CC BY
28Trudy Petrozavodskogo gosudarstvennogo universiteta
Seria “Matematika” Vypusk 13, 2006
UDK 517.54
ON COMPLEX HARMONIC TYPICALLY-REAL FUNCTIONS WITH A POLE AT THE POINT ZERO
Z. J. JAKUBOWSKI, A. SlBELSKA
Several mathematicians examined classes of meromorphic typically-real functions with a simple pole at the point zero. This article includes results concern class QH of complex harmonic typically-real functions with a pole at the point zero. There are determined the relationships between this class and the class Q'r of meromorphic typically-real funtions with a pole at the origin, which was investigated by S. A. Gelfer [4]. We present also coefficient estimates for functions of a subclass of the class QH and properties of the Hadamard product with fuctions of the class QH.
§ 1. Introduction
Meromorphic typically-real functions have been examined for a long time ( [1], [2]). In several papers authors investigated functions with a pole at the infinity ( e.g. [3]). Other matematicians (e.g. S. A. Gelfer [4], J. Zamorski [5], M. P. Remizowa [6], Z. J. Jakubowski, K. Skalska [7]) examined classes of functions, which are holomorphic typically-real in the ring P := {z e C : 0 < 1 z1 < 1}, with a simple pole at the point z = 0. Our results extend these investigations to classes of complex harmonic typically-real functions in P with a simple pole at z = 0. Our study was inspired, among others, by [8], [9], [7] and [10].
1 Key Words and Phrases: harmonic functions, typically-real functions, meromorphic functions, coefficient estimates, Hadamard product.
Mathematics Subject Classification: 30C45, 30C50, 31A05.
© Z. J. Jakubowski, A. Sibelska, 2006
§ 2. On a class of typically-real functions with a pole at the zero
S. A. Gelfer in paper [4] considered, among others, the class Q'r of functions H of the form
H(z) =-------+ ас + aiz + ..., z Є P, (1)
holomorphic typically-real in P and such that H(z) = G, z Є P.
He showed also that the following properties hold:
(a) H Є Qr ^ ImzlmH(z) > G, z Є P, z = z,
(b) H Є Qr ^ an = a„, n = G, 1, 2,...,
(c) H Є Qr ^ {Re{ 1—2 H(z)} < G Л an = an, n = G, 1, 2,... }.
Moreover, if Sr denotes the class of functions of the form (1), with real coefficients, univalent in P and H(z) = G, z Є P, then S'r С Q'r.
Definition 1. Let QH denote the class of complex functions f harmonic in P and such that
(i) f (z) = F(z) + G(z) F(z) = — 1 + S^Lc anzn, G(z) = S^Lc bnzn
z Є P,
(ii) f (z) = G, z Є P,
(iii) Imzlmf (z) > G, z Є P, z = z.
Directly from the definition we obtain limz^c f (z) = то and each function f Є QH is locally univalent in some neighbourhood of the point z = G. Moreover, if f Є QH and it is locally univalent function in P П R, then f (x), x Є P П R, is an increasing function on the intervals ( —1, G) and (G, 1). Besides, limx^_1+ f (x) > G, limx^c- f (x) = +то,
limK^c+ f (x) = —то, lim^^i- f (x) < G.
If H Є Qr, then H Є QH, of course. It is known that if a function h(z) = z + a2z2 + ... is holomorphic typically-real in the unit disc Д, then H = — h Є Qr ([11]). Consequently, this function H belongs to Q'H.
It is know that h'(x) > G, x Є Д П R. This property does not have to hold for harmonic typically-real functions.
Indeed, for the function /0(z) := z+a(z3+z3), z e A, a e (—2, — 1) we have /mz/m/(z) > 0, z = z e A and /(xa) = 0, where
Let SH denote the class of functions of the form (i), with real coefficients an, , n = 0,1, 2,..., univalent and satisfaying the condition (ii).
Then we have SH C QH.
