# Distortion Type Theorems for Functions in the Logarithmic Bloch Space

Authors: Armando J. García-Ortíz; Milton del Castillo Lesmes Acosta; Julio C. Ramos-Fernández
Journal: Journal of Function Spaces
 (2017)

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## 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 spaceBlog. As a consequence of our results we have estimations of the schlicht radius for functions in these classes.


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## Body

## 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 f in the class S of all conformal functions on D, the open unit disk of the complex plane C, normalized such that f0=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 thegrowth and thedistortion of conformal maps in the class S; more precisely, they showed that for any f∈S and z∈D the following estimations hold.(1)
Growth theorem:(1)z1+z2≤fz≤z1-z2(2)
Distortion theorem:(2)1-z1+z3≤f′z≤1+z1-z3with equality if and only if f is a rotation of the Koebe function defined by(3)Kz=z1-z2,z∈D,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 p-Bloch space Bp for all p≥3 (see Section 3 for the definition of Bp). 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 classS, it cannot be applied to the bigger class of locally schlicht functions defined on D satisfying the normalized Bloch conditions f0=0=f′0-1 (recall that a holomorphic function f is locally schlicht on D if f′z≠0 for all z∈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 f in the classical Bloch space B satisfying the conditions f0=0, f′0=1, and fB=1 (see Section 3 for the definition of Bloch space). Liu and Minda give sharp lower bounds for f′z and for Re⁡f′z 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 p-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 Blog 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 Blog satisfying a normalized Bloch condition f0=0=f′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.


## 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 byD 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 HD. A function f∈HD is said to benormalized if f0=0 and f′0=1 and f islocally schlicht or locally univalent if f′z≠0 for all z∈D. A point z0 is abranch point for f if f′z0=0. For r>0, we define (4)Δ1,r=z∈D:1-z21-z2<r.Δ1,r is known as ahorodiskD; that is, it is an Euclidean disk contained in D which is tangent to ∂D at 1. Furthermore, Δ1,r has center at 1/1+r and radius r/1+r. The closure of Δ1,r relative to D is denoted by Δ¯1,r. Observe that 1∉Δ¯1,r but 1-r/1+r∈Δ¯1,r. 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 thatw is a complex and holomorphic function on D∪1 such that w maps D into H+=z∈C:Re⁡z>0, the right half-plane, and w1=0. Then, for any r>0, the function w maps the horodisk Δ¯1,r into the Euclidean disk z∈C:z-dr<dr, where d=-w′1>0. Furthermore, a boundary point of the first disk is mapped on the boundary of the second disk if and only if w is a conformal function mapping D onto H+ and satisfying w1=0.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 1]).
Letw be a holomorphic function on D∪1. Suppose that w maps D into the right half-plane H+ and that w1=0. Then d=-w′1>0 and (5)Re⁡wx≤2d1-x1+x,for all x∈-1,1, with equality for some x∈-1,1 if and only if (6)wz=2d1-z1+z,for all z∈D.Proof.
Indeed, let us fixx∈-1,1; then r=1-x/1+x>0 and by Julia’s Lemma, w maps Δ¯1,r into the Euclidean disk Ddr,dr. In particular, since x∈Δ¯1,r, then wx∈Ddr,dr; this fact implies that Re⁡wx≤2dr. Furthermore, x∈∂Δ1,r; hence if Re⁡wx=2dr, then we conclude that wx=2dr∈∂Ddr,dr and, by Julia’s Lemma, this last fact occurs if w is the conformal map from D onto H+ such that w1=0; that is, wz=2d1-z/1+z for all z∈D. This shows the lemma.Lemma 3 ([6, corollary to Theorem 3]).
Letf be a holomorphic function on D∪1. Suppose that fD⊂D, f1=1 and that all the zeros of f have multiplicity at least n. If f′1=n, then (1)
fx≥xn for all x∈0,1, with equality for some x∈0,1 if and only if fz=zn for all z∈D;(2)
Refx≥xn for all n-1/n+1≤x<1, with equality for some x∈n-1/n+1,1 if and only if fz=zn for all z∈D.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.
