JJ II

(1)

volume 3, issue 1, article 9, 2002.

Received 11 May, 2001;

accepted 11 October, 2001.

Communicated by:S.S. Dragomir

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

ON HARMONIC FUNCTIONS BY THE HADAMARD PRODUCT

M. ÖZTÜRK, S. YALÇIN AND M. YAMANKARADENIZ

Uludag University, Faculty of Science, Department of Mathematics 16059 Bursa/Turkey.

EMail:ometin@uludag.edu.tr

c

2000School of Communications and Informatics,Victoria University of Technology ISSN (electronic): 1443-5756

040-01

(2)

On Harmonic Functions Constructed by the Hadamard

Product

Metin Öztürk,Sibel Yalçin and Mümin Yamankaradeniz

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of18

J. Ineq. Pure and Appl. Math. 3(1) Art. 9, 2002

http://jipam.vu.edu.au

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.

2000 Mathematics Subject Classification:30C45, 31A05.

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

1. Introduction

LetU denote the open unit disk in Cand letf = u+iv be a complex valued harmonic function on U. Since u andv are real parts of analytic functions, f admits a representationf =h+g for two functionshandg, analytic onU.

The Jacobian of f is given byJ_f(z) = |h⁰(z)|² − |g⁰(z)|². The necessary and sufficient conditions for f to be local univalent and sense-preserving is J_f(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.

(4)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of18

2. The Class P e

_H⁰

(α)

LetP_H denote the class of all functionsf =h+¯gso thatRef >0andf(0) = 1 wherehandgare analytic onU.

If the functionf_z+f_z =h⁰+g⁰belongs 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,

then the class of functionsf =h+g is 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).

Here f ∗(tα + tα) is the Hadamard product of the functions f and tα +tα. Therefore

F(z) = H(z) +G(z) (2.4)

= z+

∞

X

n=2

a_n

1 + (n−1)αzⁿ+

∞

X

n=2

b_n

1 + (n−1)αzⁿ

= z+

∞

X

n=2

Anzⁿ+

∞

X

n=2

Bnzⁿ, z ∈U

(5)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of18

is inPe_H⁰(α).

Conversely, if F is in the form (2.4), with an, bn being the coefficients of f ∈Pe_H⁰,thenF ∈Pe_H⁰(α).

Furthermore, if α = 0, then as F = f, we have Pe_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.

(6)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of18

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

a_n

1 + (n−1)αzⁿ+

∞

X

n=2

b_n

1 + (n−1)αzⁿ

= f(z)∗[t_α(z) +t_α(z)].

Corollary 2.2. A function F = H+Gof the form (2.4) belongs to Pe_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 Theorem2.1

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)}.

(7)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of18

Conversely, if the functionF =H+Gof the form (2.4) satisfies (2.7), then by Theorem2.1,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 Theorem2.1,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, letFn=Hn+Gn ∈Pe_H⁰(α)and letFn→ F =H+G.

By Corollary2.2,

α[zH_n⁰(z) +zG⁰_n(z)] + (1−α)[H_n(z) +G_n(z)]∈Pe_H⁰. SincePe_H⁰ is compact, [5],

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

Hence, by Theorem2.1, F = H+ ¯G∈ Pe_H⁰(α).Therefore,Pe_H⁰(α)is compact.

(8)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of18

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, if F = H +G ∈ Pe_H⁰(α), then there exists f = 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. If F = H+G ∈ Pe_H⁰(α)andReα > 0,then there exists an f ∈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,

(9)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of18

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. If Reα > 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 Corollary2.2 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).

(10)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of18

Remark 2.1. For some values of α, 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, α6=−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 each α ∈ C with Reα ∈

−^|α|₃²,0 , α 6=−1,−¹₂, . . . ,Pe_H⁰(α)−Pe_H⁰ 6=φ.