Remark 1. Let / e QH. From (in) we conclude that /(z) = /(z) if and only if z = z e P. Hence an — = an — bn, n = 0,1, 2,....
Remark 2. Let / be a function of the form (i) and an = an, = bn,
n = 0,1, 2,.... If z e P, then /(z) = /(z).
Indeed, we have
7(f) = F(z) + G(f) = F(f) + G(z) = F(!) + G(z) = /(z), z e P.
The following two theorems determine relationship beetwen the classes Qr and QH.
Theorem 1. If / = F + G e QH, then g = F — G is a typically-real function in P and exist limx^i- g(x), limx^_i+ g(x). If
then g e Qr.
Proof. Let / = F + G e QH. Let us consider the function g = F — G. It is easy to observe that g is holomorphic in P and is of the form g(z) = — 1 + (a0 — b0) + (ai — 6i)z + ..., which is required in the class Qr. According to the remark 1, g is a real function in the set P n R. Moreover, /m/ = /m(F + G) = /m(F — G) = /mg. Thus, on account of the condition (iii) from the definition of the class QH, we have /mz/mg(z) > 0, z e P, z = z. Therefore g is a typically-real function.
Let us observe that g(z) = 0, z e P.
If there existed a z0 e P, z0 = z0, such that g(z0) = 0, we would have F(z0) = G(z0), thus ImF(z0) = /mG(z0). Hence /m/(z0) = 0, which contradicts (iii).
Since g is holomorphic typically-real in P, we have
g'(x) = 0, x e ( — 1,0) U (0,1). Indeed, if there existed a x0 e P n R, such g'(x) = 0, the function g would be the double function in some neighbourhood of the point x0. This contradicts the typically-reality of g.
lim g(x) < 0, lim g(x) > 0,
(2)
The continuity of g' on the intervals ( —1, 0), (0,1) and the fact that limx—0+ g'(x) = +to, limx—0- g'(x) = +to give g'(x) > 0, x e P n R. Furthermore, limx—0+ g(x) = —to, limx—0- g(x) = +to. Hence, from (2) and since g is increasing in (—1, 0) and (0, 1), it follows that does not exist x0 e PnR, such that g(x0) =0. □
The following example shows that the assumption (2) is needed.
Example 1. Let /1 be the function of the form
/i(z) = Fi (z) + G1(z),
Fi(z) =-------+ z, Gi(z) = biz, bi < —1, z e P. (3)
z
Obviously, /i is of the form (i). For z e P n R, we have /i(z) = /i(z).
Moreover, /i(z) = 0, z e P and /m/i(z) = /mz(y^p- + 1 — bi), thus /i
belongs to QH.
It is easy to check that the condition (2) does not hold.
Let us consider the function gi(z) = Fi(z) — Gi(z), z e P. It is
a holomorphic typically-real function. But gi takes the value zero at
some points of the ring P, therefore gi e Qr.
Furthermore, we have
Theorem 2. If F and G are functions holomorphic in P of the form
.. oo oo
F(z) = — z + G(z) = bnz", z e P,
n=0 n=0
respectively, such that g = F — G e Qr and
Re{F(x) + G(x)} = 0, x e P n R, (4)
then / = F + G e QH.
Proof. Obviously, the function /(z) = — i +Y10=0 anzn + Y10=0 bnzz is of the form (i) from the definition of the class QH and it is complex harmonic in P. Since /(z) = g(z)+2ReG(z), z e P, we have /(x) = /(x) for x e P n R. Moreover, /m/(z) = /mg(z), z e P, thus (iii) holds.
Let us observe that /(z) = 0, z e P. Since /m/(z) = /mg(z) = 0 for z e P, z = z, it suffices to show that /(z) = 0 for z e P n R.
If there existed an xo G P fi R, such that f (xo) = 0, we would have
F(xo) + G(xo) = 0, i.e. Re{F(xo) + G(xo)} = 0, Im{F(xo) — G(xo)} = 0.
