MAPPING PROPERTIES OF NONLINEAR INTEGRAL OPERATORS AND PRE-SCHWARZIAN DERIVATIVES

YONG CHAN KIM, S. PONNUSAMY, AND TOSHIYUKI SUGAWA

Abstract. In this paper, nonlinear integral operators on normalized analytic functions in the unit disk are investigated in connection with the pre-Schwarzian derivative and the Hornich operation. In particular, several non-trivial relations between these operators and the class of strongly starlike functions will be deduced.

1. Introduction

H. Hornich [13] defined an unusual operation (often called the Hornich operation) on the set of locally univalent (analytic) functions in the unit disk (or, more generally, a convex domain). Without essential loss of generality, we may restrict ourselves to the set A of analytic functions f in the unit disk 𝔻 = {|z| < 1} normalized by f(0) = 0 = f′(0) − 1. Let f and g be locally univalent functions in A and α be a complex number. Then the Hornich operation is defined by

f ⊕ g(z) = ∫_{0}^{z} f′(w)g′(w)dw, and
α ⋆ f(z) = ∫_{0}^{z} {f′(w)}^{α}dw,

where the branch of (f′)^{α} = exp(α log f′) is taken so that (f′)^{α}(0) = 1. Thus, this manipulation gives a structure of vector space to the set LU of locally univalent functions in A, namely, LU = {f ∈ A : f′(z) ≠ 0 ∀z ∈ 𝔻}.

Hornich [13] also introduced a norm to a subset of LU which makes it a separable real Banach space with the above operation. After then, J. A. Cima and J. A. Pfaltzgraff [9] gave another separable real Banach space structure to a slightly different set from Hornich’s one. These spaces, however, both do not contain the whole set S of normalized univalent functions. On the other hand, D. M. Campbell, Cima and Pfaltzgraff [8] considered a complex Banach space structure on the set of locally univalent functions of finite order, where the order of a function f is defined by

1991 Mathematics Subject Classification. 30C45, 30C50.

Key words and phrases. univalent function, integral transform, Hornich operation, Teichmüller space.

A part of the present research was carried out during the third author’s visit to the University of Helsinki under the bilateral exchange program of scientists between the Academy of Finland and the JSPS. The first author was supported by grant No. R05-2001-000-00020-0 from the Basic Research Program of the Korea Science and Engineering Foundation.

ord(f) = sup_{z∈𝔻} |−z̄ + \frac{1}{2}(1 − |z|^{2})\frac{f′′(z)}{f′(z)}| ,

see [24]. This space is non-separable, but has the advantage that it contains the whole S as a closed subset with non-empty interior as we shall explain later. It is quite easy to see that ord(f) < ∞ if and only if the norm

∥f∥ = sup_{z∈𝔻}(1 − |z|^{2})|T_{f}(z)|

is finite, where T_{f} = f′′/f′ denotes the pre-Schwarzian derivative T_{f} = f′′/f′ of f. Note that the Schwarzian derivative of f is defined as S_{f} = (f′′/f′)′−(f′′/f′)^{2}/2 = T′_{f} −(T_{f})^{2}/2. It should also be noted that the norm ∥f∥ is nothing but the Bloch semi-norm of the function log f′. The above norm is not same as, but equivalent to, that considered in [8].

S. Yamashita observed in [29] that f ∈ LU is of finite order if and only if f is uniformly locally univalent in 𝔻, namely, there is a positive constant ρ for which f is univalent in every hyperbolic disk in 𝔻 of radius ρ. Earlier than this, Ch. Pommerenke obtained in [24, Satz 2.6] an explicit estimate for the radius ρ of the univalent hyperbolic disk for f in terms of ord(f).

We can now view the Hornich operation more naturally through the pre-Schwarzian derivative. Indeed, since T_{f⊕g} = T_{f} + T_{g} and T_{α⋆f} = αT_{f}, it is just the transformation of the usual linear operation under the inverse of taking pre-Schwarzian derivative. This simple fact is, however, a source of ideas developed in the present paper. Note that this point of view was used by Yamashita [28] in a more general context.

The complex Banach space B = {f ∈ LU : ∥f∥ < ∞} with the Hornich operation and the norm ∥f∥ is thus a natural object to investigate. It is well known that S is closed in B and contained in {f ∈ B : ∥f∥ ≤ 6}. The Koebe function is an example so that ∥f∥ = 6. Also, sufficient conditions for univalence and boundedness for f are known.

Theorem A.

(i) If ∥f∥ ≤ 1, then f is univalent in 𝔻 and ∥f∥ ≤ k < 1, then f has a K-quasiconformal extension to ℂ, where K = (1 + k)/(1 − k).

(ii) If ∥f∥ < 2, then f is bounded in 𝔻. The bounds 1 and 2 are sharp.

The first assertion was proved by J. Becker [4]. The sharpness of the constant 1 is due to [5]. The second assertion is immediate. The reader can find a proof of it in [15] with examples showing the sharpness of the bound.

