COEFFICIENT BOUNDS AND CONVOLUTION PROPERTIES FOR

CERTAIN CLASSES OF CLOSE-TO-CONVEX FUNCTIONS

YONG CHAN KIM^{∗)}, JAE HO CHOI^{∗)} AND TOSHIYUKI SUGAWA^{∗∗)}

Abstract. In this paper, we consider the class C(φ, ψ) of normalized close-to-convex functions which is defined by using subordination for analytic functions φ and ψ on the unit disk. Our main object is to provide bounds of the quantity a_{3} − μa_{2}^{2} for functions f(z) = z + a_{2}z^{2} + a_{3}z^{3} + ⋯ in C(φ, ψ) in terms of φ and ψ, where μ is a real constant. We also show that the class C(φ, ψ) is closed under the convolution operation by convex functions, or starlike functions of order 1/2 when φ and ψ satisfy some mild conditions.

1. Introduction

Let A denote the class of functions of the form

f(z) = z + ∑_{n=2}^{∞} a_{n}z^{n},

which are analytic in the open unit disk 𝔻 = {z ∈ ℂ : |z| < 1}. Also let S, S^{∗}(α) and K(α) denote the subclasses of A consisting of functions which are univalent, starlike of order α and convex of order α in 𝔻. In particular, the classes S^{∗}(0) = S^{∗} and K(0) = K are the familiar ones of starlike and convex functions in 𝔻, respectively. For analytic functions g and h with g(0) = h(0), g is said to be subordinate to h if there exists an analytic function ω on 𝔻 such that ω(0) = 0, |ω(z)| < 1 and g(z) = h(ω(z)) for z ∈ 𝔻. The subordination will be denoted by

g ≺ h  or  g(z) ≺ h(z) in 𝔻.

Note that g ≺ h if and only if g(0) = h(0) and g(𝔻) ⊂ h(𝔻) when h is univalent in 𝔻.

Let M be the class of analytic functions φ in 𝔻 normalized by φ(0) = 1, and let N be the subclass of M consisting of those functions φ which are univalent in 𝔻 and for which φ(𝔻) is convex. Also, for a constant α ≥ 0, set N (α) = {φ ∈ N : Re φ > α}.

Ma and Minda [6] and the authors [3] defined the subclasses K(φ), S^{∗}(φ) and C(φ, ψ) of A by

K(φ) = \{f ∈ A : 1 + \frac{zf′′(z)}{f′(z)} ≺ φ(z) in 𝔻\} ,

S^{∗}(φ) = \{f ∈ A : \frac{zf′(z)}{f(z)} ≺ φ(z) in 𝔻\} , (1.1)

and

C(φ, ψ) = \{f ∈ A : ∃h ∈ K(φ) s.t. \frac{f′(z)}{h′(z)} ≺ ψ(z) in 𝔻\}

for φ, ψ ∈ M. Note that f ∈ K(φ) if and only if zf′ ∈ S^{∗}(φ). Hence f ∈ C(φ, ψ) if and only if

∃g ∈ S^{∗}(φ) such that zf′(z)/g(z) ≺ ψ(z) in 𝔻. (1.2)

For functions φ, ψ ∈ M, if φ and e^{−iβ}ψ have positive real part in 𝔻, where β is some constant in (−π/2, π/2), then the class C(φ, ψ) is obviously a subclass of close-to-convex functions, in particular, consists of univalent functions in 𝔻. Now we recall that if f ∈ A satisfies

|arg \frac{zf′(z)}{f(z)}| < \frac{π}{2} α (z ∈ 𝔻) (1.3)

for a constant α (0 < α ≤ 1), then f(z) is said to be strongly starlike of order α in 𝔻, and we write f ∈ S_{α}^{∗}. If we set φ_{α}(z) = ((1+z)/(1−z))^{α} (0 < α ≤ 1), then, from (1.1) and (1.3), we can easily see the inclusion

S_{α}^{∗} = S^{∗}(φ_{α}) ⊂ C(φ_{α}, φ_{α}). (1.4)

For constants β ∈ (−π/2, π/2) and γ with 0 ≤ γ < cos β, we set

ψ_{β,γ}(z) = \frac{1 + (e^{iβ} − 2γ)e^{iβ}z}{1 − z} .

The function ψ_{β,γ} maps the unit disk onto the halfplane {z : Re(e^{−iβ}z) > γ}. Note that S^{∗}(α) ≡ S^{∗}(ψ_{0,α}) and K(α) ≡ K(ψ_{0,α}) for 0 ≤ α < 1. Note also that a function in S^{∗}(ψ_{β,0}) is usually called βspirallike. We set

1991 Mathematics Subject Classification. 30C45, 30C50.

Key words and phrases. univalent function, convolution, coefficient bound.

