Distortion Type Theorems for Functions in the Logarithmic Bloch Space

Abstract

We establish distortion type theorems for locally schlicht functions and for functions having branch points satisfying a normalized Bloch condition in the closed unit ball of the logarithmic Bloch space . As a consequence of our results we have estimations of the schlicht radius for functions in these classes.

1. 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 of all conformal functions on , the open unit disk of the complex plane , normalized such that 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 ; more precisely, they showed that for any and the following estimations hold.(1)Growth theorem:(2)Distortion theorem:with equality if and only if is a rotation of the Koebe function defined bywhich also belongs to the class . In particular, the distortion theorem implies that the class is contained in the closed ball with center at the origin and radius 8 of -Bloch space for all (see Section 3 for the definition of ). 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 , it cannot be applied to the bigger class of locally schlicht functions defined on satisfying the normalized Bloch conditions (recall that a holomorphic function is locally schlicht on if for all ). 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 satisfying the conditions , , and (see Section 3 for the definition of Bloch space). Liu and Minda give sharp lower bounds for and for 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 . Hence, it is important to consider certain subclasses of functions in Bloch spaces having seminorm not greater than .

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 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 satisfying a normalized Bloch condition . 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.

2. 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 the open unit disk in the complex plane , with center at the origin and radius 1; denotes the boundary of . The space of all complex and holomorphic functions on , as is usual, is denoted by . A function is said to be normalized if and and is locally schlicht or locally univalent if for all . A point is a branch point for if . For , we define is known as a horodisk ; that is, it is an Euclidean disk contained in which is tangent to at . Furthermore, has center at and radius . The closure of relative to is denoted by . Observe that but . With these notations, we can enunciate the well known Julia’s Lemma; the reader can consult the excellent book of Ahlfors [13] for its proof.

Lemma 1 (Julia’s Lemma). Suppose that is a complex and holomorphic function on such that maps into , the right half-plane, and . Then, for any , the function maps the horodisk into the Euclidean disk , where . Furthermore, a boundary point of the first disk is mapped on the boundary of the second disk if and only if is a conformal function mapping onto and satisfying .

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.

Lemma 2 ([6, corollary in Section ]). Let be a holomorphic function on . Suppose that maps into the right half-plane and that . Then and for all , with equality for some if and only if for all .

Proof. Indeed, let us fix ; then and by Julia’s Lemma, maps into the Euclidean disk . In particular, since , then ; this fact implies that . Furthermore, ; hence if , then we conclude that and, by Julia’s Lemma, this last fact occurs if is the conformal map from onto such that ; that is, for all . This shows the lemma.

Lemma 3 ([6, corollary to Theorem ]). Let be a holomorphic function on . Suppose that , and that all the zeros of have multiplicity at least . If , then (1) for all , with equality for some if and only if for all ;(2) for all , with equality for some if and only if for all .

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.

Lemma 4. Let be fixed and the Euclidean disk with center at and radius ; then

Proof. Let for simplicity and let . Since has positive real part on , the function is convex. In particular, , which proves the assertion.

3. Logarithmic Bloch Space

In this section we gather the definition and some of the properties of the logarithmic Bloch space . Let us recall that a function weight on is a bounded, positive, and continuous function defined on . Given a weight on , -Bloch space, denoted by , consists of all holomorphic functions on such that It is known that if the weight is radial, that is, for all , then is a Banach space with the norm . When , with , becomes the Bloch space which is denoted by , while when , with and fixed, we obtain -Bloch space which is denoted by .

Clearly, the function , defined by defines a weight on . Hence, the space is a Banach space with the norm We call as the logarithmic Bloch space. In the next result we are going to show that is a subspace of for all .

Proposition 5. The space is contained in , for all . Furthermore, for all function .

Proof. It is enough to show that for fixedfor all . But, this last inequality is true since the function with , is increasing and .

Also, we have the following very useful identity (see Lemma  3.3 in [12]).

Lemma 6. If , , , and , then .

Proof. Suppose that , , , and . Then, for each , we have Taylor’s theorem implies thatas . But sinceas , and as , we obtain from (15) thatas . Now, if we consider with small in (18), we conclude and we are done.

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 to denote the principal logarithmic of the complex number . Observe that the principal logarithmic is a holomorphic function on , the Euclidean disk with center at and radius :

() For each , we set where and . Clearly, for all , , and .

() For , we defineWe can see that , , and .

Also we have that the function satisfies the following properties.

Proposition 7. The function belongs to . Furthermore, but .

Proof. We see that . Indeed, we have Now, we are going to show that . Since the function is holomorphic on , then the modulus maximum principle tells us that its maximum value is attained in the boundary . But if then and henceOn the other hand, for each , we have andFurthermore, using elementary calculus, we can see that the real function is nonnegative for all (its minimum value is ). Hence for any we obtain This last implies that for all . We conclude that for any such that while for such that we have These last inequalities, (22) and (23), imply that which shows that .
Now, we are going to show that . Observe that . Also the real function with satisfies , as and it is strictly decreasing since for all . Hence we conclude that for all which shows the affirmation.

For the sequence , we have the following properties.

Proposition 8. Functions with belong to and satisfy for each . Furthermore, for each , in fact, .

Proof. Clearly, for any , the function belongs to since . We are going to show that . It is enough to show that there exists a such that , where with . Observe that, for with , we have , , and which implies that . Also, hencesince , , and and we have used that in the last equality. Thus, we conclude that and since , then there exists such that . This shows the affirmation. The other properties of ’s are clear.

