A Bohr phenomenon on the punctured unit disk
Zhen Chuan Ng and Rosihan M. Ali
Citation: AIP Conference Proceedings 1750, 050002 (2016); doi: 10.1063/1.4954590
View online: http://dx.doi.org/10.1063/1.4954590
View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1750?ver=pdfcov
Published by the AIP Publishing
Articles you may be interested in
Affine THDMR on the unit disk
AIP Conf. Proc. 1648, 160008 (2015); 10.1063/1.4912458
General properties of concave univalent functions in the unit disk
AIP Conf. Proc. 1571, 1042 (2013); 10.1063/1.4858791
Study of head takeoff phenomenon influenced by stick-slip vibration of slider on head-disk interface
J. Appl. Phys. 99, 08N104 (2006); 10.1063/1.2172533
Instability Phenomenon Associated with an Enclosed Rotating Disk
Phys. Fluids 12, 720 (1969); 10.1063/1.1692537
High‐Speed Rotating Disk Electrode Unit
Rev. Sci. Instrum. 36, 1610 (1965); 10.1063/1.1719403
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 60.49.37.129 On: Fri, 11 Nov 2016 14:28:48
A Bohr Phenomenon on the Punctured Unit Disk
Zhen Chuan Ng1, a) and Rosihan M. Ali1, b)
1
School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia
a)
Corresponding author: [email protected]
b)
[email protected]
Abstract. This paper establishes the Bohr inequality with respect to the spherical chordal distance for the class of analytic
functions mapping the unit disk in the complex plane into the punctured unit disk.
INTRODUCTION
Let
f z f
¦a z
n
n
n 0
be analytic in the unit disk U :
^ z : z 1` of the complex plane satisfying f z 1 in U . Bohr [1] in 1914 showed
that these functions satisfy
f
¦a z
n
n
d1
(1)
n 0
in the disk z d 1 6 . This result became known as the Bohr theorem. The radius 1 6 was later improved independently
to the sharp constant 1 3 by Wiener, Riesz and Schur (see [2-4]). Analogous results to the Bohr for several complex
variables have been established by replacing U with a complete Reinhardt domain [ 5], a unit ball or hypercone in
higher dimensions [6].
An equivalent form for (1) is
f
¦a
nz
n
d 1 a0
d a 0 , wU ,
n 1
where d and wU denote respectively the Euclidean distance and the boundary of U . Generally, a class of analytic
(or harmonic) functions f z ¦
f
a z
n 0 n
n
in U is said to have a Bohr phenomenon (first introduced in [ 7]) if there
exists a number 0 U 0 d 1 satisfying
f
¦a
nz
n
d d a 0 , w f U n 1
Advances in Industrial and Applied Mathematics
AIP Conf. Proc. 1750, 050002-1–050002-7; doi: 10.1063/1.4954590
Published by AIP Publishing. 978-0-7354-1407-5/$30.00
050002-1
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 60.49.37.129 On: Fri, 11 Nov 2016 14:28:48
for all z U 0 . The constant U 0 is then known as the Bohr radius for this class of functions (with respect to the
Euclidean distance). Under this definition, the Bohr radius 1 3 has been found to hold for the class of convex
functions [8] and the class of analytic functions mapping U into the exterior of the closed unit disk [ 9].
This paper discusses the Bohr phenomenon for the class H 0 consisting of analytic functions mapping U into the
punctured unit disk U 0 U \ ^ 0` . Instead of the Euclidean distance, the following spherical chordal distance will be
used:
z1 z 2
O z1 , z 2 2
1 z1
2
1 z2
.
Note that O a0 , wU 0 min ^ O a0 , 0, O a0 , 1 ` and
­! 1,
2 a0 °
°
® 1,
1 a0 °
1,
°
¯
O a 0 , 0
O a 0 , 1
Thus
¦
if a 0 2 1,
f
n 0
2 1.
if a 0
­ O a0 , 1 ,
°
°
® O a0 , 0 ,
°O a , 0 O a , 1 ,
0
0
°
¯
O a 0 , wU 0 Given an analytic function f z if a 0 ! 2 1,
if a 0 ! 2 1,
if a 0 2 1,
2 1.
if a 0
a n z n , define
f * z f
¦a
zn.
n
n 0
It is evident that both power series
¦
f
n 0
¦
a n z n and
f
a n z n have the same radius of convergence. The
n 0
¦
following properties will be needed in the sequel. Let g 1 z f
n 0
a n z n and g 2 z ¦
f
b z
n 0 n
n
be two analytic
functions. Then
g1 g 2 * z f
¦a
n
n
bn z
f
¦ a
d
n 0
bn z
n
n 0
f
¦a
n
n
z n
f
¦b
n
z
n
g1* z g 2 * z ,
n 0
n 0
and
g1 g 2 * z f
n
¦ ¦a
n
z d
n k bk
n 0 k 0
§
¨
¨
©n
f
¦a
0
§
n 0
n ·§
z ¸¨
¸¨
¹© n
f
n
f
¦b
0
n
n
¦ ¨¨© ¦ a
n·
z ¸
¸
¹
k 0
nk
· n
bk ¸ z
¸
¹
g1* z g 2 * z .