The set T ⊂ S consisting of those functions f which have quasiconformal extensions to the Riemann sphere ℂ̂ = ℂ ∪ {∞} is known to be an open set in B and considered as a model of the universal Teichmüller space. See [2] and [31]. One of the most interesting features of T is the following theorem due to I. V. Zhuravlev [31].

Theorem B. The space T decomposes into the uncountably many connected components T_{0} and T_{ζ}, ζ ∈ ∂𝔻, where

T_{0} = {f ∈ T : f(𝔻) is bounded } and T_{ζ} = {f ∈ T : f(z) → ∞ as z → ζ}.

Moreover, {f ∈ B : ∥f − L_{ζ}∥ < 1} ⊂ T_{ζ} holds for each ζ ∈ ∂𝔻, where L_{ζ}(z) = z/(1 − ζ̄z).

We can see, in particular, that {f ∈ B : ∥f∥ < 1} ⊂ T_{0} and {f ∈ B : ∥f − L_{ζ}∥ < 1} ⊂ T_{ζ} ⊂ {f ∈ B : 2 < ∥f∥ < 6} for ζ ∈ ∂𝔻. The non-separability of B follows also from the above result (cf. [8]). K. Astala and F. W. Gehring showed in [2] that the interior of S in B coincides with T but the closure of T does not coincide with S.

The above aspects can be used to consider problems about some integral operators. Let I_{α}[f] = α ⋆ f for f ∈ LU and J_{α}[f] := I_{α}[J[f]] for f ∈ ZF = {f ∈ A : f(z) ≠ 0 for all z ∈ 𝔻 \ {0}}, where α ∈ ℂ and J stands for the Alexander transformation. More explicitly,

I_{α}[f](z) = ∫_{0}^{z} \{f′(w)\}^{α}dw and J_{α}[f](z) = ∫_{0}^{z} \{\frac{f(w)}{w}\}^{α} dw.

Note that the map J = J_{1} : ZF → LU is bijective and that the inverse of J can be represented by J^{−1}[f](z) = zf′(z). It is also well known that J(S^{∗}) = K, where S^{∗} and K stand for the classes of starlike and convex functions, respectively (see [1]). These operators have been studied by many authors. We refer the reader to [10, § 8.5] and [12, Chapter 15] for basic information about these operators, and standard terminology in the theory of univalent functions, as well.

Problems concerning the operator J_{α} can reduce to ones concerning I_{α} instead, if we once know about the image under the Alexander transformation J because of the relation J_{α} = I_{α} ∘ J. For example, we know that J(S^{∗}) = K, and therefore, J_{α}(S^{∗}) = I_{α}(K). Though one can regard J as a smoothing operator, the behaviour of J is not so simple as it is known that J(S) is not contained in S. On the other hand, we certainly have a better estimate for functions in J(S) than the estimate ∥f∥ ≤ 6 for f ∈ S. In Section 2, we give a proof of the following result as well as some remarks. Compare with the inequality ∥I_{α}[f]∥ = |α|∥f∥ ≤ 6|α| for f ∈ S.

Theorem 1.1. The inequality ∥J_{α}[f]∥ ≤ 4|α| holds for every f ∈ S and every complex number α. The bound is sharp.

Pfaltzgraff [23] showed that I_{α}(S) ⊂ S if |α| ≤ 1/4. On the other hand, W. C. Royster [25] proved that, for each number α other than 1 with |α| > 1/3, there is a function f in S with I_{α}(f) ∉ S. Up to now, nothing better has been obtained in this general situation.

4 Y. C. KIM, S. PONNUSAMY, AND T. SUGAWA

The problem to find the sharp constant now reduces to find the supremum of the ratio of the outer and the inner radii of the set U(f) = {α ∈ ℂ : α ⋆ f ∈ S} (1.1)

around the origin when f runs all over the set S. Indeed, Royster’s observation was essentially that U(f_{1}) = {α ∈ ℂ : |α| ≤ 1 or |α − 2| ≤ 1}, where f_{1}(z) = log(1 + z). Note that ∥f_{1}∥ = 2 and that 2 ⋆ f_{1} = L_{−1}. Especially, it may be interesting to observe that {α ∈ ℂ : α ⋆ f_{1} ∈ T_{0}} = {|α| < 1} and {α ∈ ℂ : α ⋆ f_{1} ∈ T_{−1}} = {|α − 2| < 1} (see Theorem B). The difficulty of determining the set {α ∈ ℂ : I_{α}(S) ⊂ S} seems to come from the fact that we have only very few functions f for which the shapes of U(f) are completely determined.

By virtue of the relation J(S^{∗}) = K, Theorem 1 of E. P. Merkes [19] can read as I_{α}(K) ⊂ S for |α| ≤ 1/2, where the constant 1/2 is sharp. Relating this result, he gave in [19] the conjecture that I_{α}(K) ⊂ S for |α − 1| ≤ 1/2. We will use the above aspects to settle the conjecture by giving a complete characterization of α satisfying I_{α}(K) ⊂ S.

