Using the Salagean derivative, we introduce and study a class of Goodman- Ron- ning type harmonic univalent functions

ON THE SUBCLASS OF SALAGEAN-TYPE HARMONIC UNIVALENT FUNCTIONS

S˙IBEL YALÇIN, MET˙IN ÖZTÜRK, AND MÜM˙IN YAMANKARADEN˙IZ

ULUDA ˇGÜNIVERSITESIFENEDEBIYATFAKÜLTESI

MATEMATIKBÖLÜMÜ, 16059 BURSA, TURKEY

skarpuz@uludag.edu.tr

Received 10 October, 2006; accepted 22 April, 2007 Communicated by H.M. Srivastava

ABSTRACT. Using the Salagean derivative, we introduce and study a class of Goodman- Ron- ning type harmonic univalent functions. We obtain coefficient conditions, extreme points, dis- tortion bounds, convolution conditions, and convex combination for the above class of harmonic functions.

Key words and phrases: Harmonic, Meromorphic, Starlike, Symmetric, Conjugate, Convex functions.

2000 Mathematics Subject Classification. 30C45, 30C50, 30C55.

1. INTRODUCTION

A continuous complex valued functionf = u+iv defined in a simply connected complex domainD is said to be harmonic inD if bothu andv are real harmonic inD. In any simply connected domainD we can writef = h+ ¯g, wherehand g are analytic in D. A necessary and sufficient condition forf to be locally univalent and sense preserving inDis that|h⁰(z)|>

|g⁰(z)|, z∈D.

Denote byS_H the class of functionsf =h+ ¯gthat are harmonic univalent and sense preserv- ing in the unit diskU ={z :|z|<1}for whichf(0) =f_z(0)−1 = 0.Then forf =h+¯g ∈S_H we may express the analytic functionshandgas

(1.1) h(z) = z+

∞

X

n=2

a_nzⁿ, g(z) =

∞

X

n=1

b_nzⁿ, |b₁|<1.

In 1984 Clunie and Sheil-Small [1] investigated the classSH as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers onS_H and its subclasses. Jahangiri et al. [3] make use of the Alexander integral transforms of certain analytic functions (which are starlike or convex of positive order) with a view to investigating the construction of sense-preserving, univalent, and close to convex harmonic functions.

254-06

Definition 1.1. Recently, Rosy et al. [4], defined the subclass G_H(γ) ⊂ S_H consisting of harmonic univalent functionsf(z)satisfying the following condition

(1.2) Re

(1 +e^iα)zf⁰(z) f(z) −e^iα

≥γ, 0≤γ <1, α∈R. They proved that iff =h+ ¯gis given by (1.1) and if

(1.3)

∞

X

n=1

2n−1−γ

1−γ |a_n|+2n+ 1 +γ 1−γ |b_n|

≤2, 0≤γ <1,

thenf is a Goodman-Ronning type harmonic univalent function inU. This condition is proved to be also necessary ifhandgare of the form

(1.4) h(z) = z−

∞

X

n=2

|a_n|zⁿ, g(z) =

∞

X

n=1

|b_n|zⁿ.

Jahangiri et al. [2] has introduced the modified Salagean operator of harmonic univalent functionf as

(1.5) D^kf(z) = D^kh(z) + (−1)^kD^kg(z), k ∈N, where

D^kh(z) =z+

∞

X

n=2

n^ka_nzⁿ and D^kg(z) =

∞

X

n=1

n^kb_nzⁿ.

We letRS_H(k, γ)denote the family of harmonic functionsf of the form (1.1) such that

(1.6) Re

(1 +e^iα)D^k+1f(z) D^kf(z) −e^iα

≥γ, 0≤γ <1, α∈R, whereD^kf is defined by (1.5).

Also, we let the subclassRS_H(k, γ)consist of harmonic functionsf_k =h+ ¯g_kinRS_H(k, γ) so thathandg_kare of the form

(1.7) h(z) = z−

∞

X

n=2

|a_n|zⁿ, g_k(z) = (−1)^k

∞

X

n=1

|b_n|zⁿ.

In this paper, the coefficient condition given in [4] for the classG_H(γ)is extended to the class RS_H(k, γ)of the form (1.6). Furthermore, we determine extreme points, a distortion theorem, convolution conditions and convex combinations for the functions inRS_H(k, γ).

2. MAINRESULTS

In our first theorem, we introduce a sufficient coefficient bound for harmonic functions in RS_H(k, γ).

