CC BY
1238 Probl. Anal. Issues Anal. Vol. 4(22), No. 1, 2015, pp. 38-56
DOI: 10.15393/j3.art.2015.2910
UDC 517.57
E. G. Ganenkqya, V. V. Starkqy
ON REGULARITY THEOREMS FOR LINEARLY INVARIANT FAMILIES OF HARMONIC FUNCTIONS
Abstract. The classical theorem of growth regularity in the class S of analytic and univalent in the unit disc A functions f describes the growth character of different functionals of f G S and z G A as z tends to dA. Earlier the authors proved the theorems of growth and decrease regularity for harmonic and sense-preserving in A functions which generalized the classical result for the class S. In the presented paper we establish new properties of harmonic sense-preserving functions, connected with the regularity theorems. The effects both common for analytic and harmonic case and specific for harmonic functions are displayed.
Key words: regularity theorem, linearly invariant family, harmonic function
2010 Mathematical Subject Classification: 30C55
1. Introduction. For a function u(z), continuous in the disk A = {z E c : |z| < 1}, we denote
M(r,u) = max |u(z)| and m(r,u) = min |u(z)|.
|z|<r |z|<r
Let S be the class of all univalent analytic functions f (z) = z + ... in A. The theorem of growth regularity asserts that functions having the maximal growth in the given class, grows smoothly (regularly).
Theorem A. [1], [2], [3, pp. 104, 105], [4, pp. 8-9] Let f E S. Then there exist a ¿o E [0,1] with
lim
r —^ 1 —
M (r,f)
(1 - r)2
lim
r—1 —
M (r,f')
(1 - r)3 1 + r
= ¿c
©Petrozavodsk State University, 2015 IMlIiHI
= 1 iff f (z) = z(1 — ze-%e)—2. If = 1, then the functions under the sign of the limit increase on r.
If 5° = 0, then there exists E [0; 2n) such that
lim
r —^ 1 —
If (re^)|
(1 — r)
= lim
r—1 —
If /(rei^)|
(1 — r)' 1 + r
5°, ^ = 0, ^ = .
Here the functions under the sign of the limit are also increasing on r E E (0, 1).
In [5], Ch. Pommerenke showed that many properties of functions from the class S can be extended to linearly invariant families (LIFs) of locally univalent analytic functions in A of finite order. In [6] and [7], the theorem of growth regularity was obtained for such LIFs.
In [8], [9], the authors introduced the notion of LIF for complex-valued harmonic functions f in A. Every such function can be presented, using analytic functions h and g in A in the following way:
f (z) = h(z)+ g(z), (1)
where
h(z) = z + ^ an(f )zn and g(z) = a_n(f )zn.
n=2 n=1
As in [5], L. E. Shaubroek considered locally univalent functions in A. Moreover, these functions are sense-preserving in A, i. e. the Jacobian Jf (z) satisfies
Jf (z) = |h/(z)|2 — |g/(z)|2 > 0 Vz E A.
Definition 1. [8], [9] A set Mh of harmonic sense-preserving functions f in A of form (1) is called the linearly invariant family (LIF) if for all f E MH and for any conformal automorphism 0(z) = f+aZ, a E A, the function e—%e fa (zei6>) belongs to MH, where
= f (y(z)) — f (y(0)) (2)
fa(z)= h/(*>(0)M0) . (2)
It is assumed that the order of a family MH
2
is finite.
ord MH = sup |a2 (f)|
f eMn
In the analytic case (when g (z) = 0), the definitions of LIF and ord MH coincide with the definitions of Pommerenke [5].
In [10], for LIFs of harmonic functions, the strong order
ord Mh = sup |a2 (f) - ^ (f )|
f EMh 1 - la-l (/)T
was defined. The strong order proved to be convenient for investigation of LIFs, because it is not necessary to assume the affine invariance of a family. Moreover, for an affine LIF Mh the strong order does not exceed the old order:
ord Mh - 1 < ord Mh < ord Mh . 2
This fact allows to describe properties of affine LIFs more precisely. For a LIF M of analytic functions, ord MH = ord MH. Analogously to the analytic case in [10] the universal LIF U^ was introduced and studied. The family U^ is defined as the union of all LIFs MH such that ord MH < < a. Equivalently, U^ is the set of all harmonic sense-preserving functions / in A of the form (1) such that
oid / =f oid [e-w fa (zei0) : a e A, 6 e r} < a. It was shown in [10] that ord UH > 1.