Letx∈0,1 be fixed and Dx the Euclidean disk with center at 1-x/1+x and radius 1-x/1+x; then(7)min⁡Re⁡exp-z:z∈Dx=exp⁡-21-x1+x.Proof.
Letr=1-x/(1+x) for simplicity and let fz=erz-1. Since 1+zf′′z/f′z=1-rz has positive real part on z<1, the function f is convex. In particular, Re⁡fz>f-1=e-2r, which proves the assertion.


## 3. Logarithmic Bloch Space

In this section we gather the definition and some of the properties of the logarithmic Bloch spaceBlog. Let us recall that afunction weightμ on D is a bounded, positive, and continuous function defined on D. Given a weight μ on D, μ-Bloch space, denoted by Bμ, consists of all holomorphic functions f on D such that (8)fμ≔supz∈D⁡μzf′z<∞.It is known that if the weight μ is radial, that is, μz=μz for all z∈D, then Bμ is a Banach space with the norm fBμ=f0+fμ. When μz=1-z2, with z∈D, Bμ becomes the Bloch space which is denoted by B, while when μz=1-z2p, with z∈D and p>0 fixed, we obtain p-Bloch space which is denoted by Bp.Clearly, the functionμlog⁡, defined by (9)μlog⁡z=log⁡e1-z2-1,defines a weight on D. Hence, the space Blog⁡=Bμlog⁡ is a Banach space with the norm (10)fBlog⁡=f0+flog⁡≔f0+supz∈D⁡f′zlog⁡e/1-z2.We call Blog⁡ asthe logarithmic Bloch space. In the next result we are going to show that Blog⁡ is a subspace of Bp for all p≥1.Proposition 5.
The spaceBlog⁡ is contained in Bp, for all p≥1. Furthermore, (11)fBp≤fBlog⁡,for all function f∈Blog⁡.Proof.
It is enough to show that forp≥1 fixed(12)1-z2plog⁡e1-z2≤1,for all z∈D. But, this last inequality is true since the function (13)ht=tplog⁡et-1,with t∈0,1, is increasing and h1=0.Also, we have the following very useful identity (see Lemma  3.3 in [12]).Lemma 6.
Iff∈Blog⁡, f0=0, f′0=1, and flog⁡≤1, then f′′0=0.Proof.
Suppose thatf∈Blog⁡, f0=0, f′0=1, and flog⁡≤1. Then, for each z∈D, we have (14)f′z≤log⁡e1-z2.Taylor’s theorem implies that(15)1+f′′0z+oz2≤log2⁡e1-z2=1+z2+oz2,as z→0. But since(16)1+f′′0z+oz2=1+2Re⁡f′′0z+oz,as z→0, and (17)1+z2+oz2=1+oz,as z→0, we obtain from (15) that(18)2Re⁡f′′0z≤oz,as z→0. Now, if we consider z=rf′′0¯/f′′0 with r>0 small in (18), we conclude f′′0=0 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 uselogw to denote the principal logarithmic of the complex number w≠0. Observe that the principal logarithmic is a holomorphic function on D1,1, the Euclidean disk with center at 1 and radius 1:(1) For each n∈N, we set (19)Fnz=∫0z1-s/an1-ansn1-2log⁡1-ansds,where an=n/n+2 and z∈D. Clearly, Fn∈HD for all n∈N, Fn0=0, and Fn′0=1.(2) For z∈D, we define(20)Fz=∫0zexp⁡-2s1-s1-2log⁡1-sds.We can see that F∈HD, F0=0, and F′0=1.Also we have that the functionF satisfies the following properties.Proposition 7.
The functionF belongs to Blog⁡. Furthermore, supz∈Dμlog⁡zF′z=1 but Flog⁡>1.Proof.
