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30Труды Петрозаводского государственного университета
Серия “Математика” Выпуск 5, 1998
YJIK 517
COEFFICIENT PROBLEM FOR SOME CLASS OF FUNCTIONS NONVANISHING IN THE UNIT DISK
W. Majchrzak, A. Szwankowski
In this paper we determine the bounds of the functional 62 — otbi, a-real, for holomorphic univalent and bounded functions, symmetric with respect to the real axis, nonvanishing in the unit disk. The result generalizes the estimates of b\ and 62 for these functions, obtained by Sladkowska [4].
1. Introduction. Let 0 < |6| < 1, denote the class of all
00 __
functions / (z) = b+ ^2 bnZn, bn = bnj that are holomorphic univalent in
71=1
the unit disk A and satisfy the conditions
/(A)CA, 0£/(A).
B^b) is a normal but not compact family in the topology of locally uniform convergence in A. However, it becomes compact if the function / (z) = 6, z E A, belongs, in addition, to B^b).
The main aim of the present paper is to obtain the bounds of the functional
H(f)=b2-ab1, f G Bg(b), (1)
where a is real.
To solve this problem, we shall use the variational technique developed by Sladkowska [4] for the class B^b).
The problem posed here is connected with a coefficient problem for the class Bo(b) D Bg(b) of functions holomorphic and univalent in A satisfying the conditions / (0) = b, f (A) C A and 0 ^ / (A) (cf. [2], [1], [3]).
© W. Majchrzak, A. Szwankowski, 1998
2. Estimation of H (/). From Schiffer’s equation [4] we get that each extremal function with respect to H (/) belonging to B$(b) satisfies the following differential-functional equation
C2w'2 P(w) ^ 7 / 1\
— —2b2 + abi — bi ( C + 7 ) (2)
where
2w bi (b — w) (1 — bw) V C
P(w) = + Lw3 + Mw2 + Lw + if.
K = -265&i + 6&563 - 464M2 + 2b3h - 6b3b3 - 2b3b\
-a (4b6b2 - ±b%2 - ±b3b2) ,
L = 46% - 126% + 665M2 + 1263M2 - 462&i + 12&263 +6&2&? - 2bbxb2 + 26? - a(—Sb6b2 + 66562 + 126362 +86262 - 266f),
M = -267&i + 6b7b3 - 2b6b!b2 - 6b5bi + 186563 - 18&%&2
—1863^3 + 6636i — 6b2bib2 + 266i — 6bbs — 12bbf + 2b\b2
—a(4b7b2 - 2b6b2 + 12b6b2 - №4b2 - 12b3b2 - 6b2b2
—4:bb2 + 2b\).
The studying of solutions of equation (2) will consist of two cases: 1° K = 0, 2° K / 0.
If b E (0,1) and K — 0, then L / 0 and equation (2) has the following form:
l(w + py«« i=_fcl(i±ii! (3)
2b\ (b — w) (1 — C
or
K-+i,>y-a, = ,4)
26i (6 — w) (1 — bw) C
From the condition K — 0 we have
h = 3 ^ [62&i + 2bbib2 -h+bl+a (2b2b2 - 2bb\ - 262)] • (5)
Comparing the coefficients of the right-hand sides of (3) and (4) with
the analogous coefficients of (2), we obtain, respectively,
&2 = (1 + f)&1 or &2 = (f_1)&1' ^
Since, in the case under consideration, M = 2L, by (5) and (6) we get
, (l-42)(2-a) t (1-6») (2 +a)
*■= 2 (6 + 2) " 1,1 =--------------2 (6 + 2) • <7>
In consequence,
t (1 - 62) (2-a)2 (l-!>2)(2 + a)2
fe~al,1= 4 (5 + 2) “ h~abl= 4 (6 + 2) ' <8)
Equations (3) and (4) can be integrated and their solutions with the initial condition / (0) = b satisfy the relation
t1 + b)__~ ^__________= C+-T2 (9)
1 - b (w-b)(l-bw) ^ C
In the case when b E (0,1) and K / 0, equation (2) can take one of the following forms:
(?K (w + l)2 (w — l)2 w'2 / 1\
^^^---------------J—r = -2b2 + abi — bi ( C + 7 ) (10)
2&iw (b — w) (1 — bw) V C
or
(w + I f (w - c) («■ - I) = _2fe + ati _ bi /. IN (u)
26iw; (6 — w) (1 — bw) V C/
where c E (0,1).
