On the Spherical Derivatives of Miranda Functions
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On the Spherical Derivatives of Miranda Functions Jianming Chang1,2 Received: 30 May 2018 / Revised: 6 December 2018 / Accepted: 9 December 2018 / Published online: 4 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Let f be a holomorphic function in the unit disk (0, 1) such that f (z) = 0 and f (z) = 1. Then f # (0) =
| f (0)| < 583. 1 + | f (0)|2
Keywords Holomorphic functions · Miranda’s functions · Normal families · Spherical derivatives Mathematics Subject Classification 30D35 · 30D45
1 Introduction In the theory of normal families, Marty’s theorem says that a family F of functions meromorphic in a domain D ⊂ C is normal if and only if the set { f # : f ∈ F} of spherical derivatives is locally uniformly bounded in D, where the spherical derivative f # is defined by f # (z) :=
| f (z)| . 1 + | f (z)|2
Communicated by Lawrence Zalcman. Supported by NNSF of P. R. China (Grant nos. 11471163, 11171045).
B
Jianming Chang [email protected]
1
School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, Jiangsu, People’s Republic of China
2
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, People’s Republic of China
123
254
J. Chang
Thus for a given normal family F of functions meromorphic in the unit disk (0, 1) = {z ∈ C : |z| < 1}, there exists a constant M = M(F) such that f # (0) ≤ M for all functions f ∈ F. The constant M corresponding to the Montel’s normal family F0 , i.e., the family of holomorphic functions satisfying f = 0, 1 in the unit disk (0, 1), has been obtained 4 (1/4) . = 4.37688. by Bonk and Cherry [1]; they showed that the best value is M0 = 4π 2 Here, we focus on the constant M corresponding to the Miranda’s normal family F1 [4], i.e., the family of holomorphic functions satisfying f = 0 and f = 1 in the unit disk (0, 1). We obtain the following result: Theorem 1.1 Let f be a holomorphic function in the unit disk (0, 1) such that f (z) = 0 and f (z) = 1. Then f # (0) < 583. And for z ∈ (0, 1), (1 − |z|)2 f # (z) < 583.
(1)
We remark that from the following example given by Hayman and Storvick [3] f (z) = 2(1 − z) exp
2+z 1−z
,
(2)
one can see that there exist functions in the Miranda’s normal family F1 such that sup (1 − |z|) f # (z) = +∞.
|z| −1. We need some notation and basic results. First we define the function xy by x (x + 1) = , (x > −1, y > −1, x − y > −1), y (y + 1)(x − y + 1)
123
(3)
On the Spherical Derivatives of Miranda Functions
255
where (x) is the Gamma function defined by
+∞
(x) =
t x−1 e−t dt, (x > 0),
0
and has the property that (x + 1) = x(x). In particular, for a positive integer y, x x(x − 1) · · · (x − y + 1) = y y! is a polynomial in x (of degree y). Thus if y is a positive integer, then −x − 1 x+y = . (−1) y y y
Also note that if both x, y are positive integers and x < y, then We also use the Beta function B(x, y) =
1
u
x−1
(1 − u)
y−1
π 2
du = 2
x y
= 0.
cos2x−1 t sin2y−1 tdt
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