CERTAIN CONVEX HARMONIC FUNCTIONS

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CERTAIN CONVEX HARMONIC FUNCTIONS

YONG CHAN KIM, JAY M. JAHANGIRI, and JAE HO CHOI Received 21 December 2000 and in revised form 2 May 2001

We deﬁne and investigate a family of complex-valued harmonic convex univalent functions related to uniformly convex analytic functions. We obtain coeﬃcient bounds, extreme points, distortion theorems, convolution and convex combinations for this family.

2000 Mathematics Subject Classiﬁcation: 30C45, 30C50.

1. Introduction. A continuous complex-valued functionf =u+iv deﬁned in a simply connected complex domainᏰ⊂Cis said to be harmonic inᏰif bothuandv are real harmonic inᏰ. Consider the functionsUandVanalytic inᏰso thatu= U andv= V. Then the harmonic functionf can be expressed by

f (z)=h(z)+g(z), z∈Ᏸ^, ^(1.1) whereh=(U+V )/2 andg=(U−V )/2. We call hthe analytic part and g the co- analytic part off. If the co-analytic part offis identically zero thenfreduces to the analytic case.

The mappingzf (z)is sense-preserving and locally one-to-one inᏰif and only if the Jacobian offis positive (see [1]), that is, if and only if

J_f(z)=h(z)²−g(z)²>0, z∈Ᏸ^. ^(1.2) LetᏴdenote the family of functionsf=h+g¯which are harmonic, sense-preserving, and univalent in the open unit disk∆= {z:|z|<1}withh(0)=f (0)=f_z(0)−1=0.

Thus, we may write h(z)=z+

∞ n=2

anzⁿ, g(z)= ∞ n=1

bnzⁿ, b1<1. (1.3) Also let Ᏼ denote the subclass ofᏴ consisting of functionsf =h+g¯so that the functionshandgtake the form

h(z)=z− ∞ n=2

a_nzⁿ, g(z)= − ∞ n=1

b_nzⁿ, b1<1. (1.4) Recently, Kanas and Wisniowska [5] (see also Kanas and Srivastava [4]), studied the class ofk-uniformly convex analytic functions, denoted byk-ᐁᏯᐂ, 0≤k <∞, so that h∈k-ᐁᏯᐂif and only if

1+(z−ζ)h(z) h(z)

≥0, |ζ| ≤k, z∈∆. (1.5)

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For realφwe may letζ= −kze^iφ. Then condition (1.5) can be written as

1+

1+ke^iφzh(z) h(z)

≥0. (1.6)

Now considering the harmonic functionsf =h+g¯of the form (1.3) we deﬁne the familyᏴᏯᐂ(k,α), 0≤α <1, so thatf=h+g¯∈ᏴᏯᐂ(k,α)if and only if

1+

1+ke^iφz²h(z)+2zg(z)+z²g(z) zh(z)−zg(z)

≥α, 0≤α <1. (1.7)

Finally, we letᏴᏯᐂ(k,α)≡ᏴᏯᐂ(k,α)∩Ᏼ.

Notice that ifg≡0 andα=0 then the familyᏴᏯᐂ(k,α)defined by (1.7) reduces to the classk-ᐁᏯᐂôfk-uniformly convex analytic functions defined by (1.5). If we, further, letk=1 in (1.5), we obtain the class of uniformly convex analytic functions defined by Goodman [2]. A geometric characterization of the general familyᏴᏯᐂ(k,α) is an open question.

In Section 2, we introduce suﬃcient coeﬃcient bounds for functions to be in ᏴᏯᐂ(k,α)and show that these bounds are also necessary for functions inᏴᏯᐂ(k,α).

InSection 3, the inclusion relation between the classesk-ᐁᏯᐂandᏴᏯᐂ^(k,α)is ex- amined. Extreme points and distortion bounds forᏴᏯᐂ^(k,α)are given inSection 4.

Finally, inSection 5, we show that the classᏴᏯᐂ(k,α) is closed under convolution and convex combinations.

Here we state a result due to Jahangiri [3], which we will use throughout this paper.

