OTHER TO

VOL. 14 NO. 3

(1991)

451-456

FUNCTIONS STARLIKE WITH RESPECT TO OTHER POINTS

S. ABDUL HALIM DeDt.

^of Mathematics University of Malaya

59100

Kuala

Lumpur

Malays i a

(Received September 26, 1990 and in revised form March 5,

1991)

ABSTRACT.

In [7],

Sakaguchi introduce the class of functions starlike with respect to symmetric points. We extend this class. For 0

.<

8 i, let

S*(8)

be the

s

class of normalised analytic functions f defined in the open unit disc D such that Re

zf’(z)/[f(z)-f(-z))

> 8, for some z e D. In this paper, we introduce 2 other similar classes

S*(8),

^S

(8)

as well as give sharp results for the real

c sc

part of some function for f e

S(8) S(8)

^{and S}

⁽⁸⁾

The behaviour of certain

s sc

integral operators are also considered.

KEY WORDS AND PHRASES. Starlike functions of ^{order 8,} functions starlike with respect

to

8ynnetrical points, close-to-convex, integral operators.

AMS Subject Classification ⁽¹⁹⁸⁵ Revision): Primary

30 C 45.

i. INTRODUCTION.

Let S be the class of analytic functions f, univalent in the unit disc

D lz Izl ^i},

^with

f(z)

z + E a zn

(i.i)

n=2 n

For 0

.<

8 i, denote by

S*(8),

the class of starlike functions of order 8.

Then f e

S*(8)

if, and only if, for z e

D,

!zf’ %),]

Re

f(z)

^{> 8.}

In

[7],

Sakaguchi introduced the class

S*

of analytic functions f, normalised s

by

(1.1)

which are starlike with respect to symmetrical points.

We

begin by defining the class S* which is contained in

K,

the class of close-to-convex functions.

s’

DEFINITION 1.

A function f e S if, and only if, for z e

D,

s

Re

f(Z)2f(iz)

^{> O.}

We now extend this definition as follows:

DEFINITION 2.

A function f with normalisations

.]J

is said to be starlike of order 8, with respect to symmetric points if, and only if, for z e D and 0

.<

8 ^< l,

We denote this class by

S*(8)

and note that S*

S*(0).

s s s

In the same

manner,

we define the following new classes of close-to-convex functions, which are generalisations of the classes in E1-Ashwah and Thomas

[2].

DEFINITION 3.

A function f normalised ^by

(1.1)

is said to be starlike of order 8, with respect to conjugate points if, and only if, for z e D and 0 8 ^< l,

Re

2zf’(z) .

We denote this class by

S*(8).

c DEFINITION

4.

A function f normalised by

(1.1)

is said to be starlike of order 8, with respect to symmetric conjugate points if, and only if, for z e

D

and 0

.<

8 l,

Re

/. ^?_’.i._ .

We denote this class by S*

(8).

sc RE,%.RK.

The class S* has been studied by several authors,

(eg.

Wu

[9]

and Stankiewicz

[8]).

s

For f c

S*( 8),

Owa et al

[4]

;)roved that for 8 ^<-

s

2’

Re

f(z) f(-z)

3-48’

^{z e D.}

2. RESULTS.

THEOREM i.

Let

f e

S*(8),

then for z rei8 e.

D,

s

Re -[f(z)-f(-z)]

_z

^II(I-8) ">" ^211(1-8)

₂ ^>

^{e 8/(1-8)}

l+r The result is sharp for

fo

^{given by}

f0(z) fo(-Z) 2z(l+z2)

^8-1

To

prove Theorem i., we first require the following lemma.

LEMMA

i.

Let

g e

S*(8)

andbe odd. Then for z reie

D,

Re z >"

2"

l+r

PROOF OF

LEMM

Pinchuk

[5]

showed that if ie

F e

S*(8),

then for z re

D, ll2(1-s)

z -i

.<r.

Since g is odd, we may write

[g(z)]2 F(z 2),

so that

(2.1)

gives

z -i

.<r

(2.1)

where on squaring both sides, gives

Thus

r z

h

Ig(z)i

^{2 Re + 1}

^.<

^r

2Re + i

z

>" l-rh

_r

l_r

h

>

^{+ 1}

(i+r2)2

where we have used the inequality

[6]

ig(z)l >.

(l+r2)

^I-8

for odd starlike functions of order 8.

The Lemma now follows at once.

PROOF OF THEORY4 i.

Since f e

S*(8),

it follows that we may write s

g(z) (z)-f(-z)

2

for g an odd starlike function of order

8. An

application of

Lemma

1 proves the Theorem.

Results analogous to Theorem i THEOREM 2.

Let f

S*(8).

Then for

C

can also be found for the classes z=reiS e

D,

s(6) ^a S*s(S).

Re f(z)+f()

l/2(1-6)

l+r

2(28-1)/2(1-8)

The result is sharp for

f(z)

+

f() 2z(l+z) 2(8-I)

PROOF

Since f e

S*(8),

it is easy to see that, if

C

F(z) f(z)+f()

then F a

S*(8).

from

(2.1)

that

Using the same techniques as in the proof of

Lemma

i, it follows

Re z

The result now follows immediately.

Similarly, we have the following result, which we state without proof.

THEOREM 3.

Let f S*

(8).

Then for z re

D,

sc

112(1-8)

2112tl_S

2

(28-1) 12(1-8)

Re

f(z)-f(-)

>.

