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A n i n e q u a l i t y o f G r u n s k y
T o g g l e A n i n e q u a l i t y o f G r u n s k y s u b s e c t i o n
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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Let f be a univalent holomorphic function on the unit disc D such that f (0) = 0. Then for all r ≤ tanh π/4, the image of the disc |z| < r is starlike with respect to 0, , i.e. it is invariant under multiplication by real numbers in (0,1).
An inequality of Grunsky [ edit ]
If f (z ) is univalent on D with f (0) = 0, then
|
log
z
f
′
(
z
)
f
(
z
)
|
≤
log
1
+
|
z
|
1
−
|
z
|
.
{\displaystyle \left|\log {zf^{\prime }(z ) \over f(z )}\right|\leq \log {1+|z| \over 1-|z|}.}
Taking the real and imaginary parts of the logarithm, this implies the two inequalities
|
z
f
′
(
z
)
f
(
z
)
|
≤
1
+
|
z
|
1
−
|
z
|
{\displaystyle \left|{zf^{\prime }(z ) \over f(z )}\right|\leq {1+|z| \over 1-|z|}}
and
|
arg
z
f
′
(
z
)
f
(
z
)
|
≤
log
1
+
|
z
|
1
−
|
z
|
.
{\displaystyle \left|\arg {zf^{\prime }(z ) \over f(z )}\right|\leq \log {1+|z| \over 1-|z|}.}
For fixed z , both these equalities are attained by suitable Koebe functions
g
w
(
ζ
)
=
ζ
(
1
−
w
¯
ζ
)
2
,
{\displaystyle g_{w}(\zeta )={\zeta \over (1-{\overline {w}}\zeta )^{2}},}
where |w| = 1.
Grunsky (1932) originally proved these inequalities based on extremal techniques of Ludwig Bieberbach . Subsequent proofs, outlined in Goluzin (1939) , relied on the Loewner equation . More elementary proofs were subsequently given based on Goluzin's inequalities , an equivalent form of Grunsky's inequalities (1939) for the Grunsky matrix .
For a univalent function g in z > 1 with an expansion
g
(
z
)
=
z
+
b
1
z
−
1
+
b
2
z
−
2
+
⋯
.
{\displaystyle g(z )=z+b_{1}z^{-1}+b_{2}z^{-2}+\cdots .}
Goluzin's inequalities state that
|
∑
i
=
1
n
∑
j
=
1
n
λ
i
λ
j
log
g
(
z
i
)
−
g
(
z
j
)
z
i
−
z
j
|
≤
∑
i
=
1
n
∑
j
=
1
n
λ
i
λ
j
¯
log
z
i
z
j
¯
z
i
z
j
¯
−
1
,
{\displaystyle \left|\sum _{i=1}^{n}\sum _{j=1}^{n}\lambda _{i}\lambda _{j}\log {g(z_{i})-g(z_{j}) \over z_{i}-z_{j}}\right|\leq \sum _{i=1}^{n}\sum _{j=1}^{n}\lambda _{i}{\overline {\lambda _{j}}}\log {z_{i}{\overline {z_{j}}} \over z_{i}{\overline {z_{j}}}-1},}
where the z i are distinct points with |z i | > 1 and λi are arbitrary complex numbers.
Taking n = 2. with λ1 = – λ2 = λ, the inequality implies
|
log
g
′
(
ζ
)
g
′
(
η
)
(
ζ
−
η
)
2
(
g
(
ζ
)
−
g
(
η
)
)
2
|
≤
log
|
1
−
ζ
η
¯
|
2
(
|
ζ
|
2
−
1
)
(
|
η
|
2
−
1
)
.
{\displaystyle \left|\log {g^{\prime }(\zeta )g^{\prime }(\eta )(\zeta -\eta )^{2} \over (g(\zeta )-g(\eta ))^{2}}\right|\leq \log {|1-\zeta {\overline {\eta }}|^{2} \over (|\zeta |^{2}-1)(|\eta |^{2}-1)}.}
If g is an odd function and η = – ζ, this yields
|
log
ζ
g
′
(
ζ
)
g
(
ζ
)
|
≤
|
ζ
|
2
+
1
|
ζ
|
2
−
1
.
