Harmonic Mappings Convex in One or Every

Direction ∗

Stacey Muir †

Abstract

Given a convex complex-valued analytic mapping on the open unitdisk in C, we construct a family of complex-valued harmonic map-pings convex in the direction of the imaginary axis. We also show thatadding the condition of direction convexity preserving to the analyticmapping is a necessary and sufficient condition for the harmonic map-ping to be convex. Using analytic radial slit mappings, we provide athree parameter family of harmonic mappings convex in the directionof the imaginary axis and show in some cases that as one parametervaries continuously, the mappings vary from being convex in the di-rection of the imaginary axis to being convex. In so doing, we alsoprovide information on whether or not some analytic mappings aredirection convexity preserving. Lastly, we will provide coefficient con-ditions leading to harmonic mappings which are starlike or convex oforder α.

1 Introduction

A domain D ⊆ C is convex in the direction of φ, φ ∈ [0, π), if every lineparallel to the line through 0 and eiφ has a connected intersection with D,and a domain D that is convex in every direction is convex. In 1984, Clunieand Sheil-Small [2] introduced a technique for constructing a complex-valuedharmonic mapping (one-to-one and sense-preserving, see Section 2) on theopen unit disk D = {z ∈ C : |z| < 1} with a range that is convex in one or

∗This article is published in Computational Methods and Function Theory, 12 (2012),221 – 239.

†Department of Mathematics, University of Scranton, Scranton, PA 18510 U.S.A.,stacey.muir@scranton.edu

1

every direction. One interesting example that came from their work is theharmonic right half-plane mapping

ℓ0(z) =1

2

(z

1− z+

z

(1− z)2

)+

1

2

(z

1− z− z

(1− z)2

). (1.1)

Writing I(z) = z/(1− z), we can express ℓ0 as

ℓ0(z) =I(z) + zI ′(z)

2+

I(z)− zI ′(z)

2.

In this paper, we generalize the structure seen above to generate harmonicmappings with a range that is convex in one or every direction withoutdirectly using the construction introduced in [2]. The generalization is donein the following way. For f : D → C a univalent analytic function withf(0) = f ′(0)− 1 = 0, define

Tc[f ](z) =f(z) + czf ′(z)

1 + c+

f(z)− czf ′(z)

1 + c, z ∈ D, c > 0. (1.2)

Clearly, T1[I] = ℓ0, and in [7], it was proved that Tc[I] is a harmonic half-plane mapping of D onto {w ∈ C : Rew > −1/(1 + c)} for each c > 0.Another recent example of a harmonic mapping with the structure given in(1.2) uses Ruscheweyh and Suffridge’s [11] continuous extension of the de laVallee Poussin means Vλ : D → C, λ > 0, defined by

Vλ(z) =λz

λ+ 12F1(1, 1− λ; 2 + λ;−z) (1.3)

where 2F1 is the Gaussian hypergeometric function. In [7], Tc[Vλ] was shownto be a harmonic mapping of D onto a convex domain for each λ ≥ 1/2 andc > 0.

In the following sections, we investigate the mapping properties of Tc[f ]defined in (1.2). Our main theorem provides a necessary and sufficient condi-tion on f such that Tc[f ] is a harmonic mapping of D onto a convex domain.We will also explore some coefficient conditions on f leading to other geo-metric properties of Tc[f ](D) and provide examples throughout to illustratethe interplay between the harmonic mapping Tc[f ] and the analytic mappingf .

2 Background

Let H0(D) be the set of analytic functions on D that fix zero, and letS ⊆ H0(D) be the set of univalent functions with the added normaliza-tion f ′(0) = 1. A harmonic function f : D → C with f(0) = 0 can be

2

uniquely represented as f = h + g with h, g ∈ H0(D). Furthermore, if wewrite f(z) = f(x+ iy) = u(x, y) + iv(x, y), then f is sense-preserving if theJacobian, Jf , of the mapping (x, y) 7→ (u, v) is positive. The function f islocally univalent if Jf never vanishes on D. By a result of Lewy [6], f = h+gis locally univalent and sense-preserving if and only if |g′(z)| < |h′(z)| forall z ∈ D. In this case, we simply say f is locally univalent. In addition,we call f univalent if f is one-to-one and sense-preserving on D. Let SH

be the family of harmonic univalent functions on D of the form f = h + gnormalized by h(0) = g(0) = 0 and h′(0) = 1. Clearly, S ( SH .

