JJ II

(1)

volume 7, issue 5, article 190, 2006.

Received 17 May, 2006;

accepted 30 August, 2006.

Communicated by:N.E. Cho

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

INEQUALITIES INVOLVING MULTIPLIERS FOR MULTIVALENT HARMONIC FUNCTIONS

H. ÖZLEM GÜNEY AND OM P. AHUJA

Department of Mathematics Faculty of Science and Art Dicle University

Diyarbakir 21280, Turkey EMail:ozlemg@dicle.edu.tr Kent State University

Department of Mathematical Sciences 14111, Claridon-Troy Road

Burton, Ohio 44021, U.S.A.

EMail:oahuja@kent.edu

c

2000Victoria University ISSN (electronic): 1443-5756 145-06

(2)

Inequalities Involving Multipliers For Multivalent

Harmonic Functions H. Özlem Güney and Om P. Ahuja

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of21

J. Ineq. Pure and Appl. Math. 7(5) Art. 190, 2006

http://jipam.vu.edu.au

Abstract

We introduce inequalities involving multipliers for complex-valued multivalent harmonic functions, using two sequences of positive real numbers. By spe- cializing those sequences, we determine representation theorems, distortion bounds, integral convolutions, convex combinations and neighborhoods for such functions. The theorems presented, in many cases, confirm or generalize var- ious well-known results for corresponding classes of multivalent or univalent harmonic functions.

2000 Mathematics Subject Classification:Primary 30C45; Secondary 30C50.

Key words: Multivalent harmonic, Multivalent harmonic starlike, Multivalent har- monic convex, Multiplier, Integral convolution, Neighborhood.

The authors wish to thank the referee for suggesting certain improvements in the paper.

1. Introduction

A continuous complex-valued functionf =u+ivdefined in a simply connected complex domainDis said to be harmonic inDif bothuandvare real harmonic in D. Such functions admit the representation f = h+ ¯g, where h and g are analytic in D.In [5], it was observed thatf = h+ ¯g is locally univalent and sense preserving if and only if|g⁰(z)|<|h⁰(z)|, z ∈D.

The study of harmonic functions which are multivalent in the unit discU = {z ∈C:|z|<1}was initiated by Duren, Hengartner and Laugesen [6]. How- ever, passing from harmonic univalent functions to the harmonic multivalent functions turns out to be quite non-trivial. In view of the argument principle for harmonic functions obtained in [6], the second author and Jahangiri [1, 2]

introduced and studied certain subclasses of the family H(m), m ≥ 1, of all m-valent harmonic and orientation preserving functions in U. A functionf in H(m)can be expressed asf = h+g, where handg are analytic functions of the form

h(z) = z^m+

∞

X

n=2

an+m−1z^n+m−1, (1.1)

g(z) =

∞

X

n=1

bn+m−1z^n+m−1, |b_m|<1.

The class H(1) of harmonic univalent functions was studied by Clunie and Sheil-Small [5].

LetSH(m, α), m ≥ 1 and0 ≤ α < 1 denote the class of functions f =

(4)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of21

h+g ∈H(m)which satisfy the condition

(1.2) ∂

∂θ(arg(f(re^iθ)))≥mα,

for eachz = re^iθ, 0 ≤ θ < 2π,and0 ≤ r < 1. A functionf inSH(m, α)is called an m-valent harmonic starlike function of order α. Also, letT H(m, α), m ≥ 1, denote the class of functions f = h+g ∈ SH(m, α) so thathandg are of the form

h(z) = z^m−

∞

X

n=2

|a_n+m−1|z^n+m−1, (1.3)

g(z) =

∞

X

n=1

|bn+m−1|z^n+m−1, |b_m|<1.

The class T H(m, α) was studied by second author and Jahangiri in [1, 2]. In particular, they stated the following:

Theorem A. Let f = h+ ¯g be given by (1.3). Thenf is in T H(m, α)if and only if

(1.4)

∞

X

n=1

n−1 +m(1−α)

m(1−α) |a_n+m−1|+n−1 +m(1 +α)

m(1−α) |b_n+m−1|

≤2,

wherea_m = 1andm≥1.