Theorem 1.2. The set M = {α ∈ ℂ : I_{α}(K) ⊂ S} = {α ∈ ℂ : J_{α}(S^{∗}) ⊂ S} equals the union of the closed disk |α| ≤ 1/2 and the line segment [1/2, 3/2].

Concerning the Hornich operation, the linear structure of typical classes of univalent functions has been investigated. For convenience, we denote by [f, g] the closed line segment joining f and g, namely, [f, g] = {(1 − t) ⋆ f ⊕ t ⋆ g : 0 ≤ t ≤ 1}. It is shown in [9] that the class K of convex functions is convex, namely, [f, g] ⊂ K for any pair of functions f and g in K. Y. J. Kim and Merkes [17] proved that the class C of close-to-convex functions is also convex. In contrast, we will see that the class S^{∗} of starlike functions is not convex in Section 3. At least, however, it is reasonable to pose the following problem:

Problem 1.3. Is the class S^{∗} starlike with respect to the origin concerning the Hornich operation?

In other words, is it true that [id, f] ⊂ S^{∗} for each f ∈ S^{∗}? We consider this problem and give some partial solutions to it in Section 3.

Let α be a non-negative number. A function f ∈ A is called strongly starlike of order α if |arg(zf′(z)/f(z))| ≤ απ/2 for z ∈ 𝔻. (Here and hereafter, for a zero-free analytic function p on 𝔻 with p(0) = 1, arg p(z) is thought of a single-valued harmonic function in 𝔻 with normalization arg p(0) = 0.) We denote by SS(α) the set of strongly starlike functions in A of order α. Note that SS(α) ⊂ SS(1) = S^{∗} for 0 ≤ α ≤ 1. As is well known, for α ∈ (0, 1), each function f ∈ SS(α) is bounded (see [7]) and has a K(α)-quasiconformal extension to ℂ, where K(α) = (1 + sin(απ/2))/(1 − sin(απ/2)) (see [11]). In particular, SS(α) ⊂ T_{0} for α ∈ (0, 1). For further properties of strongly starlike functions, see, in addition, [26] and [27].

We will next consider the following problem.

NONLINEAR INTEGRAL OPERATORS AND PRE-SCHWARZIAN DERIVATIVES 5

Problem 1.4. Find a sufficient condition on f ∈ A under which I_{α}[f] ∈ SS(α) for all α ∈ [0, 1].

Note that f ∈ S^{∗} ∩ \overline{T_{0}} is a necessary condition for the above property. In particular, a function f in S^{∗} ∩ T_{ζ} (such as f = L_{ζ}) for some ζ ∈ ∂𝔻 does not have the above property. On the other hand, the condition I_{α}[f] ∈ SS(α) for all α ∈ [0, 1] clearly implies [id, f] ⊂ S^{∗}.

One may suspect that the conclusion in this problem is too strong. However, a mild condition can guarantee the positive answer to the problem as we see in the next couple of results, which will be proved in Section 4 in a slightly more general form.

Theorem 1.5. Let f ∈ A. Suppose that Re f′ > 0 in 𝔻 and that arg f′(tz) lies between 0 and arg f′(z) for each t ∈ [0, 1] and z ∈ 𝔻. Then I_{α}[f] ∈ SS[α] for α ∈ [0, 2]. In particular, f ∈ S^{∗}.

As is well known, J. W. Alexander [1] showed that if f ∈ A satisfies Re f′ > 0 in 𝔻 then f ∈ S. However, without any additional condition, Re f′ > 0 does not imply f ∈ S^{∗}. The authors learned from J. Stankiewicz that this fact was first pointed out by J. Krzyż [18]. For convenience of the reader, we will give a simplified example as well as related results in Section 4.

We give a different kind of sufficient condition for Problem 1.4:

Theorem 1.6. Suppose that f ∈ A satisfies the condition

|Im \frac{zf′′(z)}{f′(z)}| ≤ \frac{π^{2}}{8G}, z ∈ 𝔻,

where G is Catalan’s constant. Then I_{α}[f] ∈ SS[α] for each α ∈ [0, 1]. In particular, f ∈ S^{∗}.

The proof of the theorem will be given also in Section 4. We recall that Catalan’s constant G is given by

G = ∑_{n=0}^{∞} \frac{(−1)^{n}}{(2n + 1)^{2}} = 0.915965594 …. (1.2)

For interesting formulae involving Catalan’s constant, see [6, Chapter 9]. Note also that π^{2}/8G ≈ 1.346885252.

2. Proof of Theorems 1.1 and 1.2

We begin with the proof of Theorem 1.1.

Proof of Theorem 1.1. Since T_{J_{α}[f]} = αT_{J[f]}, it is enough to show the inequality ∥J[f]∥ ≤ 4 for f ∈ S. For a function f in S, the inequality due to Grunsky

|log \frac{zf′(z)}{f(z)}| ≤ log \frac{1 + |z|}{1 − |z|}

holds (see [10, p. 126]). Set w = log(zf′(z)/f(z)). Then we compute

|\frac{zf′(z)}{f(z)} − 1| = |e^{w} − 1| ≤ ∑_{k=1}^{∞} \frac{|w|^{k}}{k!} = e^{|w|} − 1 ≤ \frac{1 + |z|}{1 − |z|} − 1 = \frac{2|z|}{1 − |z|}.

