ON A CRITERION FOR MULTIVALENT HARMONIC FUNCTIONS

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ON A CRITERION FOR MULTIVALENT HARMONIC FUNCTIONS

T. HAYAMI1 §

Abstract. For normalized harmonic functions f(z) = h(z) +g(z) in the open unit disk, a criterion on the analytic parth(z) forf(z) to bep-valent and sense-preserving is discussed. Furthermore, several illustrative examples and images off(z) satisfying the obtained condition are enumerated.

Keywords: Harmonic function, Multivalent function, Univalent function. AMS Subject Classiﬁcation: 30C45, 58E20.

1. Introduction and Definitions

For a ﬁxed p (p = 1,2,3,· · ·), a meromorphic function f(z) in a domain D is said to bep-valent (or multivalent of order p) inD if for each w0 the equation f(z) = w0 has at mostproots inDwhere the roots are counted in accordance with their multiplicity and if there is somew1 such that the equationf(z) =w1 has exactlyproots inD. In particular,

f(z) is said to be univalent in Dwhen p= 1. A complex-valued harmonic function f(z) inDis given by

f(z) =h(z) +g(z) (1.1) where h(z) and g(z) are analytic in D. We call h(z) and g(z) the analytic part and co-analytic part of f(z), respectively. A necessary and suﬃcient condition for f(z) to be locally univalent and sense-preserving inDis|h′(z)|>|g′(z)|for allz∈D(see [2] or [8]). LetH(p) denote the class of functions f(z) of the from

f(z) =h(z) +g(z) =zp+ ∞ ∑

n=p+1

anzn+ ∞ ∑

n=p

bnzn (1.2)

which are harmonic in the open unit diskU={z∈C:|z|<1}. We next denote byS_H(p) the class of functionsf(z)∈ H(p) which arep-valent and sense-preserving inU. Then, we say thatf(z)∈ S_H(p) is a p-valently harmonic function in U.

In the present paper, we discuss a suﬃcient condition abouth(z) forf(z)∈ H(p) given by (1.2), satisfying

g′(z) =zm−1h′(z) (1.3) for somem(m= 2,3,4,· · ·), to be in the classS_H(p).

1 _{Department of Mathematics, Kinki Universiy, Higashi-Osaka, Osaka 577-8502, Japan,} e-mail: ha ya [email protected]

§ Submitted for GFTA’13, held in I¸sık University on October 12, 2013.

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⌊ ⌋ is the floor function, the equation

F(t) = 2kπ (2.2)

has at most a single root in[−π, π) and for all k∈K there exist exactly2p+m−1 such roots in [−π, π), then the harmonic function f(z) =h(z) +g(z) with g′(z) =zm−1h′(z)

belongs to the class S_H(p) and maps Uonto a domain surrounded by 2p+m−1 concave curves with 2p+m−1 cusps.

Remark 2.1. If we take p= 1 in Theorem 2.1, then we readily arrive at the univalence criterion for harmonic functions due to Hayami and Owa[5, Theorem 2.1](see also [10]).

3. Some Illustrative Examples and Image Domains

We discuss harmonic functions f(z) =h(z) +g(z) which satisfy the conditions of The-orem 2.1 and their image domains.

Example 3.1. Let h(z) =zp. Then we easily see that the equation (2.2)becomes

(2p+m−1)t= 2kπ

(

k= 0,±1,±2,· · · ,±⌊2p+m+ 1

2 ⌋

)

(3.1)

which satisfies the conditions of Theorem 2.1. Hence, the function

f(z) =h(z) +g(z) =zp+ p

p+m−1z

p+m−1 ₍_g′₍_z_{) =}_zm−1_h′₍_z₎₎ _(3.2)

belongs to the classS_H(p) and it mapsU onto a domain surrounded by2p+m−1concave curves with 2p+m−1 cusps. Taking p = 2 and m = 4 for example, we know that the function

f(z) =z2+2 5z

5 _(3.3)

is a 2-valently harmonic function in U and it maps U onto the domain surrounded by 7

concave curves with7 cusps as shown in Figure 1.

Remark 3.1. Since it follows that

F(t) = (2p+m−1)t+ 2Im(logH(eit)) (3.4)

where F(t) is given by (2.1), we obtain that

F′(t) =m+ 1 + 2Re (

eith′′(eit)

h′(eit₎ )

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-1 -0.5 0.5 1

-1 -0.5

0.5 1

Figure 1. The image off(z) =z2₊2

5z

5_.

which implies thatF(t) is increasing if

Re (

1 +zh ′′₍_z₎

h′(z) )

>−m−1

2 (z∈U). (3.6)

By the above remark, we derive the following exapmle.