Hence and from (4) f (xo) =0, xo G P f R. □
Example 2. Let F2(z) = — 1 + 3 + i, G2(z) = | + i. Then g2(z) = F2(z) — G2(z) is a function of the class QJ.
Let f2(z) = F2(z) + G2(z), z G P. We have f2(z) = — 1 + 2, z G P and f2( 1) = 0, therefore f2 G QH. Clearly, the condition (4) does not hold.
§ 3. Coefficient estimates
Applying the known result of S. A. Gelfer
([4], th.1), related to coefficient estimates for functions of the class QJ, in view of remark 1, for functions f G QH satisfying the condition (2), we obtain the following estimates:
—2 < «o — bo < 2,
— 1 < «1 — bi < 3,
—■4 < 4 min (sinn0sin0) < a„ — bn < 4 max (sinn0sin0) < 4, n > 2. o<0<n o<0<n
Moreover, we have Theorem 3. If f = F + G g QH, the condition (2) holds and
|G'(z)| < |F'(z)|, z G P, (5)
then
|b1| < 1, |b2 | < 2, (6)
IT . 2(n — 1)(n — 2)
|bn|<----------- -------- , n = 3, 4,..., (7)
n
9
|«1|< 4, |«21 < 2, (8)
. . 2(n2 — n + 2)
|«n| < -----------------------------, n = 3, 4,- (9)
n
The estimates (6) are sharp. In case the first estimate, extremal functions are e.g. f*(z) = — 1 + z, f2(z) = — 1 — z. In the second estimate, the equality sign occurs e.g. for the functions f3(z) = — 1 + 2z2,
f4(z) = — i — 1 z2,z G P.
Proof. Let f = F + G e QH satisfy (2) and (5). Let us set w(z) := , z e P. Since z = 0 is at least double zero of the func-
tion w, |w(z)| < 1, z e A := {z e C : |z| < 1}, by Schwarz lemma we have |w(z)| < |z|2 < |z|, z e A. From theorem 1, in view of the condition (2), we obtain g := F — G e QJ. Moreover, we have G'(z) = w(z)F'(z). Thus G'(z) = w(z)(g'(z) + G'(z)) and therefore
z2G'(z) = -^f)z2g'(z), z e A. (10)
1 — w(z)
It is know that an analytic function h is said to be subordinate to an analytic function l (written h -< l) if h(z) = 1(w(z)), |z| < 1 for some analytic function w with |w(z)| < |z| ([12], p.190). Moreover, if l and
L are given by the power series ^dnzn, ^^=0 Dnzn, convergent in
some disk |z| < R, R > 0, then we say that l is dominated by L and write l(z) C L(z), if for each integer n > 0, |dn| < Dn ([13], p.82).
We have thus -< , z G A and since <z>(z) := -r^ is a convex
1—w(z) 1 —Z7 ) 1 —z
function in A, writing 1—(Z()z) = ^fc=2 ckzk, we have |ck| < 1, k = 2, 3,...
([13]), II, p. 182). Thus
r^rr c r1-. (11>
1 — w(z) 1 — z
Obviously, z2g'(z) = 1 + 2^=1 n(a„ — 6„)zn+1, z e A, and z2G'(z) = ^^=1 n6„zn+1, z e A.
We mentioned that g e QJ, therefore by the Gelfer’s theorem ([4], th.1), we have |a1 — b1| < 3 and |a„ — bn| < 4, n = 2, 3,.... Let us consider the function ^(z) := 1 + 3z2 + 2 • 4z3 + 3 • 4z4 + • • • = 1 + 3z2 + 4J2^=2 nz”+1, z e A. From the definition of the coefficient domination C ([13], I, p. 82) we conclude that
z2g'(z) C ^(z), z e A. (12)
From (10), (11) and (12) we obtain ^L1 n6„zn+1 C 1—z^(z). Consequently, the estimates (6) and (7) hold. Hence and from the above inequalities for coefficients of functions of the class QJ we have (8) and
(9).