∗) Department of Mathematics, College of Education, Yeungnam University, 214-1 Daedong, Gyongsan 712-749, Korea.

∗∗) Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan.

The first named author was supported by Korea Research Foundation Grant (KRF-99-041-D00036). The third author was partially supported by the Ministry of Education, Grantin-Aid for Encouragement of Young Scientists, 11740088.

C_{α,γ} = ⋃_{|β|<arccosγ} C(ψ_{0,α}, ψ_{β,γ}) (1.5)

for 0 ≤ α < 1 and 0 ≤ γ < 1. A function in C_{α,γ} is called close-to-convex of order (γ, α) (cf. [2, II, p.89]). In particular, C ≡ C_{0,0} is the class of usual close-to-convex functions.

In [3], the second and third authors investigated the norm estimate of the pre-Schwarzian derivatives for the class C(φ, ψ). In this paper, we shall investigate the coefficient bounds of the class C(φ, ψ) and also give convolution properties of functions in C(φ, ψ). Here, the convolution or the Hadamard product f ∗ g of two analytic functions

f(z) = ∑_{n=0}^{∞} a_{n}z^{n}  and  g(z) = ∑_{n=0}^{∞} b_{n}z^{n}

on 𝔻 is defined by

(f ∗ g)(z) = f(z) ∗ g(z) = ∑_{n=0}^{∞} a_{n}b_{n}z^{n}.

2. Preliminary results

The following lemmas will be required in our investigation.

Lemma 2.1. Assume that η(z) = e_{1} + e_{2}z + ⋯ is analytic in 𝔻 with |η(z)| ≤ 1. Then |e_{1}|^{2} + |e_{2}| ≤ 1.

Proof. By Schwarz-Pick’s Lemma, we obtain

\frac{|η′(z)|}{1 − |η(z)|^{2}} ≤ \frac{1}{1 − |z|^{2}} ,

so that |η(0)|^{2} + |η′(0)| ≤ 1. Hence |e_{1}|^{2} + |e_{2}| ≤ 1.

Lemma 2.2 (Ma and Minda [6]). Let φ(z) = 1 + A_{1}z + A_{2}z^{2} + ⋯ be univalent in 𝔻 and let φ(𝔻) be symmetric with respect to the real axis with φ′(0) >

0. If f(z) = z + a_{2}z^{2} + a_{3}z^{3} + ⋯ ∈ K(φ), then |a_{3} − μa_{2}^{2}| ≤ K(μ, A_{1}, A_{2}), where

K(μ, A_{1}, A_{2}) (2.1)

= \begin{cases}(A_{2} − \frac{3μ}{2}A_{1}^{2} + A_{1}^{2})/6 \\ if 3A_{1}^{2}μ ≤ 2(A_{2} + A_{1}^{2} − A_{1}), \\ A_{1}/6 \\ if 2(A_{2} + A_{1}^{2} − A_{1}) ≤ 3A_{1}^{2}μ ≤ 2(A_{2} + A_{1}^{2} + A_{1}), \\ (\frac{3μ}{2}A_{1}^{2} − A_{1}^{2} − A_{2})/6 \\ if 2(A_{2} + A_{1}^{2} + A_{1}) ≤ 3A_{1}^{2}μ. \end{cases}

Lemma 2.3 (Ruscheweyh and Sheil-Small [8]). Suppose either g ∈ K, h ∈ S^{∗} or else g, h ∈ S^{∗}(1/2). Then for any analytic function G in 𝔻, we have

\frac{(g ∗ hG)(z)}{(g ∗ h)(z)} ∈ \overline{co}G(𝔻) (z ∈ 𝔻),

where \overline{co}G(𝔻) is the closed convex hull of G(𝔻).

3. Main results

We begin by proving

Theorem 3.1. Let φ(z) = 1 + A_{1}z + A_{2}z^{2} + ⋯ be univalent in 𝔻, φ(𝔻) be symmetric with respect to the real axis with φ′(0) > 0, and let ψ(z) = 1 + B_{1}z + B_{2}z^{2} + ⋯ be analytic in 𝔻. If f(z) = z + a_{2}z^{2} + a_{3}z^{3} + ⋯ ∈ C(φ, ψ), then

|a_{3} − μa_{2}^{2}| ≤ K(μ, A_{1}, A_{2}) + M(μ, A_{1}, B_{1}, B_{2}),

where K(μ, A_{1}, A_{2}) is given by (2.1) and

M(μ, A_{1}, B_{1}, B_{2})

= \begin{cases}\frac{1}{3} & (|B_{2} − \frac{3μ}{4}B_{1}^{2}| + A_{1}|B_{1}||1 − \frac{3μ}{2}|) \\ if A_{1}|B_{1}||1 − \frac{3μ}{2}| ≥ 2(|B_{1}| − |B_{2} − \frac{3μ}{4}B_{1}^{2}|), \\ \frac{|B_{1}|}{3} + \frac{(A_{1}|B_{1}||1 − \frac{3μ}{2}|)^{2}}{12(|B_{1}| − |B_{2} − \frac{3μ}{4}B_{1}^{2}|)} & otherwise. \end{cases}

