Lobachevskii Journal of Mathematics

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ISSN 1818-9962

Vol. 22, 2006, 19–26 \begin{matrix}c \\ ◯ \end{matrix} M. Darus, K .Al Shaqsi

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 + \overline{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+ ∑_{k=2}^{∞} [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.

1. Introduction

A continuous functions f = u + iv 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 ⊂ ℂ we can write f(z) = h + \overline{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]).

tive operator.

2000 Mathematical Subject Classification. 30C45.

Key words and phrases. Univalent functions, Harmonic functions, deriva-

Denote by S_{H} the class of functions f = h + \overline{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 + \overline{g} ∈ S_{H} we may express the analytic functions h and g as

h(z) = z + ∑_{n=2}^{∞} a_{n}z^{n},  g(z) = ∑_{n=1}^{∞} 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 + \overline{g} where h and g are given by

h(z) = z − ∑_{n=2}^{∞} |a_{n}|z^{n},  g(z) = − ∑_{n=1}^{∞} |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

ℜ(D_{λ}^{n}f(z))′ > 0,  z ∈ U.  (1.3)

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

D_{λ}^{n}f(z) = z + ∑_{k=2}^{∞} [1 + λ(k − 1)]C(n, k)a_{k}z^{k},  λ ≥ 0,  (1.4)

where

C(n, k) = \binom{k + n − 1}{n} = \frac{∏_{j=1}^{k−1} (j + n)}{(k − 1)!},  k ≥ 2.  (1.5)

Note that when λ = 0, we get Ruscheweyh differential operator (see[1]). Also note that the class H_{λ}^{0} ≡ HP (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 ∩ H_{λ}^{n}.

2. Coefficients Bounds

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

∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)(|a_{n}| + |b_{n}|) ≤ 2,  (2.1)

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

Proof. For |z_{1}| ≤ |z_{2}| < 1, we have by (2.1),

|f(z_{1}) − f(z_{2})| ≥ |h(z_{1}) − h(z_{2})| − |g(z_{1}) − g(z_{2})|

= |(z_{1} − z_{2}) + ∑_{k=2}^{∞} a_{k}(z_{1}^{k} − z_{2}^{k})| − | ∑_{k=1}^{∞} b_{k}(z_{1}^{k} − z_{2}^{k})|

≥ |z_{1} − z_{2}|(1 − |b_{1}| − ∑_{k=2}^{∞} k|z_{2}|^{k−1})

≥ |z_{1} − z_{2}|(1 − |b_{1}| − |z_{2}| ∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)[|a_{k}| + |b_{k}|])

≥ |z_{1} − z_{2}|(1 − |b_{1}|)(1 − |z_{2}|) > 0.

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

|h′(z)| ≥ 1 − ∑_{k=2}^{∞} k|a_{k}||z|^{k−1} > 1 − ∑_{k=2}^{∞} k|a_{k}|

≥ 1 − ∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)|a_{k}||z|

≥ ∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)|b_{k}| > ∑_{k=1}^{∞} k|b_{k}||z|^{k−1} ≥ |g′(z)|.

Now we show that f ∈ H_{λ}^{n}. Using the fact that ℜw > 0 if and only if |1 + w| ≥ |1 − w|, it suffices to show that

|1 + (D_{λ}^{n}h(z))′ + (\overline{D_{λ}^{n}h(z)})′| − |1 − (D_{λ}^{n}h(z))′ − (\overline{D_{λ}^{n}h(z)})′| = 22 MASLINA DARUS AND K .AL SHAQSI

|2+∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)a_{n}z^{n−1}+∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)b_{n}\overline{z}^{n−1}|

−|−∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)a_{n}z^{n−1}−∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)b_{n}\overline{z}^{n−1}|

≥ 2 − ∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)|a_{n}||z|^{n−1}

−∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)|b_{n}||z|^{n−1}−∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)|a_{n}||z|^{n−1}

− ∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)|b_{n}||z|^{n−1}

= 2 − 2 ∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)|a_{n}||z|^{n−1}

− 2 ∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)|b_{n}||z|^{n−1} ≥

2{1−( ∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)|a_{n}|+∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)|b_{n}|)}

≥ 0, by(2.1).

The harmonic mappings

f(z) = z + ∑_{k=2}^{∞} \frac{x_{k}}{k[1 + λ(k − 1)]C(n, k)}z^{k}

+ ∑_{k=1}^{∞} \frac{\overline{y}_{k}}{k[1 + λ(k − 1)]C(n, k)}\overline{z}^{k}

where ∑_{k=2}^{∞} |x_{k}| + ∑_{k=1}^{∞} |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

∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)(|a_{k}| + |b_{k}|) = 1 + ∑_{k=2}^{∞} |x_{k}| + ∑_{k=1}^{∞} |y_{k}| = 2.

The restriction placed in Theorem 2.1 on the moduli of the coefficients of f = h + \overline{g} enables us to conclude for arbitrary rotation of the coefficients of f that the resulting functions would still be harmonic univalent and f ∈ 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 + \overline{g} with h and g are given by (1.2). Then f ∈ T H_{λ}^{n} if and only if

∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)(|b_{k}| + |b_{k}|) ≤ 2,  (2.2)

where a_{1} = 1 and λ ≥ 0.

