_kbe deﬁned by δ_k=

∞ n=k

F(n)a_n. (4.8)

Then,

(i) if1< α≤3/2, thenf (z)∈S^∗,

(ii) if3/2≤α≤min(13/8,(3+δ3)/2), thenf (z)∈S^∗, (iii) if8/3≤α≤min(17/10,(12−δ₄+

δ²₄+48δ₄+48)/12), thenf (z)∈S^∗. Proof. For 1< α≤3/2, we know that

n−2α+1≥3−2α≥0 (n≥2), (4.9) that is,F(n)≥0(n≥2). Therefore, we have

∞ n=2

F(n)an≥0. (4.10)

If 3/2≤α≤13/8, thenF(2)=3−2α≤0 and

F(n)=2n(2−α)+1−2α≥13−8α≥0 (4.11)

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forn≥3. Further, we know that

an≤ 2(α−1)

(n−1)+|n−2α+1| (n≥2), (4.12) then|a2| ≤1. Therefore, we obtain that

∞ n=2

F(n)a_n=F(2)a2+ ∞ n=3

F(n)a_n≥3−2α+δ3≥0 (4.13) for

3

2≤α≤min 13 8 ,3+δ3

2

. (4.14)

Furthermore, if 13/8≤α≤17/10, then F(2)=3−2α≤0,

F(3)= |4−2α|+3(3−2α)=13−8α≤0,

F(n)= |n−2α+1|+(3−2α)n=4n+1−2(n+1)α≥3(n−4)

5 ≥0

(4.15)

forn≥4. Noting that|a2| ≤1 and|a3| ≤(α−1)/(3−α), we conclude that ∞

n=2

F(n)a_n=F(2)a2+F(3)a3+ ∞ n=4

F(n)a_n

≥(3−2α)+(13−8α)α−1

3−α+δ4≥0,

(4.16)

forαthat satisﬁes

6α²− 12−δ4

α+4−3δ4≤0. (4.17)

This shows that 8

3≤α≤min



17

10,12−δ4+

δ²₄+48δ4+48 12



. (4.18)

This completes the proof ofTheorem 4.1.

Finally, by virtue ofTheorem 4.1, we may suppose that iff (z)∈Asatisﬁes ∞

n=2

(n−1)+|n−2α+1|a_n≤2(α−1) (4.19)

for some 1< α <2, thenf (z)∈S^∗.

References

[1] H. Silverman,Univalent functions with negative coeﬃcients, Proc. Amer. Math. Soc.51 (1975), 109–116.

[2] B. A. Uralegaddi, M. D. Ganigi, and S. M. Sarangi,Univalent functions with positive coeﬃ- cients, Tamkang J. Math.25(1994), no. 3, 225–230.

Junichi Nishiwaki and Shigeyoshi Owa: Department of Mathematics, Kinki Univer- sity, Higashi-Osaka, Osaka577-8502, Japan

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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 at

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according to the following timetable:

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

Guest Editors

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Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; [email protected]

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Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;

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