Using the Salagean operator, we define and study a new subclass of harmonic univalent functions. In this paper, we obtain numerous sharp results including ...
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 8, 371 - 383
A New Subclass of Harmonic Univalent Functions Defined by Salagean Operator K. K. Dixita , A. L. Pathakb,1 , S. Porwala and R. Agarwalb a
b
Department of Mathematics, Janta College Bakewar, Etawah-206124 (U.P.) India
Department of Mathematics, Brahmanand College The Mall, Kanpur-208004 (U.P.) India alpathak@rediffmail.com Abstract
Using the Salagean operator, we define and study a new subclass of harmonic univalent functions. In this paper, we obtain numerous sharp results including coefficient conditions, extreme points, distortion bounds, convolution properties and convex combinations for the above class of harmonic univalent functions. The results obtained for the class reduce to the corresponding results for various well-known classes in the literature.
Mathematics Subject Classification: 30C45, 30C50, 30C55, 31A05 Keywords: Harmonic, Starlike, Convex, Univalent, Goodman-Ronning type, Salagean type functions
1
Introduction
A continuous function f = u + iv is a complex valued harmonic function in a complex domain C if both u and v are real harmonic in C. In any simply connected domain D ⊂ C, we can write f = h + g, where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f . A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h (z)| > |g (z)| in D. Denote by H the class of functions f = h + g that are harmonic univalent and sense-preserving in the unit disc Δ = {z : |z| < 1} for which h(0) = f (0) = fz (0) − 1 = 0. Then for f = h + g ∈ H we may express the analytic
372
K. K. Dixit, A. L. Pathak, S. Porwal and R. Agarwal
functions h and g as h(z) = z +
∞
n
an z , g(z) =
∞
bn z n , |b1 | < 1.
(1.1.1)
n=1
k=2
Note that H reduces to the class of normalized analytic univalent functions if the co-analytic part of its members is zero. In 1984 Clunie and Sheil-Small [2] investigated the class H as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers on H and its subclasses. Jahangiri et al. [5] 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. For more references see Ahuja [1] and Duren [3]. Definition 1.1: Recently, Rosy et al. [6], defined the subclass GH (γ) ⊂ H consisting of harmonic univalent functions f (z) satisfy the following condition iα zf (z) iα Re (1 + e ) ≥ γ, 0 ≤ γ < 1, α ∈ R. (1.1.2) −e f (z) They proved that if f = h + g is given by (1.1.1) and if ∞ 2n − 1 − γ 2n + 1 + γ |an | + |bn | ≤ 2, 0 ≤ γ < 1, 1 − γ 1 − γ n=1
(1.1.3)
then f is a Goodman-Ronning type harmonic univalent functions in U. This condition is proved to be also necessary if h and g are of the form h(z) = z −
∞
n
|an |z , g(z) =
n=2
∞
|bn |z n , |b1 | < 1.
(1.1.4)
n=1
Jahangiri et al. [4] has introduced the modified Salagean operator of harmonic univalent function f as D k f (z) = D k h(z) + (−1)k D k g(z), with N0 = N ∪ {0}, where k
D h(z) = z +
∞
nk an z n
n=2
and D k g(z) =
∞ n=1
nk bn z n .
k ∈ N0 ,
(1.1.5)
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New subclass of harmonic univalent functions
We let RH (k, γ, p, q) denote the family of harmonic functions f of the form (1.1.1) such that k+q f (z) iα D iα − pe Re (1 + pe ) k ≥ γ, D f (z)
0 ≤ γ < 1, α ∈ R, p ≥ 0, q ∈ N, (1.1.6)
where D k f is defined by (1.1.5). Also we let the subclass RH (k, γ, p, q) consist of harmonic functions fk = h + g k in RH (k, γ, p, q) so that h and gk are of the form h(z) = z −
∞
n
k
|an |z , gk (z) = (−1)
n=2
∞
|bn |z n , |b1 | < 1.
(1.1.7)
n=1
In this paper, the coefficient condition given in [9] for the class RSH (k, γ) is extended to the class RH (k, γ, p, q) of the form (1.1.6). Furthermore, we determine extreme points, a distortion theorem, convolution conditions and convex combinations for the functions in RH (k, γ, p, q).
