Coefficients of the Inverse of Strongly Starlike Functions

BULLETINof the Bull. Malaysian Math. Sc. Soc. (Second Series) 26 (2003) 63−71 MALAYSIAN

MATHEMATICAL SCIENCES SOCIETY

Coefficients of the Inverse of Strongly Starlike Functions

ROSIHAN M.ALI

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Pulau Pinang, Malaysia e-mail: [email protected]

Dedicated to the memory of Professor Mohamad Rashidi Md. Razali

Abstract. For the class of strongly starlike functions, sharp bounds on the first four coefficients of the inverse functions are determined. A sharp estimate for the Fekete-Szegö coefficient functional is also obtained. These results were deduced from non-linear coefficient estimates of functions with positive real part.

1. Introduction

An analytic function f in the open unit disk U ={z: z <1} is said to be strongly starlike of order α, 0<α ≤1, if f is normalized by f(0) = 0= f′(0)−1 and satisfies

).

( 2 ) (

)

arg ( z U

z f

z f

z ′ < πα ∈

The set of all such functions is denoted by SS*(α). This class has been studied by several authors In [5] it was shown that an univalent function belongs to

. ] 10 , 9 , 7 , 5 , 2 , 1

[ f

) (

* α

SS if and only if for every w∈ f(U) a certain lens-shaped region with vertices at the origin O and w is contained in f(U).

If

(1)

"

+ + +

= ₂ ² ₃ ³

)

(z z a z a z

f

is in the class SS*(α), then the inverse of f admits an expansion

(2)

"

+ + +

− = 3

3 2 2

1(w) w w w

f γ γ

near It is our purpose here to determine sharp bounds for the first four coefficients of

.

= 0 w

n ,

γ and to obtain a sharp estimate for the Fekete-Szegö coefficient functional γ₃ −tγ₂² .

2. Preliminary results

Let us denote by P the class of normalized analytic functions p in the unit disk Uwith positive real part so that p(0) =1and Rep(z) > 0, z∈U. It is clear that

if and only if there exists a function so that . By equating

coefficients, each coefficient of can be expressed in

terms of coefficients of a function in the class For example,

)

*( SS α f ∈ P

p∈ zf′(z)/f(z) = p^α(z)

"

+ + +

= ₂ ² ₃ ³

)

(z z a z a z

f

"

+ + + +

=1 _{1 2} ² ₃ ³

)

(z c z c z c z

p P.

1

2 c

a =α

⎥⎦⎤

⎢⎣⎡ −

−

= _{2 1}²

3 2

3 1

2 c c

a α α

(3)

⎥⎦

⎢ ⎤

⎣

⎡ − +

− + +

= _{3 1 2} ² ₁³

4 12

4 15 17 2

2 5

3 c cc c

a α α α α

Using representations (1) and (2) together with f(f⁻¹(w)) = w or , )) ( ( )) ( ( )

( ₂ ^{1 2} ₃ ^{1 3}

1 + + +"

= f⁻ w a f⁻ w a f⁻ w

w we obtain the relationships

2 2 = −a

γ (4)

2 2 3 3 =−a +2a γ

3 2 3 2 4

4 = −a +5a a −5a γ

Thus coefficient estimates for the class SS*(α) and its inverses become non-linear coefficient problems for the class P. Our principal tool is given in the following lemma.

Lemma 1 [3]. A function p(z) =1+

∑

0 2

0

2

0 1 2

1

⎪⎭ ≥

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ + −

∑

∑ ∑

=

∞

= + +

∞

= +

j k

j k k k

j k k

j c z c z

z

for every sequence {z_k} of complex numbers which satisfy lim_k→∞ sup z_k ^1/k <1. Lemma 2. If p(z) =1+

∑

{ }

⎩⎨

⎧ − ≤ ≤

=

−

≤

− c elsewhere

c 2| 1| ,

2 0

, 1 2

2 , 2 2 max

2 1

2 μ μ μ μ

If μ < 0 or μ > 2, equality holds if and only if p(z) = (1+εz) (1−εz), ε =1. If 0 < μ < 2, then equality holds if and only if p(z) = (1+εz²) (1−εz²), ε =1.

