CC BY
8
ISSN 2074-1871 Уфимский математический журнал. Том 9. № 1 (2017). С. 123-136.
growth regularity for the arguments of meromorphic in C \ {0} functions of completely
regular growth
A.YA. KHRYSTIYANYN, O.S. VYSHYNS'KYI
Abstract. We study the asymptotic behaviour for the arguments of meromorphic function in C \ {0} of completely regular growth with respect to a growth function A. We find that that the key role in the description of this behaviour is played by the function Ai (r) = f[ X(t)/tdt.
Keywords: meromorphic function, function of moderate growth, completely regular growth, growth indicator, Fourier coefficients.
Mathematics Subject Classification: 30D15, 30D35
1. Introduction
The theory of entire functions of completely regular growth with respect to the function A close to a power function was created in late 30's of the last century by B. Levin and A. Pfluger. This theory has many applications in various areas of modern complex analysis. A full exposition this theory as well as its applications can be found in [1].
Using the Fourier series method developed by L. A. Rubel and B. A. Taylor [2], A. A. Kondratyuk generalized Levin-Pfluger theory of entire functions of completely regular growth. The growth of a function was measured with respect to an arbitrary non-decreasing continuous function A satisfying the condition A(2r) ^ M\(r) for some M > 0 and all r > 0. This generalization made it possible to describe asymptotic behaviour of entire functions of completely regular growth in Lp-metrics. He also introduced the classes of meromorphic functions of completely regular growth [3], [4], [5]. One can find a thorough description of this theory in [6].
The next possible step in this field is to extend and generalize this theory for multiply connected domains. Many authors studied meromorphic functions in multiply connected domains. One of the recent approaches was proposed in [7], [8], [9]. Using a Nevanlinna type characteristic introduced in these works and the notion of finite A-density [9], the notion of holomorphic function of completely regular growth in the punctured plane C* = C \ {0} was introduced in [10] as well as its growth indicators. This work was concerned mainly with the properties of the growth indicators. Another work in this direction is [11], where using the Fourier series method under general assumptions, the problem of description of the sets of holomorphic functions in C* possessing the property of simultaneous regular growth of the logarithm of modulus and argument was solved.
In the present work we make further studies in this direction. Namely, we study the asymptotic behaviour of the arguments of meromorphic functions of completely regular growth in C*.
A.Ya. Khrystiyanyn, O.S. Vyshyns'kyi, Growth regularity of arguments of meromorphic in C \ {0} functions of completely regular growth. © Khrystiyanyn A.Ya., Vyshyns'kyi O.S. 2017.
2. Definitions, notations and main results
Definition 1 ([9]). A positive nondecreasing continuous unbounded function X in [1, is said to be a growth function.
We say that a growth function is a function of moderate growth if there exists positive M such that X(2r) ^ MX(r) for all r ^ 1. Let X be a function of moderate growth. We denote
r
Mr) := j Xf-dt. (1)
i
Let f be a meromorphic function in C* not vanishing identically. By A* we denote C* without the intervals {z = ra : r ^ 1} if |a| > 1, and {z = ra : 0 ^ r ^ 1} if |a| < 1, where a is a zero or pole of f. Let {aj} be the zeros and {bj} be the poles of f,
№ = № n (z -as )-1 n (z - ).
\aj |=1 \bd |=1
Then [9, Lemma 4.1] there exists m G Z such that for the function F(z) = z-mf (z) and for any given closed path 7 G A* we have
1
This allows us to determine a branch of the logarithm of F( ) in A*. We observe that
m =- [ ) dz,
2™ J\z\ = 1 f(z)
see [9, Lemma 4.1].
We use the following notations for the Fourier coefficients
Ik(t,F) = —[ e-ike logF(teie)d$, t> 0, k G Z, (2)
2tT J 0
2n
ck(t,F) = — f e-ike log |F(te19)d, t> 0, k G Z, (3)
2n J 0
2n
ak(t,F) = — f e-ike argF(tew)dd, t> 0, k G Z. (4)
2TT J 0
Remark 1. Note that
ck(t, F) = 1(1 k(t, F) + Lk(t, F)), ak(t, F) = 1(1 k(t, F) - — (t,F)), for > 0 and G Z.
