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19Труды Петрозаводского государственного университета
Серия “Математика” Выпуск 3, 1996
YAK 517.54
ON SOME CLASS OF FUNCTIONS WITH AN INTEGRAL REPRESENTATION INVOLVING COMPLEX MEASURES
Dorota Bartnik
In this paper we obtain some properties of functions extremal with respect to Frechet-differentiable functionals defined on V'a (see defifnition 1) and, in consequence, estimates of the functional Re{elXp(z)}, 0 / 2 G if, A G [—7r,7r), p G V'a.
§ 1. Introduction
Let V denote the well-known class of all functions of the form
p(z) = 1 + a\z + ... + CLkZk + • • • (1)
holomorphic and satisfying the condition Rep(z) > 0 in the disc K =
{z G C : |z| < 1}. As is known (e.g. [5], p. 4), a function p G V if and
only if
2n
p(z) = I P(e~it,z)dfi(t), z&K, (2)
0
where
1 A- F 7
P(z,z) = 1 _£z, \e\ = l,zeK, (3)
1991 Mathematics Subject Classification: 30 C 45
Key words and phrases: Caratheodory functions, functions generated by complex functions with bounded variation, universal linearly invariant family, estimates of functionals.
© Dorota Bartnik, 1996
fi G M, M = {/i:/iisa nondecreasing real function defined on the interval
[0,2tt], and /02?r dfi(t) = 1}.
V. Starkov ([9], see also [6]), introduced the class U'a, a > 1 of functions of the form
f'(z) = exp
2n
’ J Log (1
tz)dfi(t)
* G K,
(4)
where /i G /«. denotes the family of all complex functions /i with bounded variation, satisfying the condition
Z7T
/p2n
— 1 + J \dfi(t)\ < a.
(5)
The classes Ura appear in a natural way in the question of approximating the derivatives of functions of a universal linearly invariant family of order a by powers of the derivatives of convex functions (see [4], [9]).
Of course, if in (4) fi G M, we obtain the class Sc of convex univalent functions. Ii is a family of nondecreasing real functions such that
So’Mt)< i.
Definition 1 Let V'a, a > 1, denote the class of functions given by formula (2) where /i are elements of the class Ia.
Evidently V C V'i. The class V'a was introduced and its basic properties were studied in [3] (see also [1], [2]). In particular, we obtain
Theorem 1. The set of functions p of form (2), generated by piecewise constant functions n G Ia, is dense in V'a.
It has also been shown that the class V'a is compact in the topology of almost uniform convergence in if, convex and connected.
Theorem 2. ([3]) Let {p}k, k = 0,1,..., denote the k-th coefficient of the function p. Ifp G V'a, a > 1, then the set Vu of values of the functional H(p) = {p}k, k = 1,2,..., is the closed disc with centre at the point 0 and with radius 2a. If a > 1, then the set Vo of values of the coefficient {p}o is the ellipse
(Re A - |) (ImA)
+
a2 — 1
< 1.
(6)
If a = 1, then Vo = [0,1].
Theorem 3. ([3]) Ifp G V'a, a > 1, then
(?)
Estimate (7) is sharp.
In the proof of Theorem 2, use was made of the definition and the elementary properties of the class V'a and, in particular, of conditions (2), (3), (5). To prove Theorem 3, we use Theorem 1, condition (5) as well as certain classical inequalities.
There arises a natural question concerning the possibility of obtaining a general characterization of functions extremal with respect to a rather wide class of functionals defined on the class V'a.
§ 2. General properties of extremal functions
Let p G V'a , ao > 1. Of course, the function /i corresponding to p belongs to Iao. From (5) it follows that there can exist an a, 1 < a < ao, such that ji G Ia. The best characterization of the function ji and, consequently, of p is given by the number a* for which
Definition 2 Let p G V'ao. The number a* < «o such that p G V'a^ and p V,a^_£ for any £> 0 is called a degree of the function p. The degree of the function p is denoted by degp.
Since V'ai C V'a2 if 1 < < «2, we have
Property 1 If degp = a*, a* > 1, then p G for a > a* and p for 1 < a < a*. If degp = 1, then p G V'a for a > 1.