To discuss the mapping properties of Tc[f ] defined in (1.2) we first needto introduce a number of families of analytic and harmonic functions. Adomain D is close-to-convex if its complement can be written as the unionof non-crossing half-lines. Let C and CH denote the respective subclasses ofS and SH for which f(D) is close-to-convex. Let K(φ) and KH(φ) be therespective subclasses of S and SH for which f(D) is convex in the directionof φ. Note K(φ) ⊆ C and KH(φ) ⊆ CH . If φ = π/2, we use the phraseconvex in the direction of the imaginary axis. A domain D is starlike withrespect to a point w0 ∈ D provided for every w ∈ D, the line segmenttw + (1 − t)w0, 0 ≤ t ≤ 1 lies in D. Let S∗ and S∗

H denote the respectivesubclasses of S and SH for which f(D) is starlike with respect to the origin.Lastly, let K and KH denote the respective subclasses of S and SH for whichf(D) is convex. We will simply say an analytic function f is close-to-convex,convex in the direction of φ, starlike, or convex if f/f ′(0) is in C, K(φ), S∗, orK, respectively. Likewise, we will simply say a harmonic function f = h+ gis close-to-convex, convex in the direction of φ, starlike, or convex if f/h′(0)is in CH , KH(φ), S∗

H , or KH , respectively.The following theorems due to Clunie and Sheil-Small [2] provide some

information on the mapping properties of harmonic mappings and was usedto construct the mapping ℓ0 given in (1.1).

Theorem 2.1. A harmonic f = h+ g locally univalent on D is a univalentmapping of D onto a domain convex in the direction of the imaginary axisif and only if h + g is a conformal mapping of D onto a domain convex inthe direction of the imaginary axis.

Theorem 2.2. A function f = h + g ∈ KH if and only if the analyticfunctions

h(z)− eiφg(z), φ ∈ [0, 2π)

are convex in the direction of φ/2 and f is suitably normalized. In this case,the functions

h(z) + εg(z), |ε| ≤ 1

3

are close-to-convex. In particular, h is close-to-convex.

The concept of direction convexity preserving will come into play forour main result. Let f(z) =

∑∞n=0 anz

n and g(z) =∑∞

n=0 bnzn be analytic

on D. The function f ∗ g : D → C defined as (f ∗ g)(z) =∑∞

n=0 anbnzn

is the Hadamard product of f and g. For f ∈ S and F = H + G ∈ SH ,define f ∗F : D → C as f ∗F = f ∗ H + f ∗G. A function f ∈ S isdirection convexity preserving if it preserves the class K(φ), φ fixed, underthe Hadamard product. Let DCP be the set of direction convexity preservingfunctions. It is well-known DCP ( K. The following theorems are variationsof results from [10] that provide useful characterizations of the family DCP.

Theorem 2.3. Let f ∈ H0(D). Then f ∈ DCP if and only if for eachγ ∈ R, f(z) + iγzf ′(z) is convex in the direction of the imaginary axis.

Theorem 2.4. Let f ∈ K be analytic on D with u(θ) = Re f(eiθ) three timescontinuously differentiable. Then f ∈ DCP if and only if

u′(θ)u′′′(θ) ≤ (u′′(θ))2, θ ∈ R. (2.1)

The de la Vallee Poussin means are a well studied class of functionsrelated to the family DCP. It is shown in [11] the continuous extensionof the de la Vallee Poussin means given in (1.3) are in DCP for λ ≥ 1/2.Additionally, in Remark 5.21 in [2], V2 is used to show that not all functionsin DCP preserve KH(φ) under ∗. However, functions in DCP do preserveKH under ∗ as stated in the following theorem from [10].

Theorem 2.5. Let g ∈ H0(D). Then f ∗ g ∈ KH for all f ∈ KH if and onlyif g ∈ DCP.

Two other classes of functions that we will use in Section 5 are harmonicmappings which are starlike or convex of order α. A function f ∈ SH isstarlike of order α, α ∈ [0, 1), if

∂

∂θarg f(reiθ) > α, 0 ≤ r < 1.

Denote this set of functions by S∗H(α). It is evident S∗

H(0) = S∗H . A function

f ∈ SH is convex of order α, α ∈ [0, 1), if

∂

∂θarg

(∂

∂θf(reiθ)

)> α, 0 ≤ r < 1. (2.2)

Clearly the set of harmonic functions in SH that are convex of order 0 is justKH . The following results of Jahangiri [4, 5] connect coefficient conditionsto harmonic functions which are starlike or convex of order α.