Theorem 2.7. LetF =H+G∈Pe_H⁰(α). Then

(i) ||A_n| − |B_n|| ≤ 2

n|1 + (n−1)α|, n ≥1

(11)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of18

(ii) IfF is sense-preserving, then

|A_n| ≤ 2n−1 n

1

|1 + (n−1)α|, n= 1,2, . . . and

|B_n| ≤ 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 inPe_H⁰ given in (2.9), we obtain the coefficient inequalities forPe_H⁰(α).

(12)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of18

3. The Class P

_H

(β, α)

Letf =h+gfor analytic functions h(z) = 1 +

∞

X

n=1

a_nzⁿ and g(z) =

∞

X

n=1

b_nzⁿ

on U. The class P_H(β) of all functions with Ref(z) > β, 0 ≤ β < 1 and f(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 ∈P_H(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

(13)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of18

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. Since k₀(z) = αzk⁰_α(z) +k_α(z), for f ∈ P_H(β), using the fact that, f =f∗(k₀+k₀),

f(z) =α[f(z)∗(zk⁰_α(z) +zk⁰_α(z))] + [f(z)∗(k_α(z) +k_α(z))]

is obtained. Hence, forF ∈P_H(β, α)

f(z) =α[zF_z(z) +zF_z(z)] +F(z).

Conversely, letf =h+g ∈P_H(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 functionsH andGare in the form H(z) = 1 +

∞

X

n=1

a_n

1 +nαzⁿ =h(z)∗k_α(z),

G(z) =

∞

X

n=1

bn

1 +nαzⁿ=g(z)∗k_α(z).

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

(14)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of18

Corollary 3.2. The necessary and sufficient conditions for a functionF of form (3.3) to belong toPH(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 Theorem3.1, β < Re{f(z)}

= Re{α[zFz(z) +zFz(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 Theorem3.1, 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^α¹⁻¹ 1

1−ztdt, Re α >0,

(15)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of18

and forf =h+g ∈P_H(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)∗kα(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) ||A_n| − |B_n|| ≤ 2(1−β)

|1 +nα|, n ≥1

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

|An| ≤ (1−β)(n+ 1)

|1 +nα| and |Bn| ≤ (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

.

(16)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of18

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 forPH(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 ∈ P_H(β, α), then forX = {η :|η|= 1}and z ∈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 onXso 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)

(17)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page17of18

= Z

|η|=1

k_α(ηz) + (1−2β)ηk_α(ηz) η

dµ(η).

Theorem 3.6. If Reα ≥ 0, thenP_H(β, α) ⊂ P_H(β).Further if0 ≤ Reα₁ ≤ Reα₂,thenP_H(β, α₂)⊂P_H(β, α₁).

Proof. Let F ∈ P_H(β, α) andReα ≥ 0.Then as Re{h⁰+g⁰} > β, we have Re{H⁰+G⁰} > β andF(0) = 1.HenceF ∈PH(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 Corollary3.2,F ∈PH(β, α1).

Forf ∈PH, the classBH(α)consisting of the functionsF =f∗(kα+kα) is studied in [2]. The relation between the classesP_H(β, α)andB_H(α)is given as follows.

Proposition 3.7. ForReα≥0,PH(β, α)⊂BH(α).

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

(18)

Product

Title Page Contents

JJ II

J I

Go Back Close

Quit Page18of18

References

[1] J. CLUNIE AND T. SHEIL-SMALL, Harmonic univalent functions, Ann.

Acad. Sci. Fenn. Series A.I, Math., 9 (1984), 3–25.

[2] Z.J. JAKUBOWSKI, W. MAJCHRZAK AND K. SKALSKA, Harmonic mappings with a positive real part, Materialy XIV Konferencjiz Teorii Za- gadnien Exstremalnych, Lodz, 17–24, 1993.

[3] H. LEWY, On the non vanishing of the Jacobian in certain one to one map- pings, Bull. Amer. Math. Soc., 42 (1936), 689–692.

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

[5] S. YALÇIN, M. ÖZTÜRKANDM. YAMANKARADENIZ, On some sub- classes of harmonic functions, in Mathematics and Its Applications, Kluwer Acad. Publ.; Functional Equations and Inequalities, 518 (2000), 325–331.