We shall prove
Theorem 4. Let F be a Frechet-differentiable functional defined on V'ao, Lp its differential at the point p, and po the extremal function for the problem
2n
2tt
J dji{t) - 1 + J \dji{t)\ = a*.
o
o
max Re{F(p(n))}, 1 < «o < oo, n = 0,1,2,_ (8)
P^’c n
If there exists k > n such that L (n) (zk n) / 0, then degpo = «o-
Po
Proof. From Definition 2 it is evident that degpo < olq. Assume that degpo = ol and a < ao- Let us consider the function
Ph(z) = Po(z) + khzk + k\h\ = Po(^) + fe|ft|(l + e*arg
where k G N and ft / 0 is a sufficiently small complex number. Note that
1 + azk, |cr| = 1, is a function of the class V for any k G N. So, we have
2n 2n
Ph(z) = j P(e~tt,z)d/j,0(t) + k\h\ j P(e~%t, z)dv(t),
A^o G /«, G M, and P is defined by formula (3). Consequently,
27r
Phiz) = j P(e~tt,z)d(iM)(t)+ k\h\i/(t)).
We shall demonstrate that the function pn G for |ft| < Indeed, condition (5) is satisfied because
2n
2n
J d{p0{t) + k\h\v(t)) - 1 + J \d{p0(t) + k\h\u(t))\
o o
2n 2n 2n
J dfj,o(t) + k\h\ J du(t) — 1 + J \d(iJLo(t) + k\h\v(t))\
0 0 0
2n 2n
J dfio(t) — 1 + k\h\ + J \djjQ(£)| + k\h\
o o
i, i &Q — a
< a + 2fc ft < a + 2k——— = a0.
2k
The derivative of order n of the function pu is expressed by the formula P^\z) = p{0n)(z) + hk2(k - 1)... (k - n + 1 )zk~n, z e K.
Calculating the value of the functional F at the point p^, we get
Fip^) = F(p^] + hk2{k-l)...{k-n + l)zk~n)
= F(p(0n)) + LpM (hk2(k - 1)... (k - n + 1 )zk~n) + o(\h\)
= F(pj,n)) + hk2(k — l)...(k — n + 1 )L („,(zk~n) + o(|/i|)
where lim^i^o ° |^ = 0. By the assumption, the function p0 is extremal for problem (8), therefore, for all k > n,
L {n)(zk-n) = 0 Po K J
must take place, which contradicts the assumption. Thereby, the theorem has been proved.
To estimate some functional defined on the family V'a, we shall make use of the method described by V. Starkov in paper [6] (compare also [7]—[9]).
Let Ga be the class of functions of the form
2n
<p(z) = J g(z,t)dfi(t), z£K,
0
where n E /a, g(z,t) is a fixed function holomorphic with respect to z in the disc K and 27r-periodical and of the class C' with respect to t. The family Ga is compact in the topology of almost uniform convergence in the disc K.
Let F be a Frechet-differentiable functional defined on the set B described above. Consider the problem
max Re{F((/))} (9)
cpeGa
and denote by ipo(z) = g(z,t)d/Jo(t) an extremal function for (9) (not necessarily the only one).
Denote by Ia(n) a subset of the family Ia of piecewise constant functions which have not more than n points of discontinuity. Let us also define a suitable subset of the family Ga, Ga(n,/jbo) = {</? E Ga : viz) = fo” g(z,t)dfj.(t), fi e Ia(n), /02?r dfi(t) = dfi0(t)}. The class
Ga(n,jLLo) is compact, too.
Let us consider the following problem:
max Re{F(^)} (10)
^GGa(n,p0)
and let (fn(z) = /Q27rg(z,t)dfjLn(t) denote an extremal function for (10). From the sequence (ipn) one can choose a subsequence almost uniformly convergent in K to the function ipG Ga, with that is an extremal function for problem (9). In order to get any information about the extremal functions in the full class Ga, we may first consider analogous problems in the classes Ga(n,/jLo).