For μ = 0, equality holds if and only if

. 1 , 1 0

1 , )1 1 1 (

) 1 ( : )

( ₂ ≤ ≤ =

+

− −

− +

= +

= λ ε

ε λ ε ε

λ ε

z z z

z z p z p

For μ = 2, equality holds if and only if p is the reciprocal of p₂.

Remark. Ma and Minda [6] had earlier proved the above result. We give a different proof.

Proof. Choose the sequence {z_k} of complex numbers in Lemma 1 to be ,

1 2

0 c

z = −μ z₁ =1, and z_k = 0 if k >1. This yields

, 4 ) 1 2 (

2 1 2

1 2 2 1

2 − c + c ≤ − c +

c μ μ

that is,

. ) 2 ( 2 4

2 1 2

2 1

2 c c

c − μ ≤ +μ μ−

(5)

If μ < 0 or μ > 2, the expression on the right of inequality (5) is bounded above by . Equality holds if and only if

) 1 (

4 μ − ² c₁ = 2, i.e., p(z) = (1+z) (1−z) or its rotations. If 0 < μ < 2, then the right expression of inequality (5) is bounded above by 4. In this case, equality holds if and only if c₁ = 0 and c₂ = 2, i.e.,

) 1 ( ) 1 ( )

(z z² z²

p = + − or its rotations. Equality holds when μ = 0 if and only if ,

2 = 2

c i.e., [8, p. 41]

. 1 , 1 0

1 , )1 1 1 (

) 1 ( : )

( ₂ ≤ ≤ =

+

− −

− +

= +

= λ ε

ε λ ε ε

λ ε

z z z

z z p z p

Finally, when μ = 2, then c₂ −c₁² = 2 if and only if p is the reciprocal of p₂. Another interesting application of Lemma 1 occurs by choosing the sequence to be

} {zk 2,

2 1

0 c c

z = δ − β z₁ = −γc₁, z₂ =1, and z_k = 0 if In this case, we find that

.

> 2 k

2 2 2

1 2

1 32

1 2 1

3 ( )cc c 4 4 ( 1) c (2 )c (2 1)c

c − β +γ +δ ≤ + γ γ − + δ −γ − β −

2 2 1 2 2

1 2 2

1

2 c 4 4 ( 1) c 4 ( 1) c 2c

c −γ = + γ γ − + β β − −ν

−

4 1 2

) 1 (

)

( c

−

− − β β

βγ

δ (6)

where .

) 1 (

) ( ) 1 : (

−

− +

= −

β β

γ δ β β

ν δ

Lemma 3. Let p(z) =1+

∑

. 2

2 1 2 1³

3 − cc + c ≤

c β δ

Proof. If β = 0, then δ = 0 and the result follows since c₃ ≤ 2. If β =1, then

=1

δ and the inequality follows from a result of [4].

We may assume that 0 < β <1 so that β(β −1) < 0. With γ = β, we find from (6) that

2 2 1 2 2

1 32

1 2 1

3 2 cc c 4 4 ( 1) c 4 ( 1) c 2c

c − β +δ ≤ + β β − + β β − −ν

4 : ( )

) 1 (

)

( 4 2

1 2 2

x h cx bx

c ≤ + + =

−

− − β β

β δ

with x = c₁ ² ∈[0,4], b = 4β(β −1), and c = −(δ −β²)² β(β −1). Since it follows that provided

,

≥ 0

c h(x) ≤ h(0) h(0)− h(4) ≥ 0, i.e., b+4c ≤ 0. This condition is equivalent to δ −β² ≤ β(1−β), which completes the proof.

