In this paper the class of harmonic functions constructed by the Hadamard product in the unit disk, and properties of some of its subclasses are examined

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Volume 3, Issue 1, Article 9, 2002

ON HARMONIC FUNCTIONS CONSTRUCTED BY THE HADAMARD PRODUCT

METIN ÖZTÜRK, SIBEL YALÇIN, AND MÜMIN YAMANKARADEN˙IZ ometin@uludag.edu.tr

ULUDAGUNIVERSITY

FACULTY OFSCIENCE

DEPARTMENT OFMATHEMATICS

16059 BURSA/TURKEY.

Received 11 May, 2001; accepted 11 October, 2001.

Communicated by S.S. Dragomir

ABSTRACT. A functionf =u+ivdefined in the domainD⊂Cis harmonic inDifu,vare real harmonic. Such functions can be represented asf =h+ ¯gwhereh,gare analytic inD. In this paper the class of harmonic functions constructed by the Hadamard product in the unit disk, and properties of some of its subclasses are examined.

Key words and phrases: Harmonic functions, Hadamard product and extremal problems.

2000 Mathematics Subject Classification. 30C45, 31A05.

1. INTRODUCTION

Let U denote the open unit disk in C and let f = u+iv be a complex valued harmonic function on U. Since u and v are real parts of analytic functions, f admits a representation f =h+gfor two functionshandg, analytic onU.

The Jacobian of f is given by Jf(z) = |h⁰(z)|² − |g⁰(z)|². The necessary and sufficient conditions forf to be local univalent and sense-preserving isJf(z)>0,z ∈U [1].

Many mathematicians studied the class of harmonic univalent and sense-preserving functions onU and its subclasses [2, 5].

Here we discuss two classes obtained by the Hadamard product.

2. THECLASSPe_H⁰(α)

LetP_H denote the class of all functionsf =h+ ¯g so thatRef > 0andf(0) = 1whereh andg are analytic onU.

If the functionf_z+f_z =h⁰+g⁰ belongs toP_H for the analytic and normalized functions

(2.1) h(z) =z+

∞

X

n=2

a_nzⁿ and g(z) =

∞

X

n=2

b_nzⁿ,

ISSN (electronic): 1443-5756

c 2002 Victoria University. All rights reserved.

040-01

then the class of functionsf =h+gis denoted byPe_H⁰ [5].

The function

(2.2) t_α(z) = z+ 1

1 +αz²+· · ·+ 1

1 + (n−1)αzⁿ+· · · is analytic onU whenαis a complex number different from−1,−¹₂,−¹₃, . . . .

Forf ∈Pe_H⁰, we denote, byPe_H⁰(α),the class of functions defined by

(2.3) F =f∗(t_α+t_α).

Heref ∗(t_α+t_α)is the Hadamard product of the functionsf andt_α+t_α. Therefore F(z) = H(z) +G(z)

(2.4)

= z+

∞

X

n=2

an

1 + (n−1)αzⁿ+

∞

X

n=2

bn

1 + (n−1)αzⁿ

= z+

∞

X

n=2

A_nzⁿ+

∞

X

n=2

B_nzⁿ, z ∈U

is inPe_H⁰(α).

Conversely, ifF is in the form (2.4), witha_n, b_nbeing the coefficients off ∈Pe_H⁰,thenF ∈ Pe_H⁰(α).

Furthermore, if α = 0,then as F = f, we havePe_H⁰(0) = Pe_H⁰. Moreover Pe_H⁰(∞) = {I : I(z)≡z, z ∈U}and sinceI ∈Pe_H⁰ ,Pe_H⁰ ∩Pe_H⁰(α)6=φ.

Theorem 2.1. IfF ∈Pe_H⁰(α)then there existsf ∈Pe_H⁰ so that

(2.5) α[zF_z(z) +zF_z(z)] + (1−α)F(z) =f(z).

Conversely, for any functionf ∈Pe_H⁰, there existsF ∈Pe_H⁰(α)satisfying (2.5).