Analogous toT H(m, α)is the classKH(m, α)of m-valent harmonic convex functions of order α, 0 ≤ α < 1. More precisely, a function f = h+g,

(5)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of21

where hand g are of the form (1.3), is inKH(m, α) if and only if it satisfies the condition

∂

∂θ

arg ∂

∂θf(re^iθ)

≥mα, for eachz =re^iθ,0≤θ <2π,and0≤r <1.

Theorem B ([4]). Letf =h+ ¯g be given by (1.3). Thenf is inKH(m, α)if and only if

(1.5)

∞

X

n=1

n+m−1 m²(1−α)

(n−1 +m(1−α))|an+m−1|

+ (n−1 +m(1 +α))|bn+m−1|

≤2, wherea_m = 1andm≥1.

Inequalities (1.4) and (1.5) as well as several such known inequalities in the literature are the motivating forces for introducing a multiplier family F_m({cn+m−1},{dn+m−1})form≥1. A functionf =h+ ¯g, wherehandg are given by (1.3), is said to be in the multiplier familyF_m({cn+m−1},{dn+m−1}) if there exist sequences{cn+m−1}and{dn+m−1}of positive real numbers such that

∞

X

n=1

cn+m−1

m |an+m−1|+dn+m−1

m |bn+m−1|

≤2, (1.6)

c_m =m, d_m|b_m|< m.

(6)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of21

The multipliers{cn+m−1}and{dn+m−1}provide a transition from multivalent harmonic starlike functions to multivalent harmonic convex functions, including many more subclasses ofH(m)andH(1). For example,

(1.7) F_m

n−1 +m(1−α) 1−α

,

n−1 +m(1 +α) 1−α

≡T H(m, α),

(1.8) F_m

(n+m−1)(n−1 +m(1−α)) m(1−α)

, (n+m−1)(n−1 +m(1 +α))

m(1−α)

≡KH(m, α),

(1.9) F_m({n+m−1},{n+m−1})≡T H(m,0) := T H(m),

F_m

(n+m−1)² m

,

(n+m−1)² m

≡KH(m,0) (1.10)

:=KH(m),

(1.11) F₁({n},{n})≡T H(1,0) = T H,

(1.12) F1({n²},{n²})≡KH(1,0) :=KH,

(7)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of21

(1.13) F₁({n^p},{n^p}) := F({n^p},{n^p}), p >0,

(1.14) F₁({c_n},{d_n}) :=F({c_n},{d_n}).

While (1.7), (1.9) and (1.11) follow immediately from Theorem A, (1.8), (1.10) and (1.12) are consequences of Theorem B. Note thatT H and KH in (1.11) and (1.12) were studied in [9] as well as [10]. Also, by letting m = 1, α = 0, c_n =d_n =n^p forp > 0andb₁ = 0,the classesF₁({n^p},{n^p})were studied in [8]. Finally, (1.14) follows from (1.6) by setting m = 1which was studied in [3].

In this paper, we determine representation theorems, distortion bounds, convolutions, convex combinations and neighborhoods of functions in Fm({cn+m−1},{dn+m−1}). As illustrations of our results, the corresponding results for certain families are presented in the corollaries.

(8)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of21

2. Main Results

If(n+m−1)≤cn+m−1 and(n+m−1)≤dn+m−1, then by TheoremAwe have

F_m({cn+m−1},{dn+m−1})⊂T H(m).