In [11] and [12], the following regularity theorems for harmonic functions were proved:
Theorem B. (regularity of growth) Let f e U^. Set
$i(r) = j M(p,Jf) dp, *i(r,p) = y Jf (pe^) dp, and 0 0
Fi(r)^(1+f2I_ dp.
0
(1 - p)2
For each n > 2 successively denote
r r
(r) = J $n-i(p) dp, (r, <p) = J tfn_i(p, <p) dp, and
00
r
r
Fn(r) = Fn-i (p) dp.
Then
a) for every ^ E [0; 2n) and n E n, the functions
(1 _ r)2a+2 (1 _ r)2a+2
Jf (re*, M(r, Jf)( )
(1 + r)
2a —2 '
(r) (r,^)
and
(1 + r)
maxfn(r,
^_
Fn (r)
Fn (r)' Fn(r) '
are non-increasing on r E (0; 1);
b) there exist constants E [0; 1] and E [0; 2n) such that for 1 < n < 2a + 2,
= lim
ri
M(r, Jf) (1 - r)2a+2 Jf (0) (1 + r)2«-2
= lim
ri
Jf (reiyU) (1 - r)2a+2
Jf (0) (1 + r)
2a—2
= lim
ri
= lim
ri
M(r, dr Jf) (1 - r)2a+3 Jf (0)4(a + 1) (1 + r)2a—3
f Jf (re* )
(1 - r)
2a+3
Jf (0)4(a + 1) (1 + r)
2a — 3
= lim
= lim
r —^ 1 —
= lim
r—1 —
(r)
/or M(A dp Jf) dP (1 - r)2a+2 Jf (0) (1+ r)2a —2
/°r dp Jf (p^0 ) dP (1 - r)2«+2
Jf (0)
= lim , n^A = lim
(1 + r)2a—2
maxfn(r,
r—1— Jf (0)Fn (r) r—i— Jf (0)Fn(r) r—i— Jf (0)Fn (r)
c) = 1 for functions Qq (z) a E A, 0 E R, and
ka(z) = -1 aW 2a
= eiQka(ze—iQ) + aeiQka(ze—iQ), where
1 + z
1 - z
1
(3)
r
a
Theorem C. (regularity of decrease) Let f e U^. Set
fi fi (1 - p)2a-2 Qi(r) = j m(p,Jf) dP, Ei(r) = y (1 + p)2a+2 dp.
For each n > 2 successively denote
Qn(r) = / Qn-i(p) dp, and En(r) = / En-i(p) dp.
rr
Then
a) for every p e [0; 2n) and n e n the functions
r ( ^ (1 + r)2a+2 ( , ) (1 + r)2a+2 . Qn(r)
Jf(re'^H-tt:—^, m(r, Jf H-^—^, and ^ . ,
n y(1 - r)2«-2' v' f (1 - r)2a-2' En(r)
are non-decreasing on r e (0; 1);
b) there exist constants 50 e [1; to] and p0 e [0; 2n) such that
50 = lim
r^i-
2a+2
m(r, Jf) (1 + r) Jf (0) (1 - r)2«-2
lim
r^i-
2a+2
Jf (re'^°) (1 + r) Jf (0) (1 - r)2«-2_
= lim Qn(r)
r^i- Jf (0)En(r)' c) for p e [0; 2n) denote
i
Ri (r,p) = y Jf (pe'^) dp,
r
and for n > 2, set
i
Rn (r, p) = / Rn-i(p, p) dp
(under the assumptions of Theorem C the integrals converge). If 50 < to
E„(r)
Moreover,
Rn(r, P0)
then for n > 1 the function RE(rf,y)°) is non-decreasing on r e (0; 1).
50 = lim
r^i- Jf (0)En(r)
d) if Jf (z) is bounded in A, then for every n E n and every p E [0; 2n), the functions
Rn (r,p) and min ^n(r,p)
En(r) an En(r)
are non-decreasing on r E (0; 1) and
min Rn(r, p)
= lim
r—i— Jf (0)En(r)
e) = 1 for functions qq (z) = e ka(ze iQ) + ae ka (ze iQ), where a E A, 0 E r, and ka(z) is the function defined by (3).
