On the order of a starlike function
Authors: Holland F. , Thomas D. K. 
Published: June 2010

It is shown that if

f
∈
S

f \in S

, the class of normalised starlike functions in the unit

disc
⁡
Δ

\operatorname {disc} \Delta

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis i right-parenthesis limit Underscript r right-arrow 1 minus Endscripts StartFraction log upper P Subscript lamda Baseline left-parenthesis r right-parenthesis Over minus log left-parenthesis 1 minus r right-parenthesis EndFraction equals alpha lamda for lamda greater-than 0 semicolon">

(

i

)

lim

r
→
1
−

log
⁡

P
λ

(
r
)

−
log
⁡
(
1
−
r
)

=
α
λ

 for 

λ
>
0
;

({\text {i}})\quad \quad \lim \limits _{r \to 1 - } \frac {{\log {P_\lambda }(r)}}{{ - \log (1 - r)}} = \alpha \lambda {\text { for }}\lambda > 0;

\]

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis ii right-parenthesis limit Underscript r right-arrow 1 minus Endscripts StartFraction log StartAbsoluteValue EndAbsoluteValue f Subscript r Baseline StartAbsoluteValue EndAbsoluteValue Subscript p Baseline Over minus log left-parenthesis 1 minus r right-parenthesis EndFraction equals alpha p minus 1 for alpha p greater-than 1 semicolon">

(

ii

)

lim

r
→
1
−

log
⁡

|

|

f
r

|

|

p

−
log
⁡
(
1
−
r
)

=
α
p
−
1

 for 

α
p
>
1
;

({\text {ii}})\quad \quad \lim \limits _{r \to 1 - } \frac {{\log ||{f_r}|{|_p}}}{{ - \log (1 - r)}} = \alpha p - 1{\text { for }}\alpha p > 1;

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis iii right-parenthesis limit Underscript r right-arrow 1 minus Endscripts StartFraction log StartAbsoluteValue EndAbsoluteValue f prime Subscript r Baseline StartAbsoluteValue EndAbsoluteValue Subscript p Baseline Over minus log left-parenthesis 1 minus r right-parenthesis EndFraction equals left-parenthesis 1 plus alpha right-parenthesis p minus 1 for left-parenthesis 1 plus alpha right-parenthesis p greater-than 1 comma">

(

iii

)

lim

r
→
1
−

log
⁡

|

|

f
r
′

|

|

p

−
log
⁡
(
1
−
r
)

=
(
1
+
α
)
p
−
1

for (1 + 

α
)
p
>
1
,

({\text {iii}})\quad \lim \limits _{r \to 1 - } \frac {{\log ||f’_r||_{p}}}{{ - \log (1 - r)}} = (1 + \alpha )p - 1\quad {\text {for (1 + }}\alpha )p > 1,

\]
where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript lamda Baseline left-parenthesis r right-parenthesis equals normal upper Sigma Subscript n equals 1 Superscript normal infinity Baseline n Superscript lamda minus 1 Baseline StartAbsoluteValue a Subscript n Baseline EndAbsoluteValue Superscript lamda Baseline r Superscript n Baseline comma left-parenthesis a Subscript n Baseline right-parenthesis">

P
λ

(
r
)
=

Σ

n
=
1

∞

n

λ
−
1

|

a
n

|

λ

r
n

,
(

a
n

)

{P_\lambda }(r) = \Sigma _{n = 1}^\infty {n^{\lambda - 1}}|{a_n}{|^\lambda }{r^n},({a_n})

is the sequence of coefficients and

α
\alpha

the order of

f
f

, and where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue f Subscript r Baseline vertical-bar StartAbsoluteValue equals StartFraction 1 Over 2 pi EndFraction integral Subscript 0 Superscript 2 pi Baseline vertical-bar f left-parenthesis r e Superscript i theta Baseline right-parenthesis EndAbsoluteValue Subscript p Baseline Superscript p Baseline d theta period">

|

|

f
r

|

|

p

=

1

2
π

∫
0

2
π

|

f
(
r

e

i
θ

)

|

p

d
θ
.

||{f_r}|{|_p} = \frac {1}{{2\pi }}\int _0^{2\pi } {|f(r{e^{i\theta }})} {|^p}d\theta .

\]
The results extend work of Pommerenke. The methods of the paper yield various other results, one in particular being
\[

lim

sup

n
→
∞

log

+

n

|

a
n

|

log
⁡
n

=
α

\lim \sup \limits _{n \to \infty } \frac {{{{\log }^ + }n|{a_n}|}}{{\log n}} = \alpha

\]
, a result which has an analogy in the theory of entire functions.