We see thatFlog⁡>1. Indeed, we have (21)Flog⁡≥F′i/21-log⁡1-i/22=exp⁡2/51-log⁡3/41-log⁡542+4arctan2⁡12≈1.4014837>1.Now, we are going to show that F∈Blog⁡. Since the function exp⁡-2z/1-z is holomorphic on D, then the modulus maximum principle tells us that its maximum value is attained in the boundary ∂D. But if z=1 then 1-z2=21-Re⁡z and hence(22)supz∈D⁡exp⁡-2z1-z=supz=1⁡exp⁡2-21-Re⁡z1-z2=e.On the other hand, for each z∈D, we have arg1-z<π/2 and(23)supz∈D⁡arg⁡1-z1-log⁡1-z2≤π2.Furthermore, using elementary calculus, we can see that the real function Ht=e2-1-t1+t3 is nonnegative for all t∈0,1 (its minimum value is H1/2=e2-27/16≈5.70). Hence for any t∈0,1 we obtain (24)1-2log⁡1-t-31-log⁡1-t2=log⁡1-t1+t3e2≤0.This last implies that (25)1-2log⁡1-t1-log⁡1-t2≤3,for all t∈0,1. We conclude that for any z∈D such that 1-z2≤e(26)log⁡e/1-z21-log⁡1-z2=1-2log⁡1-z1-log⁡1-z2≤1-2log⁡1-z1-log⁡1-z2≤3,while for z∈D such that 1-z2>e we have (27)log⁡e/1-z21-log⁡1-z2=log⁡1-z2/e1-log⁡1-z2≤log⁡4-1.These last inequalities, (22) and (23), imply that (28)Flog⁡≤e3+log⁡4-1+2π2=e2+log⁡4+πwhich shows that F∈Blog⁡.
Now, we are going to show thatsupz∈Dμlog⁡zF′z=1. Observe that μlog⁡0F′0=1. Also the real function Ht=exp-2t/1-t1-2log1-t-1+log1-t2 with t∈0,1 satisfies H0=0, Ht→-∞ as t→1- and it is strictly decreasing since (29)H′t=-2t1-t2-21-t21-tlog⁡e1-t2-1exp⁡-2t1-t<0,for all t∈0,1. Hence we conclude that Ht≤0 for all t∈0,1 which shows the affirmation.For the sequenceFn, we have the following properties.Proposition 8.
FunctionsFn with n∈N belong to Blog⁡ and satisfy (30)limn→∞⁡Fn′z=F′z,for each z∈D. Furthermore, for each n∈NFnlog⁡>1, in fact, supz∈Dμlog⁡zFn′z>1.Proof.
Clearly, for anyn∈N, the function Fn belongs to Blog⁡ since Fn∈HD¯. We are going to show that supz∈Dμlog⁡zFn′z>1. It is enough to show that there exists a t0∈an,1 such that Ht0>0, where (31)Ht=t/an-11-antnlog⁡e1-ant2-log⁡e1-t2,with t∈an,1. Observe that, for tn=ran with r=2/1+an2>1, we have tn∈an,1, r-1=1-ran2, and 1-r2an2=1-ran22 which implies that Htn=0. Also, (32)H′t=nt/an-11-antn-11/an-an1-ant2log⁡e1-ant2+t/an-11-antn2an1-ant-2t1-t2;
hence(33)H′tn=n1/an-an1-ran22loge1-ran22+2an1-ran2-2ran1-r2an2=n1-ran221an-anlog⁡e1-ran22-1n2an21-an2=n1-ran221an-anlog⁡e1-ran22-1,since r-1=1-ran2, 1-r2an2=1-ran22, and ran2+1=2 and we have used that an=n/n+2 in the last equality. Thus, we conclude that H′tn>0 and since Htn=0, then there exists t0∈tn,1 such that Ht0>0. This shows the affirmation. The other properties of Fn’s are clear.


## 4. A Distortion Theorem for Locally Schlicht Functions inBlog⁡

In this section we establish a distortion theorem for locally schlicht functions in the closed unit ball ofBlog⁡ satisfying normalized Bloch conditions. We denote by βlog⁡∞ the class of all holomorphic functions f∈Blog⁡ such that f is locally schlicht, f0=0, f′0=1, and flog⁡≤1. With these notations, we have the following result.Theorem 9.