Theorem 2.1. Letf =h+ ¯gbe given by (1.1). If (2.1)

∞

X

n=1

n^k[(2n−1−γ)|an|+ (2n+ 1 +γ)|bn|]≤2(1−γ),

where a₁ = 1 and 0 ≤ γ < 1, then f is sense preserving, harmonic univalent in U, and f ∈RS_H(k, γ).

Proof. If the inequality (2.1) holds for the coefficients off = h+ ¯g, then by (1.3), f is sense preserving and harmonic univalent inU.According to the condition (1.5) we only need to show that if (2.1) holds then

Re

(1 +e^iα)D^k+1f(z) D^kf(z) −e^iα

= Re (

(1 +e^iα)D^k+1h(z)−(−1)^kD^k+1g(z) D^kh(z) + (−1)^kD^kg(z) −e^iα

)

≥γ, where 0≤γ <1.

Using the fact thatRew≥γif and only if|1−γ+w| ≥ |1 +γ−w|,it suffices to show that (2.2)

(1−γ)D^kf(z) + (1 +e^iα)D^k+1f(z)−e^iαD^kf(z)

−

(1 +γ)D^kf(z)−(1 +e^iα)D^k+1f(z) +e^iαD^kf(z) ≥0.

Substituting forD^kf andD^k+1f in (2.2) yields

(1−γ)D^kf(z) + (1 +e^iα)D^k+1f(z)−e^iαD^kf(z)

−

(1 +γ)D^kf(z)−(1 +e^iα)D^k+1f(z) +e^iαD^kf(z)

=

(2−γ)z+

∞

X

n=2

(1−γ−e^iα+n+ne^iα)n^ka_nzⁿ

−(−1)^k

∞

X

n=1

(n+ne^iα−1 +γ+e^iα)b_nzⁿ

−

γz−

∞

X

n=2

(n+ne^iα−1−γ−e^iα)n^ka_nzⁿ +(−1)^k

∞

X

n=1

(n+ne^iα + 1 +γ+e^iα)b_nzⁿ

≥(2−γ)|z| −

∞

X

n=2

n^k(2n−γ)|a_n||z|ⁿ−

∞

X

n=1

n^k(2n+γ)|b_n||z|ⁿ

−γ|z| −

∞

X

n=2

n^k(2n−γ−2)|a_n||z|ⁿ−

∞

X

n=1

n^k(2n+γ+ 2)|b_n||z|ⁿ

= 2(1−γ)|z| −2

∞

X

n=2

n^k(2n−γ−1)|a_n||z|ⁿ−2

∞

X

n=1

n^k(2n+γ+ 1)|b_n||z|ⁿ

= 2(1−γ)|z|

( 1−

∞

X

n=2

n^k(2n−γ−1)

1−γ |a_n||z|ⁿ⁻¹−

∞

X

n=1

n^k(2n+γ + 1)

1−γ |b_n||z|ⁿ⁻¹ )

>2(1−γ) (

1−

∞

X

n=2

n^k(2n−γ−1) 1−γ |a_n|+

∞

X

n=1

n^k(2n+γ+ 1) 1−γ |b_n|

!) .

This last expression is non-negative by (2.1), and so the proof is complete.

The harmonic function (2.3) f(z) = z+

∞

X

n=2

1−γ

n^k(2n−γ−1)x_nzⁿ+

∞

X

n=1

1−γ

n^k(2n+γ+ 1)y¯_nz¯ⁿ, where

∞

X

n=2

|x_n|+

∞

X

n=1

|y_n|= 1 shows that the coefficient bound given by (2.1) is sharp.

The functions of the form (2.3) are inRSH(k, γ)because

∞

X

n=1

n^k(2n−γ−1)

1−γ |a_n|+n^k(2n+γ+ 1) 1−γ |b_n|

= 1 +

∞

X

n=2

|x_n|+

∞

X

n=1

|y_n|= 2.

In the following theorem, it is shown that the condition (2.1) is also necessary for functions f_k=h+ ¯g_k,wherehandg_kare of the form (1.7).

Theorem 2.2. Letf_k =h+ ¯g_kbe given by (1.7). Thenf_k ∈RS_H(k, γ)if and only if (2.4)

∞

X

n=1

n^k[(2n−γ−1)|a_n|+ (2n+γ+ 1)|b_n|]≤2(1−γ).

Proof. SinceRS_H(k, γ)⊂RS_H(k, γ), we only need to prove the “only if” part of the theorem.

