MATEMATIQKI VESNIK

58 (2006), 7–11

UDK 517.546

originalni nauqni rad research paper

HARMONIC FUNCTIONS STARLIKE OF THE COMPLEX ORDER

Sibel Yalçin and Metin Öztürk

Abstract. The main purpose of this paper is to introduce a class T S_{H}^{∗}(γ) (γ ∈ C \ {0}) of functions which are harmonic in the unit disc. We give necessary and sufficient conditions for the functions to be in T S_{H}^{∗}(γ).

1. Introduction

A continuous complex-valued function f = u+iv defined in a simply connected domain D is said to be harmonic in D if both u and v are real harmonic in D. In any simply connected domain we can write f = h + ḡ, where h and g are analytic in D. A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h′(z)| > |g′(z)| in D.

Denote by S_{H} the class of functions f = h+ḡ that are harmonic univalent and sense preserving in the unit disc U = {z : |z| < 1} for which f(0) = f_{z}(0) − 1 = 0. Then for f = h + ḡ ∈ S_{H} we may express 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)

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 have been several related papers on S_{H} and its subclasses.

Let T S_{H} denote the family of functions f = h + ḡ that are harmonic in U with the normalization

h(z) = z − ∑_{n=2}^{∞} a_{n}z^{n},  g(z) = ∑_{n=1}^{∞} b_{n}z^{n}, a_{n}, b_{n} ⩾ 0, b_{1} < 1.  (2)

We let T S_{H}^{∗}(γ) denote the subclass of T S_{H} consisting of functions f = h+ḡ ∈ T S_{H} that satisfy the condition

Re \{1 + \frac{1}{γ} (\frac{zh′(z) − \overline{zg′(z)}}{h(z) + \overline{g(z)}} − 1)\} > 0,  γ ∈ C \ \{0\}.  (3)

AMS Subject Classification: 30 C 45, 30 C 50, 31 A 05

Keywords and phrases: Harmonic functions, starlike functions.

We further let OS_{H}^{∗}(γ) denote the subclass of T S_{H} consisting of functions f = h + ḡ ∈ T S_{H} that satisfy the condition

∑_{n=1}^{∞}[2(n − 1 + |γ|)a_{n} + (n + 1 + |n + 1 − 2γ|)b_{n}] ⩽ 4|γ|.  (4)

Denote by P S_{H}^{∗}(γ) the subclass of T S_{H} consisting of functions f = h + ḡ ∈ T S_{H} that satisfy the condition

∑_{n=1}^{∞} [(n − 1)\frac{Re(γ)}{|γ|} + |γ|] a_{n} + [(n + 1)\frac{Re(γ)}{|γ|} − |γ|] b_{n} ⩽ 2|γ|.  (5)

Recently, Avcı and Zlotkiewicz [1], Jahangiri [3], Silverman [4], and Silverman and Silvia [5] studied the harmonic starlike functions. Avcı and Slotkiewicz [1] proved that the coefficient condition

∑_{n=2}^{∞} n(|a_{n}| + |b_{n}|) ⩽ 1, b_{1} = 0

is sufficient for functions f = h + ḡ to be harmonic starlike. Silverman 4 proved that this coefficient condition is also necessary if b_{1} = 0 and if a_{n} and b_{n} in (1) are negative. Jahangiri [3] proved that if f = h + ḡ is given by (1) and if

∑_{n=2}^{∞} \frac{n − α}{1 − α}|a_{n}| + ∑_{n=1}^{∞} \frac{n + α}{1 − α}|b_{n}| ⩽ 1, 0 ⩽ α < 1,

then f is a harmonic, univalent and starlike function of order α in U. This condition proved to be also necessary if h and g are of the form (2). The case when α = 0 is given in [5] and for α = b_{1} = 0 see [4].

In this paper, we give an answer to the conjecture that T S_{H}^{∗}(γ) = OS_{H}^{∗}(γ).

2. Main results

Theorem 2.1. OS_{H}^{∗}(γ) ⊂ T S_{H}^{∗}(γ).