050002-2
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 60.49.37.129 On: Fri, 11 Nov 2016 14:28:48
MAIN RESULTS
2 1 d f 0 1 , then f * z  U 0
LEMMA 1. Let f be an analytic function mapping the unit disk U into itself. If
for z 1 3 .
PROOF. Write f z a ¦
f
a z
n 1 n
n
and without loss of generality, assume that
n
¦Z
the n -th root of unity. As U is convex and
k
2 1 d a 1 . Let Z
§ 2Si ·
exp ¨
¸ be
© n ¹
0 , the analytic function
k 1
M zn
1
n
>
¦ f Z z n
k 1
§ n
1ª
«na ¨ Z k
¨
n«
©k 1
¬
¦
@
1
f Zz f Z 2 z " f Z n z
n
k
º
·
¸a 2 z 2 " na n z n "»
¸
¹
¼»
·
§ n
¸a1 z ¨ Z k
¸
¨
¹
©k 1
¦
a a n z n a 2n z 2n " a f
¦a
mn z
mn
m 1
has its image in U . Letting z n ] , the function
M ] a f
¦a
mn ]
m
m 1
is again an analytic function mapping U into itself. Applying the Schwarz-Pick Lemma gives
M ' 0 d 1 M 0
an
2
1 a 2 .
Thus for z 1 3 ,
f * z t a f
¦a
n
n
z !a
n
z a
n 1
1 a2
2
2a 1 a 2
,
2
(2)
1 a2
2
2a 1 a 2
.
2
(3)
and
f * z d a f
¦a
n
n 1
Since a t 2 1 implies 2a 1 a 2 2 t 0 , inequality (2) gives
*
2a 1 a 2 2 1 and so inequality (3) gives f z 1 .
f * z ! 0 . On the other hand, a 1 implies
■
REMARK 2. The inequalities (2) and (3) yield the following important inequality:
2a 1 a 2
f* z
2
a
f
¦a
n
n
z a
n 1
1 a2
2
2a 1 a 2
.
2
(4)
REMARK 3. The condition in LEMMA 1 cannot be removed. Consider the Möbius transformation M in U ,
050002-3
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 60.49.37.129 On: Fri, 11 Nov 2016 14:28:48
M z za
1 az
a (1 a 2 )
f
¦ a
n 1
zn,
0a
17 3 4 1
Then
za
1 az
M * z a (1 a 2 )
f
¦a
0,
that is, for
0
a 1 2a 2 .
z
a 1 2a 2 z
.
1 az
za
1 az
2a z
a 1 2a 2 z
1 az
M * z a 1 2a 2 z
n 1 n
n 1
Note that
if and only if
2.
n 1
However,
0 a 17 3 4 | 0.28077 which implies M * z 0 0 for some z 0 1 3 .
a 1 2a 2 1 3
holds for
LEMMA 4. If f  H 0 , then for z 1 3 ,
O f 0 , f * z ­°! 1,
0 f 0 2 1,
®
O f 0, wU 0 °̄ 1,
PROOF. Write f z a ¦
f
a z
n 1 n
n
2 1 d f 0 1.
where z 1 3 . Suppose that 0 a 2 1 . Then
a
O a, wU 0 O a, 0
1 a 2
.
Note that
O a, 2a 1 a 2 2
1 a2
1 a §¨© 4 2a 1 a ·¸¹
Hence
O a , 2a 1 a 2 2
O a, wU 0 .
2 2
2
1 aa
4 2a 1 a
2 2
!1
(5)
provided 1 a 2 a 2 6 2a 1 a 2 ! 0 , which is true for all a  0, 2 1 . Thus (4) and (5) give
2
O a, f * z ! O a, 2a 1 a 2 2 ! O a, wU 0 .
Suppose now that
2 1 d a 1 Then
O a, wU 0 O a, 1
1 a
2 1 a2
.
Note that
050002-4
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 60.49.37.129 On: Fri, 11 Nov 2016 14:28:48
1 a2
O a, 2a 1 a 2 2
1 a §¨© 4 2a 1 a ·¸¹
Hence
O a, 2a 1 a 2 2
O a, wU 0 provided
4
a 4 4a 3 3
3
.
2 2
2
2 1 a 4 2a 1 a
2 2
1 a 3 3a 3a 2 a 3 ! 0 . As 3 3a 3a 2 a 3 ! 0 for
>
(6)
1
>
a  2 1, 1 ,
it follows that
a 4a 3 ! 0 for all a  2 1, 1 . Thus (4) and (6) give
O a, f * z O a, 2a 1 a 2 2 O a, wU 0 .