Let, for a fixed n E N, problem (10) be given and let tj, j = 1,..., &, k < n, be points of discontinuity of the function /in. Denote arg dnn{tj) =
0,, If k > 2 and if there exist at least two different values of Oj, then the points L(pn[g(zJtj)\ lie on the circle with centre cn and radius sn. L(fn denotes here a differential of the functional F at the point ipn. If, moreover, sn > 0, then
J \L<pn[9(z,ti)\ - cn\2 = \LVn[g(z,tj)] - cn\2 for i,j = 1,... ,k,
j {\LVn\g{z,t)\-cn\2)'t _ =0 for j = l,...,k. ^1:L)
^ t — tj
In the above case, the equalities
{LVn[g(z,tj)\-cn)e%@^ = ±sn, j = (12)
are true, too, with that the sign preceding sn is the same for all j’s.
As sn = 0, we get
LVn [d(z, ti)\ = LVn [g(z, tj)] for i,j = 1,... ,k,
Re{et&LVn[g't(z,tj)]} = 0 for j = l,...,k.
(13)
Whereas if at all points tj of discontinuity of the function fin we have arg d/jbn(tj) = 0, then
Re {e*eLVn [5^(2;, ij)]} = 0 for j = l,...,k,
Re{et@(LVn[g{z,ti)] - LVn[g(z,tj)\)} = 0 for i,j = l,...,k,
(14)
with that the first of equalities (14) is true at each point tj in all the cases under consideration.
We shall give a simple application of Theorem 4, exemplified by the following problem.
Let p G V'aJ a > 1, and let {p}&, k = 0,1,..denote, as before, the
k-th coefficient of an expansion of the function p in the power series with
centre at the point z — 0.
Consider the problem
max|{p}fc| for & = 1,2,----------- (15)
V^P'cx
Lemma 1. If po(z) = /027r P(e~lt, z)d/j,o(t), z E K, and P defined by
formula (3) is an extremal function for problem (15), then d/io(t) = 1
and Jg27r \dfjLo(t)\ = a.
Proof. Since, for p e V'a, 0 G R, p(el@ z) e V'a, therefore problem (15) is equivalent to
maxRe{p}fc for A: = 1,2,__________ (16)
peV^
Consider the function
2tt
Pa(z) = j P{e~lt,z)dn^{t), 0 < A < 1, z £ K,
0
where fiA(t) = fj,0(t) +
/, (t\ = l 0 for e
' \ —m for t e [to, 27r],
and
2n
rn = j diio(t) — 1. o
Of course, pa E V'a and
2n
Pa(z) = J P(e~tt,z)dii0(t) - mAP(e~tto,z),
0
0 < A < 1. If po is extremal for problem (16), then
Consequently,
Re(-2me“ia°) < 0, k = 1,2,.. The above inequality is true for m = 0 only. Then
2n
j dno(t) = 1. (17)
0
Let a > 1. From (5) and (17) it follows that \dfio(t)\ < a. If
2n
fo
and
d/jLo(t)\ < a, then there would exist /3 < a such that fQ w \d/jLo(t)\ = /3 JQ27r d/io(t) —1 + /Q27r |d/io(^)| = /3, which would mean that degpo < &•
This contradicts Theorem 4, so, indeed,
2n
J \dfio(t)| = a.
o
If a = 1, then from (5) we get at once JQ27r \d/jLo(t)\ = 1, which concludes the proof.