Definition 2. We say that the constant p° from Theorem B is a direction of maximal growth (d.m.g.) of a function f (z). The constant p° from Theorem C is a direction of maximal decrease (d.m.d.) of f (z).
Definition 3. The numbers from Theorem B and from Theorem C are called the Hayman numbers of a function f (z).
In the presented paper we establish new properties of U^, connected with the regularity theorems.
2. Main results. For fixed c E [0; 1) introduce the class U^ c, consisting of all functions f = h + g E U^ such that (0)| < c. That is, Jf (0) > 1 - c2 > 0 for all f E UH,c. The c lass UHc is not a LIF. Note that the family U^ is not compact in the topology inducted by locally uniform convergence in A, but for UHc the following theorem takes place.
Theorem 1. The family UHc is compact in the topology inducted by locally uniform convergence in A.
Proof. Let fn E UHc, fn = hn + gn, n E n, hn and gn be analytic functions in A. By Aa denote the set of all analytic functions h in A such that there exists an analytic function g in A and f = h + g E U^. In other words, Aa is the set of analytic parts of functions f E U^. The linearly invariance of U^ implies that Aa is a LIF of analytic functions. But for LIFs of analytic functions ord Aa = ord Aa. Therefore for all h E Aa
(1 + r)a—1 (1 - r)
lh'(z)l< + r)a+1 , |z| = r
see [5]. Since Jf(z) = |h'(z)|2 - |g'(z)|2 > 0 for all z G A and all f G Uf, we have
|g'(z)| — (1+
(1 - r)«+1'
for all f = h + g G and z G A, |z| = r. Consequently, WHc C C is uniformly bounded on compact subsets of A. According to the compactness principle, there exists a subsequence of fn (let us save the notation) which converges locally uniformly in A to a harmonic function fo. Let us show that fo G WHc.
For f G the following inequality holds (see [10])
(1 - r)2*-2 Jf (z) (1 + r)2a-2 = r
(1 + r)2«+2 - Jf (0) - (1 - r)2«+2 , |z| Therefore for fn G WHc we have
(1 _ r)2a-2
Jfn (z) > , L+2 (1 - c2) > 0.
(1 + r)
This implies Jf0 (z) > 0 for all z G A. This means that the harmonic in A function f0 is sense-preserving.
Next, we prove that ord f0 — a. Suppose not. Then, we may let ord f0 = P > a. Then, by the definition of the strong order, there exist a conformal automorphism p(z) = 1++Z of A and 6 G r such that for harmonic function
p-i0(f ) ) = fo(P(zpi0)) - fo(P(0)) = ^(A + A -k) e (fo)a(zp ) = h0(p(0))p(0)pi0 = z + A-kz ),
(A1 = 1, fo = ho + go) the inequality
IA2 - A-iA-2 p - a
' > a (4)
1 -|A-i|2
is valid.
For the automorphism p and the number 6 denote
p-i0 (fn )a(zei0) = + A-kz-k), (Aln) = 1).
k=i
From locally uniform convergence of fn to f°, the Weierstrass theorem on series of analytic functions, and inequality (4) it follows that for sufficiently large n > N
A(n) — A(n) A(n) 2 — 1 —2
1 -|Aii |2
B — a > a + „ .
Hence if n > N we have ord fn > a + and fn E UHc. This contradiction proves the theorem. □
In claim c) of Theorem B and claim e) of Theorem C some set of functions with the Hayman number 5° = 1 (or 5° = 1 for the theorem of decrease regularity) is described. These claims differ from the analytic case. In the analytic case 5° = 1 and 5° = 1 only for the functions eiQka(ze—iQ), where 0 E r, ka(z) is the function defined by (3) [7], [13], [14]. The following example shows that in the harmonic case this set has more complicated structure. We construct the example of functions f of arbitrary strong order B > 3/2 with 5° = 1. These functions are not equal to the function qq(z) from Theorem B. We use the Clunie and Sheil-Small shear construction [15] (see also [16, ch. 3.4]) to give our example. Let us note that our construction is not stable. As one can show, if we multiply the coanalytic part g of the function from our example by constant k E (0,1), then the strong order of the function changes stepwise and 5° = 1 for this function.
