Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 22, 2006, 19–26

Maslina Darus and K .Al Shaqsi
ON HARMONIC UNIVALENT FUNCTIONS DEFINED BY A GENERALIZED RUSCHEWEYH DERIVATIVES OPERATOR
(submitted by M. A. Malakhaltsev)

Abstract. Let $S_{ℋ}$ denote the class of functions $f = h + \bar{g}$ which are harmonic univalent and sense preserving in the unit disk $U$ . Al-Shaqsi and Darus[7] introduced a generalized Ruscheweyh derivatives operator denoted by $D_{λ}^{n}$ where $D_{λ}^{n} f (z) = z + \sum_{k = 2}^{\infty} [1 + λ (k - 1)] C (n, k) a_{k} z^{k}$ , where $C (n, k) = (\binom{k + n - 1}{n})$ . The authors, using this operators, introduce the class $ℋ_{λ}^{n}$ of functions which are harmonic in $U$ . Coefficient bounds, distortion bounds and extreme points are obtained.

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2000 Mathematical Subject Classiﬁcation. 30C45.

Key words and phrases. Univalent functions, Harmonic functions, derivative operator.

1. Introduction

A continuous functions $f = u + i v$ is a complex valued harmonic function in a complex domain $ℂ$ if both $u$ and $v$ are real harmonic in $ℂ$ . In any simply connected domain $D \subset ℂ$ we can write $f (z) = h + \bar{g}$ , where $h$ and $g$ are analytic in $D$ . We call $h$ the analytic part and $g$ the co-analytic part of $f$ . A necessary and sufficient condition for $f$ to be locally univalent and sense-preserving in $D$ is that $∣ h^{'} (z) ∣ > ∣ g^{'} (z) ∣$ in $D$ . See Clunie and Sheil-Small (see [2]).

Denote by $S_{ℋ}$ the class of functions $f = h + \bar{g}$ that are harmonic univalent and sense-preserving in the unit disk $U = {z : ∣ z ∣ < 1}$ for which $f (0) = h (0) = f_{z} (0) - 1 = 0$ . For $f = h + \bar{g} \in S_{ℋ}$ we may express the analytic functions $h$ and $g$ as

h (z) = z + \sum_{n = 2}^{\infty} a_{n} z^{n}, g (z) = \sum_{n = 1}^{\infty} b_{n} z^{n} ∣ b_{1} ∣ < 1 .

(1.1)

Observe that $S_{ℋ}$ reduces to $S$ , the class of normalized univalent analytic functions, if the co-analytic part of $f$ is zero.
The class $T$ is deﬁned as the subclass of $S_{ℋ}$ consisting of all functions $f = h + \bar{g}$ where $h$ and $g$ are given by

h (z) = z - \sum_{n = 2}^{\infty} ∣ a_{n} ∣ z^{n}, g (z) = - \sum_{n = 1}^{\infty} ∣ b_{n} ∣ z^{n} .

(1.2)

In 1984 Clunie and Sheil-Small [2] investigated the class $S_{ℋ}$ as well as its geometric subclasses and obtained some coefficient bounds. Since then, there has been several related papers on $S_{ℋ}$ and its subclasses such that Silverman [3], Silverman and Silvia [4] and, Jahangiri [5] studied the harmonic univalent functions.
We denote by $ℋ_{λ}^{n}$ the class of all function of the form (1.1) that satisfy the condition

\begin{array}{rcl} ℜ {(D_{λ}^{n} f (z))}^{'} > 0, z \in U . & (1.3) \end{array}

where $D_{λ}^{n} f (z) = D_{λ}^{n} h (z) + \bar{D_{λ}^{n} g (z)}$ , and $D_{λ}^{n}$ denotes the operator introduced by Al-Shaqsi and Darus[7] and is given by

\begin{array}{rcl} D_{λ}^{n} f (z) = z + \sum_{k = 2}^{\infty} [1 + λ (k - 1)] C (n, k) a_{k} z^{k}, λ \geq 0, & (1.4) \end{array}

where

\begin{array}{rcl} C (n, k) = (\binom{k + n - 1}{n}) = \frac{\prod_{j = 1}^{k - 1} (j + n)}{(k - 1)!}, k \geq 2 . & (1.5) \end{array}

Note that when $λ = 0$ , we get Ruscheweyh differential operator (see[1]). Also note that the class $ℋ_{λ}^{0} \equiv H P (α)$ the class of harmonic univalent functions studied by Yalçin and Öztürk [6]. We further denote by ${T ℋ}_{λ}^{n}$ the subclass of $ℋ_{λ}^{n}$ , where ${T ℋ}_{λ}^{n} = T \cap ℋ_{λ}^{n}$ .