§ 3. Estimation of the functional Re {e p(z)},p 6 7^
At present, we shall make use of the method described in the second section of this paper. We shall prove
Theorem 5. Ifp E V'a, a > 1, X e [—7r,7r), then the estimate
Re [e*Ap(z)] < ^ 2, [(1 + r2) cos A + 2ra + y/x] , (18)
r )
x = 'sj[a(l + r2) cos A + 2r]2 + (1 + r2)2(a2 — 1) sin2 A,
|^r| = r, z e K, takes place. The equality in (18) is obtained for functions of form (2) where ji is a piecewise constant function with one jump point to defined by the conditions
(1 — r2) sin A
sin to =
(1 + r2)a + 2r cos A ’
___________________(a2 — 1)(1 — r2)2________________________
[(1 + r2) cos A + 2rot + y/x][(l + r2)a + 2r cos A] (1 + r2) cos A + 2ra (1 + r2)a + 2r cos A
cos to =
+
(19)
with that
27r
/
d/i(i) = -
1 +
a2(l + r2) cos A + 2ra . (1 + r2)(a2 — 1) sinA
yfx
yfx
(20)
Proof. Let us determine
max Re [elXp(z)] (21)
peva
where z is any fixed point of the disc K, and A E [—7r, ir). Since if p E V'a, then p(e10z) E V'a for 0eR, thus one may adopt z = \z\ = r, 0 < r < 1. Consequently, one should examine
Z7T
J eiAP(e“i4,r)^(i)]. (22)
max Re
M £ IOL
0
As can be seen, this is a problem of type (9) for F[<p(z)] = </?(kl) where (f{z) = /Q27r g(z,t)dfjb(t), n E /«, while g(z,t) = elXP{e~lt, z). The extremal function will be denoted by ipo(z) = g(z,t)d/io(t).
First, let us deal with the problem of type (10). For any fixed n E N, determine
j g(r,t)dfi{t)
_0
with respect to fi E Ia(n) such that d/i{t) = JQ27r d/io(t). The extremal function in (21) will be denoted by ipn(z) = g(z,t)d/in(t).
Let us still notice that L(pn (•) = F(-) because the functional F is linear. It is known from the definition of the class Ia(n) that the function fin has not more than n points of discontinuity. Suppose that it has at
max
(23)
least two such points. Then one of conditions (11), (13) or (14) must be satisfied. In the problem under consideration
3 = 1,
2 < k <
So, it is evident that conditions (13) can not be satisfied. Assume that conditions (11) take place. Then all points L(pn[g(zJtj)\ lie on the circle with centre at some point cn and with radius sn > 0. It is easy to see that cn = and sn =
Consider Re Jg27r g(r,t)dfjbn(t) . The function fin belongs to the class Ia{n) of piecewise constant functions, and
k k g{r,t)dnn{t) = 'Y^jg{r,tj)aj = 'Y^LVn[g(z,tj)\aj,
Z7T
/'
0
(24)
do = GU e
3 — I 3\ — V J
Moreover, from condition (12) we have
3 = 1 3 = 1
= dfin(tj), j = 1,... ,k, l<k<
1 4- r2 " 2r k
’S^2iLVn[g(z,tj)\aj = elX _r2 + i _r2 ^2 l“il-
(25)
3 = 1
3 = 1
3 = 1
Denote Y^kj=i aj ~ A. From (5) we get that i \aj\ < ol — \A — \\. Thus
Re
Z7T
J g(r, t)dfxn{t) _0
< w
where
1 —I— T*2 2 7*
w = T^Re (eiXA) + ------------------------(a -\A-l\
1 — r2 1 — r2
It remains to determine the greatest value of the expression W.
Note that A = {pn}o where pn is the function of the class V'a corresponding to fin in the integral representation. In virtue of Theorem 2 A is a point of the domain Vo described by inequality (6). So, we may assume that A lies on some ellipse contained in Vo, with foci at the points 0 and
1. Consequently,
A _ 1 + [*?(« -!) + !] cos V’ , . y/rfia - l)2 + 2 rj{a - 1) sinV>
A — o •"1 o ’
V>e[0,2Tr], 77 G [0,1].
