CONVEX ON

Irnat. J. Math. & Math. Si.

Vol. 6 No. 2 (1983) 335-340

335

ON THE RADIUS OF UNIVALENCE OF

CONVEX COMBINATIONS OF ANALYTIC FUNCTIONS

KHALIDA I. NOOR, FATIMA M. ALOBOUDI and NAEELA ALDIHAN

Mathematics Department

Science College of Education for Girls Malaz, Sitteen Road

Riyadh, SAUDI ARABIA

(Received July i0, 1982 and in revised form February 21, 1983)

ABSTRACT. We consider for a > 0, the convex combinations f(z) (l-a)F(z)

+ azF’(z),

where F belongs to different subclasses of univalent functions and find the radius for which f is in the same class.

KEY ^WORDS AND PHRASES. Univalent funtns, alpha-quasi-convex, stake, close-to- convex functio, nvex combinons.

1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. Primary 30A32, Secondary 30A34.

1. INTRODUCTION.

Let S,

K,

S* and C denote the classes of analytic functions in the unit disc E

{z: zl

^{< i} which are} respectively univalent, close-to-convex, starlike, and convex. In

[1,2],

a new subclass C* of univalent functions was introduced and studied.

A

function

f,

analytic in

E,

^{belongs to C* if and only if there exists a} convex func- tion g such that for z ^g

E,

Re

(zf’(z))’

_>

O. (i.i)

g’(z)

Tile functions in C* are called quasi-convex and C C* K S. It is shown

[2]

that f ^g C* if and only if

zf’

^g K. Recently the functions called a-quasi-convex have been defined and their properties studied in

[3].

A function f, analytic in

E,

^{is said to} be s-quasi-convex if and only if there exists a convex function g such that, for a

real and positive

f’(z)

Re{(l

a)

’-(z) ⁺

^a

^(zf’ g’(z) ^(Z))’

^{> O, (1.2)}

It has been shown

[3]

that F is s-quasl-convex if and only if f with

f(z) (i-

e)F(z) + zF’(z)

^is close-to-convex.

(1.3)

All s-quasl-convex functions are close-to-convex.

2. MAIN RESULTS.

We shall now study the mapping properties of f:

f(z)

(i -)F(z)

+ zF’(z),

e >

0,

when F belongs to different subclasses of univalent functions.

THEOREM 2.1. Let F S* and > 0. The function

F(z)

(i-

)F(z) + zF’(z)

(2.1)

is starlike in

zl

^<

ro,

^where

This result is sharp.

ro

2

+ 2 ⁺

^{i 2}

^(2.2)

PROOF. We can write (2.1) as

1 1

2 i

f(z) z (Z

F(z))’,

and from this it follows that

1 i

z -I ^-2

F(z) z z

0 Then

1 i

z.F’(z) _{((i-

^{1 i---}c

[z

^---2

F(z)

^{)z z}

0

1 1

i

{z z--2e

^--1

{(i-

^f(z)dz

⁺

^z

0

f(z)dz

i i 1

z ---2

f(z)dz

+

f(z))/(z z

f(z)dz)}

0 i

f(z)}/{

z

f(z)dz} h(z),

(2.3)

(2.4)

where Re

h(z)

^> 0, since F e S*.

From (2.4),

we have

I

Z

i

f(z)

-

^{i) 0 zC f(z)dz h(z) 0 zCi f(z)dz. (2.5)}

Differentiating both sides of

(2.5),

we obtain

1 1 1

---2 ---i ---2

1

z(

_z ¹

zC

- ^I)

^f(z)

^{+ f’}

^(z)

-

^{i) f(z)}

^=h’(z)

^zz---2i f(z)dz+h(z)z^---2e1 0

Thus

1

(z)

h(z),+

{h’(z)

z^O

f(z) 0

i

f(z)dz}/{z

^O

f(z)}.

f(Z)o

UNIVALENCE OF CONVEX COMBINATIONS OF ANALYTIC FUNCTIONS 337

Now,

using the well-known result

[4], lh’(z) ^{-< {2Re} h(z)}/(l

r

2), Izl

^{r, we have}

z i

z-2f(z)dz

Re

zf’(z)> 2f(z)

_Re

_h(z){l-

^’0 1

}"

_(2.6) 1 r2

2 z f(z) From

(2.1)

and

(2.3),

^we have

1 i

---i ---i

F(z))’

z

f(z

z(z

I i

z-2

^{f(z)dz (z}

^1F(z))

