GEOMETRIC PROPERTIES OF NONLINEAR INTEGRAL TRANSFORMS OF CERTAIN ANALYTIC FUNCTIONS

YONG CHAN KIM, S. PONNUSAMY, AND TOSHIYUKI SUGAWA

Abstract. By using norm estimates of the pre-Schwarzian derivatives for certain analytic functions defined by a nonlinear integral transform, we shall give several interesting geometric properties of the integral transform.

1. Introduction

Let H denote the class of all analytic functions in the open unit disk 𝔻 = {|z| < 1} and A denote the class of functions f ∈ H normalized by f(0) = 0 = f′(0) − 1. Also let S denote the class of all univalent functions in A. For 0 ≤ α < 1, let S^{∗} and K denote the familiar classes of functions in A that are starlike (with respect to origin) and convex, respectively. As is well known (cf. [4]), these two classes are analytically characterized, respectively, by

f ∈ K ⇔ Re (1 + \frac{zf′′(z)}{f′(z)}) > 0, z ∈ 𝔻,

and

f ∈ S^{∗} ⇔ Re (\frac{zf′(z)}{f(z)}) > 0, z ∈ 𝔻.

Note that f ∈ S^{∗} ⇔ J[f] ∈ K, where J[f] denotes the Alexander transform [1] of f ∈ A defined by

J[f](z) = ∫_{0}^{z} \frac{f(ζ)}{ζ} dζ = ∫_{0}^{1} f(tz)\frac{dt}{t}.

In 1960, Biernacki claimed that f ∈ S implies J[f] ∈ S, but this turned out to be wrong (see [4, Theorem 8.11]). This means that the Alexander integral operator J does not preserve the class S.

A function f ∈ A is said to be close-to-convex if there exists a (not necessarily normalized) convex function g such that

Re (\frac{f′(z)}{g′(z)}) > 0, z ∈ 𝔻.

We shall denote by C the class of close-to-convex functions in 𝔻. It is well known that a close-to-convex function is univalent (cf. [4]).

1991 Mathematics Subject Classification. 30C45, 30C50.

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

The first author was supported by Yeungnam University (Grant No.105603). The second author was supported by DST Sponsored Research project (Ref No. DST/MS/092/98). The third author was partially supported by the Academy of Finland during the stay at the University of Helsinki.

In [10], Y. J. Kim and Merkes considered the nonlinear integral transform J_{α} defined by

J_{α}[f](z) = ∫_{0}^{z}(\frac{f(ζ)}{ζ})^{α}dζ

for complex numbers α and for functions f in the class

ZF = {f ∈ A : f(z) ≠ 0 for all 0 < |z| < 1}

and showed that

J_{α}(S) = {J_{α}[f] : f ∈ S} ⊂ S

when |α| ≤ 1/4. Up to now, the best constant is not known for this reult. Also, Merkes [12] proved that, for α ∈ ℂ with |α| ≤ 1/2, the inequality

J_{α}(S^{∗}) ⊂ S

holds, where 1/2 is sharp. Note also that the authors has recentely proved in [7] that J_{α}(S^{∗}) ⊂ S precisely when either |α| ≤ 1/2 or α ∈ [1/2, 3/2]. More generally, for a given constant β > 0, it may be interesting to find a subclass F of A such that J_{α}(F) ⊂ S for all α ∈ ℂ with |α| ≤ β. The main purpose of this note is to give such classes F in a concrete way.

Let f : 𝔻 → ℂ be analytic and locally univalent. The pre-Schwarzian derivative T_{f} of f is defined by

T_{f}(z) = \frac{f′′(z)}{f′(z)}.

Also, with respect to the Hornich operation [5], the quantity

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

can be regarded as a norm of the space of uniformly locally univalent analytic functions f in 𝔻 (see [7] for details). Here, an analytic function f on 𝔻 is said to be uniformly locally univalent if f is univalent on each hyperbolic disk in 𝔻 with a fixed radius. Note, in fact, that f is uniformly locally univalent if and only if ∥f∥ < ∞ (see [17]). In connection with the above norm, the following result is important to note.

Theorem A. Let f be analytic and locally univalent in 𝔻. Then

(i) if ∥f∥ ≤ 1 then f is univalent, and (ii) if ∥f∥ < 2 then f is bounded. The constants are sharp.

The part (i) is due to Becker [2] and sharpness of the constant 1 is due to Becker and Pommerenke [3]. The part (ii) is obvious (see [9, Corollary 2.4]). Note also that, recently, Kari and Per Hag [6] gave a necessary and sufficient condition for f ∈ S to have a John disk as the image in terms of the pre-Schwarzian derivative of f. Also, the norm estimates for typical subclasses of univalent functions are investigated by many authors ([18], [9], [8], and so on).