Consequently, the functionsF_m({cn+m−1},{dn+m−1})are sense-preserving, harmonic and multivalent inU. We first observe that if

f₁(z) =z^m−

∞

X

n=2

|a₁_(n+m−1)|z^n+m−1+

∞

X

n=1

|b₁_(n+m−1)|¯z^n+m−1 and

f₂(z) =z^m−

∞

X

n=2

|a₂_(n+m−1)|z^n+m−1+

∞

X

n=1

|b₂_(n+m−1)|¯z^n+m−1

are inF_m({c_n+m−1},{d_n+m−1})and0≤λ≤1, then so is the linear combina- tionλf₁+ (1−λ)f₂ by (1.6). Therefore,F_m({cn+m−1},{dn+m−1})is a convex family.

Next we determine the extreme points of the closed convex hull of the family Fm({cn+m−1},{dn+m−1}), denoted byclcoFm({cn+m−1},{dn+m−1}).

Theorem 2.1. A functionf =h+g is inclcoF_m({cn+m−1},{dn+m−1})if and only iff has the representation

(2.1) f(z) =

∞

X

n=1

(λ_n+m−1h_n+m−1(z) +µ_n+m−1g_n+m−1(z)),

(9)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of21

where

(n+m−1)≤cn+m−1, (n+m−1)≤dn+m−1, λn+m−1 ≥0, µn+m−1 ≥0,

∞

X

n=1

(λn+m−1+µn+m−1) = 1, h_m(z) =z^m, hn+m−1(z) = z^m− m

c_n+m−1z^n+m−1 and gn+m−1(z) =z^m+ m

d_n+m−1z¯^n+m−1. In particular, the extreme points of F_m({cn+m−1}, {dn+m−1}) are {hn+m−1}, {gn+m−1}.

Proof. For functionsf of the form (2.1) we have f(z) =λ_mh_m(z) +

∞

X

n=2

λn+m−1

z^m− m cn+m−1

z^n+m−1

+

∞

X

n=1

µn+m−1

z^m+ m dn+m−1

¯ z^n+m−1

=z^m−

∞

X

n=2

λn+m−1

m cn+m−1

z^n+m−1 +

∞

X

n=1

µn+m−1

m dn+m−1

¯ z^n+m−1. Then

∞

X

n=2

λ_n+m−1 m cn+m−1

c_n+m−1+

∞

X

n=1

µ_n+m−1 m dn+m−1

d_n+m−1

=

∞

X

n=2

mλn+m−1+

∞

X

n=1

mµn+m−1 ≤m,

(10)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of21

and sof ∈clcoF_m({cn+m−1},{dn+m−1}).

Conversely, supposef ∈clcoF_m({cn+m−1},{dn+m−1}). We set λn+m−1 = cn+m−1

m |an+m−1|, (n= 2,3, . . .), µ_n+m−1 = dn+m−1

m |b_n+m−1|, (n = 1,2,3, . . .), and

λ_m = 1−

∞

X

n=2

λn+m−1−

∞

X

n=1

µn+m−1. Therefore, by using routine computations,f can be written as

f(z) =

∞

X

n=1

(λ_n+m−1h_n+m−1(z) +µ_n+m−1g_n+m−1(z)).

In view of (1.7), Theorem2.1yields:

Corollary 2.2 ([2]). A functionf = h+g is inclcoT H(m, α)if and only iff can be expressed in the form (2.1), where

h_m(z) = z^m, hn+m−1(z) = z^m− m(1−α)

n−1 +m(1−α)z^n+m−1, (n = 2,3, . . .), gn+m−1(z) =z^m+ m(1−α)

n−1 +m(1 +α)z^n+m−1, (n = 1,2,3, . . .)

(11)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of21

and ∞

X

n=1

(λn+m−1+µn+m−1) = 1, λn+m−1 ≥0, µn+m−1 ≥0.

Our next result provides distortion bounds for the functions inF_m({cn+m−1}, {d_n+m−1}).