The Nevanlinna type characteristic T0(r, f) for a function f meromorphic on the annulus {z G C : < |z| < R®}, where 1 < R® ^ was introduced in [7]. Namely,
To( r, f) = mo( r, f) + No( r, f), 1 <r<R®,
where
mo ( r, f) = m( r, f)+m^1, f ^ - 2m(1, f),
2ir
l T l
m(t, î) = 1j log+ | f(t eie )l dd, — <t<Ro,
o
N»(r, /)=/ i t,
n0(t, f) is the counting function of the poles of f in the annulus 1/t ^ |z| ^ t, t ^ 1. This characteristic possesses the properties (see [7], [8], [9]) similar to the properties of the classical Nevanlinna characteristic T(r, f) [12].
Definition 2 ([9]). Let X be a growth function and f be a meromorphic function in C*. We say that f is of finite X-type ifT0( r, f) ^ AX(B r), for some positive constants A, B and for all r, r^ 1.
Definition 3. A meromorphic function f in C* is called a function of the first type completely regular growth (c.r.g.1) if f is of finite X-type and for all k G Z there exist
lim k( , /") =: ck and lim ?( 1 ^P =: c?. r^+œ A(r) r^+œ A(r)
Definition 4. A meromorphic function f in C* is called a function of the second type completely regular growth (c.r.g.2) if f is of finite X-type and for all k G Z there exist
y ck(^ f) + ck(I, f) *
lim -—--=: ck.
X(r) k
We denote by A0'1, A°'2 the classes of meromorphic functions of c.r.g.1 and c.r.g.2 in C* respectively. If f G A0'1 or f G A°'2 we say that f is of completely regular growth (c.r.g.) in C*.
Remark 2. It is obvious that A0'1 C A°'2. However, these classes do not coincide.
For example, take a growth function X and an entire function g of finite X-type, which is not of c.r.g. with respect to X in the entire complex plane C. Without loss of generality we can assume g(0) = 0 and g has no zeros on the unit circle |z| = 1. For z G C* put f(z) = g(z)/g(1/~z). Then f is obviously meromorphic in C*. We have that log lg(1/z)| is bounded as z ^ x>. And because q is not of c.r.g. in C, there exists k G Z such that the limit lim Ck(does not exist
or is infinite (see [3] or [6]). Therefore, there is no finite limit of ^^(-^p as r ^ for that same k. Hence, f G A0'1. On the other hand, ck (r, f) + ck (1, f) = 0 for all k G Z. Thus, f G A°'2.
Definition 5 ([10]). If f is of c.r.g.1 then the functions
h1 ( e, /) = £ eke , h2(e, /) = Y, &zkd
kez kez
are called the growth indicators of f; in case of c.r.g.2 the growth indicator of f is
h(d, /) =
dei ke,
kez
where ck, ck, c*k are given by Definitions 3 and 4. Our main results are the following theorems.
Theorem 1. Let X be a function of moderate growth, X1 be defined by (1), f be a meromorphic in C* function of c.r.g.1 with respect to X, and h1, h2 be the growth indicators of f. Then for all p G [1,
2-k
I \&vgF(rew) + X1(r)til(6) f)\1Jdd } =o(X1(r)), r^
2n
1 21
argF (^e^ -X1(r)h2(0, f)
o(X1(r)), r ^
Theorem 2. Let X be a function of moderate growth, X1 be defined by (1), f be a meromorphic in C* function of c.r.g.2 with respect to X, and h be the growth indicator of f. Then for all p G [1,
2n
2n.
0
arg F(reie) -arg F 1 e^ +X1 (r)h'(d, f)
d
o(X1(r)), t y +m. (7)
Theorem 3. Let X be a function of moderate growth, X1 be defined by (1), f be a meromorphic in C* function of c.r.g.1 with respect to X, and h1, h2 be the growth indicators of f. Then for all p G [1,
2-k
■1 I \logF(rel°)-X(r)h1(6, f) + X1(r)h[(9, f)\Pdd } = o(X1 (r)), r^ (8)
0
2-k
"21.