^>

z l+r

The resul is sharp for

f(z) f(-) 2z(l+z)

We now consider the results of some integral

operators. In [i]

Das and Singh, obtained analogous results of the Libera integral

operator.

They proved that for f e

S*(0),

the function h given by

s

t-l[

f(

t)-f(-t) ]dt h(zJ

70

also belongs to

S*(0).

s

The result below generalises that of Das and Singh.

THEOREM

4.

Let f e

S*(8).

Then the function

H

defined by s

H(z) ^a+__l

z

ta-l[f(t)_f(_t) ]dt,

2za

JO

(2.2)

also belongs to

S*(6) _.for

z

D

and a + 6 O.

s

We first require the following Lemma due to Miller and Mocanu

[5].

LEMMA

2.

Let

M

and

N

be analytic in

D

with

M(O) N(O)

0 and let 6 be any real number. If

N(z)

maps

D onto

a

(possibly

many

sheeted)

region which is starlike with respect to the origin, then for z e

D,

’(z) M(z)

Re

N--(z

^>

⁶

--->Re

N--

^{> 6,}

and

PROOF OF THEOREM

h.

(2.2)

_ives,

2zH’

(z) H(z)-H(-z)

Re

M’ z) M(z)

N’ z)

^<

⁶

^---->Re

N(--

^<

^6.

Iz ^ta-l[

za[ f(z)-f(-z) ]-a f(

t

)-f(-t ]dt

JO

z

ta-l[ f( ^t)-f(-t

dt 0

(z)

N-V’

^say.

Note

that

M(O) N(O)

0 and for f

S*(6),

s

zN"(z)

[--z[ _i ^{(z)+f’ (-z) ]}

i:

⁺

_{N’ ()}

⁺

Thus

N(z)

is starlike if, and only if a ^> -8.

Furthermore, since

Re N’ (z) ^Re (z)-f(-) iI

^>

"

Lemma

2 shows that

H

e

S*(8).

s

Finally, we give the following analogous results for the classes

S(8)

and S

(8).

c sc

THEOREM

5.

Let f

S*(8).

_c Then

H

defined by

H(z) a+--!l [z ta_l[f(t ⁺ f(----]dt, (2.3)

2za

Jo

also belongs to

S*(8)

for z e

D

and a + > 0.

c PROOF.

Since f e

S*(8), (2.3)

gives c

Thus

ta-l[f(t)

⁺

f(:)]dt

2 ta-I

t + ^Y. Re dt

0 0

n=2

2z

H’(z) H(z)

+

H(:)

=2

ZRea

0

n=2

ⁿ

z

ta_l _If(t)

+

f(:)]dt

0

fz ^ta_l

za[f(z)

+

f()]

a

[f(t)

+

f()]dt

0

z

ta_l[f(t + f(:)]dt

0

where

M(0) N(0)

0 and

N

e S

*

for a +

8

^> 0.

On using

Lemma

2 it follows that

H

e

S*(8)

c

THEOREM 6.

Let f e S*

(8).

Then

H

defined by sc

H(z) a+--!l [z ^ta_l[f{t f(-:)]dt,

2z^a

0

also belongs to

S* (8)

for z

D

and a +

8

> 0.

sc

PROOF.

For f e S*

(8), (2.4)

gives SC

Hi-i)

^a+l

^f-z

t

(-z)

^a 0

a-l[f(t) ^f(-)]dt

a + i

2(-z)

^a+l

(-z) ^n+a

z)a

^{a + i}

n=

2 n + a

(-n

⁺

(-l)n+lan)}

-(a+l)

_z^a

z

₀

^{ta_l[f(t f(-)]dt.}

As before, writing

2zH’(z) H(z) H(-I)

M(z) N(z)’

one can show that N e S* and hence using Lemma 2 the result follows.

REFERENCES

i.

R.N.Das

and P.Singh, On subclasses of schlicht mapping, Indian J. Pure Appl.

Math.,

8, (1977), 864-872.

2. R.M.Ei-Ashwah and D.K.Thomas, Some subclasses of close-to-convex functions, J. Ramanujan Math. Soc.

.2.(i), ⁽¹⁹⁸⁷⁾ 85-100.

3. S.S.Miller and

P.T.Mocanu,

Second order differential inequalities in the complex plane,

J...

^Math.

Ana.. ^{Appl.., 65, (1978), 289-304.}

4. S.Owa, Z.Wu

and

F.Ren,

A note on certain subclass of Sakaguchi functions, Bull. de la

Royale.

^de

Liege, 57(3), (1988), 143-150.

5.

B.Pinchuk, On starlike and convex functions of order 8, Duke Math. Journal,

35., (1968), 721-734.

6.

M.S.Robertson, On the theory of univalent functions, Ann.

Math., 3.7, (1936), 374-408.

7.

K.Sakaguchi, On a certain univalent mapping, J.

Math...Soc. ^Jpa.n,

^ii,

^(1959), 72-75.

8.

J.Stankiewicz, Some remarks on functions starlike w.r.t, symmetric points, Ann. Univ. Marie Curie

Sklodowska ¹⁹⁽ 7), (1965), 53-59.

9. Z.Wu,

On classes of Sakaguchi functions and Hadamard products, Sci. Sinica Set.

A,

30,

(1987), 128-135.

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).

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