4

Theorem 2.6. Let α ∈ [0, 1) and f : D → C be a harmonic function writtenas

f(z) =

∞∑n=1

Anzn +

∞∑n=1

Bnzn, A1 = 1. (2.3)

If∞∑n=1

(n− α

1− α|An|+

n+ α

1− α|Bn|

)≤ 2, (2.4)

then f ∈ S∗H(α). Moreover, if An ≤ 0 for n ≥ 2 and Bn ≥ 0 for n ≥ 1, the

condition given in (2.4) is necessary for f to be in S∗H(α).

If∞∑n=1

(n(n− α)

1− α|An|+

n(n+ α)

1− α|Bn|

)≤ 2, (2.5)

then f ∈ SH is convex of order α. Moreover, if An ≤ 0 for n ≥ 2 andBn ≤ 0 for n ≥ 1, the condition given in (2.5) is necessary for f to beconvex of order α.

3 Harmonic Mappings Convex in One or EveryDirection

Our first lemma establishes precisely when Tc[f ] as defined by (1.2) is locallyunivalent.

Lemma 3.1. The function Tc[f ] defined by (1.2) is locally univalent if andonly if f is convex.

Proof. Write Tc[f ] = F = H +G. Since F is locally univalent if and only if|G′(z)| < |H ′(z)| for all z ∈ D, F will be locally univalent if and only if∣∣∣∣(1− c)f ′(z)− czf ′′(z)

(1 + c)f ′(z) + czf ′′(z)

∣∣∣∣ < 1

or equivalently, since c > 0 and f ∈ S,∣∣∣∣1c −(1 +

zf ′′(z)

f ′(z)

)∣∣∣∣ < ∣∣∣∣1c +

(1 +

zf ′′(z)

f ′(z)

)∣∣∣∣ .It can be easily seen that the inequality above is equivalent to

Re

(1 +

zf ′′(z)

f ′(z)

)> 0

5

which is the analytic condition for convexity. Thus, Tc[f ] is locally univalentif and only if f is convex.

We immediately have the following.

Theorem 3.2. The function Tc[f ] defined by (1.2) is convex in the directionof the imaginary axis if and only if f is convex.

Proof. Again write Tc[f ] = H + G. Since (H + G)(z) = 2f(z)/(1 + c), byLemma 3.1 and Theorem 2.1, the proof is complete.

The concept of direction convexity preserving provides the necessary andsufficient condition for Tc[f ] to be convex. We now state and prove our maintheorem.

Theorem 3.3. The function Tc[f ] defined by (1.2) is in KH if and only iff ∈ DCP.

Proof. Let I(z) = z/(1− z). Then for f ∈ S,

Tc[f ](z) = (f ∗Tc[I])(z).

In [7], it was shown that Tc[I] is a convex harmonic mapping of D onto thehalf-plane {w : Rew > −1/(1 + c)} for each c > 0. Hence, by Theorem 2.5,Tc[f ] ∈ KH if f ∈ DCP.

To show f ∈ DCP when Tc[f ] = H + G ∈ KH we will use an approachsimilar to the proof of Theorem 2 in [10]. By Theorem 2.2 above, Tc[f ] isconvex if and only if ie−iφ/2(H − eiφG) is convex in the direction of theimaginary axis for each φ ∈ [0, 2π). Connecting this to f , we have for φ = 0and z ∈ D,

ie−iφ/2(H(z)− eiφG(z)) = −i(eiφ/2 − e−iφ/2

)1 + c

(f(z) + c

1 + eiφ

1− eiφzf ′(z)

)=

2 sin(φ/2)

1 + c

(f(z) + c

1 + eiφ

1− eiφzf ′(z)

).

Write c(1 + eiφ)/(1 − eiφ) = iγ for γ ∈ R. Since sin(φ/2)/(1 + c) ∈ R+,from the equations above, ie−iφ/2(H − eiφG) is convex in the direction ofthe imaginary axis if and only if f(z) + iγzf ′(z) is convex in the directionof the imaginary axis for each γ ∈ R. Thus, by Theorem 2.3, Tc[f ] ∈ KH

implies f ∈ DCP.

We have the following corollary as a consequence of Theorem 2.2.

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Corollary 3.4. For |ε| ≤ 1, c > 0, and f ∈ DCP,

(1 + ε)f(z) + (1− ε)czf ′(z)

is close-to-convex. In particular, f(z) + czf ′(z) is close-to-convex.

4 A Family of Harmonic Mappings Convex in Oneor Every Direction

In this section, we introduce a three parameter family of harmonic mappingswhich are convex in one or every direction constructed using Tc[f ] defined in(1.2). Using Theorem 3.3, we will also use the mapping properties of theseharmonic functions to determine whether or not certain analytic mappingsare direction convexity preserving.