Iff∈βlog⁡∞ then we have the following:(1)
f′z≥F′z=exp⁡-2z/1-z1-2log⁡1-z for all z∈D. There is not a function f0∈βlog⁡∞ such that f0′z0=F′z0 for some z0∈D∖0.(2)
Re⁡f′z≥F′z=exp-2z/1-z1-2log1-z for all z≤1/2. There is not a function f0∈βlog⁡∞ such that Re⁡f0′z0=F′z0 for some z0∈D0,1/2∖0.Proof.
(1) Suppose that f∈βlog⁡∞. Let us fix ζ=1 and we set the function (34)gu=1-2log⁡1+u2-1f′1-u2ζ,with u∈D, where logw denotes the principal logarithmic of w∈D1,1. Clearly g is holomorphic on D¯∖-1 and g1=1 because f′0=1. Since f is locally schlicht on D, we have that gz≠0 for all z∈D. Furthermore, (35)g′u=21+u1-2log⁡1+u2-2f′1-u2ζ+1-2log⁡1+u2-1f′′1-u2ζ-ζ2.In particular, g′1=1 since f′0=1 and f′′0=0 (by Lemma 6). Also, for any u∈D, we have (36)gu=1-2log⁡1+u2-1f′1-u2ζ≤1Re⁡1-2log⁡1+u/2log⁡e1-1-u/22=1log⁡e/1+u/22log⁡e/1-1-u/22<1and g maps D into D∖0. Hence, there exists a holomorphic function w mapping the unit disk D into the right half-plane H+=z:Re⁡z>0 and such that gu=exp-wu for all u∈D. Observe that w1=0 since g1=1 and d=-w′1=1. Invoking Lemma 2, it follows that (37)Re⁡wx≤21-x1+x,for all x∈-1,1. Hence (38)gx=exp⁡-wx=exp⁡-Re⁡wx≥exp⁡-21-x1+x,for all x∈-1,1. That is, (39)1-2log⁡1+x2-1f′1-x2ζ≥exp⁡-21-x1+x,for all x∈-1,1.
Making the changer=1-x/2, we obtain that r∈0,1 and (40)f′rζ≥exp⁡-2r1-r1-2log⁡1-r.Therefore, if we consider r=z with z∈D≠0 and we take ζ=z/z, we conclude that (41)f′z≥exp⁡-2z1-z1-2log⁡1-z=F′z.This shows inequality (1).
Now, if there existsf0∈βlog⁡∞ such that f0′z0=F′z0 for some z0∈D∖0, then arguing as in the proof of inequality (1), for ζ0=z0/z0, the function, (42)g0z=1-2log⁡1+z2-1f0′1-z2ζ0,maps D into D∖0. Hence, there exists a holomorphic function w mapping D into H+ such that w1=0, -w′1=1>0, and (43)g0z=exp⁡-wz,for all z∈D. In particular, for x=1-2z0∈-1,1, we have (44)exp⁡-Re⁡wx=g0x=f0′1-x/2ζ01-2log⁡1+x/2=f0′z01-2log⁡1-z0=F′z01-2log⁡1-z0=exp⁡-2z01-z0=exp⁡-21-x1+x.Thus,(45)Re⁡wx=21-x1+x,for some x∈-1,1 and, by Lemma 2, we conclude that (46)wz=21-z1+zfor all z∈D. Therefore, (47)f0′1-z2ζ0=1-2log⁡1+z2g0z=1-2log⁡1+z2exp⁡-21-z1+z,for all z∈D. Hence, changing 1-z/2 by z, which belongs to D1/2,1/2, and using the identity principle for holomorphic functions, we obtain that (48)f0′zζ0=F′z,for all z∈D. This last relation implies that Flog⁡≤1 which is a contradiction to Proposition 7. This complete the proof of item (1).