Example. Put h' (z) = (1—^+2, g' (z) = zh' (z), z E A. Let E [1, to) be fixed. If p(z) A, then for f = h + g we have
(1—z)a + 2 1+az :
a E [1, to) be fixed. If p(z) = i+a , a E A, is an automorphism of
f (z) - F(z)= HW+GM- h(p(z)) - h(p(0)) + (g(p(z)) - g(p(0))\
fa(z) =. F(z) = H(z)+G(z) = h'(p(0))p'(0) H h'(p(0))p'(0) )
where H and G are functions analytic in A,
H'(z) = h' (P(z))P'(z) and ( ) h'(p(0))p'(0)
G' = g'(p(z))p'(z) = p(z)h' (p(z))p' (z) h'(p(0))p' (0) h'(p(0))p' (0) .
Note that
Jf(z) = \H'(z)|2 - \G'(z)|2 = and, in particular,
Therefore,
,2 ^2 _ h(p(z))\2\p'(z)\2(1 - \p(z)\2)
\h'(p(0))\2(0)\2
Jf (0) = 1 -\p(0)\2.
Jf (z) _ \1 + i+
z+a 12a-2 az \
Jf (0) \1 + a\
2a—2
1a
1
z+g 1+ gaz
2a+4
22
(1 - \ a \2) \1 + az\4
■ x
1
z+a
1 + az
1
(1 - \a\2)3
1 + z
1+g
2a—2
1 - z 1=2
1g
2a+4
\1 + az\2 - \z + a\2
1- I a 12
1 + z
1+g
2a—2
1 - z 1-2
1g
2a+4
(1 - \z\2),
by generalized Schwarz's lemma. Consequently, for r G (0,1)
Jf (z) sup j
geA, Jf (0) |z|=r
(1 + r)
2a — 1
(1 - r)
2a+3 '
Therefore for P = a + 2, all a G A, and \z\ = r we get
Jf (z) (1 + r)2^-2 Jf(0) — (1 - r)2^+2 .
(5)
In [10] it was shown that for functions f harmonic and sense-preserving in A,
ord f = inf(P : — (1 + \z\)2!+2 , VF = fa, Vz G a) . (6)
Jf(0) - (1 - \z\)2^+2
From (5) and (6) we conclude that ord f — P = a + |. From Theorem B it follows that if for a function f harmonic and sense-preserving in A
lim
r —^ 1 —
Jf (z) (1 - r)2^+2 Jf (0) (1 + r)2^-2
> 0,
(7)
2
then ord f > B. For the considered function f the limit in (7) equals 1. Therefore, ord f = B and
5° = lim
r —^ 1 —
Jf (r) (1 - r)2^+2 Jf (0) (1 + r)2^-2_
= 1.
It is interesting to find out if there exist functions with 5° = 1 which are not equal to the function from the example and the functions (z).
Definition 4. A direction of intensive growth (d.i.g.) of a function f (z) is a constant p G [0; 2n) such that
lim
r—1 —
2a+2
Jf (re^ ) (1 - r) Jf (0) (1 + r)2a—2
= 5(f, p) > 0.
A direction of intensive decrease (d.i.d) of a function f (z) is a constant p G [0; 2n) such that
lim
r— 1—
2a+2
Jf (re^) (1 + r) Jf (0) (1 - r)2«—2_
= 5'(f,p) < to.
Since we study LIFs, it is important to know how d.i.g.-'s and d.i.d.-'s of a function f (z) are changed under the transformation e—fa(zei0). The case a = 0 is trivial: a d.i.g. (d.i.d.) p — 6 of the function e—iélf (zeiél) corresponds to the d.i.g. (d.i.d.) p of f (z). In this situation 5(f (z),p) = = 5(f (zei0), p - 6) (and 5'(f (z), p) = 5'(f (zei0), p - 6)). It is also interesting to find out the relationship between the Hayman numbers of the functions f and fa in general case. The following theorem concerns the non-obvious case a = 0.
Theorem 2. Let f G UH. Denote
R(r) =
re^ + a
1 + are^
Y (r) = arg
re^ + a
1 + are1^ '
a G A,
re^ = -a.