It is easy to check that the functions f *, f*, f*, f4 satisfy the assumptions of theorem 3. Thus estimates (6) are sharp. □
Remark 3. The class QH includes functions, for which the above estimates do not hold, e.g. functions of the form (3). They have the coefficient b1, which does not satisfy the first estimate (6). It is known that in this case the condition (2) does not hold.
Remark 4. Let us observe that substituting (5) for the stronger condition
|G'(z)| < |zF'(z)|, z e P (13)
and applying the same method (the subordination -< and the coefficient domination C), we do not obtain estimates better then (7)-(9).
Let us note that the functions of the form (3) from example 1 do not satisfy the condition (5), and consequently, (13) does not hold for them. For the functions f1 , f2 from theorem 3 the inequality (5) holds, but the conditon |G'(z)| < |z2F'(z)|, z e A, is false. However, the last inequality holds for the functions f*, f| e QH and, in consequence, (5) and (13) are satisfied. The question, how much the mentioned inequalities restrict the class QH is open.
The condition (5) is equivalent to locall univalence of the mapping f, of course ([14]; [15], p. 20).
Remark 5. The inequalities (6) we can obtain immediately from the definition of the function w, considering functions F, G of the form (i), such that the condition (5) holds in P, not necessarily typically-real, and comparing coefficients of the appopriate series.
Indeed, taking w(z) = ^^=2 qnzn we get
OO \ / OO \ oo
Therefore, comparing the corresponding coefficients, we have
92 = 1 • b1, 93 = 2 • b2,
nb„ = q„+1 + q„—1 • 1 • «1 + ... + 92 • (n — 2) • a„—2, n = 3, 4
The function w is a Schwarz function, thus |qn| < 1, n = 2, 3,. p.87), and finally, in view of (14), we obtain the estimates (6).
It appears that if the assumptions of theorem 3 hold, then applaying the formulas (15) and the mentioned coefficient estimates for functions of the class QJ, we get results worse, then (7)-(9).
(14)
,.... (15)
.., ([13], I,
§ 4. The Hadamard product
Let a denote a fixed real number, a = —1, — 1, — *,.... Let us consider the functions
ka (z) := z + Z—:--z2 + ... + z—-----------7T— z" + . . . , z e A, (16)
1 + a 1 + (n — 1)a
and
ha (z) := - + 1^-------- 1 + 2 z + ... + 1 + (-+ i) z" + . . . , z e P' (17)
z 1 + a 1 + 2a 1 + (n + 1)a
It is known ([16]) that the function (16) is the solution of the equation
z
azka(z) + (1 — a)ka(z) = 1-------, z e A, (18)
satisfying the conditions ka(0) = ka(0) — 1 =0 and for a > 0 it can be expressed in the form
. . 1 f 2 zt
ka(z) = — ta ----------------------dt, z e A (19)
Moreover, for a > 0 it is a typically-real function in A.
The mapping (17) (see [7]) is a solution of the equation
azha(z) + (1 + a)ha(z) = —r-——r, z e P, (20)
z(1 — z)
and for a > 0 it is given by the formula
ha(z) = 1 [ ta (1 + 1 ^dt, z e P. (21)
a Jo \zt 1 — zt J
Definition 2. The Hadamard product (f 1 * f2), of two harmonic func-
tions of the form
( 1) OO OO (2) OO OO
f 1 (z) = «z1+E )z”+]O b[)zn, f2(z) = «Z1+]O a[2)z"+53 b[2)z
n=0 n=0 n=0 n=0
is called the function
a(1)a(2) ° °
(f1 * f2)(z) = --1--1 + 5 a[1)a[2)z” + 5 b[!1)b[2)zn, z e P
z [ =0 [ =0
Is worth mentioning that the above definition came into being on the basis of a classical idea of Jacques Salomon Hadamard (1865-1963), concerning the convolution of power series ([17]). Unfortunely, so few articles include a citation of the original Hadamard’s paper on the convolutions.
