Inequalities for the Schwarzian derivative for subclasses of convex functions in the unit disc
Автор: Polatoglu Yasar, Caglar Mert, Sen Arzu
Журнал: Владикавказский математический журнал @vmj-ru
Статья в выпуске: 2 т.8, 2006 года.
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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 S_f of f by is holomorphic at z_0\in D if and only if f is locally univalent at z_0. The aim of this paper is to give sharp estimates of the Nehari norm for the subclasses of convex functions in the unit disc.
Короткий адрес: https://sciup.org/14318185
IDR: 14318185
Текст научной статьи Inequalities for the Schwarzian derivative for subclasses of convex functions in the unit disc
Let ф(z) be an analytic function defined in D = {z : | z | < 1 } . If ф(z) satisfies the condition | ф(z) | 6 1 for all z E D, then it is called a Schwarzian function. The class of Schwarzian functions is denoted by Q * .
Next, let Q be the family of functions w(z) regular in D and satisfying the conditions w(0) = 0, | w(z) | < 1 for all z E D.
Further, for arbitrary fixed complex numbers A and B denote by P(A,B ) the family of functions
p(z) = 1 + P1Z + P2Z2 +---- regular in D and such that p(z) is in P(A, B) if and only if
1 + Aw ( z )
P(Z) 1 + Bw(z)
for some w(z) E Q and every z E D. This class was introduced by W. Janowski [2].
Finally, let C(A, B) denote the family of functions f (z) = z + a2 z2 +---- regular in D such that f (z) E C(A, B) if and only if
1+ f P(z)(4)
for some p(z) E P (A, B ) and all z in D.
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 ф E Q* then uvm 1 -lф(z)l2
| ф ( z ) l 6 1 -|z| 2 (5)
for some complex number B such that | B | < 1 .
Lemma 2. The class Q * is invariant under rotations.
C It follows easily from the definition of Q * that the function ^ defined by ^(z) := e -ia ф(e ia z), ф(z) E Q * , z E D, 0 6 a 6 2n, satisfies
Kz) | = | e -ia ф(e ia z) | = | e -ia || ф(e ia z) | 6 1. B
Lemma 3. If ф(z') is an element of Q * , then
| ф ( z ) | 6
1 -| Ф(z) | 2
1 -| B | 2 |z| 2 ‘
C Using Lemma 1 and Lemma 2, and after simple calculations,we get
| B | < 1, 0 6 a 6 2n ^ etaBz E D ^
W < 1 - ^( e ia Bz ) | 2 1- \\ < B - Heia Bz ) | 2
' ф ( e B 1 - | e^Bz | 2 ^ | ф ( e Bz ) |6 1 - I B | 2 |z| 2 . '
On the other hand, since e iα Bz ∈ D, the inequality (6) can be written in the form
| *' (z) | 6 Г-Ур^ ° (1 - | B | 2 1 z | 2 ) | *' ( z) | + |ФИ| 2 6 L B P)
Theorem 1. If f belongs to the class C (A, B) then
| S f | 6
I
(A - B) | z | 2 | A+B | (A - B) | z | 2
(1 - B 2 | z | 2 ) 2 (1+B | z | ) 2
(1 - A)A | z | 2 ,
B = 0,
B = 0.
C Since f E C (A, B), we can write
1 + z7E )= p ( z )
1 + Aw(z)
1 + Bw(z) ■
From the equality (9) we have the following:
' f 00 _ p(z)-1 / f 00 V _ zp^fz^ - Cpfz)-^
f 0 = z ^ V 0 ) = z 2 ’
? £1 V = A-B h zW-w 1 - B(A - B) (W2
У f0 J (1+Bw) 2 |_ z 2 J (1+Bw) 2 у z 2
zw0 - z2
w (A - B)zp0 - (A - Bp)(p - 1)
z2(A - Bp)2 1
w 2 (p - 1) 2
z 2 z 2 (A - Bp)21
Hence,
Sf = т f A
-
2 f У
1 (P - 1) 2
2 z 2
.
B [( A - B)zp 0 - (A - Bp)(p - 1
-
A + B I |2
A-B | P - 11 •
On the other hand, we have
| p(z) - 1 | -
| z | 2 | A - p(z) | 2 = (1 — B 2 | z | 2 ) p(z)
-
1 - AB | z | 2
1 - B 2 | z | 2
-
(A - B ) 2 | z | 2 (1 - B 2 | z | 2 ) ,
and
( w(z) E Q, ф(z) E Q * ^ w(z) = zф(z) ^ Ф 0 (z) = zw ( z ) I(1 - B 2 | z | 2 ) | ф 0 (z) | + | ф(z) | 2 6 1 ^ (1 - B 2 | z | 2 ) | zw l lzl - wiz)
-
z 2
z 2
,
■ ■ 1.
Considering (10)–(16) together yields (8). B
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
I ( A - B ) zp 0 ( z ) - ( p ( z ) - 1)( A - Bp ( z )) 1 6 ,(A B.^z' 2 .
(1 - B | z | )
In this case we have the following inequalities.
(1a) A = 1, B = - 1:
12 zp 0 ( z ) + 1 - ( p ( z )) 2 1 6 (1 -|ZZ| 2 ) 2 •
This inequalities was proved by M. S. Robertson ([5]).
(1b) A = 1 - 2a (0 6 a < 1), B = - 1:
|2(1 - a)zp,(z) - (p(z) - 1)((1 - 2a) + p(z)) 1 6 4(1—a)2*z2| .
(1 - | z | )
(1c) A = 1, B = ^M - 1 (M> 1 ):
(2 - mm} zp ( z ) - ( p ( z ) - 1) (1 - (1 - M} p ( z))
(2 — i)2H2
-
(Mm - 1) WT
(1d) A = в , B = -в (0 6 в < 1):
1 zp 0 ( z ) + 1 - ( p ( z )) 2 1 6 (1^в21_.
(2) A = 1, B = - 1:
4 | z | 2
| S f 1 6 (1 - | z | 2 ) 2 ‘
This equality was obtained by M. S. Robertson [5].
-
(3) A = 1 - 2a, B = - 1:
4(1 - a)^2 _ 4a (1 - a ) \ z \ 2 ' f | 6 (1 -\ z \ 2 ) 2 (1 -\ z \ ) 2
-
(4) a = 1, в = M - 1 (M > 2 ):
| S f | 6
(2 - MV \ z \ 2 Mm (2 - M^ \ z \ 2
-
(1 - ( M - 1) 2 1 z 1 2)2 (1 + (MM - 1)1 z I f
-
(5) A = в, B = - в (0 6 в < 1):
4в 2
\ S f \ 6 (1 - в 2 \ z \ 2 ) 2 ‘
Список литературы Inequalities for the Schwarzian derivative for subclasses of convex functions in the unit disc
- Duren P. L. Univalent Functions.-New York: Springer, 1983.
- Janowski W. Some Extremal Problems for Certain Families of Analytic Functions. I//Annales Polinici Math.-1973.-V. 28.-P. 297-326.
- Kraus W. Uber den Zusammanhang Charakteristiken Eines Einfach Zusammeshangenden Bereiches mit der Kreisabbildung//Mitt. Math. Sem. Giessen.-1932.-V. 21.-P. 1-28.
- Nehari Z. The Schwarzian Derivative and Schlicht Functions//Bull. Amer. Math. Soc.-1949.-V. 55.-P. 445-551.
- Robertson M. S. Univalent Functions for which zf'(z) is spirallike//Michigan Math. J.-1969.-V. 16.-P. 97-101.