1) p is a d.i.g. (d.i.d.) of the function fa(z) iff y is a d.i.g. (d.i.d.) of f (z) and
e^ =
e
17_
a
1 - aeiY
(8)
2) for all 7 G [0, 2n)
lim
r—1 —
and,
lim
r—1 —
2a+2
Jf (reiY ) (1 - r) Jf (0) (1 + r)2«-2
2a+2
Jf (reiY ) (1 + r) Jf (0) (1 - r)2a—2
lim
r—^ 1 —
lim
r— 1—
Jf (R(r)e^(r)) (1 - R(r))2a+2 Jf (0) (1 + R(r))2«-2
Jf (R(r)eiY(r)) (1 + R(r))2a+2 Jf(0) (1 - R(r))2«-2
Here p and 7 are connected by (8).
3) if p is a d.i.g. of fa(z), y is a d.i.g. of f (z), and p is connected with y by (8), then
¿(/,7 )= ¿(/a ,P)
2N2a+2
Jf (a) (1 -H2)
Jf(0) |1+ |4a '
if p is a d.i.d. of /a(z), 7 is a d.i.d. of /(z), and p is connected with 7 by (8), then
Jf (a) |1+ äei-|4a
¿'(/,7)= ¿' (/a,p)
Jf (0) (1 - | a |2 )
2a-2 '
Proof. 1) Let p be a d.i.g. of fa (z). This means that there exists the limit
Jfa (re^) (1 - r)2a+2"
Note that
and
¿(/a ,P) = lim r—> 1 —
Jfa (Z) =
Jfa (0) (1+ r)2«-2
J ( z + a
Jf ^1+äz
|h'(a)|2|1 + äz|4,
> 0.
T (0) = Jf(a)
Jfa (0) = ïï^
\h'(a)\2'
Let us calculate the following limit, using (9) and (10),
"Jf(R(r)eiY(r)) (1 - R(r))2a+2
0 = lim
r1
= lim
r1
Jf (0) (1 + R(r))2«-2
Jfa (/e ) (a) I2 11 + are^
Jf (0)
(9) (10)
|h' (a)|211+ are*14
2a+2'
We have
lim 1 - R(r) = lim R'(r) = ~|a|2,9 . (11)
r^i- 1 - r r ^ 1 |1 + аег! |2
Using (11), we obtain
5 = f^ff1^ ^ ()2a+2 > 0' (12)
By (11), lim R'(r) > 0, therefore the function R(r) increases on an
r —^ 1 —
interval (r0,1). By Theorem B, for r0 < r < r1 < 1
Jf (R(ri)eiY(ri)) (1 - R(ri))2a+2 Jf (R(r)eiY(ri)) (1 - R(r))2a+2
Jf(0) (1 + R(ri))2«—2 - Jf(0) (1 + R(r))2a—2 '
Passing to the limit as r1 ^ 1— and using (8), we get
Jf (R(r)eiY) (1 — R(r))2a+2 5 - Jf(0) (1 + R(r))2«-2 '
Thus,
5(f ) l. f Jf (R(r)eiY ) (1 — R(r))2a+21 5 (13)
5(f, Y) = rlim4 Jf (0) (1 + R(r))2a —2 J ^ 5' (13)
Taking into account (12), we conclude that y is a d.i.g. of f (z). Now let us consider the sets
A = {eiY : y is a d.i.g. of f (z)},
B = {r++i : P is a di.g. of fa(Z^ ,
C = {ein : n is a d.i.g. of [f0](—a)(z)} .
Here [fa](—a)(z) is the transformation (2) of the function fa with the parameter —a. If n is a d.i.g. of [fa](—a)(z), then, as it was proved above,
ein =
ег| + а
1 + аег! '
where p is a d.i.g. of fa(z). This implies that C С B. Let p be a d.i.g. of fa(z). Then
ег| + а
eiY =
1 + аег! '
where y is a d.i.g. of f (z). Thus B C A. Since [fa](—a)(z) = f (z), we have A = C and, consequently, A = B. This completes the proof of the statement about d.i.g.-'s.
The statement about d.i.d.-'s is proved analogously.
2) Let us prove the first equality. If 7 is not a d.i.g. of f (z), then
lim
r—> 1 —
Jf (reiY) (1 - r)2a+2 Jf (0) (1 + r)
2a-2
= 0.