Proof. We first suppose that f ∈ T H_{λ}^{n}, then by (1.3) we have

ℜ\{(D_{λ}^{n}h(z))′ + \overline{(D_{λ}^{n}g(z))′}\}

= ℜ{1 − ∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)|a_{k}|z^{n−1}

− ∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)|b_{k}|\overline{z}^{n−1}\}

> 0.

If we choose z to be real and let z → 1^{−}, we get

1 − ∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)|a_{k}| − ∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)|b_{k}| ≥ 0,

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

ℜ\{(D_{λ}^{n}h(z))′ + \overline{(D_{λ}^{n}g(z))′}\}

= ℜ{1 − ∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)|a_{k}|z^{n−1}

− ∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)|b_{k}|\overline{z}^{n−1}\}

≥ 2 − ∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)(|b_{k}| + |b_{k}|)\overline{z}^{n−1}

> 2 − ∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)(|b_{k}| + |b_{k}|) ≥ 0.

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 ∈ T H_{λ}^{n}, for λ ≥ 0 and |z| = r > 1, then

|f(z)| ≤ (1 + b_{1})r + (1 − b_{1})r + \frac{1 − |b_{1}|}{2(1 + λ)(n + 1)}r^{2},

and

|f(z)| ≥ (1 − b_{1})r − (1 − b_{1})r + \frac{1 − |b_{1}|}{2(1 + λ)(n + 1)}r^{2}.

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

|f(z)| ≥ (1 − b_{1})r − ∑_{k=2}^{∞} (|a_{n}| + |b_{n}|)r^{n}

≥ (1 − b_{1})r − ∑_{k=2}^{∞} (|a_{n}| + |b_{n}|)r^{2}

= (1 − b_{1})r − \frac{1}{2(1 + λ)(n + 1)} ∑_{k=2}^{∞} 2(1 + λ)(n + 1)(|a_{n}| + |b_{n}|)r^{2}

≥ (1 − b_{1})r − \frac{1}{2(1 + λ)(n + 1)} ∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)(|a_{n}| + |b_{n}|)r^{2}

≥ (1 − b_{1})r − \frac{1}{2(1 + λ)(n + 1)}[1 − |b_{1}|]r^{2}.

The bounds given in Theorem 3.1 for the functions f = h + \overline{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

f(z) = z + |b_{1}|\overline{z} − \frac{1 − |b_{1}|}{2(1 + λ)(n + 1)}\overline{z}^{2}

and

f(z) = (1 − |b_{1}|)z − \frac{1 − |b_{1}|}{2(1 + λ)(n + 1)}z^{2}

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.

HARMONIC UNIVALENT FUNCTIONS 25

Corollary 3.2. If f ∈ T H_{λ}^{n}, then

\{w : |w| < \frac{1}{2(1 + λ)(n + 1)}[(1 − |b_{1}|)(2(1 + λ)(n + 1) − 1)]\} ⊂ f(U).

Theorem 3.3. f ∈ T H_{λ}^{n} if and only if f can be expressed as

f(z) = ∑_{k=1}^{∞} (γ_{k}h_{k} + μ_{k}g_{k})  (3.1)

where z ∈ U,

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)}\overline{z}^{k}, (n = 1, 2, ...),

∑_{k=1}^{∞} (γ_{k} + μ_{k}) = 1, γ_{k} ≥ 0 and μ_{k} ≥ 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

f(z) = ∑_{k=1}^{∞} (γ_{k}h_{k} + μ_{k}g_{k})
= ∑_{k=1}^{∞} (γ_{k} + μ_{k})z − ∑_{k=2}^{∞} \frac{1}{k[1 + λ(k − 1)]C(n, k)}γ_{k}z^{k}

− ∑_{k=1}^{∞} \frac{1}{k[1 + λ(k − 1)]C(n, k)}μ_{k}\overline{z}^{k}.

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

∑_{k=2}^{∞} k[1 + λ(k − 1)]C(n, k)\frac{γ_{k}}{k[1 + λ(k − 1)]C(n, k)}

− ∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)\frac{μ_{k}}{k[1 + λ(k − 1)]C(n, k)}

= ∑_{k=1}^{∞} (γ_{k} + μ_{k}) − γ_{1} = 1 − γ_{1} ≤ 1.

26 MASLINA DARUS AND K .AL SHAQSI

Conversely, suppose that f ∈ T H_{λ}^{n} . Then

∑_{k=1}^{∞} k[1 + λ(k − 1)]C(n, k)(|a_{k}| + |b_{k}|) ≤ 2.

Setting

γ_{k} = k[1 + λ(k − 1)]C(n, k)|a_{k}|, 0 ≤ γ_{k} ≤ 1, (k = 2, 3, ...),

μ_{k} = k[1 + λ(k − 1)]C(n, k)|b_{k}|, 0 ≤ μ_{k} ≤ 1, (k = 1, 2, 3, ...),

and μ_{1} = 1 − γ_{1} − ∑_{k=2}^{∞} (γ_{k} + μ_{k}) we obtain

f(z) = ∑_{k=1}^{∞} (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