0
logF(±e*)-X(r)h2(6, f)-X1(r)h2(6, f)
2
o(X1(r)), r ^
Theorem 4. Let X be a function of moderate growth, X1 be defined by (1), f be a meromorphic in C* function of c.r.g.2 with respect to X, and h be the growth indicator of f. Then for all p G [1,
2n
2n.
0
as r —> +00.
log F(reie) +log F e^ - X(r)h(6, f)+X1 (r)h'(0, f)
p 1 p
dd\ =o(X1(r)),
:10
3. Auxiliary results
Let f be a meromorphic in C* function not vanishing identically, F(z) = z-mf(z), where f and m are determined as above. Let {bj} be the poles of f, Uj = aarg bj. For k G Z we denote ([10])
n\ (r
( , ) =
1<\bj \Kr
e-ikaj.
2
nk (r
(r, f)= E
e -ikaj,
> 1,
1 <\bj\<1
and
nk (r
(r, f)= E
e-ika3
1,
1 <\bj Kr
P
v
P
P
where every pole bj is counted according to its multiplicity. In particular, no(r, f) is the counting function that appears in the definition of T0(r, f). We assume nk(1, f) = n2k(1, f) = 0 for all k G Z. Thus,
nk(r, f) = nk(r, D + nk(r, D + nk(T, D, k G Z, r ^ 1, where T = {z : |z| = 1} and nk(T, f) = nk(1, f) = ^ e-ikai.
I bj |=1
Remark 3. Note that |nk(r, /)| ^ n0(r, f), i = 1, 2, |nk(T, /)| ^ n0(T, f), and consequently |nk( r, f)| ^ n0(r, f) for all k G Z and r ^ 1.
Let
r r
Nk (r, f) = j ^^ dt, i=1, 2, Nk (r, f) = j ^^ dt, k G Z, r ^ 1. (11) 11
Since F does not have zeros and poles on the unit circle T, we have that log F is holomorphic in some annular neighbourhood of the unit circle, and therefore admits a Laurent series expansion
log F (Z) = Y®kZk (12)
kez
in that neighbourhood.
To prove our main results, we need following auxiliary lemmas.
Lemma 1. The identities
I k (r ,F ) = akrk + rkJ nl (t, 1/ftl~nl (t, f) dt, k = 0, r ^ 1, (13)
i
lo(r,F) - lo(1, F) = N(r, 1/f) -Nl(r, f), r ^ 1, (14)
hold true.
Lemma 2. The identities
1 ,-k, ,-k [*{ }) -ni (t, f)
. ~ . ; ; / M ' f k\ Ik F ) =akr-k + r~k V V_k+-dt, k = 0, 1, (15)
r I J t-k+1
1
l0 (j,^ - Io(1, F) = N2 (r, - N02(r, /), r ^ 1, (16)
hold true.
Lemma 3. The identities
r
ak(r,F) = -fc i ^^dt +
t _ (17)
+ ak - a-k r k - 1 ^ ^ 1
(n^ -nk (T, , k = 0, r^ 1,
r-
(H _
+ '^s1 (-k(Tj)k>°- '» 1
hold true.
2i 2 ki
r
ak[ 1 ,A =zk i dt +
i * (18)
Lemma 4. The identities
r
Ck r„f) = ik h-iMl dt + +
i
1 \ , nk (T, )) -nk (T f) + Nl{r,J) f) - 2krk-, k = 0, r> 1
:i9)
J1 ,f) = -i k + +
1 \ nk T, y ) - nk(T)
„-,) -mr, n
hold true.
(20)
+ Nk(r,j) -Ni(r> f) - V ^k--, k = r>
For a holomorphic function f, Lemmata 1-4 were proved in [11]. The presence of poles does not complicate the proof essentially.
Lemma 5. The identities
r t
d k( , )
N (r, f) = Ck (r, f) - k2 J dr - Ck (1, f)-
,2
ii
(21)
•t n PM nk (T, 1/f ) -nk (T, f) -1 k • ak(1,F )Iogr----log r, k = 0, r ^ 1,
r t
Nk (r, f) =<*( If yeJ djj dT- Ck i f)+
1 1 (22)
nk (T, - nk(T, f) + ik • ak(l, F)logr-----log r, k = 0, r^ 1.
hold true.