Therefore, we come to the estimate

(1 − |z|^{2})|T_{J[f]}(z)| = (1 − |z|^{2}) |\frac{f′(z)}{f(z)} − \frac{1}{z}| ≤ (1 − |z|^{2}) \frac{2}{1 − |z|} = 2(1 + |z|).

Hence, ∥J[f]∥ ≤ 4. The sharpness of the bound can be seen by the Koebe function.

The idea of using the Grunsky inequality was suggested by Vladimir Gutlyanskiĭ to one of the authors. They would like to express their sincere thanks to him.

As a consequence of the inequality ∥J[f]∥ ≤ 4 for f ∈ S, it follows that J[f] is uniformly locally univalent in 𝔻. Note that the same is no longer true for functions f in B because f(z)/z may have zeros.

Combining Theorem 1.1 with Theorem A, we obtain the assertion that J_{α}(S) ⊂ S for each α with |α| ≤ 1/4. Note that Y. J. Kim and Merkes [16] first showed this relation by using the weaker inequality (1 − |z|^{2})|zT_{J[f]}(z)| ≤ 4 for f ∈ S.

We next show Theorem 1.2.

Proof of Theorem 1.2. The part {|α| ≤ 1/2} ⊂ M was proved by Merkes [19]. On the other hand, the inclusion relation [0, 3/2] ⊂ M is due to M. Nunokawa [22]. We show that any other point does not belong to M. By definition, the set M can be described by

M = ⋂_{f∈K} U(f),

where U(f) is the set given by (1.1). First, we consider the function f_{1}(z) = log(1+z) and set f_{α} = I_{α}[f_{1}] for α ∈ ℂ. As we noted in Introduction, Royster [25] showed that U(f_{1}) = {|α| ≤ 1 or |α − 2| ≤ 1}. We observe here the simple relation U(f_{α}) = {β : αβ ∈ U(f_{1})} for α ∈ ℂ. By the relation 1+zf′′_{α}/f′_{α} = 1−αz/(1+z) and by the fact that the image of 𝔻 under z/(1 + z) is the half-plane Re w < 1/2, we see that {α ∈ ℂ : f_{α} ∈ K} = [0, 2]. Since f_{2} ∈ K, we get first M ⊂ U(f_{2}) = {|α| ≤ 1/2 or |α − 1| ≤ 1/2}. This is the reason why Merkes came to the aforementioned conjecture. In particular, we obtain M ⊂ {|α| ≤ 3/2}. Secondly, we take any number r_{0} from (1/2, 3/2]. Then f_{1/r} ∈ K, and thus, M ⊂ U(f_{1/r}) for each r ∈ (1/2, r_{0}). In particular,

{α ∈ M : |α| = r_{0}} ⊂ ⋂_{1/2<r<r_{0}}{α ∈ U(f_{1/r}) : |α| = r_{0}} = {r_{0}}.

It is now shown that {α ∈ M : |α| > 1/2} ⊂ (1/2, 3/2].

NONLINEAR INTEGRAL OPERATORS AND PRE-SCHWARZIAN DERIVATIVES 7

3. The class of starlike functions

In this section, we consider the familiar class S^{∗} of starlike functions. First, we show the next result by giving an example of two starlike functions f and g so that h = (1/2)⋆(f ⊕g) is not starlike.

Theorem 3.1. The class S^{∗} of starlike functions is not convex concerning the Hornich operation.

We remark on the subtlety of this fact. Since S^{∗} ⊂ C and C is convex [17], we can see that the segment [f, g] is contained entirely in C for f, g ∈ S^{∗}. Therefore, when we try to construct such an example as above, we cannot choose f and g so that the midpoint h is not univalent.

Proof. Putting γ = e^{πi/4}, we now define the functions f and g in S^{∗} by f′(z) = (1 + γz)/(1 − γz)^{3} and g′(z) = (1 + γ̄z)/(1 − γ̄z)^{3}. Note that f and g both are rotations of the Koebe function z/(1 − z)^{2}, and therefore, starlike functions. Then the midpoint h of f and g can be expressed in the form

h(z) = ∫_{0}^{z} \frac{(1 + γw)^{1/2}(1 + γ̄w)^{1/2}}{(1 − γw)^{3/2}(1 − γ̄w)^{3/2}} dw = ∫_{0}^{z} \sqrt{\frac{1 + \sqrt{2}w + w^{2}}{(1 − \sqrt{2}w + w^{2})^{3}}} dw,

which is a variant of the Schwarz-Christoffel transformation. Indeed, on the boundary of the unit disk, we can write

h(e^{iθ}) = \begin{cases}h(1) + \frac{i}{2} ∫_{0}^{θ} \sqrt{\frac{cos t + 1/\sqrt{2}}{(cos t − 1/\sqrt{2})^{3}}} dt, & |θ| < π/4, \\ h(−1) − \frac{i}{2} ∫_{0}^{θ−π} \sqrt{\frac{cos t − 1/\sqrt{2}}{(cos t + 1/\sqrt{2})^{3}}} dt, & |θ − π| < 3π/4. \end{cases}