Thus, by (13),
5 < lim
r—y 1 —
Jf (R(r)e^) (1 - R(r))2a+2
Jf (0) (1 + R(r))
2a-2
= 0.
This implies 5 = 0.
Now let us consider the case when 7 is a d.i.g. of f (z). We have proved above that 5(f, 7) > 5 (see (13)). It remains to show that 5(f, 7) < 5. Denote
Ri(r) =
re
17_
a
1 — areiY
Since [fa]( —a)(z) = f (z) Y is a d.i.g. of [fa]( — a) (z), i. e
,¿7
5 ([fa ](_a) ,Y)= 5(f, y) = limm
r1
J[f*](-*) (re^^) (1 — r)2a+2
Jf (0) (1 + r)
2a 2
> 0.
Arguing as in the proof of claim 1), one can note that there exists
5* =f lim r— 1—
Jf
reiY — a \ fo V1 — are^J (1 — Ri(r))2a+2
Jf* (0) (1+ Rl (r))
2a 2
Apply (13) to the function fa(z), using (9), (10), and (11):
lim
r— 1—
5* < lim
r— 1—
re* + a \
Jf* (re*) (1 — r)2a+2 Jf* (0) (1 + r)2a—2J
/ re* + a \ Jf V1 + are^y (1 — R(r))2a+2 Jf (a)|1 + are^ |4 (1 + R(r))2a—2
lim
r— 1—
1—r 1 — R(r)
2a+2
5Jf (0)
|1 + ae^|
Jf (a)11 + äe^|4 V 1 - |a|2 On the other hand, by (9),
z — a
|2 \ 2a+2
5Jf (0) |1 + ae^ |4a
Jf (a) (1 — |a|2)2«+2'
(14)
J
fa
1 — az
Jf (z)
|h'(a)|2
1 + a
za
4
1 — az
Thus, using (8), (10), and (11), we can write 5* in the form
5* = lim
r —^ 1 —
Jf (reiY)
(1 — r)
2a+2
Jf (a)
rel7 — a
1 + a feY
1 — areiY
(1 + r)
2a—2
x
x lim
r— 1—
1 — Ri(r)
1r
2a+2
= ¿(/,7)
Jf (0)
1 — |a|2 \
TToiv 14 \ h _ TToiY 12 I
2a+2
Jf (a) 11 + äe^ |4 \ |1 — ae^ | = ¿(/,Y) Jf(°) ^ r
Jf (a) (1 — |a|2)2a+2'
Substituting
= /)
Jf (0) |1 + aei^|4a Jf (a) (1 — |a|2)2a+2
in (14), we get ¿(/,7) < 5. Therefore, ¿(/,7) = 5.
The second equality of claim 2) is proved analogously.
3) The formula, connected 5(/,7) and 5(/a,p) is obtained from (12),
using 5 = 5(/,y).
The second equality is proved analogously. □
Theorem 2 implies the following
Remark. Let / G U^. For every p G [0; 2n) there exist 5(/, p) G [0; 1] and 5' (/, p) G [1; to] such that for any circle or straight line r C A, orthogonal to dA at the point e^, we have
lim
Jf (z) (1 — |z|)2a+2 Jf (0) (1 + |z|)2a—2
= 5(/,p),
4
lim
Jf (z) (1 + |z|)2a+2
= 5'(f,p),
LJf(0) (1 — |z|)2a—2_
and the constants 5(f, p), 5' (f, p) do not depend on r.
By Uf (50) denote the set of all functions from Uf with the same Hayman number 50 from Theorem B.
Let Uf (50) be the set of all functions, having the Hayman number 50 from Theorem C.
Theorem 3. 1) If f G Uf (50), 50 G (0; 1), then for every 5 G [50, 1) there exists a e A such that fa (z) e Uf (5).
2) If f e Uf (50), 50 G (1; to), then for every 5' G (1,50] there exists a G A such that fa(z) e Uf (5').