Proof. In view of (12) we have ak + a-k = 2ck(1, F) and ak — a-k = 2iak(1, F) for all k E Z. Note that
Ck (1,F) = Ck (1, f) — ^ cJ 1,1 — + ^ Ck (l, 1 — -p), k E Z. \aj \=i ^ a" \bj \=i ^
Bearing in mind that |w| = 1, one can easily compute ck(t, 1 — ^) for t = 1. Then by using the continuity of the Fourier coefficients we get that
Ck(1, F) = Ck(1,f ) + 1 ^ (T, ^ — nk(T, f)^j .
Replacing ak(t,F) in (17) by its representation (19), we arrive at obtain (21). Similarly, using (18) in (20), one gets (22). □
Lemma 6. Let X be a function of moderate growth, X1 be defined by (1), f be a function of c.r.g. with respect to X, and ck, ck, c*k be given by Definitions 3, 4. Then
(i) if f E , then for every k E Z there exist limits
lim = —,ni lim a4lFl =
r^+œ X (r) k r^+œ X (r)
(ii) if f G A °° , then for every k G Z there exists
y ak (r, F) - ak (1 ,F) *
lim -r r—--= - % fcck .
j-^+to X1(r)
Proof. (i) Let f be a function of c.r.g.1 with respect to X.
If k = 0 then from (14), (16) we obtain a0(r, F) = a0( 1, F) = a0(1, F), and obviously
r ap( r,F) v ap( 1 ,F) lim w v = lim r = 0.
j-^+to X1 (r) r^+TO X1(r)
Since f G A , we have
Ck(r, f) = ckX(r) + o(X(r)), 1, f^ = ckX(r) + o(X(r)), r ^ k G Z.
By (17) we get
ak (r, F) = - kckX1(r) + 0(X1(r)) + ak -a-k - ^fc--1 (n^ T, ^ - nk (T, /)) , k> 0.
Hence,
lim a k (T] . ) = — ikck r^+œ Ai(r) k
for all integer k > 0. Using the properties a-k(r, F) = ak(r, F) and ¿k = ck, we obtain that
lim ^ihll = —%kck, k G Z. r^+TO X (r) k
Similarly, using (18), we have
lim k (^ . ) = ik ck, k E Z.
r^+TO X (r) k
(ii) Let now f be a function of c.r.g.2 with respect to X. By (14) and (16) we obtain that a0(r, F) - a0(£, F) = 0. It follows from (17), (18) that
ak (r, F) - a^ 1, F^ = -ik ckX1 (r) + o(X1 (r)) - ^^ ^ (t, ^ - nk (T, f)^ , k> 0.
Similarly as in case (i) this implies that
r ak(r, F) - ak(l, F) *
lim -----= -i kck
r-^+TO X (r) k
for all integer k. □
Lemma 7. Let f be a meromorphic function in C* with zeros {aj} and poles {bj}, and {ak} be defined by (12). Then for k G Z\{0} and r ^ 1
1 nil r,1) - n\ (r, f) ak(r, F) =- (akrk - o-~kr-k)---+
1 ^ irk ak\ 1 ^ / rk bh
^ \ak + rk ) - 2 ik ^ \bk + rk) :\aj\Kr \ 3 / 1<\bj\<r \ 3 /
/1 \ 1 n2k( r, 7) -n2k (r, f)
*(»4 - ^)+—+
1 XT- 1 \ 1
(23)
+ E ((a^)fc +(0^) - 2ik E )fc
£ Kh'\<1 V WW i ^\<1 V
Proof. It follows from Remark 1 and Lemma 1 that
(24)
afc(r,F)=l-(aikrk -a-kr k) +
1 } ( rk tk-1 \ if 1 \ \ (25)
+ - T^Jr^^/J -nl(t, f)) ^ k = 0, 1.