To this end, for functionsf_kof the form (1.7), we notice that the condition (1.6) is equivalent to Re

(1−γ)z−P∞

n=2n^k[n−γ+ (n−1)e^iα]|an|zⁿ z−P∞

n=2n^k|a_n|zⁿ+ (−1)^2kP∞

n=1n^k|b_n|¯zⁿ

− (−1)^2kP∞

n=1n^k[n+γ+ (n+ 1)e^iα]|b_n|¯zⁿ z−P∞

n=2n^k|a_n|zⁿ+ (−1)^2kP∞

n=1n^k|b_n|¯zⁿ

= Re

1−γ−P∞

n=2n^k[n−γ+ (n−1)e^iα]|a_n|zⁿ⁻¹ 1−P∞

n=2n^k|a_n|zⁿ⁻¹+^z_z^¯(−1)^2kP∞

n=1n^k|b_n|¯zⁿ⁻¹

−

¯ z

z(−1)^2kP∞

n=1n^k[n+γ+ (n+ 1)e^iα]|b_n|¯zⁿ⁻¹ 1−P∞

n=2n^k|a_n|zⁿ⁻¹+^z_z^¯(−1)^2kP∞

n=1n^k|b_n|¯zⁿ⁻¹

≥0.

The above condition must hold for all values ofz,|z| =r <1.Upon choosing the values ofz on the positive real axis, where0≤z =r <1,we must have

Re

1−γ−P∞

n=2n^k(n−γ)|a_n|rⁿ⁻¹−P∞

n=1n^k(n+γ)|b_n|rⁿ⁻¹ 1−P∞

n=2n^k|a_n|rⁿ⁻¹+P∞

n=1n^k|b_n|rⁿ⁻¹

− e^iα P∞

n=2n^k(n−1)|a_n|rⁿ⁻¹+P∞

n=1n^k(n+ 1)|b_n|rⁿ⁻¹ 1−P∞

n=2n^k|a_n|rⁿ⁻¹+P∞

n=1n^k|b_n|rⁿ⁻¹

≥0.

SinceRe(−e^iα)≥ −|e^iα|=−1,the above inequality reduces to (2.5) 1−γ−P∞

n=2n^k(2n−γ−1)|a_n|rⁿ⁻¹−P∞

n=1n^k(2n+γ+ 1)|b_n|rⁿ⁻¹ 1−P∞

n=2n^k|a_n|rⁿ⁻¹+P∞

n=1n^k|b_n|rⁿ⁻¹ ≥0.

If the condition (2.4) does not hold then the numerator in (2.5) is negative for r sufficiently close to1.Thus there exists az0 =r0 in(0,1)for which the quotient in (2.5) is negative. This contradicts the condition forf ∈RS_H(k, γ)and hence the result.

Next we determine a representation theorem for functions inRS_H(k, γ).

Theorem 2.3. Letf_kbe given by (1.7). Thenf_k ∈RS_H(k, γ)if and only if

(2.6) f_k(z) =

∞

X

n=1

(X_nh_n(z) +Y_ng_kn(z)) whereh₁(z) =z,

h_n(z) =z− 1−γ

n^k(2n−γ−1)zⁿ (n = 2,3, . . .), g_kn(z) = z+ (−1)^k 1−γ

n^k(2n+γ+ 1)z¯ⁿ (n = 1,2, . . .),

∞

X

n=1

(X_n+Y_n) = 1,

X_n≥0, Y_n≥0.In particular, the extreme points ofRS_H(k, γ)are{h_n}and{g_kn}.

Proof. For functionsf_kof the form (2.6) we have f_k(z) =

∞

X

n=1

(X_nh_n(z) +Y_ng_kn(z))

=

∞

X

n=1

(X_n+Y_n)z−

∞

X

n=2

1−γ

n^k(2n−γ−1)X_nzⁿ + (−1)^k

∞

X

n=1

1−γ

n^k(2n+γ+ 1)Y_nz¯ⁿ. Then

∞

X

n=2

n^k(2n−γ−1) 1−γ |a_n|+

∞

X

n=1

n^k(2n+γ+ 1) 1−γ |b_n|=

∞

X

n=2

X_n+

∞

X

n=1

Y_n

= 1−X₁ ≤1 and sof_k ∈RS_H(k, γ).