2. Coefficients Bounds

Theorem 2.1. Let $f = h + \bar{g}$ with $h$ and $g$ are given by (1.1). Let

\begin{array}{rcl} \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) (∣ a_{n} ∣ + ∣ b_{n} ∣) \leq 2, & (2.1) \end{array}

where $a_{1} = 1$ and $λ \geq 0$ . Then $f$ is harmonic univalent sense preserving in $U$ and $f \in ℋ_{λ}^{n}$ .

Proof. For $∣ z_{1} ∣ \leq ∣ z_{2} ∣ < 1$ , we have by (2.1),

\begin{matrix} ∣ f (z_{1}) - f (z_{2}) ∣ \\ \geq ∣ h (z_{1}) - h (z_{2}) ∣ - ∣ g (z_{1}) - g (z_{2}) ∣ \\ = | (z_{1} - z_{2}) + \sum_{k = 2}^{\infty} a_{k} (z_{1}^{k} - z_{2}^{k}) | - | \sum_{k = 1}^{\infty} b_{k} (z_{1}^{k} - z_{2}^{k}) | \\ \geq ∣ z_{1} - z_{2} ∣ (1 - ∣ b_{1} ∣ - \sum_{k = 2}^{\infty} k ∣ z_{2} ∣^{k - 1}) \\ \geq ∣ z_{1} - z_{2} ∣ (1 - ∣ b_{1} ∣ - ∣ z_{2} ∣ \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) [∣ a_{k} ∣ + ∣ b_{k} ∣]) \\ \geq ∣ z_{1} - z_{2} ∣ (1 - ∣ b_{1} ∣) (1 - ∣ z_{2} ∣) > 0 . \end{matrix}

Consequently, $f$ is univalent in $U$ . We note that $f$ is sense preserving in $U$ . This is because

\begin{matrix} ∣ h^{'} (z) ∣ \geq 1 - \sum_{k = 2}^{\infty} k ∣ a_{k} ∣ ∣ z ∣^{k - 1} > 1 - \sum_{k = 2}^{\infty} k ∣ a_{k} ∣ \\ \geq 1 - \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ a_{k} ∣ ∣ z ∣ \\ \geq \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ b_{k} ∣ > \sum_{k = 1}^{\infty} k ∣ b_{k} ∣ ∣ z ∣^{k - 1} \geq ∣ g^{'} (z) ∣ . \end{matrix}

Now we show that $f \in ℋ_{λ}^{n}$ . Using the fact that $ℜ w > 0$ if and only if $∣ 1 + w ∣ \geq ∣ 1 - w ∣$ , it suffices to show that

\begin{matrix} | 1 + {(D_{λ}^{n} h (z))}^{'} + {(\bar{D_{λ}^{n} h (z)})}^{'} | - | 1 - {(D_{λ}^{n} h (z))}^{'} - {(\bar{D_{λ}^{n} h (z)})}^{'} | = \end{matrix}

\begin{matrix} | 2 + \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) a_{n} z^{n - 1} + \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) b_{n} {\bar{z}}^{n - 1} | \\ - | - \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) a_{n} z^{n - 1} - \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) b_{n} {\bar{z}}^{n - 1} | \\ \geq 2 - \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ a_{n} ∣ ∣ z ∣^{n - 1} \\ - \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ b_{n} ∣ ∣ z ∣^{n - 1} - \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ a_{n} ∣ ∣ z ∣^{n - 1} \\ - \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ b_{n} ∣ ∣ z ∣^{n - 1} \\ = 2 - 2 \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ a_{n} ∣ ∣ z ∣^{n - 1} \\ - 2 \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ b_{n} ∣ ∣ z ∣^{n - 1} \geq \\ 2 \{1 - (\sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ a_{n} ∣ + \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ b_{n} ∣) \} \\ \geq 0, by (2.1) . \end{matrix}

The harmonic mappings

\begin{matrix} f (z) = z + \sum_{k = 2}^{\infty} \frac{x_{k}}{k [1 + λ (k - 1)] C (n, k)} z^{k} \\ + \sum_{k = 1}^{\infty} \frac{{\bar{y}}_{k}}{k [1 + λ (k - 1)] C (n, k)} {\bar{z}}^{k} & (2.2) \end{matrix}

where $\sum_{k = 2}^{\infty} ∣ x_{k} ∣ + \sum_{k = 1}^{\infty} ∣ y_{k} ∣ = 1$ , show that the coefficient bound given by (2.1) is sharp.
The functions of the form (2.2) are in $ℋ_{λ}^{n}$ because

\begin{array}{rcl} \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) (∣ a_{k} ∣ + ∣ b_{k} ∣) = 1 + \sum_{k = 2}^{\infty} ∣ x_{k} ∣ + \sum_{k = 1}^{\infty} ∣ y_{k} ∣ = 2 . \end{array}

The restriction placed in Theorem 2.1 on the moduli of the coefficients of $f = h + \bar{g}$ enables us to conclude for arbitrary rotation of the coefficients of $f$ that the resulting functions would still be harmonic univalent and $f \in ℋ_{λ}^{n}$ . We next show that the condition (2.1) is also necessary for functions in ${T ℋ}_{λ}^{n}$ .