We denote the integral in (25) by I\. Integrating by parts, we obtain
r
h=- 4/H '•)) ~nl (t+9=
1
1 f (rk tk\ , f 1 ( 1
=-(i+S ■■+¿/6+( ,j)-n. , ,) =
t=1 1
1 ^ (£+3d (* H)--H • k=0,
ik V^ V fcV 7J-Iik
1
Now, using a property of the Stieltjes integral [13], we represent the last integral as a sum and get (23) for > 1. If = 1 then (23) is implied by Remark 1 and (12). Similarly, in view of (15) and Remark 1, we get
ak (1, F) =1(akr-k - 0-krk) +
2
+ (^H - 7+)(nl (t, ^ -n2k (t, f)) dt, k = 0, r> 1. ( )
1
Again, denoting the integral in (26) by I2 and integrating by parts we obtain
J2 i(nl (t, ^ -n2k (t, /))d(^ + ^
2ikj v ky'f
1
nl (^ 7) -nl(r, f) 1 } (tk rk \ ioi 1 \ / I I I \ I I +
ik - 2tkj v^ ^rv ^ 7) - k
1
,y) -nl (t, /))
K + 1 d [ nk \ t,-\ -ni(t, /) , k = 0, r > 1
Using the same property of the Stieltjes integral as before, we have (24) for r > 1. If r = 1 it follows from Remark 1 and (12). □
Remark 4. Note that X(r) = 0(X1(r)) as r ^ Indeed,
er er
X1(er) = J ^dt > j ^dt > X(r), r > 1 1
and, taking into account that X is a function of moderate growth,
2r 2 2r 2 r
x,(2t) = / fit = J M* + / Xfit = J fdt + / Xfit i
112 11
2 r
^ X(t)dt + M J ^dt ^ M'X1(r), r > 1. 11
Lemma 8. Let X be a function of moderate growth, X1 be defined by (1), and f be a holo-morphic function of finite X-type. Then
(3A> 0) (Vr > 1) (Vfc G Z) : |ak(r,F^ + |ak(^,F)| ^ . (27)
r |fc| + 1
Proof. In view of (23), for k = 0
—2^ — a?(r, F)= 2k+ii ^(2f)k — (2r)~k) — 1 (akrk — â-kr~k) —
nk(2 ^ 1) — nj (2r, /) nj( ^ 1) — nj( ^ /) ^ / r^^ j*
¥ik + ik + 2ik\ ^ âk— ^ V? I +
\r-<|aj |<2r 3 r<lbj |<2r 3
l i ST^ âTjk v^ \ l I v-^ ûjk sr^ bjk
+_1_I V V I- — I V__
22 k+17' k 1 Z—t r-k rk I 2i k 1 '
22k+1ik l rk rk I 2ik l rk rk
1<|«j|^2r 1<Ibj |^2r ) \1<K 1<|kj I^
Hence, in view of Remark 3
11 / 1 N |a-k| n0 (2r, I) + ^(2r, f)
k(r, F)| |ak(2r, F)| + - 1 — — ^ +
0 — 2^)
2 k k 2 22 k k 2 k
+ n1 (r, )) +n0(r, f) + n0 (2r, )) - n1 (r, )) + n0(2r, f) -n1(r, f) + (28) fc 2 fc 2 fc nj hr,)) + nj(2r, f ) n0 (r, }) + n0(r, f )
+ V 2U-+ -' k = °- r> 1
The fact that f is of finite X-type implies [9] that
< 1 •<)
, k 6 Z, (29)
| k( , )| +
for some B1 > 0 and for all r > 1. Furthermore
(3B2 > 0) (Vr > 1) : n0(r, f) + n0(r, f) ^ n0(r, f) ^ B2X(r). (30)
First consider a positive integer k. In this case (17), (29), and (30) together with Remark 4 yield
2
lak (2 r ,F )| Kk i 1 Ck (t, f) dt +
i^ - i- (D
1
(31)
K—B1X1(2r) + lak(1, F)| + ^^ K ClXl(r)
k + 1 1 1V ' 1 ' 2k for some C1 > 0 and for all r > 1. Now using (30), (31) in (28), we obtain that there exists C2 > 0 such that
C2X1(r)
lak(r,F)| K
(32)
| k| + 1
for all r > 1 and k > 0. As we have noticed in the proof of Lemma 6, a0(r, F) is constant. Using this fact and the property a-k(r,F) = ak(r,F) , we obtain that (32) holds for all integer k, possibly with a constant different from C2. Similarly, in view of (24),
ak (£ ,F) /1 akw
(H
2k+1i
1 : (ak(2 r)-k - a—k(2r)fc) - t1 (akr-k - a—^rk) +
+
nl (2r, 1) -nl (2 r, f) nl( r, 1) -nl (r, f)
2ki k
k
1
+ 2ik
E )k - E )k I +
KK'\< 1
K\bj\< -
\ U1
+
1
22k+1i k
E
(a3r)k
E
27 K\b,\<1y '
(bj r)k
Therefore, 1
1 2k
y —
^ (ajr)k
\<1
y —
^ (bj r)k
k = 0, r> 1.