Proof. LetF ∈Pe_H⁰(α). Iff ∈Pe_H⁰,then since

αzt⁰_α(z) + (1−α)t_α(z) =t₀(z), asF =f∗(t_α+t_α)we obtain that

f(z) =α[f(z)∗(zt⁰_α(z) +zt⁰_α(z))] + (1−α)[f(z)∗(t_α(z) +t_α(z))].

Therefore,

f(z) = α[zF_z(z) +zF_z(z)] + (1−α)F(z).

Conversely, forf ∈Pe_H⁰, from (2.1), (2.2) and (2.5), z+

∞

X

n=2

a_nzⁿ+

∞

X

n=2

b_nzⁿ=z+

∞

X

n=2

[1 + (n−1)α]A_nzⁿ+

∞

X

n=2

[1 + (n−1)α]B_nzⁿ. From these one obtains

(2.6) A_n= a_n

1 + (n−1)α and B_n= b_n 1 + (n−1)α. Therefore,

F(z) = z+

∞

X

n=2

an

1 + (n−1)αzⁿ+

∞

X

n=2

bn

1 + (n−1)αzⁿ

= f(z)∗[t_α(z) +t_α(z)].

Corollary 2.2. A functionF =H+Gof the form (2.4) belongs toPe_H⁰(α),if and only if (2.7) Re{z(αH⁰⁰(z) +αG⁰⁰(z)) +H⁰(z) +G⁰(z)}>0, z ∈U.

Proof. IfF =H+G∈Pe_H⁰(α),then from Theorem 2.1

α[zH⁰(z) +zG⁰(z)] + (1−α)[H(z) +G(z)] =h(z) +g(z)∈Pe_H⁰ andh⁰ +g⁰ ∈P_H.Hence

0 < Re{h⁰(z) +g⁰(z)}

= Re{αzH⁰⁰(z) +αH⁰(z) + (1−α)H⁰(z) +αzG⁰⁰(z) +αG⁰(z) + (1−α)G⁰(z)}

= Re{z(αH⁰⁰(z) +αG⁰⁰(z)) +H⁰(z) +G⁰(z)}.

Conversely, if the functionF = H+Gof the form (2.4) satisfies (2.7), then by Theorem 2.1, h⁰+g⁰ ∈P_H and the function

f(z) =h(z) +g(z) = α[zH⁰(z) +zG⁰(z)] + (1−α)(H(z) +G(z))

is from the classPe_H⁰.Hence by Theorem 2.1,F =H+G∈Pe_H⁰(α).

Proposition 2.3. Pe_H⁰(α)is convex and compact.

Proof. LetF₁ =H₁+G₁, F₂ =H₂+G₂ ∈Pe_H⁰(α)and letλ∈[0,1].Then Re{z[α(λH₁⁰⁰(z) + (1−λ)H₂⁰⁰(z) ¯α(λG⁰⁰₁(z) + (1−λ)G⁰⁰₂(z))]

+λ[H₁⁰(z) +G⁰₁(z)] + (1−λ)[H₂⁰(z) +G⁰₂(z)]}

=λRe{z[αH₁⁰⁰(z) + ¯αG⁰⁰₁(z)] +H₁⁰(z) +G⁰₁(z)}

+ (1−λ) Re{z[αH₂⁰⁰(z) + ¯αG⁰⁰₂(z)] +H₂⁰(z) +G⁰₂(z)}

>0.

Hence, from Corollary 2.2,λ F1+ (1−λ)F2 ∈Pe_H⁰(α).Therefore,Pe_H⁰(α)is convex.

On the other hand, letF_n =H_n+G_n∈Pe_H⁰(α)and letF_n →F =H+G.By Corollary 2.2, α[zH_n⁰(z) +zG⁰_n(z)] + (1−α)[Hn(z) +Gn(z)]∈Pe_H⁰.

SincePe_H⁰ is compact, [5],

α[zH⁰(z) +zG⁰(z)] + (1−α)[H(z) +G(z)]∈Pe_H⁰.

Hence, by Theorem 2.1,F =H+ ¯G∈Pe_H⁰(α).Therefore,Pe_H⁰(α)is compact.