■
Define two subclasses of H 0 as follows:
H1
^f  H
0
: 0 f 0 2 1
H2
^f  H
0
: 2 1 d f 0 1 ,
and
`
`
where H 1 ‰ H 2 H 0 and H 1 ˆ H 2 I . As mentioned earlier, the Bohr phenomenon stated in the following results
considers the spherical chordal distance O instead of the usual Euclidean distance.
THEOREM 5. The class H 2 has Bohr phenomenon and the Bohr radius is 1 3 . The Bohr phenomenon does not
occur in the class H 1 .
PROOF. The result follows from LEMMA 4. To show the Bohr radius is 1 3 , consider the function
1 z ·
§
f z exp ¨ a
¸,
1 z ¹
©
for some real a so that
z U
2 1 d e a 1 . The Taylor series expansion for f is
f
§
·
1 z ·
§
exp ¨ a
¸ exp ¨ a 2a z n ¸
¨
¸
1 z ¹
©
n 1
©
¹
f
§
·
¨ 2a z n ¸
exp
a
¨
¸
e
n 1
©
¹
1
¦
2a m §¨
¦
m
·
zn ¸
a
a
e
e m 1 m! ¨© n 1 ¸¹
f
·
2a m f §¨
1
1
¸ zn,
1
e a e a m 1 m! n m ¨© s1 s2 "sm n ¸¹
1
1
f
¦
¦
f
¦
¦
¦
where ‫ݏ‬ଵ ǡ ‫ݏ‬ଶ ǡ ǥ ǡ ‫ݏ‬௠ are positive integers and the latter sum is taken over all m -tuples s1 , s 2 , !, s m satisfying
s1 s 2 " s m n . Thus
050002-5
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 60.49.37.129 On: Fri, 11 Nov 2016 14:28:48
1
f z ea
1
ea
f
1
f
¦¦
ea
2a m §¨ n - 1 ·¸ z n
m!
m 1n m
ª
f
1
n
¦ «¬«¦
ea
n 1
¨ m - 1¸
©
¹
2a m §¨ n - 1 ·¸º z n ,
¨ m - 1¸»
©
¹¼»
m!
m 1
which yields
f* z
1
e
a
1
e
a
ª
f
n
¦ «¬«¦
n 1
2a m §¨ n - 1 ·¸º
m!
m 1
¨ m - 1¸» z
©
¹¼»
n
2
f z .
ea
Thus for z U ,
d e a , f z ,
d e a , f * z
f * z e a t e a f z
and it follows that
O e a , f * z t O e a , f z by the definition of O . Let b
(7)
f 0 e a ! 0 . Since a log b , f can be written as
1 z
§1 z
·
§ 2z
·
log b ¸ b 1 z .
log b ¸ b exp ¨
f z exp ¨
©1 z
¹
©1 z
¹
Let z
(8)
r . Note that
§
1 r
·
O ¨ b, b 1 r ¸
¨
©
O b, 1
2r
¸
¹
2b
1 r
1 b 1 r
1 b 1 r
2r
u
o
!1
1 b
1 r
if and only if r ! 1 3 as b o 1 . Consequently, together with (7) and (8),
§
1 r
·
O f * r , b t O ¨ b, b 1 r ¸ ! O b, 1 O b, wU 0 ¨
©
¸
¹
for r ! 1 3 as b o 1 . Thus the value 1 3 is best possible.
■
COROLLARY 6. The Bohr phenomenon does not occur in the class H 0 .
PROOF. THEOREM 5 and the fact that H 1  H 0 . ■
ACKNOWLEDGEMENT
The work presented here was supported in parts by the 1001/PMATHS/811280 research grant from Universiti Sains
Malaysia.
REFERENCES
1.
2.
H. Bohr, Proc. Lond. Math. Soc. (3) 13, 1–5 (1914).
V. I. Paulsen, G. Popescu, D. Singh, Proc. Lond. Math. Soc. (3) 85, 493–512 (2002).
050002-6
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 60.49.37.129 On: Fri, 11 Nov 2016 14:28:48
3.
4.
5.
6.
7.
8.
9.
S. Sidon, Math. Z. 26, 731–732 (1927).
M. Tomić, Math. Scand. 11, 103–106 (1962).
L. Aizenberg, “Generalization of Carathéodory’s inequality and the Bohr radius for multidimensional power series” in
Selected Topics in Complex Analysis, edited by V. Y. Eiderman,and M. V. Samokhin (Birkhäuser, Basel, 2005), pp. 87–94.
L. Aizenberg, Proc. Amer. Math. Soc. 128, 1147–1155 (2000).
C. Bénéteau, A. Dahlner, D. Khavinson, Comput. Methods Funct. Theory 4, 1–19 (2004).
L. Aizenberg, Stud. Math. 180, 161–168 (2007).
Y. Abu Muhanna, R. M. Ali, J. Math. Anal. Appl. 379, 512--517 (2011).
050002-7
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions IP: 60.49.37.129 On: Fri, 11 Nov 2016 14:28:48