Comparing the coefficients of equation (10) with those of equation (2), we get
b2 — abi =
(l - bi and bi = ^ (a - 2) b (l - 62) (12)
or
b2 — abi = (l + ^ &i and ^ (a + 2) b (l — b2) . (13)
In consequence, we have
b2 - ah = ~ (2 - a)2 b (l - b2) (14)
or
b2 — abi = —- (2 + a)2 b (l — 62) .
(15)
If an extremal function satisfies (11), it must map A onto A \ [—1,0]. The function that transforms A onto this set is of the form
(y/w - y/b^j (l + y/b^/w'j
(V57 + Vb) (l - Vby/rf)
= ±C-
(16)
From (16) and from the fact that the right-hand side of (11) has two distinct roots we obtain
-86(1 - 6) (62 + 26-1) 46(1-6)
62 — abi =----------:-----5----------a-
(1 + b?
where a satisfies the inequality 6-4b- 2 b2
a —
(1 + bf
a —
2-12 b- 6 b2 (1 + bf
1 + 6
> 0
or
-86(1 - 6) (62 + 26 - 1) 46(1 - 6)
b2 — Oib\ — --------------;------—----------- + OL~
(1 + b)a
where a satisfies
262 + 46 -6 (1 + bf
662 + 126 -2 (1 + bf
1 + 6
> 0.
(17)
(18) (19)
Assume now that a G (—00,0). Note that, for such an a, the coefficient 61 of a function maximizing the functional under consideration is positive and, for a minimizing function, 61 is negative.
Hence and by taking account the well-known estimate of the coefficient 61 [4], after comparing relations (8), (14), (15), (17) and (19) we obtain the following results:
Theorem 1. If f £ 6^(6), then
-86 (1 - 6) (62 + 26^1)__q;46 (1 - 6)
(l + 6)a
(1 - 62) (2 - a)2 4 (6 + 2)
1 + 6 ’
(6, a) G Di U D2,
(b,a) G D3
(20)
where
and
Di = {(b,a) D2 = {(b,a) D3 = {(b,a)
0 <b<b*, a < 0} ,
b* < b < 1, a<a0 (b)} .
b* < b < 1, cto(b) < a < 0}
b* =
2^3
- 1, cn0 (b) = (2 - 126 - 6b2) / (1 + b)2 .
Estimate (20) is sharp. The equality for (b, a) E D\ U D2 is realized
by some function described in (16) and, for (6, a) E D3, by some function
defined in (9).
Theorem 2. If f e B^b), then, for b E (0,1)
(2 - a)2 b (1 - b2) , a G [ai (6); 0],
b2 — otbi > <
-8b (1 - b) (b2 + 2b-l) | JJb (1 - b)
a + &r
1 + 6 ’
a E (-oo, ai (6)] ,
(21)
where ai (6) = (2b2 + 46 — 6) / (1 + 6)2.
Estimate (21) is sharp. For a E [ai (6), 0], the minimum is realized by a function satisfying equation (10), while, for a E (—oo,ai (6)], by some function described in (16).
Note next that, for a E (0, +oo), the coefficient b\ of a function maximizing the functional H (/) is negative and it is positive for a minimizing function. Moreover, if / (z) E B^b), then / (—z) E B^ib).
Taking account of these properties and proceeding similarly as in the case included in Theorems 1 and 2, we get the estimates of H (/) for a E [0, +oo) and b E (0,1).
It is also well known that if / E B^b), b E (0,1) then, (—/) E B^b), b E (—1,0). Hence, from Theorems 1 and 2 we can directly derive the estimates of H (/) for b E (—1,0).
Finally, remark that from (20) and (21) we can immediately get the well-known estimates of the coefficients b\ and b2 [4] for the class B^b).
References
[1] Hummel J.A., Scheinberg S. , Zalcman L. A coefficient problem for bounded nonvanishing functions// J. Analyse Math. 1977. V. 31. P. 169-190.
[21 Krzyz J. Coefficient problem for bounded nonvanishinq functions// Ann. Polon. Math. 1968. V. 70. P. 314.
[3] Prokhorov D.V., Szynal J. Coefficient estimates for bounded nonvanishing functions// Bull. Acad. Polon. Sci. Ser. Sci. Math. 1981. V. 29. P. 223-230.
[4] Sladkowska J. On the univalent, bounded, non-vanishing and symmetric functions in the unit disk// Ann. Polon. Math. 1996. V. 64. No. 3. P. 291-299.
Dept, of Special Functions Lodz University 90-238 S. Banacha, Lodz, Poland