Theorem 2.2. Let $f = h + \bar{g}$ with $h$ and $g$ are given by (1.2). Then $f \in {T ℋ}_{λ}^{n}$ if and only if

\begin{array}{rcl} \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) (∣ b_{k} ∣ + ∣ b_{k} ∣) \leq 2, & (2.3) \end{array}

where $a_{1} = 1$ and $λ \geq 0$ .

Proof. We ﬁrst suppose that $f \in {T ℋ}_{λ}^{n}$ , then by (1.3) we have

\begin{matrix} ℜ \{{(D_{λ}^{n} h (z))}^{'} + \bar{{(D_{λ}^{n} g (z))}^{'}} \} \\ = ℜ \{1 - \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ a_{k} ∣ z^{n - 1} \\ - \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ b_{k} ∣ {\bar{z}}^{n - 1} \} \\ > 0 . \end{matrix}

If we choose $z$ to be real and let $z \to 1^{-}$ , we get

\begin{array}{rcl} 1 - \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ a_{k} ∣ - \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ b_{k} ∣ \geq 0, \end{array}

which is precisely the assertion (2.3) of Theorem 2.2.
Conversely, suppose that the inequality (2.3) holds true. Then we ﬁnd from the deﬁnition (1.3) that

\begin{array}{rcl} ℜ \{{(D_{λ}^{n} h (z))}^{'} + \bar{{(D_{λ}^{n} g (z))}^{'}} \} \\ = ℜ \{1 - \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ a_{k} ∣ z^{n - 1} \\ - \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) ∣ b_{k} ∣ {\bar{z}}^{n - 1} \} \\ \geq 2 - \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) (∣ b_{k} ∣ + ∣ b_{k} ∣) {\bar{z}}^{n - 1} \\ > 2 - \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) (∣ b_{k} ∣ + ∣ b_{k} ∣) \geq 0 . \end{array}

provided that the inequality (2.3) is satisﬁed.

3. Distortion Bounds and Extreme Points.

In this section, we shall obtain distortion bounds for functions in ${T ℋ}_{λ}^{n}$ and also provide extreme points for the class ${T ℋ}_{λ}^{n}$ .

Theorem 3.1. If $f \in {T ℋ}_{λ}^{n}$ , for $λ \geq 0$ and $∣ z ∣ = r > 1$ , then

\begin{array}{rcl} ∣ f (z) ∣ \leq (1 + b_{1}) r + (1 - b_{1}) r + \frac{1 - ∣ b_{1} ∣}{2 (1 + λ) (n + 1)} r^{2}, \end{array}

and

\begin{array}{rcl} ∣ f (z) ∣ \geq (1 - b_{1}) r - (1 - b_{1}) r + \frac{1 - ∣ b_{1} ∣}{2 (1 + λ) (n + 1)} r^{2} . \end{array}

Proof. We only prove the second inequality. The argument for ﬁrst inequality is similar and will be omitted. Let $f \in {T ℋ}_{λ}^{n}$ . Taking the absolute value of $f$ , we obtain

\begin{matrix} \begin{aligned} ∣ f (z) ∣ \geq (1 - b_{1}) r - \sum_{k = 2}^{\infty} (∣ a_{n} ∣ + ∣ b_{n} ∣) r^{n} \\ \geq (1 - b_{1}) r - \sum_{k = 2}^{\infty} (∣ a_{n} ∣ + ∣ b_{n} ∣) r^{2} \\ = (1 - b_{1}) r - \frac{1}{2 (1 + λ) (n + 1)} \sum_{k = 2}^{\infty} 2 (1 + λ) (n + 1) (∣ a_{n} ∣ + ∣ b_{n} ∣) r^{2} \\ \geq (1 - b_{1}) r - \frac{1}{2 (1 + λ) (n + 1)} \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) (∣ a_{n} ∣ + ∣ b_{n} ∣) r^{2} \\ \geq (1 - b_{1}) r - \frac{1}{2 (1 + λ) (n + 1)} [1 - ∣ b_{1} ∣] r^{2} . \end{aligned} \end{matrix}

The bounds given in Theorem 3.1 for the functions $f = h + \bar{g}$ of the form (1.2) also hold for functions of the form (1.1) if the coefficient condition (2.1) is satisﬁed. The functions

\begin{array}{rcl} f (z) = z + ∣ b_{1} ∣ \bar{z} - \frac{1 - ∣ b_{1} ∣}{2 (1 + λ) (n + 1)} {\bar{z}}^{2} \end{array}

and

\begin{array}{rcl} f (z) = (1 - ∣ b_{1} ∣) z - \frac{1 - ∣ b_{1} ∣}{2 (1 + λ) (n + 1)} z^{2} \end{array}

for $∣ b_{1} ∣ < 1$ show that the bounds given Theorem 3.1 are sharp.