Theorem1.1. Letf=h+g¯withhandgof the form (1.3). If ∞

n=2

n(n−α) 1−α an+

∞ n=1

n(n+α)

1−α bn≤1, 0≤α <1, (1.8) thenfis harmonic, sense-preserving, univalent in∆, andfis convex harmonic of order αdenoted byᏴ᏷(α). Condition (1.8) is also necessary iff∈Ᏼ᏷^(α)≡Ᏼ᏷^(α)∩Ᏼ.

2. Coefficient bounds. First we state and prove a sufficient coefficient bound for the classᏴᏯᐂ^(k,α).

Theorem2.1. Letf=h+g¯be of the form (1.3). If0≤k <∞,0≤α <1, and ∞

n=2

n(n+nk−k−α) 1−α an+

∞ n=1

n(n+nk+k+α)

1−α bn≤1, (2.1) thenf is harmonic, sense-preserving, univalent in∆, andf∈ᏴᏯᐂ(k,α).

Proof. Sincen−α≤n+nk−k−αand n+α≤n+nk+k+αfor 0≤k <∞, it follows from Theorem 1.1 thatf ∈Ᏼ᏷^(α)^{and hence}^f is sense-preserving and convex univalent in∆. Now, we only need to show that if (2.1) holds then

zh(z)+

1+ke^iφ

z²h(z)+

1+2ke^iφ

zg(z)+

1+ke^iφ

z²g(z) zh(z)−zg(z)

=A(z) B(z)≥α.

(2.2)

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Using the fact that(w)≥αif and only if|1−α+w| ≥ |1+α−w|it suﬃces to show that

A(z)+(1−α)B(z)−A(z)−(1+α)B(z)≥0, (2.3) whereA(z)=zh(z)+(1+keîφ)z²h(z)+(1+2keîφ)zg(z)+(1+keîφ)z²g(z)and B(z)=zh(z)−zg(z). Substituting forA(z)andB(z)in (2.3), we obtain

A(z)+(1−α)B(z)−A(z)−(1+α)B(z)

=

(2−α)z+ ∞ n=2

n n+1−α+k(n−1)e^iφ anzⁿ +

∞ n=1

nn−1+α+k(n+1)e^iφb¯nz¯ⁿ

− −αz+

∞ n=2

n n−1−α+k(n−1)e^iφ anzⁿ +

∞ n=1

n n+1+α+k(n+1)e^iφ¯bnz¯ⁿ

≥(2−α)|z|−

∞ n=2

n n(k+1)+1−k−αan|z|ⁿ

− ∞ n=1

n n(k+1)−1+k+αbn|z|ⁿ

−α|z|−

∞ n=2

n n(k+1)−1−k−αan|z|ⁿ

− ∞ n=1

n n(k+1)+1+k+αb_n|z|ⁿ

≥2(1−α)|z|

1− ∞ n=2

n n(k+1)−k−α 1−α a_n

− ∞ n=1

n n(k+1)+k+α

1−α b_n

≥0, by (2.1).

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The harmonic functions f (z)=z+

∞ n=2

1−α

n(nk+n−k−α)x_nzⁿ+ ∞ n=1

1−α

n(nk+n+k+α)y¯_nz¯ⁿ, (2.5) where_∞

n=2|xn|+_∞

n=1|yn| =1, show that the coeﬃcient bound given inTheorem 2.1 is sharp.

The functions of the form (2.5) are inᏴᏯᐂ^(k,α)^because ∞

n=2

n(n+nk−k−α) 1−α a_n+

∞ n=1

n(n+nk+k+α) 1−α b_n=

∞ n=2

x_n+ ∞ n=1

y_n=1. (2.6)

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Next we show that the bound (2.1) is also necessary for functions inᏴᏯᐂ(k,α).

Theorem2.2. Letf=h+g¯withhandgof the form (1.4). Thenf∈ᏴᏯᐂ(k,α)if and only if

∞ n=2

n(n+nk−k−α) 1−α an+

∞ n=1

n(n+nk+k+α)

1−α bn≤1. (2.7) Proof. In view ofTheorem 2.1, we only need to show thatf∉ᏴᏯᐂ(k,α)if condition (2.7) does not hold. We note that a necessary and sufficient condition forf=h+g¯ given by (1.4) to be inᏴᏯᐂ^(k,α)is that the coefficient condition (1.7) to be satisfied.