(2) Arguing as in the proof of part (1), for ζ=1 fixed, we set the function (49)gu=1-2log⁡1+u2-1f′1-u2ζ,with u∈D. We have shown that there exists a holomorphic function w such that gu=exp⁡-wu for all u∈D. Furthermore, w satisfies the hypothesis of Julia’s Lemma (Lemma 1); that is, w is a holomorphic function in D∪1, which maps D into H+, w1=0, and -w′1=d=1. Hence, for r=1-x/1+x with x∈0,1 fixed, w maps the horodisk Δ¯1,r into the open Euclidean disk with center at 1-x/1+x and radius 1-x/1+x. In particular, since x∈Δ¯1,r, we have that wx∈Dx=D1-x/1+x,1-x/1+x. Thus, Lemma 4 allows us to write (50)Re⁡gx=Re⁡exp⁡-wx≥min⁡Re⁡exp⁡-z:z∈Dx=exp⁡-21-x1+x,for all x∈0,1. This last inequality is equivalent to writing (51)Re⁡1-2log⁡1+x2-1f′1-x2ζ≥exp⁡-21-x1+x,for all x∈0,1 and from here we have (52)Re⁡f′1-x2ζ≥exp⁡-21-x1+x1-2log⁡1+x2.Making the change r=1-x/2, we obtain r∈0,1/2 since x∈0,1 and also(53)Re⁡f′rζ≥exp⁡-2r1-r1-2log⁡1-r.We conclude, as before, that (54)Re⁡f′z≥exp⁡-2z1-z1-2log⁡1-z,for all z≤1/2. This shows the inequality in the second part of the theorem.
Now, if there exists a functionf0∈βlog⁡∞ such that Re⁡f′z0=F′z0 for some z0∈D0,1/2, then we can define ζ0=z0/z0 and the function (55)g0z=1-2log⁡1+z2-1f0′1-z2ζ0,which maps D into D∖0. Hence, as before, there exists a holomorphic function w mapping D into H+ such that w1=0, -w′1=1>0, and g0z=exp⁡-wz for all z∈D. In particular, for x=1-2z0∈0,1, we have (56)Re⁡exp⁡-wx=Re⁡g0x=Re⁡f0′1-x/2ζ01-2log⁡1+x/2=Re⁡f0′z01-2log⁡1-z0=F′z01-2log⁡1-z0=exp⁡-2z01-z0=exp⁡-21-x1+x;that is, wx is the value in Dx where Re⁡exp⁡-wx attain its minimum value, but by the proof of Lemma 4, we know that this happens if Imwx=0 and Re⁡wx=-21-x/1+x. Now, by Lemma 2, we conclude that wz=21-z/1+z for all z∈D. As before, this last fact implies that f0′zζ0=F′z for all z∈D and therefore Flog≤1 which is a contradiction to Proposition 7. This completes the proof of item (2).


## 5. Distortion Theorems for Complex Functions inBlog⁡ Having Branch Points

In this section we establish a distortion theorem for functions in the closed unit ball ofBlog having branch points and satisfying a normalized Bloch conditions. More precisely, for each n∈N, we denote by βlogn the class of all holomorphic functions f∈Blog such that f0=0, f′0=1, flog≤1 and if f′b=0 for some b∈D then fkb=0 for all k=1,2,…,n. Clearly we have (57)βlog⁡∞=⋂n=1∞βlog⁡n.With these notations, we have the following result.Theorem 10.
Forn∈N fixed, we set an=n/n+2. Then for every f∈βlog⁡n we have the following:(1)
f′z≥Fn′z=an-z/an-an2zn1-2log⁡1-anz,  for all z≤an. There is not a function f0∈βlog⁡n such that f0′z0=Fn′z0 for some z0∈D¯0,an∖0.(2)
Ref′z≥Fn′z=an-z/an-an2zn1-2log⁡1-anz, for all z≤nn+2/2n+1. Furthermore, there is not a function f0∈βlog⁡n such that Re⁡f0′z0=Fn′z0 for some z0∈D¯0,nn+2/2n+1∖0.Proof.