In the present paper, first we estimate the norm of J_{α}[f] for a function f in a subclass of A and then make use of Theorem A to obtain boundedness and univalence of the nonlinear integral transform J_{α}[f] of f. We give also conditions for J_{α}[f] to be in typical subclasses of univalent functions such as S^{∗}, K and C.

2. Main Results

For a constant 0 < λ ≤ 1, consider the class U(λ) defined by

U(λ) = {f ∈ A : |f′(z)(z/f(z))^{2} − 1| < λ, z ∈ 𝔻}.

The class U(λ) looks natural through the transformation F (ζ) = 1/f(1/ζ), where |ζ| > 1. In fact, F′(1/z) = f′(z)(z/f(z))^{2} and therefore, f ∈ U(λ) if and only if |F′(ζ) − 1| < λ in |ζ| > 1. Note that f ∈ U(λ) has no zeros in 𝔻^{∗} = 𝔻 \ {0}, namely, U(λ) ⊂ ZF, because z^{2}f′(z)/f(z)^{2} is analytic in 𝔻. It is known [16] that U(λ) ⊂ S for 0 < λ ≤ 1 and that every f ∈ U(λ) admits a K-quasiconformal extension to the Riemann sphere when K = (1 + λ)/(1 − λ) < ∞ (see [11]). In particular, the Bieberbach theorem yields that |a_{2}| = |f′′(0)/2| ≤ 2 for f ∈ U(1). Set

U_{σ}(λ) = {f ∈ U(λ) : |f′′(0)| ≤ 2σ}

for σ ≥ 0. Recently, the class U(λ) and its related classes have been studied extensively by M. Obradović and Ponnusamy [14]. Furthermore, it is shown in [15] that U_{0}(λ) ⊂ S^{∗} for 0 < λ ≤ 1/\sqrt{2}, and that, for 1/\sqrt{2} < λ ≤ 1, every function in U_{0}(λ) is starlike in |z| < 1/\sqrt{2λ}.

Theorem 2.1. Let λ, μ and σ be non-negative numbers with μ = σ + λ ≤ 1. For a function f ∈ U_{σ}(λ), one obtains the estimate

∥J_{α}[f]∥ ≤ \frac{2|α|μ}{1 + \sqrt{1 − μ^{2}}} (2.2)

for every α ∈ ℂ, where equality holds precisely when f(z) = z/(1 − az) with |a| = μ. In particular, J_{α}[f] ∈ S whenever |α| ≤ (1 + \sqrt{1 − μ^{2}})/2μ.

Proof. Taking a logarithmic differentiation, we obtain J_{α}[f] = αJ[f] and thus

∥J_{α}[f]∥ = |α|∥J[f]∥.

Hence it suffices to show the inequality (2.2) in the case α = 1. Let f(z) = z + a_{2}z^{2} + ⋯ be in U_{σ}(λ) and set F = J[f]. Since f′(z)(z/f(z))^{2} = 1 + (a_{3} + 3a_{2}^{2})z^{2} + ⋯, we can write

f′(z) (\frac{z}{f(z)})^{2} = 1 + λz^{2}ω(z),

where ω is an analytic function in 𝔻 with |ω(z)| ≤ 1. If we set g(z) = 1/f(z) − 1/z, then we see that g is analytic in 𝔻 and g(0) = −a_{2}. Using the identity

g′(z) = −\frac{f′(z)}{f^{2}(z)} + \frac{1}{z^{2}} = −λω(z),

we get the representation

\frac{z}{f(z)} = 1 − a_{2}z − λz^{2} ∫_{0}^{1} ω(tz) dt, (2.3)

of f. (Conversely, for an arbitrary analytic function ω : 𝔻 → ℂ with |ω(z)| ≤ 1, the function f given by (2.3) belongs to the class U(λ) as long as the right-hand side of (2.3) does not vanish in 𝔻. The requirement that f ∈ ZF is guaranteed by |a_{2}| ≤ σ = 1 − λ.)

Since |a_{2}| + λ ≤ 1, by (2.3), we get

|\frac{z}{f(z)} − 1| ≤ |a_{2}z| + λ|z|^{2} < μ.

This implies that F′(z) = f(z)/z is subordinate to the function p(z) = 1/(1 + μz). By the Schwarz-Pick lemma, we easily obtain

∥F ∥ ≤ sup_{z∈𝔻}(1 − |z|^{2}) |\frac{p′(z)}{p(z)}| ,

see [9, Theorem 4.1]. Since

\frac{p′(z)}{p(z)} = −\frac{μ}{1 + μz} ,

a computation shows that

sup_{z∈𝔻}(1 − |z|^{2}) |\frac{p′(z)}{p(z)}| = μ sup_{0<t<1} \frac{1 − t^{2}}{1 − μt} = \frac{2μ}{1 + \sqrt{1 − μ^{2}}} ,

where the supremum is attained by z = t = μ/(1 + \sqrt{1 − μ^{2}}). (This calculation has been done in [8, Lemma 4.2] in a general situation.) Thus inequality (2.2) follows. The case of equality can be easily analyzed in the above.