Notice we may write Tc[f ] as

Tc[f ](z) =2

1 + cRe f(z) + i

2c

1 + cIm(zf ′(z)). (4.1)

and as we saw in the introduction, for I(z) = z/(1−z), T1[I] is the harmonichalf-plane mapping ℓ0 given in (1.1). Observe zI ′(z) = z/(1−z)2 is the one-slit Koebe mapping. Generalizing this and utilizing the structure in equation(4.1), we consider the mapping properties of Tc[f ] for rotations of f ∈ Ksuch that zf ′(z) is a radial n-slit map.

We begin with some background on the analytic functions we wish touse. For each n ∈ N, let gn : D → C be the radial n-slit mapping given by

gn(z) =z

(1− zn)2/n,

and for each n ∈ N, let fn : D → C be the solution of

zf ′n(z) = gn(z), fn(0) = 0. (4.2)

Each fn ∈ K by Alexander’s Theorem [1]. For the function gn, the slits forman angle of (2k+1)π/n, k = 0, . . . , n− 1 with the positive real axis and thetips of the slits which are given by gn(e

πi(2k+1)/n), k = 0, . . . , n− 1 lie on acircle of radius 2−2/n. From equation (4.2),

f1(z) =z

1− z, f2(z) =

1

2log

1 + z

1− z,

and

fn(z) = z 2F1

(1

n,2

n; 1 +

1

n; zn), n ≥ 3.

7

Clearly, f1 maps D onto a right half-plane and f2 maps D onto the strip{w ∈ C : | Imw| < π/4}. For n ≥ 3, it is known fn maps D onto the interiorof a regular n-gon whose vertices form angles of 2πk/n, k = 0, . . . , n − 1with the positive real axis and lie on a circle with radius

21−4/nΓ2(1/2− 1/n)

2n sin(π/n)Γ(1− 2/n)

where Γ is the gamma function.For n ∈ N and α ∈ [0, 2π/n), define fn,α : D → C by fn,α(z) =

e−iαfn(eiαz). We will study the three parameter family of mappings given

by

Tc[fn,α](z) =e−iαfn(e

iαz) + czf ′n(e

iαz)

1 + c+e−iαfn(eiαz)− czf ′

n(eiαz)

1 + c. (4.3)

Note it is because of the rotational symmetry of fn and gn that it sufficesto restrict α to the interval [0, 2π/n) when determining the geometry ofTc[fn,α](D). Immediately, we have the following corollary to Theorems 3.2and 3.3.

Corollary 4.1. For each n ∈ N, α ∈ [0, 2π/n), and c > 0, the functionTc[fn,α] given by (4.3) is in KH(π/2). Moreover, Tc[fn,α] ∈ KH if and onlyif fn,α ∈ DCP.

For n = 1, 2, we will explicitly determine Tc[fn,α](D) below. However,for n ≥ 3, this becomes more difficult because of the involvement of thehypergeometric function. Nonetheless, we will provide sufficient detail toconclude that Tc[fn,α] ∈ KH for n ≥ 3 and thereby show the polygonalmappings fn,α, n ≥ 3 are not in DCP by Corollary 4.1.

Before we begin the analysis for specific values of n, we make the follow-ing general observations. For n ∈ N and α ∈ [0, 2π/n), define φn,α : R → Rby

φn,α(k) =2πk

n− α (4.4)

For each n ∈ N and α ∈ [0, 2π/n), let En,α = {eiφn,α(k) : k = 0, . . . , n − 1}.Then Tc[fn,α] is continuous to ∂D \ En,α. Furthermore, En,α is the set ofinfinite boundary discontinuities of Tc[fn,α]. By equations (4.1) and (4.2),whenever e−iαgn(e

iαz) has slit(s) which lie on the real axis, Tc[fn,α] willcollapse arc(s) of ∂D to single point(s) on the real axis. If n is odd, only one

8

slit may lie on the real axis, and this occurs when α = 0 or when α = π/n.Specifically, if n is odd and θ ∈ (φn,0((n− 1)/2), φn,0((n+ 1)/2)),

Tc[fn,0](eiθ) = − 2

1 + c2F1

(1

n,2

n; 1 +

1

n;−1

). (4.5)

On the other hand, if n is odd and θ ∈ (φn,π/n(0), φn,π/n(1)),

Tc[fn,π/n](eiθ) =

2

1 + c2F1

(1

n,2

n; 1 +

1

n;−1

). (4.6)