Proposition 2.4. IfF =H+G∈Pe_H⁰(α)and|z|=r <1then

−r+ 2 ln(1 +r)≤Re{α[zH⁰(z) +zG⁰(z)] + (1−α)[H(z) +G(z)]}

≤ −r−2 ln(1−r).

Equality is obtained for the function (2.3) where

f(z) = 2z+ ln(1−z)−3z−3 ln(1−z), z ∈U.

Proof. From Theorem 2.1, ifF =H+G∈Pe_H⁰(α),then there existsf =h+ ¯g ∈Pe_H⁰ so that α[zH⁰(z) +zG⁰(z)] + (1−α)[H(z) +G(z)] =f(z).

Since by [5, Proposition 2.2]

−r+ 2 ln(1 +r)≤Ref(z)≤ −r−2 ln(1−r),

the proof is complete.

Proposition 2.5. IfF =H+G∈Pe_H⁰(α)andReα >0,then there exists anf ∈Pe_H⁰ so that

(2.8) F(z) = 1

α Z 1

0

ζ^α1−2f(zζ)dζ, z ∈U.

Proof. Since

t_α(z) = 1 α

Z 1

0

ζ^α1−1 z

1−zζ dζ, |ζ| ≤1, Reα >0, and forf =h+g ∈Pe_H⁰

h(z)∗ z

1−zζ = h(zζ)

ζ , g(z)∗ z

1−zζ = g(zζ) ζ , we have

H(z) =h(z)∗t_α(z) = 1 α

Z 1

0

ζ^α1−2h(zζ)dζ and

G(z) = g(z)∗t_α(z) = 1 α

Z 1

0

ζ^α1−2g(zζ)dζ.

HenceF is type (2.8).

Theorem 2.6. IfReα >0,thenPe_H⁰(α)⊂Pe_H⁰.Further, for any0<Reα₁ ≤Reα₂, Pe_H⁰(α₂)⊂ Pe_H⁰(α1).

Proof. LetF ∈Pe_H⁰(α)andRe α >0.Then there existsf ∈Pe_H⁰ so that F =H+G=f ∗(t_α+t_α) = (h∗t_α) + (g∗t_α).

Hence,0 <Re{h⁰ +g⁰} = Re{h⁰ +g⁰}and sinceReα > 0,Re{H⁰ +G⁰}> 0,andH(0) = 0, H⁰(0) = 1, G(0) =G⁰(0) = 0and henceF =H+G∈Pe_H⁰.

For0<Reα₁ ≤Reα₂,ifF ∈Pe_H⁰(α₂),from Corollary 2.2

0 < Re{z(α₂H⁰⁰(z) +α₂G⁰⁰(z)) +H⁰(z) +G⁰(z)}

≤ Re{z(α₁H⁰⁰(z) +α₁G⁰⁰(z)) +H⁰(z) +G⁰(z)}

we getF ∈Pe_H⁰(α1).

Remark 2.7. For some values ofα,Pe_H⁰(α)⊂Pe_H⁰ is not true. It is known [5, Corollary 2.5] that the sharp inequalities

(2.9) |a_n| ≤ 2n−1

n and |b_n| ≤ 2n−3 n are true. Hence, for example, the function

f(z) = z+

∞

X

n=2

2n−1 n zⁿ+

∞

X

n=2

2n−3 n zⁿ belongs toPe_H⁰.In this case

F(z) =z+

∞

X

n=2

2n−1

n[1 + (n−1)α]zⁿ+

∞

X

n=2

2n−3

n[1 + (n−1)α]zⁿ belongs to the classPe_H⁰(α)forα∈C, α 6=−1/n, n∈N.However, forReα ∈

−^|α|₃²,0 , α6=

−1,−¹₂, . . . as the coefficient conditions ofPe_H⁰ given in (2.9) are not satisfied,F /∈Pe_H⁰.Hence for eachα∈CwithReα ∈

−^|α|₃²,0

, α6=−1,−¹₂, . . . ,Pe_H⁰(α)−Pe_H⁰ 6=φ.

