CC BY
82Trudy Petrozavodskogo gosudarstvennogo universiteta
Seria “Matematika” Vypusk 5, 1998
UDK 517.54
HIGHER ORDER SCHWARZIAN DERIVATIVES FOR CONVEX UNIVALENT FUNCTIOS
We observe that in contrast to the class S, the extremal functions for the bound of higher order Schwarzian derivatives for the class Sc of convex univalent functions are different. We prove the sharp bound for three first consecutive derivatives.
Let S denote the class of holomorphic and univalent functions in the unit disk D = z : |z| < 1 of the form
denote the Schwarzian derivative for f. Let us denote a3(f) = S(f) and let the higher order Schwarzian derivative be defined inductively (see [5]) as:
In [5] it was proved that the upper bound for |an(f )|, f G S is attained for the Koebe function for each n = 3,4,....
In this note we show that situation is different when we deal with the class of convex univalent functions. Because of linear invariance of the class Sc one can restrict the considerations to an(f)(0) := Sn. We have the following
M. DORFF, J. SZYNAL
f(z) — z + a2z + a3z + ..., z G D,
and Sc C S the class consisting of convex functions. Let
, f ^n+l (f) = (o-n(f)) - (n - 1)^n(f) • f, n > 4
(1)
© M. Dorff, J. Szynal, 1998
Theorem 1. If f G Sc, then the following sharp estimates hold:
|S31 = |6(as - a2)|< 2,
|S41 = 24|a4 — 3a302 + 2a|| < 4,
|S51 = 24|5a5 — 200402 — 9a| + 4803a2 — 24a41 < 12.
The extremal functions (up to rotations) have the form
fn(z) = [ (1 - tn-1)-^dt, n = 3,4, 5, (2)
(3)
respectively.
Proof. From (1) one can easily find
».(/) = ‘y - 6‘‘r + 6( ‘ )3,
»5<f) = ‘ - I0‘f - 6(f)2 +48‘ - 36(f)4.
Note that in [5] there are two misprints in the last formula.
Therefore we have from (3):
S3 = 6(03 - a2),
S4 = 24(04 - 30203 + 20^), (4)
S5 = 24(505 - 200402 - 903 + 48aзa2 - 2402).
We are going to use the connection of the class Sc and functions with positive real part in D, as well as the functions satisfying the Schwarz lemma conditions.
Namely we have
f G S- « 1 + = p(z) = 1+^, z G D, (5)
f 'z 1 - w(z)
where p(z) = 1+ p1z + p2z2 +..., Re{p(z)} > 0, z G D (i.e., p G P, the class of functions with positive real part) and w(z) = c1z + c2z2 + ..., |w(z)| < 1, z G D (i.e., w G 0, the class of Schwarz functions).
From (5) we find
S3 = 2c2
0
and because |c2| < 1 - |c112,
|S3| < 2
which is as well the well-known result of Hummel [1]. The extremal function is
fz dt 1 1 +z f3(z)^0 ~ = 2logT-T.
The functional S4 has a special form of the functional |04 + s0203 + ua3|, u, s G R which was estimated sharply for each s,u G R in [4] and therefore the result follows by taking s = -3, u = 2 in Theorem 1 in [4].
The extremal function is determined by taking w(z) = z3 in (5) which gives (2). Finally in order to get the bound for |S5| which is complicated we transform it to the class 0 of Schwarz functions w(z).
By equating the coefficients in (5) one can find the relations:
02 = Cl,
03 = 3(C2 + 3c1),
04 = 1(c3 + 5C1C2 + 6c + 13),
6
1 , 14 43 2 „ 2 _ 4n
05 = ^0 (c4 + C3C1 + C2Ci + 2c2 + 10c1),
witch transform S5 as given by (4) to a nicer form
S5 = 12(C4 - 2C3C1 + C2C2). (6)
Now we can try to estimate (6) by the use of the Caratheodory inequalities applied to the class 0 as it was done in [4]. However, this leads to very complicated calculations. But one can observe that within the class 0 the functional |c4 - 2c3c1 + c2c2| and |c4 + 2c3c1 + c2c1| have the same upper bound, because if w(z) G 0, then w1(z) = -w(-z) G 0.
On the other hand, comparing the coefficients pk and ck in (5) one gets
P1 = 2c1,
P2 = 2(c2 + c1),
P3 = 2(c3 + 2C1C2 + C3),
P4 = 2(c4 + 2C1C3 + c2 + 3c1c2 + c4),
from which we obtain that
2(C4 + 2C3C1 + C2C1) = P4 - ^P2.
Leutwiller and Schober [3] gave the precise bound for |p4 -1 p2 | < 2, which implies that |c4 + 2c3c1 + c3| = |c4 - 2c3c1 + c3| < 1. This completes the proof. The extremal function is obtained by taking w(z) = z4 in (5).
Note that writing S5 with the coefficients of pk leads to another ”bad” expression
Remark 1. We conjecture that for every n = 6,... the maximal value of |Sn| is attained by the function given by (2).
Remark 2. The application of the general approach to the bound S4 and S5 would lead within the class P to consideration of functions of the form
or 5, which is very difficult to handle because it involves long and tedious calculations.
Remark 3. One can observe that the bound for |<rn(f )| given in [5] follows directly from the formula (1) in [5] and the result of R.Klouth and K.-J.Wirths[2].
[1] Hummel J. A. A variational method for starlike functions / J. A. Hummel // Proc. Amer. Math. Soc. 9(1952). 82-87.
[2] Klouth R. Two new extremal properties of the Koebe-function / R. Klouth,
K.-J. Wirths // Proc. Amer. Math. Soc. 80(1980). No. 4. 594-596.
[3] Leutwiller H. Toeplitz forms and the Grunsky-Nehari inequalities /
H. Leutwiller, G. Schober // Michigan Math. J. 20(1973). 129-135.
[4] Prokhorov D. V. Inverse coefficients for (a, !3)-convex functions /
D. V. Prokhorov, J. Szynal // Ann. Univ. Mariae Curie Sklodowska. Sect.
n
p(z) = A
k=1
Bibliography
A. 35(1981). 125-143.
[5] Schippers E. Distortion theorems for higher order Schwarzian derivatives of univalent functions / E. Schippers // Proc. Amer. Math. Soc. 128(2000). 3241-3249.
Department of Mathematics.
Brigham Young University, Provo, UT 84602, USA E-mail: [email protected] Department of Applied Mathematics
Faculty of Economics, Maria Curie Skiodowska University,
20-031 Lublin, Poland
E-mail: [email protected]