(1) Let us fixζ=1, n∈N and an=n/n+2<1. We set the function (58)gu=1-2log⁡1-an2+2log⁡1-an2u-1f′an-anu1-an2uζ,with u∈D, where log⁡w denotes the principal logarithmic of the complex number w∈D1,1. Clearly the function g is holomorphic on D∪1 and g1=1 because f′0=1. Also, we have (59)g′u=-1-2log⁡1-an2+2log⁡1-an2u-2-2an21-an2uf′an-anu1-an2uζ+1-2log⁡1-an2+2log⁡1-an2u-1f′′an-anu1-an2uζanan2-11-an2u2ζ.And hence g′1=2an2/1-an2=n since f′0=1 and f′′0=0 (by Lemma 6). Furthermore, since f∈βlog⁡n and gu0=0 if and only if f′an-anu0/1-an2u0ζ=0, we conclude that all the zeros of the function g have multiplicity at less n.
On the other hand, sinceflog⁡≤1 we have (60)gu≤1-2log⁡1-an2+2log⁡1-an2u-1log⁡e1-an-anu/1-an2u2≤1Re⁡1-2log⁡1-an2+2log⁡1-an2ulog⁡e1-an-anu/1-an2u2=log⁡e1-an2/1-an2u2-1log⁡e1-an-anu/1-an2u2<1,for all u∈D, since (61)1-an21-an2u2<1-an-anu1-an2u2,for all u∈D. Hence, we have shown that gD⊂D. Invoking Lemma 3, we conclude that gx≥xn for all x∈0,1. Therefore, for each x∈0,1, the following estimation holds: (62)1-2log⁡1-an2+2log⁡1-an2x-1f′an-anx1-an2xζ≥xn.That is, (63)f′an-anx1-an2xζ≥xn1-2log⁡1-an2+2log⁡1-an2x,for all x∈0,1 since an∈0,1.
Next, we make the changer=an-anx/1-an2x. Then r∈0,an since x∈0,1, x=1/anan-r/1-anr and we can write (64)f′rζ≥1anan-r1-anrn1-2log⁡1-an2+2log⁡1-an21anan-r1-anr=1anan-r1-anrn1-2log⁡1-anr.Finally, if we set ζ=z/z and r=z, we conclude that (65)f′z≥1anan-z1-anzn1-2log⁡1-anz=1-z/an1-anzn1-2log⁡1-anz=Fn′z,for all z∈0,an. This shows the inequality in part (1).
The proof of the second part is similar to part (1) of Theorem 9. If there exists a function f0∈βlog⁡n such that f0′z0=Fn′z0 for some z0∈D¯0,an∖0, then we set ζ=z0/z0 and the function (66)g0u=1-2log⁡1-an2+2log⁡1-an2u-1f0′an-anu1-an2uζwith u∈D. We have showed that g0 satisfies all the hypothesis of Lemma 3. Furthermore, choosing x∈0,1 such that (67)an-anx1-an2x=z0we obtain (68)g0an-z0an-an2z0=g0x=1-2log⁡1-an2+2log⁡1-an21-anz0-1f0′z0=Fn′z01-2log⁡1-anz0=an-z0an-an2z0n.By Lemma 3, we conclude that g0z=zn for all z∈D. Hence, (69)f0′an-anz1-an2zζ=1-2log⁡1-an2+2log⁡1-an2zzn,for all z∈D. Changing an-anz/1-an2z by z, we obtain that (70)f0′zζ=1-2log⁡1-anzan-zan-an2zn=Fn′z,for all z∈D0,an and consequently for all z∈D. This last equality implies that Fnlog⁡=f0log⁡≤1 which is a contradiction to Proposition 8.
(2) As before, for n∈N, we set an=n/n+2, we fix ζ=1, and we define the function (71)gu=1-2log⁡1-an2+2log⁡1-an2u-1f′an-anu1-an2uζ,with u∈D. In the first part we have shown that this function satisfies the hypothesis of Lemma 3. Hence Re⁡gx≥xn for all n-1/n+1≤x<1. Therefore (72)Re⁡1-2log⁡1-an2+2log⁡1-an2x-1f′an-anx1-an2xζ≥xn,and thus we have (73)1-2log⁡1-an2+2log⁡1-an2x-1Re⁡f′an-anx1-an2xζ≥xn,which is the same as (74)Re⁡f′an-anx1-an2xζ≥xnlog⁡e1-an2/1-an2x2.As before, we make the change r=an-anx/1-an2x; then 0<r≤nn+2/2n+1, x=1/anan-r/1-anr, and (75)Re⁡f′rζ≥1anan-r1-anrnlog⁡e1-an2/1-an21/anan-r/1-anr2=1anan-r1-anrn1-2log⁡1-anr.Setting ζ=z/z and r=z, we conclude that (76)Re⁡f′z≥1anan-z1-anzn1-2log⁡1-anz,for all z≤nn+2/(2n+1). This shows the inequality in part (2).