Proof. Let f ∈ OS_{H}^{∗}(γ). According to the condition (2) we only need to show that if (4) holds then

Re \{\frac{(γ − 1)(h(z) + \overline{g(z)}) + zh′(z) − \overline{zg′(z)}}{γ(h(z) + \overline{g(z)})}\} > 0,

where γ ∈ C \ {0}. Using the fact that Re w > 0 if and only if |1 + w| > |1 − w|, it suffices to show that

|(2γ − 1)(h(z) + \overline{g(z)}) + zh′(z) − \overline{zg′(z)}| − |h(z) + \overline{g(z)} − zh′(z) + \overline{zg′(z)}| > 0. (6)

Substituting for h(z) and g(z) in (6) yields

|(2γ − 1)(h(z) + \overline{g(z)}) + zh′(z) − \overline{zg′(z)}| − |h(z) + \overline{g(z)} − zh′(z) + \overline{zg′(z)}|

= |2γz − ∑_{n=2}^{∞}(2γ − 1 + n)a_{n}z^{n} − ∑_{n=1}^{∞}(n + 1 − 2γ)b_{n}z̄^{n}|−

− | ∑_{n=2}^{∞}(n − 1)a_{n}z^{n} + ∑_{n=1}^{∞}(n + 1)b_{n}z̄^{n}|

⩾ 2|γ||z| − ∑_{n=2}^{∞} 2(n − 1 + |γ|)a_{n}|z|^{n} − ∑_{n=1}^{∞}(n + 1 + |n + 1 − 2γ|)b_{n}|z|^{n}

> 2|γ| − ( ∑_{n=2}^{∞} 2(n − 1 + |γ|)a_{n} + ∑_{n=1}^{∞}(n + 1 + |n + 1 − 2γ|)b_{n}) ⩾ 0.

The functions

f(z) = z − ∑_{n=2}^{∞} \frac{|γ|}{n − 1 + |γ|}x_{n}z^{n} + ∑_{n=1}^{∞} \frac{2|γ|}{n + 1 + |n + 1 − 2γ|}y_{n}z̄^{n}  (7)

where x_{n}, y_{n} are non-negative and

∑_{n=2}^{∞} x_{n} + ∑_{n=1}^{∞} y_{n} = 1,

show that the coefficient bound given by (4) is sharp. The functions of the form (7) are in T S_{H}^{∗}(γ) because

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

= 2|γ|(1 + ∑_{n=2}^{∞} x_{n} + ∑_{n=1}^{∞} y_{n}) = 4|γ|.

Theorem 2.2. T S_{H}^{∗}(γ) ⊂ P S_{H}^{∗}(γ).
Proof. Let f ∈ T S_{H}^{∗}(γ). From (3) we have

Re \{\frac{1}{γ} (\frac{− ∑_{n=2}^{∞}(n − 1)a_{n}z^{n} − ∑_{n=1}^{∞}(n + 1)b_{n}z̄^{n}}{1 − ∑_{n=2}^{∞} a_{n}z^{n} + ∑_{n=1}^{∞} b_{n}z̄^{n}} )\} > −1.

If we choose z on the real axis and z → 1^{−} we get

\frac{∑_{n=2}^{∞}(n − 1)a_{n} + ∑_{n=1}^{∞}(n + 1)b_{n}}{1 − ∑_{n=2}^{∞} a_{n} + ∑_{n=1}^{∞} b_{n}} Re (\frac{1}{γ} ) ⩽ 1,

whence

\frac{∑_{n=2}^{∞}(n − 1)a_{n} + ∑_{n=1}^{∞}(n + 1)b_{n}}{1 − ∑_{n=2}^{∞} a_{n} + ∑_{n=1}^{∞} b_{n}} \frac{Re(γ)}{|γ|^{2}} ⩽ 1,

and so

∑_{n=2}^{∞}(n − 1)a_{n} + ∑_{n=1}^{∞}(n + 1)b_{n} ⩽ \frac{|γ|^{2}}{Re(γ)}(1 − ∑_{n=2}^{∞} a_{n} + ∑_{n=1}^{∞} b_{n})

which is equivalent to (5). Thus f ∈ P S_{H}^{∗}(γ).

10 S. Yalçin, M. Öztürk

Theorem 2.3. If γ ∈ (0, 1], then OS_{H}^{∗}(γ) = T S_{H}^{∗}(γ) = P S_{H}^{∗}(γ).

Proof. If γ ∈ (0, 1], then the inequalities (4) and (5) are equivalent; hence OS_{H}^{∗}(γ) = P S_{H}^{∗}(γ). By using Theorem 2.1 and Theorem 2.2, from this assertion we obtain the conclusion of the present theorem.

Theorem 2.4. If −1/2 > Re(γ) ⩽ 0 or γ ∈ (3/2, +∞), then

P S_{H}^{∗}(γ) ⊈ T S_{H}^{∗}(γ).