Example 3.2. Let h(z) = zp+ c

p+ 1z p+1

(

|c| ≤p− 2p

2p+m+ 1 )

. Then the equation

(2.2)becomes

F(t) = (2p+m−1)t+ 2 arg(p+ceit). (3.7)

Noting

Re (

1 +zh ′′₍_z₎

h′(z) )

> p+ 1− p

p− |c| ≥ − m−1

2 (z∈U), (3.8)

F(−π) =−(2p+m−1)π−2 arctan (

|c|sinθ p− |c|cosθ

)

(3.9)

and

F(π) = (2p+m−1)π−2 arctan (

|c|sinθ p− |c|cosθ

)

(3.10)

where 0 ≤ θ = arg(c) < 2π, we see that F(t) satisfies the conditions of Theorem 2.1. Hence, the function

f(z) =h(z) +g(z) =zp+ c

p+ 1z

p+1₊ p

p+m−1z

p+m−1₊ c

p+mz

p+m _(3.11)

belongs to the classS_H(p) and it mapsU onto a domain surrounded by2p+m−1concave

curves with2p+m−1 cusps. Putting p= 2, m= 4 and c= 2 3i

(

|c| ≤ 14

9 )

, we know

that the function

f(z) =z2+2i 9z

3₊ 2 5z

5₊ i 9z

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-1.5 -1 -0.5 0.5 1

-1.5 -1 -0.5

Figure 2. The image off(z) =z2₊2i

9z

3₊2

5z

5₊ i

9z

6_.

In consideration of the process of proving Theorem 2.1, we obtain the following inter-esting example.

Example 3.3. If we consider special functions h(z) and g(z) given by

h′(z) = pz p−1

1 +z2p+m−1 and g′(z) =

pzp+m−2

1 +z2p+m−1 (

g′(z) =zm−1h′(z)), (3.13)

then the function

f(z) =h(z) +g(z) = ∫ z

0

pζp−1

1 +ζ2p+m−1dζ+ ∫ z

0

pζp+m−2

1 +ζ2p+m−1dζ (3.14)

is a member of the class S_H(p) and it maps U onto a domain surrounded by 2p+m−1

straight lines with2p+m−1 cusps. Indeed, setting p= 2 and m= 2, we know that

f(z) = ∫ z

0 2ζ

1 +ζ5dζ+ ∫ z

0 2ζ2

1 +ζ5dζ (3.15)

is a 2-valently harmonic function and it maps U onto a star as shown in Figure 3. Fur-thermore, if we take p= 1 in (3.14), then we see that the function

fm+1(z) =h(z) +g(z) = ∫ z

0 1

1 +ζm+1dζ+ ∫ z

0

ζm−1

1 +ζm+1dζ (3.16)

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-1 -0.5 0.5 1

-1 -0.5

0.5 1

Figure 3. The image off(z) =

∫ z

0

2ζ

1 +ζ5dζ+ ∫ z

0

2ζ2

1 +ζ5dζ

4. Appendix

Finally, we recall here the following theorem due to Mocanu [9].

Theorem 4.1. Let h(z) and g(z) be analytic functions in a domain D. If h(z) is convex in D and |g′(z)| < |h′(z)| for z ∈ D, then the harmonic function f(z) = h(z) +g(z) is univalent and sense-preserving inD.

In other words, if h(z) and g(z) satisfy

g′(z) =w(z)h′(z) (z∈D) (4.1)

and

Re (

1 +zh ′′₍_z₎

h′(z) )

>0 (z∈D) (4.2)

for some analytic functionw(z)in Dsatisfying|w(z)|<1 (z∈D), thenf(z)is univalent and sense-preserving inD.

Bshouty and Lyzzaik [1] have shown the next theorem which is closely related to The-orem 2.1 and Remark 3.1 with p= 1 and m = 2 as the stronger result of the conjecture of Mocanu [10].

Theorem 4.2. If h(z) and g(z) are analytic in U, with h′(0)̸= 0, which satisfy

g′(z) =zh′(z) (4.3)

and

Re (

1 +zh ′′₍_z₎

h′(z) )

>−1

2 (4.4)

for allz∈U, then the harmonic function f(z) =h(z) +g(z) is univalent close-to-convex in U.

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References

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