On a subclass of spiral-like functions
Authors: Silvia E. M. 
Published: June 2010

Let

α
≧
0
,
0
≦
β
>
1
,

|

λ

|

>
π

/

2

\alpha \geqq 0,0 \leqq \beta > 1,|\lambda | > \pi /2

and suppose that

f
(
z
)
=
z
+

∑

n
=
2

∞

a
n

z
n

f(z) = z + \sum \nolimits _{n = 2}^\infty {{a_n}{z^n}}

is holomorphic in

U
=
{
z
:

|

z

|

>
1
}

U = \{ z:|z| > 1\}

. If
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R e left-bracket e Superscript i lamda Baseline StartFraction z f prime left-parenthesis z right-parenthesis Over f left-parenthesis z right-parenthesis EndFraction plus alpha left-parenthesis StartFraction z f left-parenthesis z right-parenthesis Over f prime left-parenthesis z right-parenthesis EndFraction plus 1 minus StartFraction z f prime left-parenthesis z right-parenthesis Over f left-parenthesis z right-parenthesis EndFraction right-parenthesis right-bracket greater-than beta cosine lamda">

Re
⁡

[

e

i
λ

z

f
′

(
z
)

f
(
z
)

+
α

(

z
f
(
z
)

f
′

(
z
)

+
1
−

z

f
′

(
z
)

f
(
z
)

)

]

>
β
cos
⁡
λ

\operatorname {Re} \left [ {{e^{i\lambda }}\frac {{zf’(z)}}{{f(z)}} + \alpha \left ( {\frac {{zf(z)}}{{f’(z)}} + 1 - \frac {{zf’(z)}}{{f(z)}}} \right )} \right ] > \beta \cos \lambda

\]
for

z
∈
U

z \in U

, then

f
(
z
)

f(z)

is said to be

α
\alpha

-

λ
\lambda

-spiral-like of order

β
\beta

and we write

f
(
z
)
∈

S
α
λ

(
β
)

f(z) \in S_\alpha ^\lambda (\beta )

. The author shows that for each

α
≧
0
,
α

\alpha \geqq 0, \alpha

-

λ
\lambda

-spiral-like functions of order

β
\beta

are

λ
\lambda

-spiral-like of order

β
\beta

. The following representation theorem is obtained: The function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis element-of upper S Subscript alpha Superscript lamda Baseline left-parenthesis beta right-parenthesis left-parenthesis alpha greater-than 0 comma 0 less-than-over-equals beta greater-than 1 comma StartAbsoluteValue lamda EndAbsoluteValue greater-than pi slash 2 right-parenthesis">

f
(
z
)
∈

S
α
λ

(
β
)
(
α
>
0
,
0
≦
β
>
1
,

|

λ

|

>
π

/

2
)

f(z) \in S_\alpha ^\lambda (\beta )(\alpha > 0,0 \leqq \beta > 1,|\lambda | > \pi /2)

, if and only if there exists a function

F
(
ζ
)
λ

F(\zeta )\lambda

-spiral-like of order

β
\beta

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis z right-parenthesis equals left-bracket left-parenthesis e Superscript i lamda Baseline slash alpha right-parenthesis integral Subscript 0 Superscript z Baseline upper F left-parenthesis zeta right-parenthesis Superscript e Super Superscript i lamda Superscript slash alpha Baseline zeta Superscript negative 1 Baseline d zeta right-bracket Superscript alpha e Super Superscript minus i lamda Superscript Baseline period">

F
(
z
)
=

[

(

e

i
λ

/

α
)

∫
0
z

F

(
ζ
)

e

i
λ

/

α

ζ

−
1

d
ζ

]

α

e

−
i
λ

.

F(z) = {\left [ {({e^{i\lambda }}/\alpha )\int _0^z {F{{(\zeta )}^{{e^{i\lambda }}/\alpha }}{\zeta ^{ - 1}}d\zeta } } \right ]^{\alpha {e^{ - i\lambda }}}}.

\]
A distortion theorem for

log
⁡

|

f
(
z
)

/

z

|

\log |f(z)/z|

and a rotation theorem for

arg
⁡
f
(
z
)

/

z

\arg f(z)/z

are also proved for functions

f
(
z
)
∈

S
0
λ

(
β
)

f(z) \in S_0^\lambda (\beta )

.