Proof. Let

f(z) = z − z^{2}.  (8)

Then f ∈ P S_{H}^{∗}(γ) when γ ∈ C \ {0} and Re(γ) < 0, because

∑_{n=1}^{∞} [(n − 1)\frac{Re(γ)}{|γ|} + |γ|] a_{n} + [(n + 1)\frac{Re(γ)}{|γ|} − |γ|] b_{n}

= |γ| · 1 + \frac{Re(γ)}{|γ|} + |γ| = 2|γ| + \frac{Re(γ)}{|γ|} ⩽ 2|γ|.

Now let r = Re(γ) < 0 and let s be a negative real number such that 1+2r(1−s) > 0. If we choose z = \frac{γ(1 − s)}{1 + γ(1 − s)}, then z ∈ U and for f given by (8) we have

1 + \frac{1}{γ} (\frac{zh′(z) − \overline{zg′(z)}}{h(z) + \overline{g(z)}} − 1) = s < 0,

hence f ∉ T S_{H}^{∗}(γ).

Similarly, let

f(z) = z + z̄^{2}.  (9)

Then f ∈ P S_{H}^{∗}(γ) when γ ∈ (3/2, +∞), because

∑_{n=1}^{∞} [(n − 1)\frac{Re(γ)}{|γ|} + |γ|] a_{n} + [(n + 1)\frac{Re(γ)}{|γ|} − |γ|] b_{n}

= |γ| · 1 + (3\frac{Re(γ)}{|γ|} − |γ|) · 1 = 3\frac{Re(γ)}{|γ|} ⩽ 2|γ|.

Now let γ ∈ (3/2, +∞) and let s be a negative real number such that γ+γ(s−1) < 0. If we choose z = −\frac{γ(s − 1)}{3 + γ(s − 1)}, then z ∈ U and for f given by (9) we have

1 + \frac{1}{γ} (\frac{zh′(z) − \overline{zg′(z)}}{h(z) + \overline{g(z)}} − 1) = s < 0,

hence f ∉ T S_{H}^{∗}(γ).

Theorem 2.5. If γ ∈ (−∞, −1) ∪ (−1/2, 0), then

T S_{H}^{∗}(γ) ⊈ OS_{H}^{∗}(γ).

Proof. Let γ ∈ (−∞, −1); we verify that the functions

f_{λ}(z) = z − λz^{2}  (10)

belong to T S_{H}^{∗}(γ) for λ > \frac{γ}{1+γ} and that f ∉ OS_{H}^{∗}(γ).

Indeed we have

∑_{n=1}^{∞}[2(n − 1 + |γ|)a_{n} + (n + 1 + |n + 1 − 2γ|)b_{n}] = 2|γ| + 2(1 + |γ|)λ > 4|γ|,

because λ > \frac{γ}{1+γ} > 1.

We also have

Re \{1 + \frac{1}{γ} (\frac{zh′_{λ}(z) − \overline{zg′_{λ}(z)}}{h_{λ}(z) + \overline{g_{λ}(z)}} − 1)\} = Re \{1 + \frac{λz}{γ(λz − 1)}\} > 0, z ∈ U, (11)

for λ > \frac{γ}{1+γ} and γ < −1, hence f_{λ} ∈ T S_{H}^{∗}(γ).

Let now γ ∈ (−1/2, 0), and let f_{λ} be defined by (10), where

−\frac{γ}{1 − γ} < λ < −\frac{γ}{1 + γ} .

Then λ > −\frac{γ}{1−γ} implies f_{λ} ∉ OS_{H}^{∗}(γ) and for λ < −\frac{γ}{1 + γ} the inequality (11) is

also verified, hence f_{λ} ∈ T S_{H}^{∗}(γ).

Acknowledgments. The authors warmly thank the referee and editors for their suggestions and criticisms which have essentially improved our original paper.

REFERENCES

[1] Avcı, Y. and Zlotkiewicz, E., On harmonic univalent mappings, Ann. Univ. Mariae Curie-Sklodowska, Sect. A, 44 (1990), 1–7.

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

[3] Jahangiri, J. M., Harmonic functions starlike in the unit disk, J. Math. Anal. Appl., 235 (1999), 470–477.

[4] Silverman, H., Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl., 220 (1998), 283–289.

[5] Silverman, H. and Silvia, E. M., Subclasses of harmonic univalent functions, N. Z. J. Math., 28 (1999), 275–284.

(received 24.09.2003, in revised form 24.04.2005)

Uludaǧ Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, 16059, Bursa, Turkey

E-mail: skarpuz@uludag.edu.tr