4. A Distortion Theorem for Locally Schlicht Functions in

In this section we establish a distortion theorem for locally schlicht functions in the closed unit ball of satisfying normalized Bloch conditions. We denote by the class of all holomorphic functions such that is locally schlicht, , , and . With these notations, we have the following result.

Theorem 9. If then we have the following:(1) for all . There is not a function such that for some .(2) for all . There is not a function such that for some .

Proof. () Suppose that . Let us fix and we set the function with , where denotes the principal logarithmic of . Clearly is holomorphic on and because . Since is locally schlicht on , we have that for all . Furthermore, In particular, since and (by Lemma 6). Also, for any , we have and maps into . Hence, there exists a holomorphic function mapping the unit disk into the right half-plane and such that for all . Observe that since and . Invoking Lemma 2, it follows that for all . Hence for all . That is, for all .
Making the change , we obtain that and Therefore, if we consider with and we take , we conclude that This shows inequality (1).
Now, if there exists such that for some , then arguing as in the proof of inequality (1), for , the function, maps into . Hence, there exists a holomorphic function mapping into such that , , and for all . In particular, for , we have Thus,for some and, by Lemma 2, we conclude that for all . Therefore, for all . Hence, changing by , which belongs to , and using the identity principle for holomorphic functions, we obtain that for all . This last relation implies that which is a contradiction to Proposition 7. This complete the proof of item (1).
() Arguing as in the proof of part (1), for fixed, we set the function with . We have shown that there exists a holomorphic function such that for all . Furthermore, satisfies the hypothesis of Julia’s Lemma (Lemma 1); that is, is a holomorphic function in , which maps into , , and . Hence, for with fixed, maps the horodisk into the open Euclidean disk with center at and radius . In particular, since , we have that . Thus, Lemma 4 allows us to write for all . This last inequality is equivalent to writing for all and from here we have Making the change , we obtain since and alsoWe conclude, as before, that for all . This shows the inequality in the second part of the theorem.
Now, if there exists a function such that for some , then we can define and the function which maps into . Hence, as before, there exists a holomorphic function mapping into such that , , and for all . In particular, for , we have that is, is the value in where attain its minimum value, but by the proof of Lemma 4, we know that this happens if and . Now, by Lemma 2, we conclude that for all . As before, this last fact implies that for all and therefore which is a contradiction to Proposition 7. This completes the proof of item (2).

5. Distortion Theorems for Complex Functions in Having Branch Points

In this section we establish a distortion theorem for functions in the closed unit ball of having branch points and satisfying a normalized Bloch conditions. More precisely, for each , we denote by the class of all holomorphic functions such that , , and if for some then for all . Clearly we have With these notations, we have the following result.

Theorem 10. For fixed, we set . Then for every we have the following:(1),  for all . There is not a function such that for some .(2), for all . Furthermore, there is not a function such that for some .

Proof. (1) Let us fix , and . We set the function with , where denotes the principal logarithmic of the complex number . Clearly the function is holomorphic on and because . Also, we have And hence since and (by Lemma 6). Furthermore, since and if and only if , we conclude that all the zeros of the function have multiplicity at less .
On the other hand, since we have for all , since for all . Hence, we have shown that . Invoking Lemma 3, we conclude that for all . Therefore, for each , the following estimation holds: That is, for all since .
Next, we make the change . Then since , and we can write Finally, if we set and , we conclude that for all . This shows the inequality in part ().
The proof of the second part is similar to part () of Theorem 9. If there exists a function such that for some , then we set and the function with . We have showed that satisfies all the hypothesis of Lemma 3. Furthermore, choosing such that we obtain By Lemma 3, we conclude that for all . Hence, for all . Changing by , we obtain that for all and consequently for all . This last equality implies that which is a contradiction to Proposition 8.
() As before, for , we set , we fix , and we define the function with . In the first part we have shown that this function satisfies the hypothesis of Lemma 3. Hence for all . Therefore and thus we have which is the same as As before, we make the change ; then , , and Setting and , we conclude that for all . This shows the inequality in part ().
If there exists a function such that for some , then we set and the function with . We have showed that satisfies all the hypothesis of Lemma 3. Furthermore, choosing such that we obtain and since we can see that . Furthermore, By Lemma 3, we conclude that for all . Hence, as before, this last fact implies that which is a contradiction to Proposition 8.

6. 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 and , denote the radius of the largest schlicht disk on the Riemann surface centered at (a schlicht disk on centered at means that maps an open subset of containing conformally onto this disk). With this notation, we have the following results.

Corollary 11. If , then

Proof. From the definition of , it follows the fact that there exists a simply connected domain containing the zero such that maps conformally onto an Euclidean disk with center at and radius . This Euclidean disk must meet the boundary of because, in other cases, the boundary of the set is a Jordan curve in the interior of and we can find an open set where is univalent; hence contain an Euclidean disk with center at and radius greater than , which contradict with the definition of . We conclude then that there is a radial segment jointing to the boundary of . Let be inverse image of under ; then joint the point to the boundary of . Thus, from Theorem 9, it follows that where we have used Cauchy-Schwarz’s inequality in the fourth line, is the scalar product of and , and we have made the change , where as . This shows the result.

While for functions in the class we have the following.

Corollary 12. Suppose that is fixed. If , then

Proof. Indeed, arguing as in the proof of Corollary 11, from Theorem 10, it follows that where we have used that as . This shows the result.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.