In [18] (p. 248) it is given that J. S. Hadamard was in contact
with Polish mathematicians (among others with Waclaw Sierpinski) and was a member of the Polish Academy of Sciences. Professor Zygmunt Charzynski many times expressed hopes that we would wait with Hada-mard to celebrate the centenary of his birthday.
We hope that this historical note compensates a bit for ”our faults”.
Definition 3. Let QH := {f e QH : b0 = 0}.
Definition 4. QH (a) := {u : u = f * (ha + ka) : f e QH}, a e R,
a = —1, — 2, — *,..., where ha, ka, are expressed by (16) and (17), respectively.
According to the forms of the functions ka and ha we assume in the definition 3 that b0 = 0 and we leave a coefficient -0, as in the form (i) of the function f.
Remark 6. We have k0(z) = , z e A, h0(z) = 1 + , z e P,
thus f * (h0 + k0) = f, hence QH(0) = QH. The function k0 is univalent in A. However, h0 is neither typically-real nor univalent in P.
Let us observe that for any admissible a, the functions f*,f* (from th.3) belong to the class QH (a). Hence QH = 0 and QH H QH (a) = 0.
Moreover, in view of (16) and (17), assuming that k+0(z) = k—0(z) = z
h+o(z) = h—o(z) = 1 we have QH(+to) = QH(—to) = { —1 + b1z}, where the coresponding function f e QH is of the form
f(z) = F(z) + G(z),F(z) = — 1 + E0O=o-«z”, G(z) = E°=1 bnz[
z e P. If b1 = b1, then u = f * (ha + ka),a = —to, +to belong to QH (+to) = QH (—to) and there are not typically-real.
Directly from definition 4 and theorem 3 we obtain
Corolary 1. Let u e QH (a), a > 0, and let the corresponding function f e QH satisfy the assumption of theorem 3. If u is of the form
u(z) = — 1 + ^0=0 ®«z" + 20=1 b«z", then the estimates
|b1| < 1, N < 1
2(1 + a)
- . 2(n — 1)(n — 2)
|bn| < (1 + (-1^r, n > 3,
n(1 + (n — 1)a)
49
M < , |—2| <
1 + 2a’ 1 21 - 2(1 + 3a) ’
. „ . 2(n2 — n + 2)
| —n | < (1 + ( ), n > 3
n(1 + (n — 1)a)
hold. Extremal functions for |b1| are e.g. f*, f2*. The equality sign for |b2 | occurs e.g. for f3 * (ha + ka) = —1 + 2(1+a) z2,
f4 * (ha + ka) = —1 — 2(1 + a) z2, a > 0.
Moreover, we have
Theorem 1. If u e QH(a), a e R, a = —1, — 2, — 1,..., u = s + r, then there exists f = F + G e QH, such that the system
|azs/(z) + (1 + a)s(z) = F (z); (22)
|^azr/(z) + (1 — a)r(z) = G(z); z e P.
holds. Conversely, for any f = F + G e QH, the solution u = s + r, where s(z) = F(z) * ha(z), r(z) = G(z) * ka(z), z e P, of the system (22) belongs to the class QH (a).
The proof follows by definitions 2, 4, the formulas (18),(20) and remark 4.
From theorem 1 and in view of (19), (21) it follows
Corolary 2. If u e QH (a), a > 0, then there exists f = F + G e QH, such that
1
u(z) = — J ta ^F(zt) + t—2G(zt^ dt, z e P. (23)
0
Conversely, if f e QH, f = F + G, then u of the form (23) belongs to the class QH (a).
We do not know, if or when for finite a the functions of the form (23) are typically-real in P.
The main results of this paper were presented during the XI-th Environmental Mathematically-Informatical Conference in Chelm (30.06.2005-03.07.2005)([19]).
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Chair of Nonlinear Analysis Faculty of Mathematics University of Lodz
ul. S. Banacha 22, 90-238 Lodz, Poland E-mail: zjakub@math.uni.lodz.pl koszeka@math.uni .lodz.pl
