Distortion Type Theorems for Functions in the Logarithmic Bloch Space

One of the most important results in the area of geometric theory of functions of a complex variable is the celebrated distortion’s theorem established by Koebe and Bieberbach [1, 2] at the beginning of the twentieth century. Koebe and Bieberbach showed that the range of any function f in the class S of all conformal functions on D, the open unit disk of the complex plane C, normalized such that f(0) = 0 = f󸀠(0) − 1 contain the Euclidean disk with center at the origin and radius 1/4. This last result is today known as Koebe 1/4 Theorem and, in particular, shows that Bloch’s constant (see [3]) is greater than or equal to 1/4. Koebe and Bieberbach found sharp lower and upper bounds for the growth and the distortion of conformal maps in the class S; more precisely, they showed that for any f ∈ S and z ∈ D the following estimations hold.


Introduction
One of the most important results in the area of geometric theory of functions of a complex variable is the celebrated distortion's theorem established by Koebe and Bieberbach [1,2] at the beginning of the twentieth century.Koebe and Bieberbach showed that the range of any function  in the class S of all conformal functions on D, the open unit disk of the complex plane C, normalized such that (0) = 0 =   (0) − 1 contain the Euclidean disk with center at the origin and radius 1/4.This last result is today known as Koebe 1/4 Theorem and, in particular, shows that Bloch's constant (see [3]) is greater than or equal to 1/4.Koebe and Bieberbach found sharp lower and upper bounds for the growth and the distortion of conformal maps in the class S; more precisely, they showed that for any  ∈ S and  ∈ D the following estimations hold.
(1) Growth theorem: (2) Distortion theorem: 3 ≤        ()      ≤ 1 + || (1 − ||) 3  (2) with equality if and only if  is a rotation of the Koebe function defined by which also belongs to the class S. In particular, the distortion theorem implies that the class S is contained in the closed ball with center at the origin and radius 8 of -Bloch space B  for all  ≥ 3 (see Section 3 for the definition of B  ).For more properties of conformal maps and distortion theorem, we recommend the excellent books [4,5].
Although the distortion theorem gives sharp bounds for the modulus of the derivative of functions in the class S, it cannot be applied to the bigger class of locally schlicht functions defined on D satisfying the normalized Bloch conditions (0) = 0 =   (0) − 1 (recall that a holomorphic function  is locally schlicht on D if   () ̸ = 0 for all  ∈ D).Many authors have obtained distortion type theorems or lower bounds for the modulus or real part of the derivative of locally schlicht functions in Bloch-type spaces.The pioneer work about this subject appears in 1992 and is due to Liu and Minda [6].They established distortion theorems for locally schlicht functions  in the classical Bloch space B satisfying the conditions (0) = 0,   (0) = 1, and ‖‖ B = 1 (see Section 3 for the definition of Bloch space).Liu and Minda give sharp lower bounds for |  ()| and for Re   () and as consequence of their results they obtain a lower bound for Bloch's constant.Determination of the (locally schlicht) Bloch constant is still an open problem.By Landau's reduction, it is enough to consider those functions with Bloch seminorm not greater than 1.Hence, it is important to consider certain subclasses of functions in Bloch spaces having seminorm not greater than 1.
The results of Liu and Minda in [6] have been extended to other classes of locally schlicht functions or to functions having branch points in the Bloch space by Yanagihara [7], Bonk et al. [8,9], and Graham and Minda [10].The extension of the above results to -Bloch spaces was obtained by Terada and Yanagihara [11] and by Zheng and Wang [12].It is an open problem to obtain distortion type theorems for locally schlicht functions in other spaces of analytic functions.
In this article we extend the results of Liu and Minda [6] to the logarithmic Bloch space B log which we define in Section 3; we obtain lower bounds for the modulus and the real part of the derivative of locally schlicht functions and for functions having branch points in the closed unit ball of B log satisfying a normalized Bloch condition (0) = 0 =   (0) − 1.Our results will be showed in Sections 4 and 5, as consequence of our results, in Section 6, we obtain lower bounds for the schlicht radius of functions in these classes.

Some Preliminaries: Julia's Lemma
In this section we gather some notations, definitions, and results that we will need through this note.We denote by D the open unit disk in the complex plane C, with center at the origin and radius 1; D denotes the boundary of D. The space of all complex and holomorphic functions on D, as is usual, is denoted by (D).A function  ∈ (D) is said to be normalized if (0) = 0 and   (0) = 1 and  is locally schlicht or locally univalent if   () ̸ = 0 for all  ∈ D. A point  0 is a branch point for  if   ( 0 ) = 0.For  > 0, we define In 1992, Liu and Minda [6] established distortion theorem for functions in the Bloch space; they showed the following results which are consequences of Julia's Lemma.We include the proof of the first one to illustrate the application of Julia's Lemma.
We finish this section by establishing the following elementary property of the complex exponential.We thank the reviewer for providing us the following simple demonstration of this fact.  1.Since 1 +   ()/  () = 1 −  has positive real part on || < 1, the function  is convex.In particular, Re(()) > (−1) =  −2 , which proves the assertion.