Equivalently, we must have (1−α)zh(z)+

1+ke^iφ

z²h(z)+

1+α+2ke^iφ

zg(z)+

1+ke^iφ

z²g(z)

zh(z)−zg(z) ≥0.

(2.8) Upon choosing the values ofzon the positive real axis where 0≤z=r <1, the above inequality reduces to

1−α−_∞

n=2n(nk+n−k−α)an+_∞

n=1n(nk+n+k+α)bnrⁿ⁻¹ 1−_∞

n=2na_nrⁿ⁻¹+_∞

n=1nb_nrⁿ⁻¹ ≥0. (2.9) If condition (2.7) does not hold then the numerator in (2.9) is negative forr suﬃ- ciently close to 1. Thus there existsz0=r0in (0,1)for which the quotient (2.9) is negative. This contradicts the required condition forf∈ᏴᏯᐂ^(k,α)and so the proof is complete.

3. Inclusion relations. As mentioned earlier in the proof ofTheorem 2.1, the functions inᏴᏯᐂ(k,α)are convex harmonic in∆. In the following example we show that this inclusion is proper.

Example3.1. Consider the harmonic functions f_n(z)=z−1

2z¯− 1

2n²z¯ⁿ, z∈∆, n=2,3,.... (3.1) Fora_n≡0 andb_n= −1/2n², we observe that

∞ n=2

n²a_n+ ∞ n=1

n²b_n=1

2+n² 1 2n²

=1 2+1

2=1. (3.2)

Therefore, byTheorem 1.1,f_n∈Ᏼ᏷^(0).

On the other hand, 2k+1+α

1−α −1

2

+n(nk+n+k+α) 1−α

− 1 2n

=2k+1+α

2(1−α) +nk+n+k+α

2n(1−α) >1. (3.3) Thus, byTheorem 2.2,f∉ᏴᏯᐂ(k,α).

More generally, we can prove the following theorem.

Theorem3.2. Let0≤k <∞,0≤α <1, and0≤β <1. Ifk > β/(1−β)then the proper inclusion relationᏴᏯᐂ(k,α)⊂Ᏼ᏷(β).

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Proof. Letf∈ᏴᏯᐂ(k,α), then, byTheorem 2.2, ∞

n=2

n(nk+n−k−α) 1−α an+

∞ n=1

n(nk+n+k+α)

1−α bn≤1. (3.4) Since (n−β)/(1−β) < (nk+n−k−α)/(1−α) and (n+β)/(1−β) < (nk+n+ k+α)/(1−α), byTheorem 1.1, we conclude thatf∈Ᏼ᏷^(β).

To show that the inclusion is proper, consider the harmonic functions f_n(z)=z− 1−β

2(1+β)z¯− 1−β

2n(n+β)z¯ⁿ, z∈∆, n=2,3,.... (3.5) ByTheorem 1.1,fn∈Ᏼ᏷(β), because

∞ n=2

n(n−β) 1−β an+

∞ n=1

n(n+β)

1−β bn=1+β 1−β

1−β

2(1+β)+n(n+β) 1−β

1−β

2n(n+β)=1. (3.6) On the contrary, byTheorem 2.2,fn∉ᏴᏯᐂ(k,α), because

∞ n=1

n(nk+n+k+α)

1−α bn=1+α+2k 1−α

1−β 2(1+β)+n

n+α+(n+1)k 1−α

1−β 2n(n+β)

= 1−β 2(1−α)

1+α+2k

1+β +n+α+(n+1)k n+β

> 1−β 2(1−α)

1+α+2β/(1−β)

1+β +n+α+(n+1)β/(1−β) n+β

= 1 2(1−α)

2+α(1−β)(n+1+2β) (1+β)(n+β)

≥1.

(3.7)

4. Extreme points and distortion bounds. Using deﬁnition (1.7), and according to the arguments given in [3], we obtain the following extreme points of the closed convex hulls ofᏴᏯᐂ^(k,α)denoted by clcoᏴᏯᐂ^(k,α).