Theorem 2.3. Let{cn+m−1}and{dn+m−1}be increasing sequences of positive real numbers so that

c_m+1≤d_m+1, (n+m−1)≤cn+m−1 and (n+m−1)≤dn+m−1

for alln ≥2. Iff ∈F_m({cn+m−1},{dn+m−1}),then (1− |b_m|)r^m−

m−d_m|b_m| cm+1

r^m+1

≤ |f(z)| ≤(1 +|b_m|)r^m+

m−d_m|b_m| cm+1

r^m+1. The bounds given above are sharp for the functions

f(z) =z^m± |b_m|¯z^m+

m−d_m|b_m| c_m+1

¯

z^m+1, d_m|b_m|<1.

Proof. Letf ∈ F_m({cn+m−1},{dn+m−1}). Taking the absolute value off, we

(12)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of21

obtain

|f(z)|=

z^m−

∞

X

n=2

|an+m−1|z^n+m−1 +

∞

X

n=1

|bn+m−1|¯z^n+m−1

≤r^m+

∞

X

n=2

|a_n+m−1|r^n+m−1+

∞

X

n=1

|b_n+m−1|r^n+m−1

= (1 +|b_m|)r^m+

∞

X

n=2

(|an+m−1|+|bn+m−1|)r^n+m−1

≤(1 +|b_m|)r^m+ 1 c_m+1

∞

X

n=2

c_m+1(|an+m−1|+|bn+m−1|)r^m+1

≤(1 +|bm|)r^m+ 1 c_m+1

∞

X

n=2

(cm+1|an+m−1|+dm+1|bn+m−1|)r^m+1

≤(1 +|b_m|)r^m+ 1 c_m+1

∞

X

n=2

(c_n+m−1|a_n+m−1|+d_n+m−1|b_n+m−1|)r^m+1

≤(1 +|b_m|)r^m+ 1

c_m+1(m−d_m|b_m|)r^m+1.

We omit the proof of the left side of the inequality as it is similar to that of the right side.

Corollary 2.4. Iff ∈T H(m, α), then

|f(z)| ≤(1 +|b_m|)r^m+ m(1−α−(1 +α)|b_m|) 1 +m(1−α) r^m+1,

(13)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of21

and

|f(z)| ≥(1− |b_m|)r^m− m(1−α−(1 +α)|b_m|) 1 +m(1−α) r^m+1, where|z|=r <1.

The following covering result follows from the left hand inequality in Theo- rem2.3.

Corollary 2.5. Letf be as in Theorem2.3. Then

w:|w|< 1

c_m+1(c_m+1−m−(c_m+1−d_m)|b_m|)

⊂f(U).

Corollary 2.6. Iff ∈T H(m, α), then

w:|w|< 1 + (2mα−1)|b_m| 1 +m(1−α)

⊂f(U).

Remark 1. Forα= 0, the corresponding results in Corollary2.4and Corollary 2.6were also found in [1].

In the next result, we find the convex combinations of the members of the familyF_m({cn+m−1},{dn+m−1}).

Theorem 2.7. If (n +m − 1) ≤ cn+m−1 and (n +m − 1) ≤ dn+m−1 for all n + m − 1 ≥ 2, then F_m({c_n+m−1},{d_n+m−1}) is closed under convex combinations.

(14)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of21

Proof. Consider

f_i(z) = z^m−

∞

X

n=2

|a_i_n+m−1|z^n+m−1+

∞

X

n=1

|b_i_n+m−1|¯z^n+m−1 fori= 1,2, . . . .Iff_i ∈F_m({cn+m−1},{dn+m−1})then

(2.2)

∞

X

n=2

cn+m−1|a_i_n+m−1|+

∞

X

n=1

dn+m−1|b_i_n+m−1| ≤m, i= 1,2, . . . . ForP∞

i=1t_i = 1, 0≤t_i ≤1,we have

∞

X

i=1

t_if_i(z)

=z^m−

∞

X

n=2

∞

X

i=1

t_i|a_i_n+m−1|

!

z^n+m−1+

∞

X

n=1

∞

X

i=1

t_i|b_i_n+m−1|

!