Conversely, suppose thatf_k∈RS_H(k, γ).Setting X_n = n^k(2n−γ−1)

1−γ a_n, (n= 2,3, . . .), Yn = n^k(2n+γ+ 1)

1−γ bn, (n= 1,2, . . .), whereP∞

n=1(X_n+Y_n) = 1,we obtain f_k(z) =

∞

X

n=1

(X_nh_n(z) +Y_ng_kn(z))

as required.

The following theorem gives the distortion bounds for functions inRS_H(k, γ)which yields a covering result for this class.

Theorem 2.4. Letf_k ∈RS_H(k, γ).Then for|z|=r <1we have

|f_k(z)| ≤(1 +|b₁|)r+ 1 2^k

1−γ

3−γ − 3 +γ 3−γ|b₁|

r²

and

|f_k(z)| ≥(1− |b₁|)r− 1 2^k

1−γ

3−γ − 3 +γ 3−γ|b₁|

r².

Proof. We only prove the right hand inequality. The proof for the left hand inequality is similar and will be omitted. Letf_k ∈RS_H(k, γ). Taking the absolute value off_kwe obtain

|f_k(z)| ≤(1 +|b₁|)r+

∞

X

n=2

(|a_n|+|b_n|)rⁿ

≤(1 +|b₁|)r+

∞

X

n=2

(|a_n|+|b_n|)r²

≤(1 +|b₁|)r+ 1−γ 2^k(3−γ)

∞

X

n=2

2^k(3−γ)

1−γ (|a_n|+|b_n|)r²

≤(1 +|b₁|)r+ 1−γ 2^k(3−γ)

∞

X

n=2

n^k(2n−γ−1)

1−γ |a_n|+ n^k(2n+γ + 1) 1−γ |b_n|

r²

≤(1 +|b₁|)r+ 1−γ 2^k(3−γ)

1− 3 +γ 1−γ|b₁|

r²

≤(1 +|b₁|)r+ 1 2^k

1−γ

3−γ − 3 +γ 3−γ|b₁|

r².

The following covering result follows from the left hand inequality in Theorem 2.4.

Corollary 2.5. Letf_kof the form (1.7) be so thatf_k ∈RS_H(k, γ).Then

w:|w|< 3.2^k−1−(2^k−1)γ

2^k(3−γ) −3(2^k−1)−(2^k+ 1)γ 2^k(3−γ) |b₁|

⊂f_k(U).

For our next theorem, we need to define the convolution of two harmonic functions. For harmonic functions of the form

fk(z) = z−

∞

X

n=2

|an|zⁿ+ (−1)^k

∞

X

n=1

|bn|¯zⁿ and

F_k(z) =z−

∞

X

n=2

|A_n|zⁿ+ (−1)^k

∞

X

n=1

|B_n|¯zⁿ we define the convolution offkandFkas

(f_k∗F_k)(z) =f_k(z)∗F_k(z) (2.7)

=z−

∞

X

n=2

|a_n||A_n|zⁿ+ (−1)^k

∞

X

n=1

|b_n||B_n|¯zⁿ.

Theorem 2.6. For 0 ≤ β ≤ γ < 1, let f_k ∈ RS_H(k, γ) and F_k ∈ RS_H(k, β). Then the convolution

f_k∗F_k ∈RS_H(k, γ)⊂RS_H(k, β).

Proof. Then the convolutionf_k∗F_kis given by (2.7). We wish to show that the coefficients of f_k∗F_ksatisfy the required condition given in Theorem 2.2. ForF_k ∈RS_H(k, β)we note that

|A_n| ≤1and|B_n| ≤1.Now, for the convolution functionf_k∗F_k,we obtain

∞

X

n=2

n^k(2n−β−1)

1−β |a_n||A_n|+

∞

X

n=1

n^k(2n+β+ 1)

1−β |b_n||B_n|

≤

∞

X

n=2

n^k(2n−β−1) 1−β |a_n|+

∞

X

n=1

n^k(2n+β+ 1) 1−β |b_n|

≤

∞

X

n=2

n^k(2n−γ−1) 1−γ |a_n|+

∞

X

n=1

n^k(2n+γ+ 1)

1−γ |b_n| ≤1

since0≤β ≤γ <1andf_k∈RS_H(k, γ).Thereforef_k∗F_k∈RS_H(k, γ)⊂RS_H(k, β).

Next we discuss the convex combinations of the classRS_H(k, γ).

Theorem 2.7. The familyRS_H(k, γ)is closed under convex combination.

Proof. Fori= 1,2, . . . ,suppose thatf_ki ∈RS_H(k, γ),where fki(z) =z−

∞

X

n=2

|a_in|zⁿ+ (−1)^k

∞

X

n=1

|b_in|¯zⁿ.