For n even, two slits may lie on the real axis and this occurs only for α = π/n.Specifically, if n is even and θ ∈ (φn,π/n(0), φn,π/n(1)),

Tc[fn,π/n](eiθ) =

2

1 + c2F1

(1

n,2

n; 1 +

1

n;−1

). (4.7)

and if θ ∈ (φn,π/n(n/2), φn,π/n(n/2 + 1)),

Tc[fn,π/n](eiθ) = − 2

1 + c2F1

(1

n,2

n; 1 +

1

n;−1

). (4.8)

Additionally, for n ≥ 3, fn maps D onto the interior of a regular n-gon.Thus, by equation (4.1), for n ≥ 3 and z ∈ D,

2

1 + cmin{Re fn,α(z) : z ∈ En,α} < ReTc[fn,α](z)

<2

1 + cmax{Re fn,α(z) : z ∈ En,α}.

(4.9)

To determine Tc[fn,α](D) for n = 1, 2, we will perform a change of vari-ables using

w =1 + zeiα

1− zeiα. (4.10)

Thus, as z varies in D, we examine Tc[fn,α](z(w)) = g(w) = u + iv asw = x+ iy varies in the right half-plane.

For n = 1,

Tc[f1,α](z) =2

1 + cRe

(z

1− eiαz

)+ i

2c

1 + cIm

(z

(1− eiαz)2

)and using (4.10) gives

u = Re g(w) =1

1 + c((x− 1) cosα+ y sinα) (4.11)

9

and

v = Im g(w) =c

2(1 + c)(2xy cosα− (x2 − y2 − 1) sinα). (4.12)

We will consider α = 0, π first as these values of α result in an arc of ∂Dcollapsing to a single point on the real axis. In this case, Tc[f1,0] = Tc[I]where again I(z) = z/(1 − z). In [7], it was shown that Tc[I] maps Dunivalently onto {w ∈ C : Rew > −1/(1 + c)} for each c > 0 using thechange of variables above. An argument similar to the one in [7] showsTc[f1,π] maps D univalently onto the half-plane {w : Rew < 1/(1 + c)}. Bydirectly showing Tc[f1,0], Tc[f1,π] ∈ KH , we have used a harmonic functionapproach to show the analytic functions f1,0, f1,π ∈ DCP using Corollary4.1. In this instance, since f1,0(z) = z/(1 − z) is the identity under ∗for functions in H0(D), it is clearly in DCP and it is not difficult to showf1,π(z) = z/(1+z) ∈ DCP. However, it is interesting how a result regardingharmonic functions informs our understanding of analytic functions in thisway.

For α = 0, π and x = k ≥ 0 fixed, we see that g = u + iv defined byequations (4.11) and (4.12) gives the parabola

v =c(1 + c)

2 sinα

(u+

cosα

1 + c

)2

+c

2(1 + c)(1− k2 csc2 α) sinα (4.13)

Thus, Tc[f1,α](∂D \ {e−iα}) is the parabola given by (4.13) when k = 0 andTc[f1,α] maps D to the exterior of this parabola. From this, we concludef1,α ∈ DCP for α = 0, π. See Figure 1 for an example.

For n = 2,

Tc[f2,α](z) =1

1 + cRe

(e−iα log

(1 + zeiα

1− zeiα

))+ i

2c

1 + cIm

(z

1− e2iαz2

).

We will use the same change of variables in (4.10) to determine Tc[f2,α](D)where

u = Re g(w) =

(cosα ln

√x2 + y2+

sinα tan−1(y/x))/(1 + c)if x > 0

(cosα ln y + (π/2) sinα) /(1 + c) if x = 0, y > 0(cosα ln(−y)− (π/2) sinα) /(1 + c) if x = 0, y < 0

(4.14)and for x, y = 0

v = Im g(w) =c

2(1 + c)

(cosα

(y +

y

x2 + y2

)− sinα

(x− x

x2 + y2

)).

(4.15)

10

Figure 1: Graphs of T2[f1,π/4](z) for |z| = 0.5, 0.75 and |z| = 1, z = e−iπ/4.

By (4.7) and (4.8), when α = π/2, the two arcs of the unit circle {eiθ :θ ∈ (−π/2, π/2)} and {eiθ : θ ∈ (π/2, 3π/2)} collapse to the two points(−π/(2(1 + c)), 0) and (π/(2(1 + c)), 0), respectively. Clearly, by (4.14), forα = π/2,

− π

2(1 + c)< u <

π

2(1 + c).