Because ∥J_{α}[f]∥ ≤ 2|α|μ/(1 + \sqrt{1 − μ^{2}}) ≤ 1, Becker’s univalence criterion (Theorem A) yields the second assertion.

Letting a_{2} = 0 in Theorem 2.1, we obtain the following corollary.

Corollary 2.4. Let 0 < λ ≤ 1, and α ∈ ℂ with |α| ≤ (1 + \sqrt{1 − λ^{2}})/2λ. Then, J_{α}(U_{0}(λ)) ⊂ S holds.

We may rewrite the last corollary in the following equivalent form.

Corollary 2.5. For β ≥ 0, set F_{β} = U_{0}(4β/(1 + 4β^{2})). Then J_{α}(F_{β}) ⊂ S holds for all α ∈ ℂ with |α| ≤ β.

When α is real, we can deduce a stronger conclusion.

Theorem 2.6. Let f be a function in U_{1−λ}(λ) for some λ ∈ (0, 1]. Then J_{α}[f] is a closeto-convex function for each α ∈ [−1, 1].

Proof. By (2.3), we have

Re \frac{z}{f(z)} > 1 − (|a_{2}| + λ) ≥ 0, z ∈ 𝔻.

Therefore, both J_{−1}[f] and J[f] = J_{1}[f] are close-to-convex functions. Convexity of the class C with respect to the Hornich operation (cf. [13]) implies that J_{α}[f] ∈ C for α ∈ [−1, 1].

Next, we consider a function f ∈ A satisfying the condition |f′′(z)/2| ≤ μ, z ∈ 𝔻, for a positive constant μ. As we see below, if μ ≤ 1/2, then f is starlike, and thus, univalent. Otherwise, however, f may not be locally univalent as the example f(z) = z + μz^{2} shows.

Theorem 2.7. Let f be a function in A such that |f′′(z)| ≤ 2μ, z ∈ 𝔻, holds for some constant 0 < μ ≤ 1. Then f ∈ ZF and the sharp inequality

∥J_{α}[f]∥ ≤ \frac{2|α|μ}{1 + \sqrt{1 − μ^{2}}} (2.8)

holds for each α ∈ ℂ. If, in addition, μ < 1, then equality holds above precisely when f(z) = z + az^{2} for a constant a with |a| = μ. Moreover,

(i) J_{α}[f] ∈ S if |α| ≤ (1 + \sqrt{1 − μ^{2}})/2μ (ii) J_{α}[f] ∈ K if |α| ≤ (1 − μ)/μ.

Note that (1 + \sqrt{1 − μ^{2}})/2μ > (1 − μ)/μ holds for all μ > 0.

Proof. We may write f′′(z) = 2μω(z), where |ω| ≤ 1. By integration, we have

f′(z) = 1 + 2μz ∫_{0}^{1} ω(tz) dt and f(z) = z + 2μz^{2} ∫_{0}^{1}(1 − t)ω(tz) dt.

Since | ∫_{0}^{1}(1 − t)ω(tz) dt| ≤ 1/2, we conclude that |f(z)/z − 1| ≤ μ|z| < 1. In particular, f ∈ ZF. Furthermore,

\frac{zf′(z)}{f(z)} − 1 = \frac{2μz ∫_{0}^{1} tω(tz) dt}{1 + 2μz ∫_{0}^{1}(1 − t)ω(tz) dt},

and hence,

|\frac{zf′(z)}{f(z)} − 1| ≤ \frac{μ|z|}{1 − μ|z|}.

In particular, it turns out that f is starlike when μ ≤ 1/2. Since

1 + \frac{z(J_{α}[f])′′(z)}{(J_{α}[f])′(z)} = 1 + α (\frac{zf′(z)}{f(z)} − 1) ,

we obtain the convexity of J_{α}[f] under the assumption |α|μ/(1 − μ) ≤ 1. In addition, we have the estimate

∥J[f]∥ ≤ sup_{0<t<1} μ\frac{1 − t^{2}}{1 − μt} = 2\frac{1 − \sqrt{1 − μ^{2}}}{μ} = \frac{2μ}{1 + \sqrt{1 − μ^{2}}}

in the same way as in the proof of Theorem 2.1, where the supremum is taken by t_{0} = μ/(1 + \sqrt{1 − μ^{2}}). When μ < 1, this point is contained in 𝔻. Therefore, we can examine the equality case through the above proof. The univalence of J_{α}[f] under the hypothesis |α| ≤ (1+\sqrt{1 − μ^{2}})/2μ follows from Theorem A (i) because ∥J_{α}[f]∥ = |α| ∥J[f]∥ ≤ 1.

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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