Then by Theorem 2.2, (2.8)

∞

X

n=2

n^k(2n−γ−1)

1−γ |a_in|+

∞

X

n=1

n^k(2n+γ+ 1)

1−γ |b_in| ≤1.

ForP∞

i=1t_i = 1, 0≤t_i ≤1,the convex combination off_ki may be written as

∞

X

i=1

t_if_ki(z) = z−

∞

X

n=2

∞

X

i=1

t_i|a_in|

!

zⁿ+ (−1)^k

∞

X

n=1

∞

X

i=1

t_i|b_in|

!

¯ zⁿ. Then by (2.8),

∞

X

n=2

n^k(2n−γ−1) 1−γ

∞

X

i=1

t_i|a_in|

! +

∞

X

n=1

n^k(2n+γ+ 1) 1−γ

∞

X

i=1

t_i|b_in|

!

=

∞

X

i=1

t_i

∞

X

n=2

n^k(2n−γ−1)

1−γ |a_in|+

∞

X

n=1

n^k(2n+γ+ 1) 1−γ |b_in|

!

≤

∞

X

i=1

t_i = 1 and therefore

∞

X

i=1

tifki(z)∈RSH(k, γ).

Following Ruscheweyh [5], we call theδ−neighborhood off the set N_δ(f_k) =

(

F_k :F_k(z) =z−

∞

X

n=2

|A_n|zⁿ+ (−1)^k

∞

X

n=1

|B_n|¯zⁿ and

∞

X

n=2

n(|a_n−A_n|+|b_n−B_n|) +|b₁−B₁| ≤δ )

. Theorem 2.8. Assume that

f_k(z) =z−

∞

X

n=2

|a_n|zⁿ+ (−1)^k

∞

X

n=1

|b_n|¯zⁿ belongs toRS_H(k, γ).If

δ≤ 1 3

(1−γ)

1− 1 2^k

−

3 + 2γ− 1

2^k(3 +γ)

|b₁|

, thenN_δ(f_k)⊂RS_H(0, γ).

Proof. Let

F_k(z) =z−

∞

X

n=2

|A_n|zⁿ+ (−1)^k

∞

X

n=1

|B_n|¯zⁿ belong toN_δ(f_k).We have

∞

X

n=2

2n−γ−1

1−γ |A_n|+

∞

X

n=1

2n+γ+ 1 1−γ |B_n|

≤

∞

X

n=2

2n−γ−1

1−γ |a_n−A_n|+

∞

X

n=1

2n+γ+ 1

1−γ |b_n−B_n| +

∞

X

n=2

2n−γ−1 1−γ |a_n|+

∞

X

n=1

2n+γ+ 1 1−γ |b_n|

≤ 3 1−γ

∞

X

n=2

n(|a_n−A_n|+|b_n−B_n|) + 3

1−γ|b₁|+ γ 1−γ|b₁| + 1

2^k

∞

X

n=2

n^k(2n−γ−1)

1−γ |a_n|+ 1 2^k

∞

X

n=2

n^k(2n−γ−1)

1−γ |b_n|+ 3 +γ 1−γ|b₁|

≤ 3δ

1−γ + γ

1−γ|b₁|+ 1 2^k

1− 3 +γ 1−γ|b₁|

+3 +γ

1−γ|b₁| ≤1.

ThusF_k∈RS_H(0, γ)for δ≤ 1

3

(1−γ)

1− 1 2^k

−

3 + 2γ− 1

2^k(3 +γ)

|b₁|

.

REFERENCES

[1] J. CLUNIE ANDT. SHEIL-SMALL, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math., 9 (1984), 3–25.

[2] J.M. JAHANGIRI, G. MURUGUSUNDARAMOORTHYANDK. VIJAYA, Salagean-type harmonic univalent functions, South. J. Pure and Appl. Math., Issue 2 (2002), 77–82.

[3] J.M. JAHANGIRI, CHAN KIM YONGAND H.M. SRIVASTAVA, Construction of a certain class of harmonic close to convex functions associated with the Alexander integral transform, Integral Transform. Spec. Funct., 14(3) (2003), 237–242.

[4] T. ROSY, B. ADOLPH STEPHENANDK.G. SUBRAMANIAN, Goodman Ronning type harmonic univalent functions, Kyungpook Math. J., 41 (2001), 45–54.

[5] St. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 81 (1981), 521–527.