¯ z^n+m−1. In view of the above equality and (2.2), we obtain

∞

X

n=2

cn+m−1

∞

X

i=1

t_i

a_i_n+m−1 +

∞

X

n=1

dn+m−1

∞

X

i=1

t_i

|b_i_n+m−1|

=

∞

X

i=1

t_i ( _∞

X

n=2

cn+m−1|a_i_n+m−1|+

∞

X

n=1

dn+m−1|b_i_n+m−1| )

≤

∞

X

i=1

tim =m.

HenceP∞

i=1tifi ∈Fm({cn+m−1},{dn+m−1}), by an application of (1.6).

(15)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of21

In view of relations (1.7) to (1.10), we have the following results:

Corollary 2.8. The family T H(m, α), KH(m, α), T H(m) and KH(m) are closed under convex combinations.

For harmonic functions (2.3) f(z) = z^m−

∞

X

n=2

|an+m−1|z^n+m−1+

∞

X

n=1

|bn+m−1|¯z^n+m−1

and

(2.4) F(z) = z^m−

∞

X

n=2

|An+m−1|z^n+m−1+

∞

X

n=1

|Bn+m−1|¯z^n+m−1 define the integral convolution off andF as

(2.5) (fF) (z) = z^m−

∞

X

n=2

|an+m−1An+m−1|

n+m−1 z^n+m−1 +

∞

X

n=1

|bn+m−1Bn+m−1|

n+m−1 z¯^n+m−1. In the following result, we show the integral convolution property of the class F_m({cn+m−1},{dn+m−1}).

Theorem 2.9. Let(n+m−1)≤ c_n+m−1 and(n+m−1)≤ d_n+m−1 for all n+m−1≥2.Iff andF are inF_m({cn+m−1},{dn+m−1}), then so isfF.

(16)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of21

Proof. Since F_m({cn+m−1},{dn+m−1}) ⊂ T H(m) and F ∈ F_m({cn+m−1}, {dn+m−1}), it follows that |An+m−1| ≤ 1and |Bn+m−1| ≤ 1. Then f F ∈ F_m({cn+m−1},{dn+m−1})because

∞

X

n=2

cn+m−1

m(n+m−1)|an+m−1An+m−1|+

∞

X

n=1

dn+m−1

m(n+m−1)|bn+m−1Bn+m−1|

≤

∞

X

n=2

cn+m−1

m(n+m−1)|an+m−1|+

∞

X

n=1

dn+m−1

m(n+m−1)|bn+m−1|

≤

∞

X

n=2

cn+m−1

m |an+m−1|+

∞

X

n=1

dn+m−1

m |bn+m−1| ≤2.

Corollary 2.10. IffandF are inT H(m, α),KH(m, α),T H(m)andKH(m), then so isfF.

Theδ−neighborhood of the functionsf =h+¯ginF_m({(n+m−1)cn+m−1}, {(n+m−1)d_n+m−1})is defined as the setN_δ(f)consisting of functions

F(z) =z^m+B_mz¯^m+

∞

X

n=2

(A_n+m−1z^n+m−1+B_n+m−1z¯^n+m−1) such that

∞

X

n=2

[(n+m−1)(|a_n+m−1−A_n+m−1|+|b_n+m−1−B_n+m−1|)]

+m|b_m−B_m| ≤δ, δ >0.

(17)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page17of21

Our next result guarantees that the functions in a neighborhood of F_m({(n+m−1)cn+m−1}, {(n+m−1)dn+m−1}) are multivalent harmonic starlike functions.

Theorem 2.11. Let {cn+m−1} and{dn+m−1} be increasing sequences of real numbers so that cm+1 ≤ dm+1, (n+m−1) ≤ cn+m−1 and(n+m−1) ≤ dn+m−1 for alln≥2. If

δ= m

c_m+1(c_m+1−1−(c_m+1−d_m)|b_m|), then

N_δ(F_m({(n+m−1)cn+m−1},{(n+m−1)dn+m−1}))⊂T H(m).