Proof. By Theorem B, for any p e [0; 2n) there exists
lim
r— 1—
Jf (re*) (1 — r)2a+2 Jf (0) (1+ r)2a —2_
= 5(f, p).
re* — a
Let us fix a e A p e [0;2n). Denote z = --=——, |z| = R(r) and
consider the limit
1 — are^
5* (p) =f lim
r— 1—
Jf* (z) (1 — R(r))2a+2 Jf* (0) (1 + R(r))2a—2
Let us calculate 5*(p), using (9) and (10)
5*(p) = lim
r— 1—
Jf (re*)
(1 — R(r))2a+2
Jf(a)
1 + aa
rei^— a
1 — areiv
(1 + R(r))2a—2
lim
r1
By (11),
Jf (re*) (1 — r)2a+2 Jf (0W1 — R(r) Jf (0) (1 + r)2a—2 Jf (a^ 1 — r
2a+2'
1 + aa
rei^— a
1 — areiv
5*(p) = 5(f, p)
Jf(0) (1 — |a|2)2a+2 |1 — ae*|4 Jf (a) |1 — ae^ |4a+4 (1 — |a|2)4
= 5(f, p) ¿£(2)(1—avr! <
Jf (a) |1 — ae^|
4a —
4
1
— lim
R(r) —1 —
M(R(r), Jf) (1 - R(r))2a+2
Jf (0) (1 + R(r))
2a — 2
def ,
= 5«.
Let p be equal to d.m.g. p° of f (z) and a = pei(^ . Then 5(f, p) = and
Jf (0) (1 - p2)2a—2
Jf (pe^0) (1 - p)
4a
= 5°
Jf (0) (1 + p) Jf (peV0 )(1 - p)
2a—2
2a—2 —
< 5a.
(15)
By Theorem B, there exists a d.m.g. p1 e [0; 2n) of fa(z) such that
5a = lim
r1
Jf„ (reiy0) (1 - r)2a+2 Jfa (0) (1+ r)2a-2
lim
r1
t f re'yi +q
(1 - r)
2a+2
Jf (a) 11 + are^i |4 (1 + r)2a-2
re^^i + a
Denote R1 (r)eiYl(r) = -:—, where Y1(r) is a real-valued function.
1W 1 + are^i v y
Then, using (11) for R(r) = R1 (r), we obtain
5a — lim
r1
= lim
r—^ 1 —
M(R (r), Jf) (1 - r)2a+2 Jf (a) 11 + are^14 (1 + r)2a—2
M(R1(r), Jf) (1 - R1 (r))2a+2
Jf (0)
x
x
Jf (0)
(1 + R1 (r))2a —2
X 2a+2
1r
= 5° Jf(0)
• lim . . .
r—1— \ 1 - R1(r)
1 f |1 + ae^i |2 Va+2_5° Jf (0) |1 + aei^i |4a
Jf (a) |1 + ae^^i |4 V 1 - |a|2
Jf (a) (1 - |a|2)2a+2
<
2a- 2
5
0 Jf (0) (1 + p)4 _
Jf (a) (1 - p2)2a+2 " Jf (a) (1 - p)2a+2 •
=5
0 Jf (0) (1+ p)
Taking into account inequality (15), we get
5° Jf (0) (1 + p)2a—2 = 5
°Jf(peV°)(1 - p)2a+2
Since the continuous function J Jp(0)o) (1—p)2a+2 decreases on p, equals 1
as p = 0, and tends to zero as p ^ 1 — , then we can find p e [0; 1) such that 5a takes preassigned value from [50; 1).
Claim 2 of the theorem is proved analogously. □
In [7] (see also [17], [14]) it was proved that the set of all d.i.g.-'s and d.i.d.-'s of a given analytic function is at most countable. The following theorem shows that this statement is true for set of d.i.g.-'s of harmonic function too. But we don't know whether this fact is true for set of d.i.d.-'s.
Theorem 4. Let f e U(
countable.
H a '
Then the set of all d.i.g.-'s of f is at most Proof. If f = h + g G UH, then ord h < a. Since
Jf(z) = |h'(z)|2 — |g'(z)|2 < |h'(z)|2 for all z G A, then for p e [0, 2n) and r e [0,1)
Jf (re*) (1 — r)2a+2 Jf (0) (1 + r)
2a — 2 <
|h'(re* )|
(1 — r)
a+1
(1 + r)
a — 1
Jf(0)"
(16)
By Theorem B and theorem of growth regularity from [7], there exist the limits
" Jf (re*) (1 — r)2a+2'
and
5(f, p) = lim r— 1—
<5(h, p) = lim
r— 1—
Jf (0) (1 + r)2a —2
(1 — r)2a+2
|h' (re*)|
(1 + r)
2a — 2
From (16) we get 5(f, p) < J(h(0)'). If p is a d.i.g. of f, then 5(f, p) > 0.