Therefore, we see that h(𝔻) = {z ∈ ℂ : Re z < h(1)} \ E, where E is the closed half parallel strip given by {z ∈ ℂ : Re z ≤ h(−1) and |Im z| ≤ a} and a is the positive number defined by

a = \frac{1}{2} ∫_{0}^{π/4} \sqrt{\frac{cos t − 1/\sqrt{2}}{(cos t + 1/\sqrt{2})^{3}}} dt.

In particular, the image h(𝔻) is not starlike with respect to the origin.

We next consider Problem 1.3. So far, we have no complete solution to it. We have yet the following sufficient condition for a function f ∈ S^{∗} to satisfy [id, f] ⊂ S^{∗}.

Theorem 3.2. Let f ∈ A. Suppose that the function zf′′(z)/f′(z) takes no values in the set E_{0} = \{x + yi : x ≤ −1 and |y| ≥ −\sqrt{3}x\}. Then f_{α} = I_{α}[f] ∈ S^{∗} for 0 ≤ α ≤ 1.

8 Y. C. KIM, S. PONNUSAMY, AND T. SUGAWA

This result can be obtained as an immediate consequence of the following special case of the Open Door Lemma due to P. Mocanu [21]. Here, E_{1} denotes the closed subset \{−1 + yi : |y| ≥ \sqrt{3}\} of ℂ consisting of two rays.

Lemma 3.3 (Open Door Lemma). Let f be a function in A. If zf′′(z)/f′(z) ∈ ℂ \ E_{1} for every z ∈ 𝔻, then f is starlike.

Proof of Theorem 3.2. For α ∈ [0, 1], zf′′_{α}(z)/f′_{α}(z) = αzf′′(z)/f′(z) ∈ {αw : w ∈ E_{0}} ⊂ ℂ \ E_{1}. Now Lemma 3.3 yields that f_{α} ∈ S^{∗}.

Note that, for the Koebe function K(z) = z/(1 − z)^{2}, the function zK′′(z)/K′(z) = 2z(2 + z)/(1 − z^{2}) is known to map the unit disk conformally onto the slit domain ℂ \ E_{1}. Lemma 3.3 means exactly that if zf′′(z)/f′(z) is subordinate to zK′′(z)/K′(z) then zf′(z)/f(z) is subordinate to zK′(z)/K(z) = (1 + z)/(1 − z) for a function f ∈ A. We also see that the Koebe function does not satisfy the hypothesis of Theorem 3.2 because the set E_{0} is larger than E_{1}. Though Theorem 3.2 does not imply [id, K] ⊂ S^{∗}, this claim itself can be proved directly (see Proposition 4.5 in the next section). Since the Koebe function is extremal in various aspects, the validity of the statement [id, K] ⊂ S^{∗} may be thought as supporting evidence for the affirmative answer to Problem 1.3. For sufficient conditions of different types for f ∈ S^{∗} to satisfy [id, f] ⊂ S^{∗}, see also results concerning Problem 1.4.

4. Strongly starlike functions

In this section, we concentrate on Problem 1.4. We first prove Theorem 1.5 by showing a slightly more general result. For θ_{0}, θ_{1} ∈ ℝ, we set

Γ[θ_{0}, θ_{1}] = {z ∈ ℂ : z ≠ 0, arg z = (1 − t)θ_{0} + tθ_{1} for some t ∈ [0, 1]}

and

Γ(θ_{0}, θ_{1}) = {z ∈ ℂ : z ≠ 0, arg z = (1 − t)θ_{0} + tθ_{1} for some t ∈ (0, 1)}.

It is important in the sequel to note that the set Γ[θ_{0}, θ_{1}] is convex if |θ_{1} − θ_{0}| < π and that Γ(θ_{0}, θ_{1}) is convex if |θ_{1} − θ_{0}| ≤ π. Then one can prove the following result, from which Theorem 1.5 follows.

Theorem 4.1. Let β be a positive constant and f ∈ A. If |arg f′(z)| < βπ/2 and if f′(tz) ∈ Γ[0, arg f′(z)] for each t ∈ [0, 1] and z ∈ 𝔻, then the function f_{α} = I_{α}[f] is strongly starlike of order αβ provided that αβ ≤ 2.