With δ = β in Lemma 3, we obtain an extension of Libera and Zlotkiewicz [4]

result that c₃ −2c₁c₂ +c₁³ ≤ 2.

Corollary 1. If p(z) =1+

∑

2

2 _{1 2 1}³

3 − cc + c ≤

c β β

When β = 0, equality holds if and only if

( )

∑

− =

−

= +

= ³

1

k 2 /3

3 / 2

3 , 1

1 ) 1

( : )

( ε

ε λ ε _π^π

z e

z z e

p z

p ik

ik k

,

≥ 0

λk with λ1 +λ2 +λ3 =1. If β =1, equality holds if and only if p is the reciprocal of p₃. If 0 < β <1, equality holds if and only if

, 1

|

| , ) 1 ) 1 ( )

(z = +εz −εz ε =

p or p(z) = (1+εz³) (1−εz³), ε =1.

Proof. We only need to find the extremal functions. If β = 0, then equality holds if and only if c₃ = 2, i.e., p is the function p₃ [8, p. 41]. If β =1, then equality holds if and only if p is the reciprocal of p₃. When 0 < β <1,we deduce from (6) that

2 2 1 2 2

1 32

1 2 1

3 2

) 1 1 ( 4 )

1 ( 4 4

2 cc c c c c

c − β +β ≤ + β β − + β β − −

. 4 ) 1 ( )

1 ( 4 4 )

1

( − ₁ ⁴ ≤ + − ₁ ² − − ₁ ⁴ ≤

−β β c β β c β β c

Equality occurs in the last inequality if and only if either c₁ = 0 or c₁ = 2. If c₁ = 0, then c₂ = 0, i.e., p(z) = (1+εz³) (1−εz³), ε =1. If c₁ = 2, then

. 1 ), 1 ( ) 1 ( )

(z = +εz −εz ε = p

Lemma 4. If = +

∑

1 ,

1 )

(z _k c_kz^k P

p t

{ }

⎩⎨

⎧ − ≤ ≤

=

−

≤ + +

− cc c elsewhere

c 2 2 1 ,

1 0

, 1 2

2 2 , 2 max )

1

( _{1 2 1}³

3 μ μ μ μ μ

Proof. For 0 ≤ μ ≤1, the inequality follows from Lemma 3 with δ = μ, and .

1

2β = μ + For the second estimate, choose β = μ, γ =1,and δ = μ in (6). Since ,

0 ) 1 (μ − >

μ we conclude from (5) and (6) that

. ) 1 2 ( 4 )

1 ( 4 4 )

1

( _{1 2 1}^{3 2} _{2 1}^{2 2 2}

3 − μ+ cc +μc ≤ + μ μ− c −c ≤ μ −

c

3. Coefficient bounds

Theorem 1. Let f ∈SS^*(α) and f⁻¹(w) = w+γ₂w² +γ₃w³ +". Then ,

2 2α

γ ≤

with equality if and only if

. 1 1 ,

1 ) (

)

( ⎟ =

⎠

⎜ ⎞

⎝

⎛

−

= +

′ ε

ε ε ^α

z z z

f z f

z (7)

Further

⎪⎩

⎪⎨

⎧

≤

≤

≤

≤ <

1 ,

5

0 ,

5 2 1

5 1

3 α α

α γ α

For α >1/5, extremal functions are given by (7). If 0 <α <1/5, equality holds if and only if

, 1 , 1

1 ) (

) (

2

2⎟⎟⎠ =

⎜⎜ ⎞

⎝

⎛

−

= +

′ ε

ε

ε ^α

z z z

f z f

z (8)

while if α =1/5, equality holds if and only if

. 1 0

, 1 1 ,

)1 1 1 (

) 1 ) (

( ) (

2 ⎟ = ≤ ≤

⎠

⎜ ⎞

⎝

⎛

+

− −

− +

= +

′ = ₋ ⁻

λ ε ε

λ ε ε

λ ε ^α

α

z z z

z z z p

f z f z

Moreover,

( )