Also, for x = k > 0, fixed, g = u+ iv defined by equations (4.14) and (4.15)gives the curve

v = − c

2k(1 + c)(k2 − cos2((1 + c)u)) (4.16)

Thus, as k varies in (0,∞), we see that Tc[f2,π/2] is a strip mapping takingD onto {w : |Rew| < π/(2(1 + c))}, and by Corollary 4.1, we conclude theanalytic mapping f2,π/2(z) = −i log((1 + iz)/(1− iz)) is in DCP.

For α = π/2, setting x = 0 in g = u+ iv defined by equations (4.14) and(4.15) shows Tc[f2,α](∂D \ E2,α) maps to the curves given by

v =c

1 + ccosα cosh

((1 + c)u− π

2 sinα

cosα

), y > 0 (4.17)

v = − c

1 + ccosα cosh

((1 + c)u+ π

2 sinα

cosα

), y < 0 (4.18)

11

Figure 2: Graphs of T2[f2,π/4](z), |z| = 0.5, 0.75, 0.9 and T2[f2,π/4](∂D \E2,π/4).

By Corollary 4.1, we know Tc[f2,α](D) is convex in the direction of the imagi-nary axis. Furthermore, Tc[f2,α] is continuous to ∂D\E2,α. Thus, Tc[f2,α](D)fills the region between the curves given by equations (4.17) and (4.18), andf2,α ∈ DCP for α = π/2. See Figure 2 for an example.

Recall for n ≥ 3 and α fixed, fn,α maps D onto the interior of a regularn-gon and thus, by equation (4.1), ReTc[fn,α](D) is bounded as described inequation (4.9). However, ImTc[fn,α](z) becomes unbounded as z approachesvalues in En,α. Using the mapping properties of fn,α and its derivative,equation (4.1) provides the following information about Tc[fn,α](∂D \En,α).First assume α = 0, π/n if n is odd and α = π/n if n is even, and letk ∈ {0, . . . , n− 1} be fixed. If

nα− π

2π< k <

nα+ π(n− 1)

2π,

then ReTc[fn,α](eiθ) decreases for θ ∈ (φn,α(k), φn,α(k + 1)) where φn,α is

given by (4.4) while ImTc[fn,α](eiθ) decreases for θ ∈ (φn,α(k), φn,α((2k +

1)/2)) and then increases for θ ∈ (φn,α((2k + 1)/2), φn,α(k)). For the re-maining values of k, ReTc[fn,α](e

iθ) increases for θ ∈ (φn,α(k), φn,α(k + 1))while ImTc[fn,α](e

iθ) increases for θ ∈ (φn,α(k), φn,α((2k + 1)/2)) and thendecreases for θ ∈ (φn,α((2k+1)/2), φn,α(k+1)). The relative extrema occur

12

Figure 3: The solid curves above represent T2[f5,π/4](∂D \ E5,π/4). Thesecurves along with the dashed curves form ∂T2[f5,π/4](D).

at the points(2 cosφn,α(k + 1/2)

1 + c2F1

(1

n,2

n; 1 +

1

n;−1

),21−2/nc

1 + csinφn,α(k + 1/2)

).

(4.19)As in the n = 2 case, using the geometry given in Corollary 4.1, we seeTc[fn,α] maps D onto the region trapped by the curves described above butnow with the real part bounded by the values given in (4.9). See Figures3 and 4 for examples.

For n odd and α = 0, π/n and for n even and α = π/n, arc(s) of theunit circle now collapse to point(s) on the real axis. The above mappingbehavior still holds with the exceptions given in (4.5), (4.6), (4.7), and (4.8).Compare Figures 3 and 4 with Figures 5 and 6 for examples with and withoutcollapsing. Thus, we see for n ≥ 3, the polygonal mappings fn,α ∈ DCP forany α.

As the rotation of a function in DCP is typically not again in DCP,looking at Tc[fn,α] given by (4.3) with c > 0 fixed and n = 1 or n = 2,we obtain an interesting example of a family of harmonic mappings thatas α varies continuously, the functions move between being convex in thedirection of the imaginary axis and convex in every direction. It is worthnoting that Pokhrel [8] introduced a subclass of DCP consisting of those

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Figure 4: The solid curves above represent T2[f8,π/4](∂D \ E8,π/4). Thesecurves along with dashed curves form ∂T2[f8,π/4](D).

Figure 5: The solid curves and point on the real axis above representT2[f5,0](∂D \ E5,0). These along with dashed curves form ∂T2[f5,0](D).

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Figure 6: The solid curves and two points on the real axis above representT2[f8,π/8](∂D\E8,π/8). These along with dashed curves form ∂T2[f8,π/8](D).

functions in DCP which are rotation invariant. Using such functions, onecould use a structure like that in (4.3) to construct other convex harmonicfunctions.