Proof. Let α ≤ 2/β. By assumption, f′_{α}(tz) = (f′(tz))^{α} ∈ Γ[0, α arg f′(z)] for t ∈ [0, 1] and z ∈ 𝔻. Therefore, the average

\frac{f_{α}(z)}{z} = ∫_{0}^{1} f′_{α}(tz)dt (4.1)

of f′_{α}(tz) belongs to the convex set Γ[0, α arg f′(z)] for each z ∈ 𝔻. Hence, zf′_{α}(z)/f_{α}(z) ∈ Γ[0, α arg f′(z)] ⊂ Γ[−αβπ/2, αβπ/2]. Now the conclusion follows.

Without the additional assumption about f′(tz) in Theorem 4.1, we would only have the conclusion that f_{α}(z)/z ∈ Γ[−αβπ/2, αβπ/2] for α ∈ [0, 1/β] merely from the above argument. Therefore, we still have the inequality |arg(zf′_{α}(z)/f_{α}(z))| ≤ αβπ for α ∈ [0, 1/β]. We record it as a proposition for a future reference.

Proposition 4.2. Suppose that f ∈ A satisfies |arg f′(z)| ≤ βπ/2 in 𝔻. Then f_{α} ∈ SS(2αβ) for α ∈ [0, 1/β]. In particular, if f′ has positive real part, |arg(zf′_{α}(z)/f_{α}(z))| ≤ απ for z ∈ 𝔻 and α ∈ [0, 1].

Example 4.3. We consider the case when f(z) = −2 log(1−z)−z. This function clearly satisfies the hypothesis in Theorem 1.5. Therefore, the function F_{α} defined by

F_{α}(z) = I_{α}[f](z) = ∫_{0}^{z} (\frac{1 + w}{1 − w})^{α} dw (4.2)

is strongly starlike of order α for α ∈ [0, 2], and hence, univalent in 𝔻 for α ∈ [0, 1]. On the other hand, F_{α} is not univalent in 𝔻 for α > 1, see [15].

We next show the following assertion, from which Theorem 1.6 follows as a corollary.

Theorem 4.4. Suppose that a function f ∈ A satisfies the inequality

|Im \frac{zf′′(z)}{f′(z)}| ≤ \frac{πβ}{2}

in |z| < 1, where β is a positive constant Then f_{α} = I_{α}[f] ∈ SS(4Gαβ/π) holds as long as 4Gαβ ≤ π, where G is Catalan’s constant given in (1.2).

Proof. Set v(z) = Im (zf′′(z)/f′(z)). Then v is a bounded harmonic function in 𝔻 with |v| ≤ πβ/2. Since v(0) = 0, the harmonic Schwarz lemma (cf. [3, Chapter 6]) yields the inequality |v(z)| ≤ 2β arctan |z|, z ∈ 𝔻. (4.3)

Next we observe the formula

\frac{d}{ds} log f′(sz) = \frac{zf′′(sz)}{f′(sz)}

for z ∈ 𝔻 and positive parameter s ≤ 1. Taking the imaginary part of the both sides and integrating it in s over the interval [t, 1], we obtain

arg \frac{f′(z)}{f′(tz)} = ∫_{t}^{1} Im (\frac{zf′′(sz)}{f′(sz)}) ds = ∫_{t}^{1} \frac{v(sz)}{s}ds.

By (4.3), we have

|arg \frac{f′(z)}{f′(tz)}| ≤ 2β ∫_{t}^{1} \frac{arctan(s|z|)}{s}ds

< 2β ∫_{0}^{1} \frac{arctan s}{s}ds

for each t ∈ [0, 1]. Since arctan x = ∑_{n=0}^{∞}(−1)^{n}x^{2n+1}/(2n + 1), we have the well-known relation

∫_{0}^{1} \frac{arctan x}{x}dx = ∑_{n=0}^{∞} \frac{(−1)^{n}}{2n + 1} ∫_{0}^{1} x^{2n}dx = ∑_{n=0}^{∞} \frac{(−1)^{n}}{(2n + 1)^{2}} = G.

We now conclude that the value f′(z)/f′(tz) belongs to the set Γ(−2Gβ, 2Gβ). This implies that (f′(tz)/f′(z))^{α} lies in Γ(−2Gαβ, 2Gαβ). Therefore, the average

\frac{f_{α}(z)}{zf′_{α}(z)} = ∫_{0}^{1} (\frac{f′(tz)}{f′(z)})^{α} dt

belongs to the convex set Γ(−2Gαβ, 2Gαβ) as long as 2Gαβ ≤ π/2. This means that f_{α} = I_{α}[f] ∈ SS(4Gαβ/π) when 4Gαβ ≤ π.

We are now ready to show the fact that [id, K] ⊂ S^{∗} by giving a more refined relation.

Proposition 4.5. Let K(z) = z/(1 − z)^{2} be the Koebe function. Then K_{α} = I_{α}[K] ∈ SS(min{1, 3α}) for α ∈ [0, 1].

Proof. We first note that arg K′(tz) ∈ Γ[0, arg K′(z)] for t ∈ [0, 1]. Indeed, when Im z ≥ 0, we have 0 ≤ arg((1 + tz)/(1 − tz)) ≤ arg((1 + z)/(1 − z)) and 0 ≤ arg(1/(1 − tz)) ≤ arg(1/(1 − z)). Since arg K′(z) = arg((1 + z)/(1 − z)) + 2 arg(1/(1 − z)), the required claim follows when Im z ≥ 0. When Im z ≤ 0, we show it in the same way as above.