0

from which it follows that

1 1

l{z-if(z)/ Iz

⁰

^z-2

i i

---i ---2

1 z

z{z

^c

F’(z) +

-

ⁱ⁾

^F(z)}

i

(z

^c F(z)

zF’

(z) 1 1

Fz) ⁺

ⁱ

-

^{i) i h(z)}

⁺

¹

^(-

^{r i),}

f(z)dz} >- Re{h(z) + -i)} ^>- (-i) ⁺

₁

₊

r (2.7)

Using

(2.7),

^we have from

(2.6)

2 Re

zf’(z)

_{> Re}

h(z){l-

^{2 r}

+

r

1

)}

2 1

+ (_

^2)r

f(z) 1 r

c

i i i

le

^4r-

- ^2)r2)}/{(1

^r)(

⁺ - ^2r)}

Re hi

){(_.z._.

(2.8)

The right hand side of (2.8) is positive for r < r where r is given by (2.2). ^This

o o

result is sharp as can be seen by

1 i

f (z)

{C(z(

- ^{- 2)z))}/(l}

^z)

o

(i- o)F (z)

+

OzF

’(z),

o o (2.9)

where

F

(z)

^z

o (i- z)² S*

RE}L%RK 2.1. Let f e C, then f, given by

(2.1),

is convex for

zl

^{< r}_o ^{where r}_o

is given by

(2.2).

The proof follows on the same lines as in Theorem 2.1. See also

[5]

and

[6].

REMARK

2.2. In

[6],

Nikolaeva and Repnina treated the same problem, with a dif- ferent notation, for the convex and starlike functions of order

8.

Theorem 2.1 follows from their result when we take

’8

0 for 0 ^{< <} i. On the other

hand,

our proof of Theorem 2.1 is much simpler and the result holds for all > 0.

THEOREM 2.2. Let F e K and f(z) (i

e)F(z) + zF’(z),

^>

O.

Then f is close-to-convex in

zl

^{< r}₀ ^{r is given by (2 2). The function f}₀ ^{in (2 9) shows}

that this result is sharp.

PROOF. Since F E

K,

there exists a G e S* such that, for z E

E,

Re

zF’(z)

G(z) ^> 0.

Now let g(z) (i

)G(z) + zG’(z).

Then by Theorem 2.1 g is starlike for

Izl ^<r

O r_O is defined by (2.2). Using the same technique of Theorem 2 1 we can easily show that Re

zf’(z

_{g(z) > 0 for}

zl

^{< r}_o

REMARK 2.3. For a

1/2

this result has been proved in

[7].

As an easy consequence of

(1.3)

^and Theorem 2.2, we have the following.

COROLLARY 2.1. Let F e K and f(z) (i

e)F(z) + azF’(z),

^{a >} O. Then F is e- quasi-convex in

zl

^{< r This means} that the radius of a-quasi-convexity for close-

O

to-convex functions is given by (2.2).

THEOREM 2.3. Let F ^g C* and a > 0. Let f(z) (i

a)F(z) + azF’(z).

Then f is in

C*,

for

Izl

^<

ro,

^{r is given by}

^(2.2).

PROOF. Since F

C*,

there exists a G e C such that for z e

E,

Re

(zF’(z))’

G’ (z)

^{> 0,} Now let g(z) (i )G(z)

+ zG’(z),

then g is convex in

zl

^{< r}_O ^{We can write}

1 1

2 i

f(z) (i a)F(z)

+ azF’(z)

z

F(z))’

and i i

2 1

g(z) (i

e)G(z) + ezG’(z)

z

G(z))’o

Thus

i i i i

2-E

z

E-I 2- z-

¹

(zf’ (z) )’

g’(z) ((z(z F(z))’)’)’)/(z (( O(z))’)’)

Now

1 1

2 1

1 1

z

2F"

(z(z (z F(z))’)’)’ (z((-I)F(z) + zF’(z))’)’ zF’(z) + (z))

1 i i i i

2 i

o.

1 o

(Z Z

(2.10)

F’(z) +z F"(z)))’ (z (z zF’ (z)) ’) ’o

Let

zF’(z) H(z),

then from

(2.10),

we have

1 1 1 1

2 i 2 i

(zf’(z))’

(z e

z a

za

g’

(z) H(z))

’)’

/ (z G(z))

’)’

1 1

2 1

Since from Theorem 2.2, the function (l-)tt(z)

+ ztt’(z) It(z))’

be--.

longs to Kwith respect to a convex function g: g(z) (l-e)G(z)

+ ezG’(z)

in

UNIVALENCE OF CONVEX COMBINATIONS OF ANALYTIC FUNCTIONS ₃₃₉

zl

^{< r}₀ ^{so f is in C* for}

zl

^{< r}₀ ^{where r}₀ ^{is given by}

^(2.2).