Proof. Suppose

f =h+ ¯g ∈F_m({(n+m−1)cn+m−1},{(n+m−1)dn+m−1}).

LetF =H+ ¯G∈N_δ(f)where H(z) = z^m+

∞

X

n=2

An+m−1z^n+m−1 and G(z) =

∞

X

n=1

Bn+m−1z^n+m−1. We need to show that F ∈ T H(m). It suffices to show that F satisfies the condition

M(F) :=

∞

X

n=2

(n+m−1)(|An+m−1|+|Bn+m−1|) +m|Bm| ≤m

(18)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page18of21

Note that M(F)≤

∞

X

n=2

(n+m−1)[|A_n+m−1−a_n+m−1|+|B_n+m−1−b_n+m−1|]

+m|B_m−b_m|+

∞

X

n=2

(n+m−1)(|an+m−1|+|bn+m−1|) +m|b_m|

≤δ+m|b_m|+

∞

X

n=2

(n+m−1)(|an+m−1|+|bn+m−1|)

=δ+m|b_m|+ 1 c_m+1

∞

X

n=2

c_m+1(n+m−1)|an+m−1| +c_m+1(n+m−1)|b_n+m−1|

≤δ+m|b_m|+ 1 cm+1

∞

X

n=2

(n+m−1)cn+m−1|an+m−1| + (n+m−1)dn+m−1|bn+m−1|

≤δ+m|b_m|+ 1

c_m+1(m(1−d_m|b_m|)).

But, the last expression is never greater thanmprovided that δ≤m−m|b_m| − 1

c_m+1(m(1−d_m|b_m|))

= m

c_m+1(c_m+1−1−(c_m+1−d_m)|b_m|).

(19)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page19of21

Corollary 2.12. If

δ= m−(m−2m²α)|b_m| 1 +m(1−α) , thenN_δ(KH(m, α))⊂T H(m).

Lettingα = 0and m = 1, Corollary2.12 yields the following interesting result.

Corollary 2.13. N¹

2(1−|b₁|)(KH)⊂T H.

(20)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page20of21

References

[1] O.P. AHUJAANDJ.M. JAHANGIRI, Multivalent harmonic starlike func- tions, Ann.Univ. Mariae Curie-Sklodowska, Section A, 55(1) (2001), 1–

13.

[2] O.P. AHUJA AND J.M. JAHANGIRI, Errata to "Multivalent harmonic starlike functions" [Ann.Univ. Mariae Curie-Sklodowska, Vol. LV, 1 Sectio A 55(2001), 1-13], Ann.Univ. Mariae Curie-Sklodowska, Sectio A, 56(1) (2002), 105.

[3] O.P. AHUJAANDJ.M. JAHANGIRI, Certain multipliers of univalent har- monic functions, Appl. Math. Letters, 18 (2005), 1319–1324.

[4] O.P. AHUJAAND J.M. JAHANGIRI, Multivalent harmonic convex func- tions, submitted for publication.

[5] J. CLUNIEAND T. SHEIL-SMALL, Harmonic univalent functions, Ann.

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

[6] P.L. DUREN, W. HENGARTNERANDR.S. LAUGESEN, The argument principles harmonic functions, Amer. Math. Monthly , (5) 103 (1996), 411–

425.

[7] A. GABRIELA AND A. LAZINSKA, On some generalization of coef- ficient conditions for complex harmonic mappings, Demonstratio Math., 37(2) (2004), 317–326.

[8] A. GANCZAR, On harmonic univalent mappings with small coefficients, Demonstratio Math., 34(3) (2001), 549–558.

(21)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page21of21

[9] J.M. JAHANGIRI, Harmonic functions starlike in the unit disc, J. Math.

Anal. Appl., 235 (1999), 470–477.

[10] H. SILVERMAN AND E.M. SILVIA, Subclasses of harmonic univalent functions, New Zealand J. Math., 28 (1999), 275–284.