Consequently, <(h, p) > 0 and p is a d.i.g. of h. Therefore the set V of all d.i.g.-'s of f is contained in the set W of all d.i.g.-'s of h. As it was proved in [7], W is at most countable. Hence V is at most countable too. □
Acknowledgment. This work was supported by RFBR (projects N 14-01-00510f, N 14-01-92692). The authors thank S. Yu. Graf and S. Pon-nusamy for valuable comments on improving the paper.
2
1
References
[1] Hayman W. K. Some applications of the transfinite diameter to the theory of functions. J. Anal. Math., 1951, no. 1, pp. 155-179.
[2] Krzyz J. On the maximum modulus of univalent function. Bull. Pol. Acad. Sci. Math., 1955, vol. CI, no. 3, pp. 203-206.
[3] Bieberbach L. Einfuhrung in die konforme Abbildung. Sammlung Goschen, Band 768/786a, 1967.
[4] Hayman W. K. Multivalent functions. Cambridge University Press, 1994.
[5] Pommerenke Ch. Linear-invariante Familien analytischer Funktionen. I. Math. Ann., 1964, vol. 155, pp. 108-154. DOI: 10.1007/BF01344077.
[6] Campbell D. M. Applications and proofs of a uniqueness theorems for linear invariant families of finite order. Rocky Mountain J. Math., 1974, vol. 4, no. 4, pp. 621-634. DOI: 10.1216/RMJ-1974-4-4-621.
[7] Starkov V. V. Regularity theorems for universal linearly invariant families of functions. Serdica Math. J., 1985, vol. 11, no. 3, pp. 299-318. (in Russian).
[8] Schaubroeck L. E. Subordination of planar harmonic functions. Complex Variables, 2000, vol. 41, is. 2, pp. 163-178. DOI: 10.1080/17476930008815245
[9] Sheil-Small T. Constants for planar harmonic mappings. J. Lond. Math. Soc. (2), 1990, vol. 42, pp. 237-248. DOI: 10.1112%2Fjlms%2Fs2-42.2.237.
[10] Sobczak-Knec M., Starkov V. V., Szynal J. Old and new order of linear invariant family of harmonic mappings and the bound for Jacobian. Ann. Univ. Mariae Curie-Sklodowska Sect. A, 2012, vol. LXV, no. 2, pp. 191202. DOI: 10.2478/v10062-011-0024-3
[11] Ganenkova E. G., Starkov V. V. Regularity theorems for harmonic functions. J. Appl. Anal., 2014, vol. 21, is. 1, pp. 25-36. DOI: 10.1515/jaa-2015-0003.
[12] Graf S. Yu. Regularity theorems for Jacobian in linearly and affine invariant families of harmonic mappings. Application of the functional analysis in the approximation theory, 2014, pp. 10-21. (in Russian).
[13] Ganenkova E. G. A theorem of decrease regularity in linearly invariant families of functions. Tr. Petrozavodsk. Gos. Univ. Ser. Mat, 2006, vol. 13, pp. 46-59 (in Russian).
[14] Ganenkova E. G. A theorem on the regularity of decrease in linearly invariant families of functions. Russian Mathematics (Izvestiya VUZ. Mate-matika), 2007, vol. 51, no. 2, pp. 71-74. DOI: 10.3103/S1066369X07020090.
[15] Clunie J., Sheil-Small T. Harmonic univalent functions. Ann. Acad. Sci. Fenn. Math., 1984, vol. 9, pp. 3-25. DOI: 10.5186/aasfm.1984.0905#sthash.GEE2gFK0.dpuf.
[16] Duren P. Harmonic mappings in the plane. Cambridge university press, 2004.
[17] Godula J., Starkov V. V. Linear invariant families. Tr. Petrozavodsk. Gos. Univ. Ser. Mat., 1998, vol. 5, p. 3-96 (in Russian).
Received May 14, 2015.
In revised form, September 3, 2015.
Petrozavodsk State University
33, Lenina st., 185910 Petrozavodsk, Russia
E-mail: [email protected], [email protected]