If there exists a functionf0∈βlog⁡n such that Re⁡f0′z0=Fn′z0 for some z0∈D¯0,nn+2/(2n+1)∖0, then we set ζ=z0/z0 and the function (77)g0u=1-2log⁡1-an2+2log⁡1-an2u-1f0′an-anu1-an2uζ,with u∈D. We have showed that g0 satisfies all the hypothesis of Lemma 3. Furthermore, choosing x∈0,1 such that (78)an-anx1-an2x=z0,we obtain (79)x=1anan-z01-anz0,and since 0<z0≤ann+2/2n+1 we can see that n-1/n+1≤x<1. Furthermore, (80)Re⁡g0an-z0an-an2z0=Re⁡g0x=1-2log⁡1-an2+2log⁡1-an21-anz0-1Re⁡f0′z0=Fn′z01-2log⁡1-anz0=an-z0an-an2z0n.By Lemma 3, we conclude that g0z=zn for all z∈D. Hence, as before, this last fact implies that Fnlog⁡=f0log⁡≤1 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 Sections4 and 5. We recall that if f is a holomorphic function on D and z0∈D, rsz0,f denote the radius of the largest schlicht disk on the Riemann surface fD centered at fz0 (a schlicht disk on fD centered at fz0 means that f maps an open subset of D containing z0 conformally onto this disk). With this notation, we have the following results.Corollary 11.
Iff∈βlog⁡∞, then (81)rs0,f≥∫01F′zdz=∫01exp⁡-2t1-t1-2log⁡1-tdt.Proof.
From the definition ofrs0,f, it follows the fact that there exists a simply connected domain E⊂D containing the zero such that f maps E conformally onto an Euclidean disk with center at f0 and radius rs0,f. This Euclidean disk must meet the boundary of fD because, in other cases, the boundary of the set E is a Jordan curve in the interior of D and we can find an open set W⊂D where f is univalent; hence fW contain an Euclidean disk with center at f0 and radius greater than rs0,f, which contradict with the definition of rs0,f. We conclude then that there is a radial segment Γ jointing f0 to the boundary of fD. Let γ be inverse image of Γ under f; then γ joint the point 0 to the boundary of D. Thus, from Theorem 9, it follows that (82)rs0,f=∫Γdw=∫γf′zdz≥∫01F′γtγ′tdt=∫01F′γtγtγ′tγtdt≥∫01F′γtγt·γ′tγtdt=∫01F′τdτ=∫01exp⁡-2τ1-τ1-2log⁡1-τdτ≈0.4104136111,where we have used Cauchy-Schwarz’s inequality in the fourth line, γt·γ′t is the scalar product of γt and γ′t, and we have made the change τ=γt=γt·γt, where τ→1- as t→1-. This shows the result.While for functions in the classβlog⁡n we have the following.Corollary 12.
Suppose thatn∈N is fixed. If f∈βlog⁡n, then (83)rs0,f≥∫0n/n+2Fn′tdt=∫0n/n+21-t/an1-antn1-2log⁡1-antdt.Proof.
Indeed, arguing as in the proof of Corollary11, from Theorem 10, it follows that (84)rs0,f=∫Γdw=∫γf′zdz≥∫0n/n+2Fn′τdτ=∫0n/n+2an-τan-an2τn1-2log⁡1-anτdt,where we have used that τ=γt→an as t→1-. This shows the result.



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*Source: 8694516-2017-04-16.xml*