5 Coefficient Conditions for Harmonic MappingsConvex or Starlike of Order α

By Theorem 3.2, we know that if f ∈ K, Tc[f ] ∈ KH(π/2) ⊆ S∗H . In this

section, we will use Theorem 2.4 to provide a coefficient condition on f thatresults in Tc[f ] ∈ S∗

H(α) ∩ KH(π/2) ( CH or Tc[f ] ∈ SH being convex oforder α. Thus, this latter condition also provides a criterion for f to bedirection convexity preserving.

Theorem 5.1. Let α ∈ [0, 1), c ≥ 1, and f : D → C be an analytic with theform

f(z) =∞∑n=1

anzn, a1 = 1. (5.1)

If∞∑n=1

cn2 − α

(1− α)(1 + c)|an| ≤ 1 (5.2)

15

then Tc[f ] ∈ S∗H(α) and f ∈ K. Furthermore, if an ≤ 0 for n ≥ 2, then (5.2)

is necessary for Tc[f ] to be in S∗H(α).

Proof. Using (5.1) and (1.2), we have

Tc[f ](z) =1

1 + c

∞∑n=1

(1 + cn)anzn +

1

1 + c

∞∑n=1

(1− cn)anzn.

Thus, substituting An = (1 + cn)an/(1 + c) and Bn = (1 − cn)an/(1 + c)into (2.4) gives (5.2) and by Theorem 2.6, Tc[f ] ∈ SH(α). It follows thenthat Tc[f ] is locally univalent and thus, by Lemma 3.1, f ∈ K.

Lastly, if an ≤ 0 for n ≥ 2, then (5.2) is necessary for Tc[f ] ∈ S∗H(α)

because then An ≤ 0 for n ≥ 2 and Bn ≥ 0 for n ≥ 1.

As a corollary to this theorem, we use a starlike condition for harmonicfunctions to provide an alternate proof to Goodman’s [3] classical resultfor analytic functions that states if f(z) = z +

∑∞n=2 anz

n ∈ H0(D) and∑∞n=2 n

2|an| ≤ 1, then f ∈ K.

Corollary 5.2. Let f(z) = z +∑∞

n=2 anzn ∈ H0(D) and c ≥ 1. If

∞∑n=2

n2|an| ≤1

c,

then Tc[f ] ∈ S∗H and f ∈ K.

We also have the following.

Corollary 5.3. Let c ≥ 1 and suppose f ∈ S has the form f(z) = z −∑∞n=2 c|an|zn. Then f ∈ K if and only if Tc[f ] ∈ S∗

H .

Proof. If Tc[f ] ∈ S∗H , then Tc[f ] is locally univalent, and hence by Lemma

3.1, f ∈ K. On the other hand, setting Bn ≡ 0 in inequality (2.5) in Theorem2.6, we see that f ∈ K if and only if

∑∞n=2 cn

2|an| ≤ 1. Thus, by Corollary5.2, f ∈ K implies Tc[f ] ∈ S∗

H .

Inequality (2.5) in Theorem 2.6 leads to the following theorem whichprovides a coefficient condition for Tc[f ] to be convex of order α and thus,by Theorem 3.3, also provides a condition for f to be in DCP. The proof ofthis theorem is similar to that of Theorem 5.1 and will be omitted.

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Theorem 5.4. Let α ∈ [0, 1), c ≥ 1, and f ∈ K be written as in (5.1). If

∞∑n=1

n(cn2 − α)

(1− α)(1 + c)|an| ≤ 1, (5.3)

then Tc[f ] is convex of order α and f ∈ DCP.

Interestingly, in the case of α = 0 or c ≤ 1/α−1, inequality (5.3) impliesfor fφ(z) = e−iφf(eiφz), φ ∈ R, Tc[fφ] is a function in KH for each φ (see[8]).

We will end this section with an application illustrating the results. Letk = 2, 3, . . . and a ∈ C. Define gk,a : D → C by

gk,a(z) = z + azk. (5.4)

If |a| ≤ (1 − α)/(k2 − α), then gk,a is convex of order α (see (2.5) withBn ≡ 0) and by Theorem 5.1, T1[gk,a] ∈ S∗

H(α). By Theorem 5.4, if |a| ≤(1 − α)/(k(k2 − α)), then T1[gk,a] ∈ SH is convex of order α and hence,gk,a ∈ DCP. Additionally, by a result in [8], eiφgk,a ∈ DCP for φ ∈ R. Soone could produce functions in KH using rotations of gk,a as in done in (4.3).

