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Владикавказский математический журнал Апрель-июнь, 2006, Том 8, Выпуск 2
YffK 517.98
INEQUALITIES FOR THE SCHWARZIAN DERIVATIVE FOR SUBCLASSES OF CONVEX FUNCTIONS IN THE UNIT DISC1
Y. Polatoglu, M. Caglar, A. Sen
Nehari norm of the Schwarzian derivative of an analytic function is closely related to its univalence. The famous Nehari-Kraus theorem ([3], [4]) and Ahlfors-Weill theorem [1] are of fundamental importance in this direction. For a non-constant meromorphic function f on D the unite disc, the Schwarzian derivative Sf of f by is holomorphic at z0 £ D if and only if f is locally univalent at z0. The aim of this paper is to give sharp estimates of the Nehari norm for the subclasses of convex functions in the unit disc.
1. Introduction
Let 0(z) be an analytic function defined in D = {z : |z| < 1}. If 0(z) satisfies the condition |0(z)| ^ 1 for all z G D, then it is called a Schwarzian function. The class of Schwarzian functions is denoted by fi*.
Next, let fi be the family of functions w(z) regular in D and satisfying the conditions w(0) = 0, |w(z)| < 1 for all z G D.
Further, for arbitrary fixed complex numbers A and B denote by P(A, B) the family of functions
p(z) = 1 + Piz + P2z2 +--------(1)
regular in D and such that p(z) is in P(A, B) if and only if
= 1 + Aw(z) (2)
P(z)=1 + Bw(z) (2)
for some w(z) G fi and every z G D. This class was introduced by W. Janowski [2]. Finally, let C(A, B) denote the family of functions
f (z) = z + 0,2 z2 + ... (3)
regular in D such that f (z) G C(A, B) if and only if
1 + zf(z)= p(z) (4)
for some p(z) G P(A, B) and all z in D.
© 2006 Polatoglu Y., Caglar M., Sen A.
12000 Mathematics Subject Classification: Primary 30C45.
2. Schwarzian Derivative Inequality For the Class C(A, B)
The following lemma which is due to Caratheodory [1] is fundamental for our present aim. Lemma 1. If $ £ Q* then
W(z)| < (5)
for some complex number B such that |B | < 1.
Lemma 2. The class Q* is invariant under rotations.
< It follows easily from the definition of Q* that the function p defined by p(z) : =
e-ia$(eiaz), $(z) £ Q*, z £ D, 0 < a < 2n, satisfies
|p(z)| = |e-ia$(eiaz)| = |e-ia||$(eiaz)| < 1. > Lemma 3. If $(z) is an element of Q*, then
№z)\ < 1 -|$(z)|2 (z)| < 1 - |B|2|z|2•
< Using Lemma 1 and Lemma 2, and after simple calculations,we get
|B| < 1, 0 < a < 2n — eiaBz £ D —
(^Bz)|< - "™< • (6)
On the other hand, since eiaBz £ D, the inequality (6) can be written in the form
|$'(z)| < ^ (1 -|B|2|z|2Mz)| + |$(z)|2 < 1. > (7)
Theorem 1. If f belongs to the class C(A, B) then
( (A-B)\z\2 \A+B\(A-B)\z\2 B =0 If (1-B2\z\2)2 (1+B\z\)2 > B =0' (g)
f \(1 - A)A|z|2, B = 0.
< Since f £ C(A, B), we can write
f "(z) 1 + Aw(z)
From the equality (9) we have the following:
r = p(z)-1 — (' = zp'(z)-(p(z)-1) /
f' - z — I f' I = zr-
f"Y = A-B
f 'J = (1+Bw)2
B(A-B) w2 (1+Bw)^
zw' - w = (A - B )zp' - (A - Bp)(p - 1)
z2 = z2(A - Bp)2 :
w2 (p - 1)2 ^ = z2(A - Bp)2'
(10)
(11) (12)
zw w
z2
2-52
Y. Polatoglu, M. Caglar, A. Sen
1 ( f"\2_ 1(P - 1)2
2 V f V 2 z2
Hence,
Sf = 1
A-B
On the other hand, we have
AB
|p(z) - 1|2 - |z|2|A - p(z)|2 = (1 - B2|z|2) p(z) - 1 AB|z|2
(A - B)2|z|2 (1 - B2|z|2)
1 - B2|z|2 and
(w(z) G fi, 0(z) G fi* ^ w(z) = z0(z) ^ 0'(z) = 1(1 - B2|z|2)|^'(z)| + |^(z)|2 < 1 ^ (1 - B2|z|2)
z2
2
zw' (z)-w(z) |
z2 +
(w(z))2
z2
< 1.
Considering (10)-(16) together yields (8). >
3. Special Cases
For special values of A and B, we obtain the following inequalities. (1) From Lemma 2 and the equality (11) we have
| (A - B)zp'(z) - (p(z) - 1)(A - Bp(z))| < ^/^pp -
In this case we have the following inequalities. (1a) A = 1, B = -1:
|2zp'(z) + 1 - (p(z))2| <
4|z|2
(1 -|z|2)2'
This inequalities was proved by M. S. Robertson ([5]). (1b) A = 1 - 2a (0 < a < 1), B = -1:
12(1 - a)zp'(z) - (p(z) - 1)((1 - 2a) + p(z)) | < ^ -az)|22l)z2|2 -
(1c) A = 1, B = MM - 1 (M> 2):
(2 - zp'(z) - (p(z) - 1) (1 - (1 - p(z)) (1d) A = £, B = (0 < £ < 1):
< (2- M |z|2
" (1 - (MT - 1) |z|2) =
|zp'(z) + 1 - (P(z))2| < (1 -4£|2z|lz2|2)2 -
(2) A = 1, B = -1:
|Sf|< 4|z|2
(1 - |z|2)2
(13)
(A - B)zp - (A - Bp)(p - 1) - 4+4|p - 1|2- (14)
(15)
(16)
This equality was obtained by M. S. Robertson [5]. (3) A = 1 - 2a, B = -1:
| | < 4(1 - a)|z|2 4a(1 - a)|z|2
(1 - |z|2)2 (1 -|z|)2 (4) A = 1, B = MM - 1 (M> 2):
|Sf | <
(2 - MM)2 |z|2 M (2 - M) |z|2
(1 - (MM -1) V)2 (1 + (MM - ^ N)2'
(5) A = ß, B = -ß (0 < ß < 1):
4ß2
|Sf | <
(1 - ß2|z|2)2'
References
1. Duren P. L. Univalent Functions.—NY: Springer-Verlag New York Inc., 1983.—??? p.
2. Janowski W. Some Extremal Problems for Certain Families of Analytic Functions, I // Annales Polinici Math.—1973.—V. 28.—P. 297-326.
3. Kraus W. Uber den Zusammanhang Charakteristiken Eines Einfach Zusammeshangenden Bereiches mit der Kreisabbildung // Mitt. Math. Sem. Giessen.—1932.—V. 21.—P. 1-28.
4. Nehari Z. The Schwarzian Derivative and Schlicht Functions // Bull. Amer. Math. Soc.—1949.—V. 55.— P. 445-551.
5. Robertson M. S. Univalent Functions for which zf '(z) is spirallike // Michigan Math. J.—1969.—V. 16.— P. 97-101.
Received by the editors July 21, 2006. yasar polatoglu
Istanbul, Turkey, Istanbul Kultur University E-mail: [email protected]
Mert Caglar, Ph. D.
Istanbul, Turkey, Istanbul Kultur University E-mail: [email protected]
Arzu Sen
Istanbul, Turkey, Istanbul Kultur University E-mail: [email protected]