Proof. If f ∈ C(φ, ψ), from the definition of the class C(φ, ψ) there exists a function h ∈ K(φ) such that f′/h′ ≺ ψ. We set

h(z) = z + d_{2}z^{2} + d_{3}z^{3} + ⋯

and

g(z) = \frac{f′(z)}{h′(z)} = 1 + b_{1}z + b_{2}z^{2} + ⋯ = ψ(ω(z)), (3.1)

where ω is an analytic function on 𝔻 such that |ω(z)| ≤ |z| for z ∈ 𝔻. Then a simple calculation shows b_{1} = 2(a_{2} −d_{2}) and b_{2} = 3(a_{3} −d_{3})−4d_{2}(a_{2} −d_{2}), so that a_{2} = b_{1}/2 + d_{2} and a_{3} = d_{3} + b_{2}/3 + (2/3)b_{1}d_{2}. Thus we have

a_{3} − μa_{2}^{2} (3.2)

= (d_{3} − μd_{2}^{2}) + \frac{1}{3} (b_{2} − \frac{3μ}{4}b_{1}^{2}) + (\frac{2}{3} − μ) b_{1}d_{2}.

By Lemma 2.2, we have

|d_{3} − μd_{2}^{2}| ≤ K(μ, A_{1}, A_{2}). (3.3)

We write ω(z) = e_{1}z + e_{2}z^{2} + · · ·. Then, from (3.1) we have b_{1} = B_{1}e_{1} and b_{2} = B_{1}e_{2} + B_{2}e_{1}^{2}. Since 1+ \frac{zh′′(z)}{h′(z)} ≺ φ(z) in 𝔻, Rogosinski’s result [7] implies |d_{2}| ≤ \frac{1}{2}A_{1}. Therefore, we get

|\frac{1}{3} (b_{2} − \frac{3μ}{4}b_{1}^{2}) + (\frac{2}{3} − μ) b_{1}d_{2}|

≤ \frac{|B_{1}|}{3}|e_{2}| + \frac{1}{3} |B_{2} − \frac{3μ}{4}B_{1}^{2}| |e_{1}|^{2}

+ |\frac{2}{3} − μ| |d_{2}B_{1}||e_{1}|

≤ \frac{|B_{1}|}{3}|e_{2}| + \frac{1}{3} |B_{2} − \frac{3μ}{4}B_{1}^{2}| |e_{1}|^{2}

+ |\frac{1}{3} − \frac{μ}{2}| A_{1}|B_{1}||e_{1}|.

Taking η(z) = ω(z)/z in Lemma 2.1, we obtain |e_{2}| ≤ 1 − |e_{1}|^{2}, so that

|\frac{1}{3} (b_{2} − \frac{3μ}{4}b_{1}^{2}) + (\frac{2}{3} − μ) b_{1}d_{2}| ≤ P (|e_{1}|),

where P (x) = ax^{2} + bx + c and a = \frac{1}{3}(|B_{2} − \frac{3μ}{4}B_{1}^{2}| − |B_{1}|), b = A_{1}|B_{1}||\frac{1}{3} − \frac{μ}{2}| and c = |B_{1}|/3. Since b ≥ 0 and 0 ≤ |e_{1}| ≤ 1, we have

P (|e_{1}|) ≤ \begin{cases}P (−b/2a) = c − b^{2}/4a \\ if a < 0 and − b/2a < 1, \\ P (1) = a + b + c otherwise. \end{cases}

Thus we conclude

|\frac{1}{3} (b_{2} − \frac{3μ}{4}b_{1}^{2}) + (\frac{2}{3} − μ) b_{1}d_{2}| (3.4)

≤ M(μ, A_{1}, B_{1}, B_{2}).

Hence, making use of (3.3) and (3.4) in equality (3.2), we obtain the desired result.

Corollary 3.2. If f(z) = z + a_{2}z^{2} + a_{3}z^{3} + ⋯ ∈ C(ψ_{0,0}, ψ_{0,0}), then

|a_{3} − μa_{2}^{2}| ≤ \begin{cases}3 − 4μ  if μ ≤ 1/3, \\ 1/3 + 4/9μ  if 1/3 ≤ μ ≤ 2/3, \\ (16 − 21μ + 9μ^{2})/3(4 − 3μ) \\ if 2/3 ≤ μ ≤ 1 \\ 3μ − 5/3  if 1 ≤ μ ≤ 4/3, \\ 4μ − 3  if 4/3 ≤ μ. \end{cases}

Remark . ¿From (1.5) it is clear that C(ψ_{0,0}, ψ_{0,0}) ⊂ C. For the cases of 0 ≤ μ ≤ 1/3 and 1/3 ≤ μ ≤ 2/3, the above estimates agree with those of Koepf [4].