⎪⎪

⎩

⎪⎪⎨

⎧

≤

≤ +

≤

<

≤

1 3

, 1 1 9 62

2

31 0 1

3 , 2

4 2

α α α

α α γ

For α ≥1/ 31, extremal functions are given by (7), while for 0<α ≤1 31, equality holds if and only if

. 1 , 1

1 ) (

) (

3

3⎟⎟⎠ =

⎜⎜ ⎞

⎝

⎛

−

= +

′ ε

ε

ε ^α

z z z

f z f z

Proof. The following relations are obtained from (3) and (4):

1

2 αc

γ = −

⎥⎦⎤

⎢⎣⎡ − +

−

= _{2 1}²

3 2

5 1

2 c α c

γ α (9)

E c

c c

c : 3

6 2 15 ) 31

5 1 3 (

3 1 2

2 1 3

4

α α

α α

γ α ⎥ = −

⎦

⎢ ⎤

⎣

⎡ + +

+ +

−

−

=

The bound on γ₂ follows immediately from the well-known inequality c₁ ≤2. Lemma 2 with μ =1+5α yields the bound on γ3 and the description of the extremal functions.

For the fourth coefficient, we shall apply Lemma 3 with 2β =1+5α and .

6 ) 2 15 31

( ² + +

= α α

δ The conditions on β and δ are satisfied if α ≤1 31. Thus γ₄ ≤ 2α 3, with equality if and only if ^{zf′(z)/ f(z) =}

[ (

) (

) ]

For 1 31 <α ≤1/5, Corollary 1 yields

(

)

2 6

1 31 2

5 ) 1

5 1

( ₁ ^{3 2}

3 2 1 2

1

3 − + + + + − ≤ +

≤ c α cc αc α c α

E

It remains to determine the estimate for 1/5<α ≤1. Appealing to Lemma 4 with ,

5α

μ = and because 31α² −15α +2> 0in (0,1], we conclude that

) 1 10 ( 6 2

2 15 5 31

) 5 1

( ₁³

3 2 1 2 1

3 − + ≤ −

+ +

+

−

≤ c α cc αc α α c α

E

(

)

3 ) 2 2 15 31 3(

4 2 − + = 2 +

+ α α α

Theorem 2. Let f ∈SS^*(α) and f⁻¹(w) = w+γ₂w² +γ₃w³ +". Then

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

≥ +

−

≤ +

− ≤

≤ −

−

≤

−

4 1 , 5

) 5 4 (

4 1 5 4

1 , 5

4 1 , 5

) 4 5 (

2 2

2 2 3

α α

α α α

α α

γ γ

t t

t t

t t

If ⁵⁻¹₄^/α < t < ⁵⁺¹₄^/α, equality holds if and only if f is given by (8). If t < ⁵⁻¹₄^/α or

4 ,

/ 1 5+ α

t > equality holds if and only if f is given by (7). If t = ⁵⁺¹₄^/α , equality holds if and only if ^z_f^f₍^′(_z^z₎⁾ = p₂(z)^α, while if t = ⁵⁻¹₄^/α , then equality holds if and only if

. )

2(

) (

)

( −α

′_z = p z

f z f z

Proof. From (9), we obtain

2 . ) 4 5 ( 1 2

2 1 2

2 2

3 ⎥⎦⎤

⎢⎣⎡ + −

−

−

=

− t c

c

tγ α α

γ

The result now follows from Lemma 2.

Remark. An equivalent result for the Fekete-Szegö coefficient functional over the class )

(

* α

SS was also given by Ma and Minda [5].

Acknowledgement. This research was supported by a Universiti Sains Malaysia Fundamental Research Grant. The author is greatly indebted to Professor R.W. Barnard for his helpful comments in the preparation of this paper.

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