In [8], it was shown g3,a ∈ DCP for 0 ≤ a ≤ 1/27 and that this inequalityis tight for any a ≥ 0. Hence, T1[g3,a] ∈ S∗

H \ KH for 1/27 < a ≤ 1/9. SeeFigure 7. In fact, for k ≥ 3, T1[gk,1/k2 ] ∈ S∗

H \ KH and T1[gk,1/k3 ] ∈ KH .However, for k ≥ 4 and a ≥ 0, T1[gk,a] changes from a function in S∗

H to afunction in KH for some a ∈ (1/k3, 1/k2). To see why T1[gk,1/k2 ] ∈ KH ifk ≥ 3, we will use Theorem 2.4 to show gk,1/k2 ∈ DCP.

Set u(θ) = Re gk,1/k2(eiθ) = cos θ + (cos kθ)/k2. Then for k ≥ 3,

3π/(2k) ∈ (0, π/2] and((u′′(3π

2k

))2

− u′(3π

2k

)u′′′(3π

2k

))= 2−

(k +

1

k

)sin

(3π

2k

)≤ 2− k sin

(3π

2k

)(5.5)

Let p(k) be the last expression above. Then p′(k) = 3π/(2k) cos(3π/(2k))−sin(3π/(2k)). Since x cosx−sinx < 0 for x ∈ (0, π/2], we see that inequality(2.1) is violated when θ = 3π/(2k) and gk,1/k2 ∈ DCP for k ≥ 3.

6 Concluding Remarks

While proofs of the geometric properties of Tc[f ] defined in (1.2) rely on theresults of Clunie and Sheil-Small [2], Theorem 3.3 provides an interesting

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Figure 7: Above are the images of ∂T1[g3,1/9](D), ∂T1[g3,2/27](D), and∂T1[g3,1/27](D).

way of constructing convex harmonic mappings without directly using theshear construction. Moreover, both Lemma 3.1 and Theorem 3.3 presentways of using mapping properties of a harmonic function to determine if ananalytic function is in K or DCP. In a recent paper, Ruscheweyh and Salinas[9] conjectured for 0 < a ≤ b that 1F1(a; b; 1 + z) ∈ DCP and perhaps thisconjecture could be explored from a perspective of harmonic functions usingTheorem 3.3. It would also be interesting to ask what subset of K leadsto Tc[f ] ∈ S∗

H and Theorem 3.3 and Theorem 5.1 give a partial answer tothis. The subset of DCP that leads to Tc[f ] being convex of a certain ordermight be of interest too. It appears for k ≥ 4, there exist α near one suchthat T1[gk,(1−α)/(k2−α)] is actually convex where gk,a is given by (5.4). Sincegk,(1−α)/(k2−α) ∈ S∗(β) for some β ≥ α, perhaps the order of starlikeness ofan analytic function f can be used to determine an order of convexity forthe harmonic function Tc[f ].

References

[1] J. W. Alexander, Functions which map the interior of the unit circleupon simple regions, Ann. of Math., 17 (1915-1916), 12–22.

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[2] J. Clunie and T. Sheil-Small, Harmonic univalent functions, AnnalesAcad. Sci. Fenn., 9 (1984), 3–25.

[3] A. W. Goodman, Univalent functions and nonanalytic curves, Proc.Amer. Math. Soc., 8 (1957), 598 – 601.

[4] J. Jahangiri, Coefficient bounds and univalence criteria for har-monic functions with negative coefficients, Ann. Univ. Mariae Curie-Sk lodowska Sect. A, 2 (1998), 57 – 66.

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

[6] H. Lewy, On the non-vanishing of the Jacobian in certain one-to-onemappings, Bull. Amer. Math. Soc., 42 (1936), 689 –692.

[7] S. Muir, Weak subordination for convex univalent harmonic functions,J. Math. Anal. Appl., 348 (2008), 862 –871.

[8] C. Pokhrel, On a subclass of DCP, Nepali Math Sci. Rep., 22 (2004),67 – 85.

[9] S. Ruscheweyh and L. Salinas, On the DCP radius of exponential partialsums, J. Comput. Anal. Appl., 7 (2005), 271 –288.

[10] S. Ruscheweyh and L. Salinas, On the preservation of direction-convexity and the Goodman-Saff conjecture, Ann. Acad. Sci. Fenn.,14 (1989), 63 – 73.

[11] S. Ruscheweyh and T. Suffridge, A continuous extension of the de laVallee Poussin means, J. Anal. Math., 89 (2003), 155–167.

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