A Refinement of Schwarz – Pick Lemma for Higher Derivatives

In this paper, a Schwarz–Pick estimate of a holomorphic self map f of the unit disc D having the expansion f (w) = c0 + cn(w− z)n + ... in a neighborhood of some z in D is given. This result is a refinement of the Schwarz–Pick lemma, which improves a previous result of Shinji Yamashita.


Introduction
For the open unit disc D of the complex plane and the boundary ∂D of D, the following Schwarz-Pick lemma(see [1], Lemma 1.2) is well-known.
Theorem 1.Let f : D −→ D be holomorphic and z 0 ∈ D.Then, and (2) Equality in (1) holds at some point z = z 0 or equality in (2) holds if and only if for some c ∈ ∂D and a ∈ D.
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, | f | < 1, in D, and let z ∈ D. Suppose that f (w) = c 0 + c n (w − z) n + c n+1 (w − z) n+1 + ... in a neighborhood of z, where n ≥ 1 depends on z and c n = 0 is possible.Then, The inequality ( 4) is sharp in the sense that equality holds for the function For f holomorphic in D, 0 ≤ r < 1, and 0 ≤ p ≤ ∞, as it is commonly used we denote M p (r, f ) by the p-mean of f on ∂D, that is, If f is holomorphic, then M p (r, f ) is an increasing function of p : 0 ≤ p ≤ ∞ as well as an increasing function of r : 0 ≤ r < 1 (see [3]).
For a ∈ D, let ϕ a be defined by ϕ a satisfies ϕ a (ϕ −a (z)) = z for all z ∈ D. It is well-known that ϕ a (∂D) = ∂D and that the set of automorphisms, i.e., bijective biholomorphic mappings, of D consists of the mappings of the form αϕ a (z), where a ∈ D and |α| = 1.Extending (2) in terms of M p (r, f ), there is another result of Shinji Yamashita(see [4], Theorem 2): for all w ∈ D and r : 0 < r < 1, where If the equality r −1 M p (r, f w ) = 1 holds in (5) for w ∈ D and 0 < r < 1, then f is of the form (3).
Note that n = 1 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 n = 1.This is our objective of this note.

Result
The following is our corresponding result: in a neighborhood of z, then f (w) and for all r : 0 < r < 1, and for all p : 0 ≤ p ≤ ∞, where Equality in (7) holds at some point w ∈ D, w = z if and only if Equality in the first inequality or in the second inequality of (8) holds for some r : 0 < r < 1 and p : 0 < p ≤ ∞ if and only if f is of the form (10) (with |α| ≤ 1 or |α| = 1, respectively).
for all w ∈ D. This can be verified as follows: Since any automorphism, i.e., bijective holomorphic mapping, of D is of the form of the right-hand side of (11), it suffices to show that the left-hand side of (11), denote Φ(w), is an automorphism of D. That Φ(w) is holomorphic and into D is obvious.We show Φ(w) is bijective: If Φ(w 1 ) = Φ(w 2 ), then ϕ −z (w 1 ) = ϕ −z (w 2 ), and the injectivity of ϕ −z shows w 1 = w 2 .Thus, Φ(w) is injective.Next, for any ζ ∈ D, by the surjectivity of For this η, there is ξ ∈ D such that η = ϕ −z (ξ), whence Φ(w) is surjective.
(2) Fix z ∈ D and self-map f of D. Then, applying Littlewood's inequality (see [3,5,6]), it follows that with equality holding only if f is an inner function.Equation (7) follows directly from (12).In addition, the inequality of (8) can be obtained as a one stroke limit from (7): as w → z (by applying L'Hospital's rule).
The point of Theorem 4 lies in its connection with M p (r, •) and in clarifying the condition of equality to make Yamashita type theorem complete.
After proving Theorem 4 in Section 3, applications of Theorem 4 to some coefficient problems will be given in Section 4.

Proof of Theorem 4
We may assume c n = 0. ( 7) can be expressed as By (6), f (w) − f (z) has a zero of order n at w = z so that is holomorphic in D whose modulus at w ∈ ∂D is not greater than 1, so that the maximum principle gives (7).Next, to verify inequality (8), take δ > 0 such that (6) holds for w : |w − z| < δ.Then, by (6), for w : |w| < δ 1+|z| .This is because Thus, f z (w) defined by (9) has a zero of order n at w = 0. Hence, is holomorphic in D. Since h(0) = 0 in a neighborhood of 0, log |h| is harmonic in the neighborhood, hence there exists r 0 such that for r : r < r 0 .
On the other hand, by (15), In order to calculate the final term of (17), let's put By (14), whence Noting from ( 6) that c n = f (n) (z) n! , we have, by ( 15), ( 16) and ( 18), for r < r 0 .Now, the first inequality of (8) follows from the fact that M p (r, h) is an increasing function of p : 0 ≤ p ≤ ∞ and also an increasing function of r : 0 < r < 1.
In addition, since M p (r, h) ≤ M ∞ (r, h) and |h| < 1 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 w ∈ D, w = z; then, (13) is a constant function of modulus 1 by virtue of the maximum principle, which gives (10) with |α| = 1 by (20).To see that f (w) of (10) with |α| = 1 gives the equality in (7) is straightforward also by (20).

Applications
Theorem 4 immediately gives the following estimate: for all r : 0 < r < 1, and for all p : 0 ≤ p ≤ ∞, where f z is defined by (9).Equality in the first inequality or in the second inequality of (21) holds for some r : 0 < r < 1 and p : 0 < p ≤ ∞ if and only if f is of the form (10) (with |α| ≤ 1 or |α| = 1, respectively).
For the case z = 0, we can also obtain Corollary 2. Let f be a function holomorphic and bounded, for all r : 0 < r < 1, and for all p : 0 ≤ p ≤ ∞, where Equality in the first inequality or in the second inequality of (22) holds for some r : 0 < r < 1 and p : 0 < p ≤ ∞ if and only if g is of the form (with |α| ≤ 1 or |α| = 1, respectively).

Proof of Corollary 2.
As was frequently used (see [7] for example), we make use of the facts that ∑ n k=1 e i2πjk/n = n if j is a multiple of n, and 0 if j is otherwise.Noting that for all r : 0 < r < 1, and for all p : 0 ≤ p ≤ ∞.
In addition, by Corollary 1, equality in the first inequality or in the second inequality of (22) holds for some r : 0 < r < 1 and p : 0 < p ≤ ∞ if and only if g is of the form (23) (with |α| ≤ 1 or |α| = 1, respectively).

Conclusions
With imperative applications to particular situations, various forms of Schwarz Lemma have been called for.In this paper, we presented Schwarz-Pick Lemma for higher derivatives in connection with p-mean M p (r, f )(see Theorem 4).It refined a previous result of Shinji Yamashita and clarified the condition of equality.As an immediate consequence, the result could be applied to refine well-known estimates for n-th Taylor coefficient of holomorphic self maps of D (see Corollary 1 and 2).We are expecting its further extensions and applications.

Theorem 4 .
Let f be a function holomorphic and bounded, | f | < 1, in D and let z ∈ D. If f