Since |arg K′(z)| < 3π/2 in 𝔻, we conclude that K_{α} ∈ SS(3α) for 0 ≤ α ≤ 1/3 from Theorem 4.1.

Next, we consider the case when 1/3 < α < 1. Then the desired conclusion is that p_{α}(z) = zK′_{α}(z)/K_{α}(z) has positive real part in 𝔻. First note that p_{α} extends to a holomorphic function in some open neighbourhood of \overline{𝔻} \ \{1, −1\}. We now examine the behaviour of p_{α}(z) around z = 1. Since (1 + z)^{α} = 2^{α}(1 − (1 − z)/2)^{α} = 2^{α}[1 − α(1 − z)/2 + α(α − 1)(1 − z)^{2}/8 + ⋯] as z → 1, one obtains the asymptotic expansion of K′_{α} near to z = 1 in 𝔻 :

K′_{α}(z) = 2^{α}(1 − z)^{−3α} − α2^{α−1}(1 − z)^{1−3α} + α(α − 1)2^{α−3}(1 − z)^{2−3α} + O(1).

Integrating the above gives us the expansion

K_{α}(z) = \frac{2^{α}}{3α − 1}(1 − z)^{1−3α} − \frac{α2^{α−1}}{3α − 2}(1 − z)^{2−3α} + O(1)

as z → 1 in 𝔻, where the second term should be eliminated when 2 − 3α ≥ 0. Therefore,

p_{α}(z) = (3α − 1)\frac{z}{1 − z} (1 + O(|1 − z|)) = c \frac{1 + z}{1 − z} + O(1)

as z → 1 in 𝔻, where c = (3α − 1)/2 > 0.

On the other hand, p_{α}(z) is bounded around z = −1. Therefore, the real part of p_{α}(z) can be written in the form h(z) + c(1 − |z|^{2})/|1 − z|^{2}, where h is a bounded harmonic function in 𝔻. Now it remains to show that h > 0 in 𝔻. To this end, it suffices to see that P_{α}(θ) := 1/p_{α}(−e^{−iθ}) has non-negative real part for each θ ∈ (0, π). We have used here the symmetric property P_{α}(−θ) = P_{α}(θ) of P_{α}.

Since

K′_{α}(−e^{−iθ}) = \frac{(1 − e^{−iθ})^{α}}{(1 + e^{−iθ})^{3α}} = 2^{−2α}e^{iα(θ+π/2)} sin^{α}(θ/2) cos^{−3α}(θ/2), θ ∈ (0, π),

we compute

K_{α}(−e^{−iθ}) = (∫_{0}^{−1} + ∫_{−1}^{−e^{−iθ}}) \frac{(1 + w)^{α}}{(1 − w)^{3α}} dw

= − ∫_{0}^{1} \frac{(1 − x)^{α}}{(1 + x)^{3α}} dx + ie^{iαπ/2}2^{−2α} ∫_{0}^{θ} e^{i(α−1)t} sin^{α}(t/2) cos^{−3α}(t/2)dt

for θ ∈ (0, π). Therefore, we obtain

\frac{cos^{3α}(θ/2)}{sin^{α}(θ/2)} P_{α}(θ) = 2^{2α}e^{iα(θ+π/2)−iθ}C_{α} + i ∫_{0}^{θ} e^{i(1−α)(θ−t)} sin^{α}(t/2) cos^{−3α}(t/2)dt,

where C_{α} = ∫_{0}^{1}(1 − x)^{α}(1 + x)^{−3α}dx > 0, and thus, the real part of P_{α}(θ) has the same signature as

2^{2α}C_{α} cos {(1 − α)θ − απ/2} + ∫_{0}^{θ} sin(1 − α)(θ − t) sin^{α}(t/2) cos^{−3α}(t/2)dt.

It is clear that the integrand of the second term is nonnegative. On the other hand, since (1 − α)θ − απ/2 ≤ (1 − α)π − απ/2 = π/2 + (1 − 3α)π/2 ≤ π/2, the first term is also nonnegative. The proof has been completed.

Remark. It is easy to see that K_{α} = I_{α}[K] is unbounded when α ≥ 1/3. Recalling the fact that strongly starlike functions of order < 1 are bounded, we see that f = K_{α} does not satisfy the conclusion in Problem 1.4 for α > 1/3. On the other hand, the above proposition asserts that the function f = K_{1/3} does!

As we noted in Introduction, the condition Re f′ > 0 does not imply starlikeness of f. J. Krzyż [18] constructed a counterexample in a clever but a little complicated way. Since

12 Y. C. KIM, S. PONNUSAMY, AND T. SUGAWA

the reference [18] may be difficult to access for the reader, we give a somewhat simplified example for convenience.