If we take φ = ψ = φ_{α} = z + 2αz^{2} + 2α^{2}z^{3} + ⋯ in Theorem 3.1, we obtain Corollary 3.3. If f(z) = z + a_{2}z^{2} + a_{3}z^{3} + ⋯ ∈ C(φ_{α}, φ_{α}), then

|a_{3}−μa_{2}^{2}| ≤ \begin{cases}(3 − 4μ)α^{2} if 3αμ ≤ 2α − 1 \\ (1 − μ)α^{2} + \frac{α}{3} \{2 + \frac{(2 − 3μ)^{2}α^{2}}{2 − (2 − 3μ)α}\} \\ if 2α − 1 ≤ 3αμ ≤ 3α − 1 \\ α \{1 + \frac{(2 − 3μ)^{2}α^{2}}{3(2 − 2α + 3αμ)}\} \\ if 3α − 1 ≤ 3αμ ≤ 2α \\ α \{1 + \frac{(2 − 3μ)^{2}α^{2}}{3(2 − 3αμ + 2α)}\} \\ if 2α ≤ 3αμ ≤ 2α + 1 \\ (3μ − 2)α^{2} + α/3 \\ if 2α + 1 ≤ 3αμ ≤ 3α + 1 \\ (4μ − 3)α^{2} if 3α + 1 ≤ 3αμ. \end{cases}

Noting the relation S_{α}^{∗} ⊂ C(φ_{α}, φ_{α}), we would have an estimate for strongly starlike functions of order α. When 3αμ ≤ 2α − 1 or 3αμ ≥ 3α + 1, that estimate incidentally coincides with the sharp estimate for strongly starlike functions of order α obtained previously by Ma and Minda [5].

Now, by using Lemma 2.3, we investigate convolution properties of functions in C(φ, ψ). First, we recall results due to Ma and Minda. The following form is slightly different from the original one, so we include its proof here.

Proposition 3.4 ([6]).

(a) Let φ ∈ N (0). For g ∈ K and h ∈ S^{∗}(φ), we have g ∗ h ∈ S^{∗}(φ).

(b) Let φ ∈ N (1/2). For g ∈ S^{∗}(1/2) and h ∈ S^{∗}(φ), we have g ∗ h ∈ S^{∗}(φ).

Proof. First, we prove (a). Set G = zh′/h ≺ φ. Since z(g ∗ h)′ = g ∗ (zh′) = g ∗ (Gh), from Lemma 2.3, we see

\frac{z(g ∗ h)′(z)}{(g ∗ h)(z)} = \frac{(g ∗ Gh)(z)}{(g ∗ h)(z)} ∈ \overline{co}G(𝔻) ⊂ \overline{φ(𝔻)}.

Hence, we have z(g ∗ h)′/g ∗ h ≺ φ. Assertion (b) can be shown similarly.

With the aid of the above result, we can now prove the following.

Theorem 3.5.

(a) Let φ ∈ N (0) and ψ ∈ N . Then, for g ∈ K and f ∈ C(φ, ψ), we have g ∗ f ∈ C(φ, ψ).

(b) Let φ ∈ N (1/2) and ψ ∈ N . Then, for g ∈ S^{∗}(1/2) and f ∈ C(φ, ψ), we have g ∗ f ∈ C(φ, ψ).

Proof. We show only (a). We can handle (b) in the same fashion. Let φ ∈ N (0) and ψ ∈ N . If f ∈ C(φ, ψ), there is a function h ∈ S^{∗}(φ) such that zf′/h ≺ ψ. Set G(z) = zf′(z)/h(z). Then G(𝔻) ⊂ ψ(𝔻) and z(g ∗f)′ = g ∗(zf′) = g ∗Gh. Since ψ(𝔻) is convex and since z(g ∗ f)′/(g ∗ h) is analytic, Lemma 2.3 implies that

\frac{z(g ∗ f)′(z)}{(g ∗ h)(z)} = \frac{(g ∗ Gh)(z)}{(g ∗ h)(z)}

lies in ψ(𝔻), in other words, z(g ∗ f)′/g ∗ h ≺ ψ. Now Proposition 3.4 ensures g ∗ h ∈ S^{∗}(φ). Hence we find from definition (1.2) that g ∗ f ∈ C(φ, ψ), which completes the proof of Theorem 3.5.

Remark . If we apply the above theorem to the case φ = ψ_{0,0} and ψ = ψ_{β,0} for |β| < π/2, then Theorem 3.5 would immediately yield that f ∗ g ∈ C for f ∈ C and g ∈ K (see [1, Theorem 8.7]).

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