Hence
W =
1 -hr2 /1 + [rj(a — 1) + 1] cos^
cos A
^r]2(a — l)2 + 2r](a — 1) simp
sin A
+
+
2 r
1 — r2
a —
[ry(a — 1) + 1] cos?/’ — 1
. i/?72(a — l)2 + 2r](a — 1) sin^
2
{(1 + r2) cos A + 2r[2a — r](a — 1) — 1]
2(1 — r2)
+ [(1 + r2)[r](a — 1) + 1] cos A + 2r] cos^
— (1 + r2)^rf(a — l)2 + 2r](a — 1) sin A sin
VF is a function of the variables r] and ^ defined on the set [0,1] x [0, 2ir]. Using an elementary method, one can demonstrate that its greatest value is equal to
W( l,^o) =
{(1 + r2) cos A + 2ra
2(1 — r2)
+ yj[a(l + r2) cos A + 2r]2 + (1 + r2)2(a2 — 1) sin2 A j
where ifio is defined by conditions
— (1 + r2)Va2 — 1 sin A
(27)
sin 'IpQ = cos ^0 =
^[a( 1 + r2) cos A + 2r]2 + (1 + r2)2(a2 — 1) sin2 A a( 1 + r2) cos A + 2r ^/[a(l + r2) cos A + 2r]2 + (1 + r2)2(a2 — 1) sin2 A
The reasoning carried out implies that the maximum of W is attained at the boundary point of ellipse (6) from Theorem 2. From the proof of this theorem included in paper [3], it follows that to the boundary points
of the set Vo there correspond functions fin E Ia such that arg dfin(t) = const. = 0 for all t for which dfin(t) / 0. Hence we infer that, for the extremal functions (fnj the functions fin cannot have two distinct values
e3.
However, if all 0j = 0 = const., then the jump points of the function satisfy conditions (14). The first of these equations is an algebraic equation of the second degree with respect to elt. The application of the other equation from (14) and of Rolle’s theorem lessens the number of the roots twice. So, in consequence, we have that, in all the cases, the function nn{t) corresponding to the extremal function Lpn has only one jump to. Note that L(fn [g(z, to)] lies on the circle described earlier with centre cn and with radius sn and the estimate obtained is true in this case, too. Moreover dfin(to) = A0 where A0 is defined by formula (26) in which r] = 1, ^ = t/jq; hence condition (20) follows. It can easily be verified that the point to satisfies conditions (19).
As has been observed earlier, from the sequence ((pn) of extremal functions one may choose a subsequence almost uniformly convergent in K to the extremal function in problem (22). Thereby, estimate (18) is true. Of course, to the functions (pn there correspond functions pn = e~lX(pn of the class V'a.
Remark 1 We were first seeking for the forms of the extremal functions in the subclasses of the family V'a, generated by piecewise constant functions ii. Of course, they are extremal in the full class V'a, as well.
Remark 2 It turns out that Theorem 5 can be proved also in the way which was applied in paper [3].
Since the class V'aJ a > 1, is convex and, for each zi E K, there exists a function pi E V'a such that pi(zi) = 0, with that Re pi(zi) >
(see (7)), therefore Theorem 5 implies COROLARY 1. L
et z / 0 be a fixed point of the disc K. Then the set Aa, a > 1, of values of the functional H(p) = p(z), p E V'a, is a set of the plane (w) whose boundary is given by the equation
A\
W =
+
2(1 -r2)
[(1 + r2) cos A + 2ra
y![a(l + r2) cos A + 2r]2 + (1 + r2)2(a2 — 1) sin2 A
(28)
\z\ = r, A G [—7r,7r], with that 0 E Ax-
Putting A = 0 and A = —7r in (18), we evidently obtain Theorem 3, whereas substituting A = if, we have
COROLARY 2. I
f p G V'a, a > 1, then the sharp estimate
2 + a + \/(l + r2)2a2 — (1 — r2)2 2(1 -r2)
2 + a + i/(l + r2)2a2 — (1 — r2)2
< Im p(z) (29)
2(1-r2) 5 1 1
holds.
Whereas Corollary 1 and the form of equation (28) imply COROLARY 3. I
n the class V'aJ a > 1, the sharp estimate
1^)1 - Y^r ' \z\ = r, (30)
takes place.
Remark 3 Passing to the limit with a —>• 1+ in the estimate from above of (7) as well as in (27) and (28), we obtain estimates of the corresponding functionals in the class the bounds of these functionals being, as can be seen, analogous to those in the class V. For obvious reasons, we have lost the estimates of Re p{z) and \p{z) \ from below. Of course, the formal justifications can be carried out by using, for instance, the fact that for a = 1, the set Vo = [0,1].
The present paper has been written within the framework of Professor Z. Jakubowski’s seminar conducted in the University of Lodz.
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