{\displaystyle \left|\log {\zeta g^{\prime }(\zeta ) \over g(\zeta )}\right|\leq {|\zeta |^{2}+1 \over |\zeta |^{2}-1}.}
Finally if f is any normalized univalent function in D , the required inequality for f follows by taking
g
(
ζ
)
=
f
(
ζ
−
2
)
−
1
2
{\displaystyle g(\zeta )=f(\zeta ^{-2})^{-{1 \over 2}}}
with
z
=
ζ
−
2
.
{\displaystyle z=\zeta ^{-2}.}
Proof of the theorem [ edit ]
Let f be a univalent function on D with f (0) = 0. By Nevanlinna's criterion , f is starlike on |z| < r if and only if
ℜ
z
f
′
(
z
)
f
(
z
)
≥
0
{\displaystyle \Re {zf^{\prime }(z ) \over f(z )}\geq 0}
for |z| < r . Equivalently
|
arg
z
f
′
(
z
)
f
(
z
)
|
≤
π
2
.
{\displaystyle \left|\arg {zf^{\prime }(z ) \over f(z )}\right|\leq {\pi \over 2}.}
On the other hand by the inequality of Grunsky above,
|
arg
z
f
′
(
z
)
f
(
z
)
|
≤
log
1
+
|
z
|
1
−
|
z
|
.
{\displaystyle \left|\arg {zf^{\prime }(z ) \over f(z )}\right|\leq \log {1+|z| \over 1-|z|}.}
Thus if
log
1
+
|
z
|
1
−
|
z
|
≤
π
2
,
{\displaystyle \log {1+|z| \over 1-|z|}\leq {\pi \over 2},}
the inequality holds at z . This condition is equivalent to
|
z
|
≤
tanh
π
4
{\displaystyle |z|\leq \tanh {\pi \over 4}}
and hence f is starlike on any disk |z| < r with r ≤ tanh π/4.
References [ edit ]
Duren, P. L. (1983), Univalent functions , Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, pp. 95–98, ISBN 0-387-90795-5
Goluzin, G.M. (1939), "Interior problems of the theory of univalent functions" , Uspekhi Mat. Nauk , 6 : 26–89 (in Russian)
Goluzin, G. M. (1969), Geometric theory of functions of a complex variable , Translations of Mathematical Monographs, vol. 26, American Mathematical Society
Goodman, A.W. (1983), Univalent functions , vol. I, Mariner Publishing Co., ISBN 0-936166-10-X
Goodman, A.W. (1983), Univalent functions , vol. II, Mariner Publishing Co., ISBN 0-936166-11-8
Grunsky, H. (1932), "Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche (inaugural dissertation)" , Schr. Math. Inst. U. Inst. Angew. Math. Univ. Berlin , 1 : 95–140, archived from the original on 2015-02-11, retrieved 2011-12-07 (in German)
Grunsky, H. (1934), "Zwei Bemerkungen zur konformen Abbildung" , Jber. Deutsch. Math.-Verein. , 43 : 140–143 (in German)
Hayman, W. K. (1994), Multivalent functions , Cambridge Tracts in Mathematics, vol. 110 (2nd ed.), Cambridge University Press, ISBN 0-521-46026-3
Nevanlinna, R. (1921), "Über die konforme Abbildung von Sterngebieten", Öfvers. Finska Vet. Soc. Forh. , 53 : 1–21
Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen , Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Grunsky%27s_theorem&oldid=961784481 "
C a t e g o r y :
● T h e o r e m s i n c o m p l e x a n a l y s i s
● T h i s p a g e w a s l a s t e d i t e d o n 1 0 J u n e 2 0 2 0 , a t 1 1 : 3 5 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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