The following covering result follows from the second inequality in Theorem 3.1.

Corollary 3.2. If $f \in {T ℋ}_{λ}^{n}$ , then

\begin{array}{rcl} \{w : ∣ w ∣ < \frac{1}{2 (1 + λ) (n + 1)} [(1 - ∣ b_{1} ∣) (2 (1 + λ) (n + 1) - 1)] \} \subset f (U) . \end{array}

Theorem 3.3. $f \in {T ℋ}_{λ}^{n}$ if and only if $f$ can be expressed as

f (z) = \sum_{k = 1}^{\infty} (γ_{k} h_{k} + μ_{k} g_{k})

(3.1)

where $z \in U$ ,

\begin{array}{rcl} h_{1} (z) = z, h_{k} (z) = z - \frac{1}{k [1 + λ (k - 1)] C (n, k)} z^{k}, (k = 2, 3, . . .), \\ g_{k} (z) = z - \frac{1}{k [1 + λ (k - 1)] C (n, k)} {\bar{z}}^{k}, (n = 1, 2, . . .), \end{array}

$\sum_{k = 1}^{\infty} (γ_{k} + μ_{k}) = 1, γ_{k} \geq 0$ and $μ_{k} \geq 0$ .

In particular, the extreme points of ${T ℋ}_{λ}^{n}$ are ${h_{k}}$ and ${g_{k}}$ .

Proof. Note that for $f$ we may write

\begin{array}{rcl} f (z) & = & \sum_{k = 1}^{\infty} (γ_{k} h_{k} + μ_{k} g_{k}) \\ = & \sum_{k = 1}^{\infty} (γ_{k} + μ_{k}) z - \sum_{k = 2}^{\infty} \frac{1}{k [1 + λ (k - 1)] C (n, k)} γ_{k} z^{k} \\ - \sum_{k = 1}^{\infty} \frac{1}{k [1 + λ (k - 1)] C (n, k)} μ_{k} {\bar{z}}^{k} . \end{array}

Now the ﬁrst part of the proof is complete, since by Theorem 2.2

\begin{array}{rcl} \sum_{k = 2}^{\infty} k [1 + λ (k - 1)] C (n, k) \frac{γ_{k}}{k [1 + λ (k - 1)] C (n, k)} \\ - \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) \frac{μ_{k}}{k [1 + λ (k - 1)] C (n, k)} \\ = \sum_{k = 1}^{\infty} (γ_{k} + μ_{k}) - γ_{1} = 1 - γ_{1} \leq 1 . \end{array}

Conversely, suppose that $f \in {T ℋ}_{λ}^{n}$ . Then

\begin{array}{rcl} \sum_{k = 1}^{\infty} k [1 + λ (k - 1)] C (n, k) (∣ a_{k} ∣ + ∣ b_{k} ∣) \leq 2 . \end{array}

Setting

\begin{array}{rcl} γ_{k} = k [1 + λ (k - 1)] C (n, k) ∣ a_{k} ∣, 0 \leq γ_{k} \leq 1, (k = 2, 3, . . .), \\ μ_{k} = k [1 + λ (k - 1)] C (n, k) ∣ b_{k} ∣, 0 \leq μ_{k} \leq 1, (k = 1, 2, 3, . . .), \end{array}

and $μ_{1} = 1 - γ_{1} - \sum_{k = 2}^{\infty} (γ_{k} + μ_{k})$ we obtain

$f (z) = \sum_{k = 1}^{\infty} (γ_{k} h_{k} + μ_{k} g_{k})$ as required.

References

[1] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49, (1975),109-115.

[2] J. Clunie and T. Shell-Small, Harmonic univalent functions, Ann. Acad. Aci. Fenn. Ser. A I Math. 9, (1984), 3-25.

[3] H. Silverman, Harmonic univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51, (1998), 283-289.

[4] H. Silverman and E. M. Silvia, Subclasses of harmonic univalent functions, New Zeal. J. Math. 28, (1999), 275-284.

[5] J. M. Jahangiri, Harmonic functions starlike in the unite disk, J. math. Anal. Appl. 235, (1999), 470-477.

[6] S. Yalçin and M. Öztürk, A new subclass of complex harmonic functions, Math. Ineq. Appl. 7, (2004), 55-61.

[7] K.A.Shaqsi and M. Darus, On univalent functions with respect to $K$ -symmetric points given by a generalised Ruscheweyh derivatives operator, submitted

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D. Ehsan, Malaysia

E-mail address: ommath@hotmail.com

E-mail address: maslina@pkrisc.cc.ukm.my

Received March 30, 2006