K\bj\<1
1
Iak(-,F)| K^
r 2 K
+ 1 f, M KI + no [2r, f
+ 2 V 22k) rk + 2kk
(2r, 1) + n0(2r, f)
+
(£- )|+2 (1 - i)
(r, 7) - n0(r, f) + n0 (2r, )) - nl (r, )) + nl(2r, f) - n0(r, /) , (33) + 2 k +
2 k
nl (2r, 1) + n0(2r, f) nl (r, 1) + n2(r, f)
+----'-h-TT,--1--v "y ,-, k = 0, r> 1.
2k+1k 2k ' ^ '
If k is a positive integer then from (18), using (29), (30), and Remark 4 we obtain
2
+ 2L((2r)-* _ 1K(D
K-jJkiB1X1(2r) + Iak(1, F)| + B2kr1 K C^r)
1
1
1
1
for some C3 > 0 and for all r > 1. Again, using (30), the previous inequality and the property a-k(1, F) = ak(1, F) in (33), we get
^ ( PF)
< C4Ai(r)
|fc| + 1
for all r > 1 and k G Z. □
4. Connection between the indicators of completely regularly growing
MEROMORPHIC function IN C*
Let f be a meromorphic in C* function of c.r.g. with respect to X. It does not matter what type of c.r.g. the function f actually is of, it is assumed to be of finite X-type in C* anyway. This implies that the growth indicators, h1 and h2 in the case of c.r.g.1 or h in the case of c.r.g.2, belong to L2[0, 2tt\. For holomorphic functions f this was proved in [10]. The assumption that f is meromorphic in C* give rise to no significant changes in the proof of that result.
We note that by the definition, the indicators h1, h2, and h are the growth indicators of log |f| with respect to X. However, Lemmata 6 and 8 allow us to introduce the notion of the growth indicators of argF with respect to X1. We denote
1 r ak(^ F) 2 r ak(1 ,F) , -o,1
ak = lim A . . , ak = lim A . . if t G AA , k X1 (r) k X1(r)
* v ak( r,F ) — ak ( 1 ,F ) 2 ak = Iim----- if / e A A .
k r-^+œ Ai(r) J H
Using Lemma 8, we obtain
A I 2 I Îk I ^
1,1-7, Iak I < TTi-7, lakI < "¡rn— -,
|k| + 1' 1 k | k| + 1' 1 k |k | + 1
141 141 |akI k e Z. (34)
i k
Thus, by the Riesz-Fischer Theorem [14, p. 79] there exist unique functions g1(9,f ) = ake
kez
g2(Q, f)=Yl a2keik0, or g(6, f) = ^ a*keikd, which belong to L2[0, 2n}. We call these functions
kez kez
the growth indicators of argF with respect to A1. By Lemma 6 we have
ak = —ikc\, o? = ikck, a*k = — ikc*k, k e Z. (35)
Let
a2(r) = jT f/'Airl^, r> 1
and A A be the class of holomorphic functions of c.r.g.1 in C*. Using the inverse formulas for the Fourier coefficients ck(r, f) + ck (f) from [15] it was proved in [10, Theorem 3] that if f e A A then the sum of the growth indicators h1 + h2 is w-trigonometrically convex [1] for u e [k, p], where
„2 M_______ A(0
k = lim inf , p = lim sup
r^+œ A2(rY r^+œ A2(r)
The presence of the poles of does not complicate the proof essentially.