Logarithmic Bloch Space
In this section we gather the definition and some of the properties of the logarithmic We call B log as the logarithmic Bloch space.In the next result we are going to show that B log is a subspace of B  for all  ≥ 1.
for all function  ∈ B log .
Proof.It is enough to show that for  ≥ 1 fixed for all  ∈ D. But, this last inequality is true since the function with  ∈ (0, 1], is increasing and ℎ(1) = 0.
Also, we have the following very useful identity (see Lemma 3.3 in [12]).
The following functions play a very important role in our work; they will be used to get lower bounds for locally schlicht functions and for functions having branch points in certain classes in the logarithmic Bloch space.From now, we use log() to denote the principal logarithmic of the complex number  ̸ = 0. Observe that the principal logarithmic is a holomorphic function on (1, 1), the Euclidean disk with center at 1 and radius 1: (1) For each  ∈ N, we set where   = √/( + 2) and  ∈ D. Clearly,   ∈ (D) for all  ∈ N,   (0) = 0, and    (0) = 1.
Furthermore, using elementary calculus, we can see that the real function Hence for any  ∈ [0, 1) we obtain This last implies that for all  ∈ [0, 1).We conclude that for any  ∈ D such that while for  ∈ D such that |1 − | 2 >  we have These last inequalities, ( 22) and (23), imply that which shows that  ∈ B log .Now, we are going to show that sup satisfies (0) = 0, () → −∞ as  → 1 − and it is strictly decreasing since for all  ∈ [0, 1).Hence we conclude that () ≤ 0 for all  ∈ [0, 1) which shows the affirmation.
For the sequence {  }, we have the following properties.
Proposition 8. Functions   with  ∈ N belong to B log and satisfy Proof.Clearly, for any  ∈ N, the function   belongs to B log since   ∈ (D).We are going to show that sup It is enough to show that there exists a  0 ∈ (  , 1) such that ( 0 ) > 0, where with  ∈ (  , 1).Observe that, for , and ( 2  + 1) = 2 and we have used that   = √/( + 2) in the last equality.Thus, we conclude that   (  ) > 0 and since (  ) = 0, then there exists  0 ∈ (  , 1) such that ( 0 ) > 0. This shows the affirmation.The other properties of   's are clear.

Distortion Theorems for Complex Functions in B log Having Branch Points
In this section we establish a distortion theorem for functions in the closed unit ball of B log having branch points and satisfying a normalized Bloch conditions.More precisely, for each  ∈ N, we denote by  () log the class of all holomorphic functions  ∈ B log such that (0) = 0,   (0) = 1, ‖‖ log ≤ 1 and if   () = 0 for some  ∈ D then  () () = 0 for all  = 1, 2, . . ., .Clearly we have With these notations, we have the following result.

Some Estimations for the Schlicht Radius
In this section we present some consequences of the results obtained in Sections 4 and 5.We recall that if  is a holomorphic function on D and  0 ∈ D,   ( 0 , ) denote the radius of the largest schlicht disk on the Riemann surface (D) centered at ( 0 ) (a schlicht disk on (D) centered at ( 0 ) means that  maps an open subset of D containing  0 conformally onto this disk).With this notation, we have the following results.(81) Proof.From the definition of   (0, ), it follows the fact that there exists a simply connected domain  ⊂ D containing the zero such that  maps  conformally onto an Euclidean disk with center at (0) and radius   (0, ).This Euclidean disk must meet the boundary of (D) because, in other cases, the boundary of the set  is a Jordan curve in the interior of D and we can find an open set  ⊂ D where  is univalent; hence () contain an Euclidean disk with center at (0) and radius greater than   (0, ), which contradict with the definition of   (0, ).We conclude then that there is a radial segment Γ jointing (0) to the boundary of (D).Let  be inverse image of Γ under ; then  joint the point 0 to the boundary of D. Thus, from Theorem 9, it follows that where we have used that  = |()| →   as  → 1 − .This shows the result.

Corollary 12 .
used Cauchy-Schwarz's inequality in the fourth line, () ⋅   () is the scalar product of () and   (), and we have made the change  = |()| = √() ⋅ (), where  → 1 − as  → 1 − .This shows the result.While for functions in the class  () log we have the following.Suppose that  ∈ N is fixed.If  ∈  () log , then   (0, ) ≥ ∫ log (1 −   )) .(83) Proof.Indeed, arguing as in the proof of Corollary 11, from Theorem 10, it follows that   (0, ) = ∫ Γ || = ∫         ()      || log (1 −   )) , Bloch space B log .Let us recall that a function weight  on D is a bounded, positive, and continuous function defined on D. Given a weight  on D, -Bloch space, denoted by B  , consists of all holomorphic functions  on D such that defines a weight on D. Hence, the space B log = B  log is a ∈D       () (0, ) = ∫