Theorem 2.8. LetF =H+G∈Pe_H⁰(α). Then (i) ||A_n| − |B_n|| ≤ 2

n|1 + (n−1)α|, n≥1 (ii) IfF is sense-preserving, then

|A_n| ≤ 2n−1 n

1

|1 + (n−1)α|, n= 1,2, . . . and

|Bn| ≤ 2n−3 n

1

|1 + (n−1)α|, n= 2,3, . . . Equality occurs for the functions of type (2.3) where

f(z) = 2z

1−z + ln(1−z)− 3¯z−z¯²

1−z¯ −3 ln(1−z),¯ z ∈U.

Proof. By (2.6),

||A_n| − |B_n||= 1

|1 + (n−1)α|||a_n| − |b_n||. Also by [5, Theorem 2.3], we have

||a_n| − |b_n|| ≤ 2 n the required results are obtained.

On the other hand, from (2.6) and from the coefficient relations in Pe_H⁰ given in (2.9), we

obtain the coefficient inequalities forPe_H⁰(α).

3. THECLASSP_H(β, α) Letf =h+g for analytic functions

h(z) = 1 +

∞

X

n=1

a_nzⁿ and g(z) =

∞

X

n=1

b_nzⁿ

onU.The classP_H(β)of all functions withRef(z)> β, 0≤ β < 1andf(0) = 1is studied in [5].

Let us consider the function (3.1) k_α(z) = 1 + 1

1 +αz+· · ·+ 1

1 +nαzⁿ+· · · , α∈C, α6=−1,−1 2, . . . which is analytic onU.

Forf ∈PH(β), let us denote the class of functions

(3.2) F =f∗(k_α+k_α) = (h∗k_α) + (g∗k_α) = H+G, byP_H(β, α). Ifα= 0,then sinceF =f,P_H(β,0) =P_H(β).

Therefore,

F(z) = H(z) +G(z) (3.3)

= 1 +

∞

X

n=1

a_n

1 +nαzⁿ+

∞

X

n=1

b_n 1 +nαzⁿ

= 1 +

∞

X

n=1

A_nzⁿ+

∞

X

n=1

B_nzⁿ, z ∈U

Theorem 3.1. IfF ∈P_H(β, α)then there exists anf ∈P_H(β), so that (3.4) α[zF_z(z) +zF_z(z)] +F(z) =f(z).

Conversely, forf ∈P_H(β), there is a solution of (3.4) belonging toP_H(β, α).

Proof. Sincek₀(z) = αzk⁰_α(z) +k_α(z),forf ∈P_H(β), using the fact that,f =f∗(k₀+k₀), f(z) = α[f(z)∗(zk⁰_α(z) +zk_α⁰(z))] + [f(z)∗(k_α(z) +k_α(z))]

is obtained. Hence, forF ∈P_H(β, α)

f(z) = α[zFz(z) +zFz(z)] +F(z).

Conversely, letf =h+g ∈PH(β)be given by (3.4). Hence, we can write (3.5) h(z) =αzH⁰(z) +H(z), g(z) = αzG⁰(z) +G(z).

From the system (3.5) the analytic functionsHandGare in the form H(z) = 1 +

∞

X

n=1

a_n

1 +nαzⁿ=h(z)∗k_α(z),

G(z) =

∞

X

n=1

b_n

1 +nαzⁿ=g(z)∗k_α(z).

Hence the functionF =H+Gbelongs to the classP_H(β, α).

Corollary 3.2. The necessary and sufficient conditions for a functionF of form (3.3) to belong toP_H(β, α)are

(3.6) Re{z(αH⁰(z) +αG⁰(z)) +H(z) +G(z)}> β, z ∈U.

Proof. IfF ∈P_H(β, α)then by Theorem 3.1, β < Re{f(z)}

= Re{α[zF_z(z) +zF_z(z)] +F(z)}

= Re{z(αH⁰(z) +αG⁰(z)) +H(z) +G(z)}, z∈U.

Conversely, if a functionF =H+Gof form (3.3) satisfies (3.6), then zαH⁰(z) +H(z) +αzG⁰(z) +G(z)∈P_H(β).

