1. Introduction
For the open unit disc
D of the complex plane and the boundary
of
D, the following Schwarz–Pick lemma(see [
1], Lemma 1.2) is well-known.
Theorem 1. Let be holomorphic and Then,and Equality in (1) holds at some point or equality in (2) holds if and only iffor some and . Among those interesting extensions of (2), there is a result of Shinji Yamashita(see [
2], Theorem 1):
Theorem 2. Let f be a function holomorphic and bounded, , in D, and let . Suppose thatin a neighborhood of z, where depends on z and is possible. Then, The inequality (4) is sharp in the sense that equality holds for the functionof w. For
f holomorphic in
D,
, and
, as it is commonly used we denote
by the
p-mean of
f on
, that is,
If
f is holomorphic, then
is an increasing function of
as well as an increasing function of
(see [
3]).
For
, let
be defined by
satisfies
for all
. It is well-known that
and that the set of automorphisms, i.e., bijective biholomorphic mappings, of
D consists of the mappings of the form
, where
and
.
Extending (2) in terms of
, there is another result of Shinji Yamashita(see [
4], Theorem 2):
Theorem 3. Let f be a function holomorphic and bounded, , in D and let Thenfor all and , where If the equality holds in (5) for and , then f is of the form (3).
Note that in (4) reduces to (2) and that (5) refines (2). As the same manner, it is expected that there might be a refinement of Theorem 2 which reduces to Theorem 3 when . This is our objective of this note.
3. Proof of Theorem 4
We may assume
(7) can be expressed as
By (6),
has a zero of order
n at
so that
is holomorphic in
D whose modulus at
is not greater than 1, so that the maximum principle gives (7).
Next, to verify inequality (8), take
such that (6) holds for
Then, by (6),
for
This is because
Thus,
defined by (9) has a zero of order
n at
. Hence,
is holomorphic in
D. Since
in a neighborhood of 0,
is harmonic in the neighborhood, hence there exists
such that
for
On the other hand, by (15),
In order to calculate the final term of (17), let’s put
and
Then,
Noting from (6) that
, we have, by (15), (16) and (18),
for
.
Now, the first inequality of (8) follows from the fact that is an increasing function of and also an increasing function of .
In addition, since and by the maximum principle, the second inequality of (8) follows.
We next check the conditions of equality. Elementary calculation shows that
Thus, if equality in (7) holds, at some point , ; then, (13) is a constant function of modulus 1 by virtue of the maximum principle, which gives (10) with by (20). To see that of (10) with gives the equality in (7) is straightforward also by (20).
If, for some p, , and for some the first inequality of (8) becomes equality, then, by (19), for , so that =constant, a.e. . Since h is holomorphic and =constant, a.e. for , it follows that h is a constant function. Letting with and solving this, as in (20), gives (10).
Finally, suppose the second inequality of (8) becomes equal for some
so that
. Then,
for
. Since
is a convex function of
(see [
3]) and
for
it follows that
for
whence
for all
Thus,
Since h maps D into D, this forces, by the maximum principle, that is a constant, , with . Hence, (20) gives (10).
Conversely, by (20), of (10) with makes h in (15) constant, so that the two inequalities in (8) become equalities.