REMARK

2.4. For F e C* and

1/2,

Theorem 2.3 has been proved in

[1].

We now deal with a generalized form of (i.i) by taking g to be starlike and prove the following.

(zF’ (z))’

THEOREM 2.4. Let F be analytic in E and let for z e

E,

Re

G’(z)

>0, GeS*.

Let f(z) (l-)F(z)

+ zF’(z)

and g(z) (l-)G(z)

+ zG’ (z),

with e > 0.

Re

(.zf’ (z))’

g’(z)

^{> 0 for}

Izl

^<

_rl,

^where

Then

rI

3

+ 49

²

⁺

^{1- 2}

For e

1/2,

the problem has been solved in

[8].

Thus

PROOF. From

(2.3),

we can write

I 1

1

i-

iz

^---2C

F(z) z

0

f(z)dz

i i

1 ^i-- _i

[z

²

j

^z

zF

(.)

z

((1-)

0

i f(z)dz

+

za

f(z))

1 1

z za

f’

(z) dz) 0

(zF’ (z))’

G’(z)

1 1

I

fz

^I

(zef ’(z) (-i)

^z

f’(z)dz)

0

z 1 a ⁱ z 0

g’

(z)dz

h

(z),

(2.ii)

where Re h(z) > 0, z E.

From

(2.11),

we write

1 1 1

f,

^{i z --I}

I

^{z --I}

z (z)

(-i)

^z

^f’(z)dz

^{h(z) z}

0 0

g’

(z)dz

Differentiating both sides, and simplifying, we obtain

(zf’ (z))’

g’(z)

^=h(z)

+

i

h’ (z)(

z

J0

g’

(z) dz) 1

z

g’(z)

(2.12)

Using

lh’ ^(z)]

^{< 2Re h(z)}

_2-- ^(2.12)

^gives

1 r

Re

(zf’ (z))_’

g’Cz)

^{> Re}

h(z)[l

i

i r2 0

za

i

g’ (z)

dz) /(z g (z)dz) ]o (2.13)

Now

i i

I

^z

^E-

(zg (z))/(

z

g’

(z)dz) 0

(i/)G’(z) + zG"(z) G’(z)

i

(zG’ (z))’

(-i) ⁺ _G’(z)

^(2.14)

Since G e

S*,

so

(zG’

(z))’

G’(z)

l-4r+r2

2

(2.15)

From

(2.13), (2.14)

and

(2.15),

we obtain

Re

(zf’ (z))’

g’(z)

2 r(l- r

2)

>-

Re

h(z)[l

2 1

4r ^{1 2}

1 r

-

^2)r

1 6r-

(I 2)r

2 Re h(z) 2

1 4er (i 2)r

and this positive for

zl

^<

rl,

^where

r 1 1

3c

+ /9a

²

⁺

^{1 2a}

ACKNOWLEDGEMENT. The authors are grateful for the referee’s helpful comments and suggestions on the earlier version of this paper. In particular, the reference to Nikolaeva and Repnina was kindly supplied by him.

REFERENCES

I. NOOR,

K.

INAYAT.

On a subclass of close-to-convex functions, Comm. Math. Univ.

St. Pauli

2__9 (1980),

^25-28.

2.

NOOR,

K. INAYAT and

THOMAS,

D.K. On quasi-convex univalent functions, Inter. J.

Math. and Math. Sci. 3

(1980),

255-266.

3.

NOOR,

K. INAYAT and

ALOBOUDI, F.M.

Alpha-quasi-convex functions, to appear.

4.

LIBERA,

R.J. Some radius of convexity problems, Duke Math. J. 31

(1964),

143-158.

5.

CAMPBELL,

D.M. A survey of problems of the convex combination of univalent func- tions, Rockey Mount J. Math. 5

(1975),

475-492.

6.

NIKOLAEVA,

R.V. and

REPNINA,

L.G.

A

certain generalization of theorems due to Livingston, (Russian), Ukrain

Mat.

Z. 24

(1972),

268-273, MR45 # 5336.

7.

LIVINGSTON,

A.E. On the radius of univalence of certain analytic functions, Proc.

Amer.

Math. Soc. 17

(1966),

352-357.

8.

NOOR,

K INAYAT and

ALDIHAN,

N. A subclass of close-to-convex functions, to appear.

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