Hence, from Theorem 3.1, we haveF =H+ ¯G∈P_H(β, α).

Proposition 3.3. IfF ∈P_H(β, α),Reα >0then there exists anf ∈P_H(β)so that

(3.7) F(z) = 1

α Z 1

0

t^α1−1f(zt)dt, z ∈U.

The converse is also true.

Proof. Since

k_α(z) = 1 α

Z 1

0

t^1α−1 1

1−ztdt, Re α >0, and forf =h+g ∈PH(β),

h(z)∗ 1

1−zt =h(zt) and g(z)∗ 1

1−zt =g(zt), we obtain

H(z) =h(z)∗k_α(z) = 1 α

Z 1

0

t^α1−1h(zt)dt and

G(z) = g(z)∗k_α(z) = 1 α

Z 1

0

t^α1−1g(zt)dt.

Therefore,F =H+Gis of type (3.7).

Theorem 3.4. LetF ∈P_H(β, α). Then (i) ||An| − |Bn|| ≤ 2(1−β)

|1 +nα|, n≥1

(ii) IfF is sense- preserving, then forn = 1,2, . . .

|A_n| ≤ (1−β)(n+ 1)

|1 +nα| and |B_n| ≤ (1−β)(n−1)

|1 +nα| . Equality is valid for the functions of type (3.2) where

(3.8) f(z) = Re

1 + (1−2β)z 1−z

+iIm

1 +z 1−z

.

Proof. LetF ∈P_H(β, α). Then from (3.3), as the coefficient relation forP_H(β)is

||a_n| − |b_n|| ≤2(1−β) [5, Proposition 3.4], the required inequalities are obtained.

On the other hand, from (3.3), as the coefficient relations forP_H(β)are

|a_n| ≤(1−β)(n+ 1) and |b_n| ≤(1−β)(n−1)

the required inequalities are obtained.

Proposition 3.5. IfF =H+G∈PH(β, α), then forX ={η:|η|= 1}andz ∈U, H(z) +G(z) = 2(1−β)

Z

|η|=1

k_α(ηz)dµ(η).

Hereµis the probability measure defined on the Borel sets onX.

Proof. From [5, Corollary 3.3] there exists a probability measureµdefined on the Borel sets on Xso that

h(z) +g(z) = Z

|η|=1

1 + (1−2β)zη

1−zη dµ(η).

Taking the Hadamard product of both sides byk_α(z), we get H(z) +G(z) =

Z

|η|=1

k_α(z)∗ 1 1−zη

+ (1−2β)η

k_α(z)∗ z 1−zη

dµ(η)

= Z

|η|=1

kα(ηz) + (1−2β)ηk_α(ηz) η

dµ(η).

Theorem 3.6. If Reα ≥ 0, then P_H(β, α) ⊂ P_H(β). Further if 0 ≤ Reα₁ ≤ Reα₂, then P_H(β, α₂)⊂P_H(β, α₁).

Proof. LetF ∈P_H(β, α)andReα≥0.Then asRe{h⁰+g⁰}> β,we haveRe{H⁰+G⁰}> β andF(0) = 1.HenceF ∈P_H(β).Further as0≤Reα₁ ≤Reα₂,forF ∈P_H(β, α₂)

β < Re{z(α₂H⁰(z) +α₂G⁰(z)) +H(z) +G(z)}

< Re{z(α₁H⁰(z) +α₁G⁰(z)) +H(z) +G(z)}.

Therefore, by Corollary 3.2,F ∈PH(β, α1).

Forf ∈P_H, the classB_H(α)consisting of the functionsF =f∗(k_α+k_α)is studied in [2].

The relation between the classesP_H(β, α)andB_H(α)is given as follows.

Proposition 3.7. ForReα ≥0,P_H(β, α)⊂B_H(α).

Proof. If F ∈ P_H(β, α) then there exists an f ∈ P_H(β) so thatF = f ∗(k_α +k_α). Since Ref(z) > β, f(0) = 1and 0 ≤ β < 1, Ref(z) > 0. Hence,f ∈ P_H. By the definition of

B_H(α),F ∈B_H(α).

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