Theorem 4.1. Let f be the form of (1.4). Then f ∈clcoᏴᏯᐂ^(k,α) if and only if f (z)= _∞

n=1(Xnhn+Yngn) where h1(z)= z, hn(z)=z−((1−α)/n(n+nk− k−α))zⁿ(n=2,3,...), g_n(z)=z−((1− α)/n(n+nk+k+α))¯zⁿ(n=1,2,3,...), _∞

n=1(X_n+Y_n)=1,X_n≥0andY_n≥0. In particular, the extreme points ofᏴᏯᐂ^(k,α) are{hn}and{gn}.

Similarly, follows the distortion bounds for functions inᏴᏯᐂ(k,α).

Theorem4.2. Iff∈ᏴᏯᐂ(k,α)then f (z)≤

1+b1r+1 2

1−α

2+k−α−1+2k+α 2+k−α b1

r², |z| =r <1, f (z)≥

1−b1r−1 2

1−α

2+k−α−1+2k+α 2+k−α b1

r², |z| =r <1.

(4.1)

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If we letr→1 in the left-hand inequality ofTheorem 4.2and collect the like terms, we obtain the following theorem.

Theorem4.3. Iff∈ᏴᏯᐂ(k,α)then{w:|w|< (3+2k−α)/2(2+k−α)−3(1−α)/

2(2+k−α)|b1|} ⊂f (∆).

5. Convolutions and convex combinations. For harmonic functions f (z)=z− _∞

n=2|an|zⁿ−_∞

n=1|bn|z¯ⁿ andF(z)=z−_∞

n=2|Anzⁿ−_∞

n=1|Bn|z¯ⁿ, we deﬁne the convolution off andF as

(f∗F)(z)=f (z)∗F(z)=z− ∞ n=2

anAnzⁿ− ∞ n=1

bnBnz¯ⁿ. (5.1)

In the following theorem we examine the convolution properties of the classᏴᏯᐂ(k,α).

Theorem5.1. For0≤α≤β <1, letf∈ᏴᏯᐂ(k,β)andF∈ᏴᏯᐂ(k,α)then f∗F∈ᏴᏯᐂ^(k,β)⊂ᏴᏯᐂ^(k,α). ^(5.2) Proof. Express the convolution off andF as that given by (5.1) and note that

|A_n| ≤1 and|B_n| ≤1. Now the theorem follows upon the application of the required condition (2.7).

The convex combination properties of the classᏴᏯᐂ(k,α)is given in the following theorem.

Theorem5.2. The classᏴᏯᐂ^(k,α)is closed under convex combinations.

Proof. Fori=1,2,...,suppose thatfi∈ᏴᏯᐂ(k,α)wherefiis given byfi(z)= z−_∞

n=2|ain|zⁿ−_∞

n=1|bin|z¯ⁿ. For_∞

i=1ti=1, 0≤ti≤1, the convex combinations off_imay be written as

∞ i=1

tifi(z)=z− ∞ n=2

_∞

i=1

tiain zⁿ−

∞ n=1

_∞

i=1

tibin

z¯ⁿ. (5.3)

Now, the theorem follows by (2.7) upon noting that_∞

i=1t_i=1.

Acknowledgment. The research of the ﬁrst author was supported by Yeungnam University, Korea.

References

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

[2] A. W. Goodman,On uniformly convex functions, Ann. Polon. Math.56(1991), no. 1, 87–92.

[3] J. M. Jahangiri,Coeﬃcient bounds and univalence criteria for harmonic functions with negative coeﬃcients, Ann. Univ. Mariae Curie-Skłodowska Sect. A52(1998), no. 2, 57–66.

[4] S. Kanas and H. M. Srivastava,Linear operators associated withk-uniformly convex func- tions, Integral Transform. Spec. Funct.9(2000), no. 2, 121–132.

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[5] S. Kanas and A. Wisniowska,Conic regions andk-uniform convexity, J. Comput. Appl. Math.

105(1999), no. 1-2, 327–336.

Yong Chan Kim: Department of Mathematics Education, Yeungnam University, Gyongsan712-749, Korea

E-mail address:[email protected]

Jay M. Jahangiri: Department of Mathematics, Kent State University, Burton, OH 44021-9500, USA

E-mail address:[email protected]

Jae Ho Choi: Department of Mathematics Education, Yeungnam University, Gyongsan712-749, Korea

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