Example 4.6. Let Ω be the Jordan domain {z ∈ ℂ : Re z > 0, Im z > −1} and p : 𝔻 → Ω be the conformal homeomorphism determined by p(0) = 1 and p(1) = 0. We can give an explicit form of p by p(z) = (1+i)\sqrt{(α − z)/(α + z)}−i, where α = (3+4i)/5, although it does not matter below. Then f(z) = ∫_{0}^{z} p(ζ)dζ is such a function, namely, Re f′ > 0 while zf′(z)/f(z) takes a value with negative real part for some z ∈ 𝔻. Note that p analytically extends to the point z = 1. We now use the following claim. We can see it directly but we prefer to give a geometric proof below in order to clarify what conditions are essential.

Claim. Im p(x) > 0 for 0 < x < 1.

Set q(z) = f(z)/z. Since q(1) can be written as the average ∫_{0}^{1} p(x)dx of p(x) over the interval [0, 1], the above claim implies Im q(1) > 0. By continuity, Im q(e^{iθ}) > 0 holds for sufficiently small θ > 0. On the other hand, p(e^{iθ}) takes the form i P (θ), where P (θ) < 0 for sufficiently small θ > 0. Therefore, Re (zf′(z)/f(z)) = Re (p(z)/q(z)) = P (θ)|q(e^{iθ})|^{−2}Im q(e^{iθ}) < 0 for z = e^{iθ} with θ > 0 small enough.

Proof of Claim. Note that the image of the segment (0, 1) under p is the hyperbolic geodesic in Ω joining 1 and 0. Therefore, the claim follows from the inequality ρ_{Ω}(z̄) < ρ_{Ω}(z) for z ∈ Ω^{−} = {z ∈ Ω : Im z < 0}, where ρ_{Ω} denotes the hyperbolic (or Poincaré) density of the domain Ω. The last inequality is implied by the reflection principle of hyperbolic metric due to D. Minda [20, Theorem 3].

We end this section with a small remark on Problem 1.4. If I_{α}[f] ∈ SS(α) for some α < 1, then the function I_{α}[f] is necessarily univalent and bounded in 𝔻. This conclusion itself can be deduced only from the assumption Re f′ > 0. Indeed, we have a stronger result as in the following.

Theorem 4.7. Suppose that e^{iβ}f′ has positive real part in 𝔻 for some f ∈ A and a real constant β. Then, I_{α}[f] ∈ S whenever |α| ≤ 1/2 or α ∈ [0, 1] and I_{α}[f] is bounded whenever |α| < 1. The latter bound is sharp.

Theorem 4.7 follows immediately from the next more general theorem up to the assertion that I_{α}[f] ∈ S for α ∈ [0, 1], which is, however, a direct consequence of the convexity of C because f ∈ C in this case.

A function p analytic in 𝔻 with p(0) = 1 is called Gelfer if p(z)+p(w) ≠ 0 for every pair of points z, w ∈ 𝔻. In particular, if e^{iβ}p has positive real part for some real constant β, then p is Gelfer. We refer the reader to [30] for interesting properties of Gelfer functions.

Theorem 4.8. Suppose that the derivative of a function f ∈ A is Gelfer. Then I_{α}[f] ∈ S whenever |α| ≤ 1/2 and I_{α}[f] is bounded whenever |α| < 1. The latter bound is sharp.

Proof. Gelfer’s theorem implies that ∥f∥ = sup(1 − |z|^{2})|f′′(z)/f′(z)| ≤ 2 (see [14] for a simple proof). Therefore, ∥I_{α}[f]∥ = |α|∥f∥ ≤ 1 for |α| ≤ 1/2. Becker’s theorem (Theorem A) now yields the univalence of I_{α}[f]. The boundedness follows from the inequality ∥I_{α}[f]∥ < 2 for |α| < 1. We see that this bound is sharp by considering the function F_{α} = I_{α}[f] defined in (4.2).

Since J_{α} = I_{α} ∘ J, the above theorem is translated into the following equivalent form through the Alexander transformation J.

Corollary 4.9. Suppose that the function f(z)/z is Gelfer for some f ∈ A. Then J_{α}[f] ∈ S whenever |α| ≤ 1/2 and J_{α}[f] is bounded whenever |α| < 1. The latter bound is sharp.

Acknowledgements. In December, 2001, the third author had a chance to give a seminar talk on the present research at the Technical University of Rzeszów, Poland. Then, Professor Jan Stankiewicz drew his attention to a paper [18] of J. Krzyż. The authors are grateful to him as well as Doctor Stanisława Kanas, who arranged the third author’s stay there, and all the participants of the seminar there.

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Department of Mathematics Education, Yeungnam University, 214-1 Daedong Gyongsan 712-749, Korea

  E-mail address: kimyc@yu.ac.kr

Department of Mathematics, Indian Institute of Technology, IIT-Madras, Chennai-600 036, India

  E-mail address: samy@acer.iitm.ernet.in

Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526 Japan

  E-mail address: sugawa@math.sci.hiroshima-u.ac.jp