It turns out that for our purpose we need both h1 and h2 to possess the property of w-trigonometrical convexity separately in the case f e A0,1. And if f e A^2 we need this property for h as well. Analyzing the proof of Theorem 3 from [10], we conclude that the key role in showing w-trigonometrical convexity is played by the inverse formulae for the Fourier coefficients. While such formulae from [15] for the sum ck (r, f) + ck (1, f ) can be obtained in the case f e A^2, we need formulae (21) and (22) for the case f e A°,:L. So, once we have them, in
the similar way as in [10] by taking a meromorphic in C* function f either of c.r.g.1 or c.r.g.2, and using the inverse formulae for the Fourier coefficients ck(r, f), ck, f) given by Lemma 5, one can prove that the indicators h\, h2, or h are ш-trigonometrically convex for ш E [к,р] in the appropriate case.
Each trigonometrically convex function is differentiable almost everywhere [1]. It follows from (35) that the Fourier coefficients of the functions g\ and —h[, g2 and h'2, g and —h' coincide and
9l(6, f) = —h\(e, f), д2(в, f) = h'2(e, f), g(6, f) = — h(9, f) (36)
almost everywhere.
5. Proof of the main results
Proof of Theorem 1. Let gg2 be the functions defined in the previous section. The Fourier
f\ — ak(r,F) „1 gk(±,F) „2
(r)
k Z.
coefficients of ^ —дг(в, Д ше ¡(Г) ] —д2{в, /) are —a\ and ^r(^ -a2k respectively,
By the Parseval's identity [14] we have
2ж
1
2tÎ
arg F (re )
X, (r)
— g i(o, f)
£
Kkez
ak (r, F)
X (r)
— ak
2 ^ 2
r > 1,
(37)
and
1
2/n
2ж
arg F (Г e* )
r - 92(0 J)
X (r)
d0
{E
Kkez
ak (Г ,F )
X (r)
— aL
}2
r > 1.
(38)
It follows from the convergence of the series -¡^ that for any e > 0 there exists k0 G N such
that
£
k=l
1 e2
<
(39)
( k + 1)2 32A2 '
k=k0+1 y '
where A is the constant from (27). Applying Minkowski inequality [16], and using (27), (39), we obtain
L
|k|>ko
k( , F)
X (r)
— a.
k( , F)
€2 i £
X (r)
A2
| k|> ko
+ 1)2
+ iE I ak I
| k|> ko
al\2\ €
2^2 A
IkS-i (I*I + !)2}
(40)
< 2-
Similarly,
ak (Г ,F )
X (r)
— ab
By Lemma 6 there exist r0 > 1 such that
ak (r, F)
X( )
— ak
<
ak (Г ,F )
4kn
X( )
< 2-
n2 — ak
<
4 ko
for all r > r0 and Щ € k0.
(41)
(42)
2
2
2
2
2
2
2
2
2
2
2
2
Thus, for any e > 0 and for r > ro, in view of (37), (40), and (42), we get
2тг
1
2К
arg F (r e )
Ai (r)
-g i(o, f)
de
|fc|^fco
ak (r, F)
Ai (r)
— ak
+
+ { T,
|k|>ko
k( , F)
A (r)
a
I—;- fc fc fc fc
«V2^ 4M0 + 2< 2 + 2
Similarly, from (38), (41), and (42) it follows that for any e > 0 and for r > r0
1
2K
2ж
argF (\e* )
r - д2(о, Л
A (r)
d
< £.
Together with (36) this proves (5), (6) for p = 2. Furthermore, in view of (27), (34), and (35), by applying Hausdorff-Young Theorem [14] we obtain (5), (6) for p > 2. Taking into account the monotonicity of the pth integral means, we establish that relations (5), (6) hold for all p G [1, +ro). □
Proof of Theorem 2. One can prove this theorem by considering the Fourier coefficients of argF(re (re ) _ g(Q, y), using appropriate part of Lemma 6, and arguing just as in the
proof of Theorem 1. □
Proof of Theorems 3, 4. Relations (8), (9), (10) are implied immediately by Theorem 2 in [10] and Theorems 1, 2 by Minkowski inequality [16] and Remark 4. □
2
2
2
2
2
2
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Andriy Yaroslavovych Khrystiyanyn, Ivan Franko National University of Lviv, 1 Universytetska st., 79000, Lviv, Ukraine E-mail: [email protected]
Oleg Stepanovych Vyshyns'kyi,
Ivan Franko National University of Lviv
1 Universytetska st.
79000, Lviv, Ukraine
E-mail: [email protected]