2
Main Results
In our first theorem, we introduce a sufficient coefficient bound for harmonic functions in RH (k, γ, p, q). Theorem 2.1: let f = h + g be given by (1.1.1). If ∞ n=1
k
q
n {n (p + 1) − (γ + p)}|an | +
∞
nk {nq (p + 1) − (−1)q (γ + p)}|bn | ≤ 2(1 − γ),
n=1
(2.2.1) where a1 = 1, k ∈ N0 , 0 ≤ γ < 1, p ≥ 0 and q ∈ N, then f is sense preserving, harmonic univalent in U, and f ∈ RH (k, γ, p, q). Proof. If z1 = z2 , then f (z1 ) − f (z2 ) ≥ 1 − g(z1 ) − g(z2 ) h(z1 ) − h(z2 ) h(z1 ) − h(z2 ) ∞ bn (z1n − z2n ) n=1 = 1 − ∞ an (z1n − z2n ) (z1 − z2 ) + n=2
374
K. K. Dixit, A. L. Pathak, S. Porwal and R. Agarwal ∞
n|bn |
n=1 ∞
> 1−
1−
∞
n|an |
n=2
nk {nq (p + 1) − (−1)q (γ + p)}
n=1
|bn | (1 − γ) ≥1− ∞ nk {nq (p + 1) − (γ + p)} |an | 1− (1 − γ) n=2 ≥0 which proves univalence. Now we see that f is sense preserving in U. This is because
|h (z)| ≥ 1 −
∞
n|an ||z|
n−1
>1−
n=2
≥ ≥
n=1 ∞
∞
(1 − γ)
n=2
∞ nk {nq (p + 1) − (−1)q (γ + p)}
n=1
≥
∞ nk {nq (p + 1) − (γ + p)}
(1 − γ)
|an |
|bn |
nk {nq (p + 1) − (−1)q (γ + p)} |bn ||z|n−1 (1 − γ) n|bn ||z|n−1 = |g (z))|.
n=1
Using the fact that Rew ≥ γ if and only if |1 − γ + ω| ≥ |1 + γ − ω|, it suffices to show that |(1 − γ)D k f (z) + (1 + peiα )D k+q f (z) − peiα D k f (z)| −|(1 + γ)D k f (z) − (1 + peiα )D k+q f (z) + peiα D k f (z)| ≥ 0. Substituting for D k f and D k+q f in (2.2.2) yields, |(1 − γ)D k f (z) + (1 + peiα )D k+q f (z) − peiα D k f (z)| −|(1 + γ)D k f (z) − (1 + peiα )D k+q f (z) + peiα D k f (z)| ∞ ∞ nk an z n + (−1)k nk bn z n = (1 − γ − peiα ) z + n=1 n=2 ∞ ∞ + (1 + peiα ) z + nk+q an z n + (−1)k+q nk+q bn z n n=2
n=1
(2.2.2)
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New subclass of harmonic univalent functions
∞ ∞ iα k n k k − (1 + γ + pe ) z + n an z + (−1) n bn z n n=2 n=1 ∞ ∞ iα k+q n k+q k+q n n an z + (−1) n bn z − (1 + pe ) z + n=2 n=1
∞ = (2 − γ)z + nk (1 − γ − peiα + nq + nq peiα )an z n n=2 +(−1)k
∞
k iα q iα q n n (1 − γ − pe ) + (−1) (1 + pe )n bn z
n=1 ∞ − γz + nk (1 + γ + peiα − nq − nq peiα )an z n n=2
−(−1)k
∞ n=1
≥2(1 − γ)|z| − 2
∞
nk [nq (p + 1) − (γ + p)]|an ||z|n
n=2 ∞
−
k q q iα iα n n (−1) n (1 + pe ) − (1 + γ + pe ) bn z
k n [(1 − γ − peiα ) + (−1)q (1 + peiα )nq ] |bn | |z|n
n=1 ∞
−
k n [(−1)q nq (1 + peiα ) − (1 + γ + peiα )] |bn | |z|n
n=1
⎧ ∞ ⎪ ⎪ ⎪ 2(1 − γ)|z| − 2 nk [nq (p + 1) − (γ + p)]|an | |z|n ⎪ ⎪ ⎪ ⎪ n=2 ⎪ ∞ ⎪ ⎪ ⎪ ⎪ −2 nk [nq (1 + p) + γ + p]|bn | |z|n , if q is odd ⎪ ⎨ =
n=1
∞ ⎪ ⎪ ⎪ nk [nq (p + 1) − (γ + p)]|an | |z|n 2(1 − γ)|z| − 2 ⎪ ⎪ ⎪ ⎪ n=2 ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ −2 nk [nq (p + 1) − (γ + p)]|bn | |z|n , if q is even ⎩ n=1
=2(1 − γ)|z| 1 −
∞ nk [nq (p + 1) − (γ + p)] n=2
−
(1 − γ)
|an | |z|n−1
∞ nk [nq (p + 1) − (−1)q (γ + p)] n=1
(1 − γ)
|bn | |z|n−1
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K. K. Dixit, A. L. Pathak, S. Porwal and R. Agarwal
>2(1 − γ) 1 −
∞ nk [nq (p + 1) − (γ + p)]|an |
+
n=2 ∞ n=1
(1 − γ) nk [nq (p + 1) − (−1)q (γ + p)] |bn | (1 − γ)
.
This last expression is non-negative by (2.2.1), and so the proof is complete. The harmonic univalent functions f (z) = z+
∞ n=2
+
(1 − γ) xn z n nk [nq (p + 1) − (γ + p)]
∞ n=1
nk [nq (p
(1 − γ) yn z n , + 1) − (γ + p)]
(2.2.3)
where, k ∈ N0 , p ≥ 0, q ∈ N and ∞
|xn | +
n=2
∞
|yn | = 1,
n=1
show that the coefficient bound given by (2.2.1) is sharp. The functions of the form (2.2.3) are in RH (k, γ, p, q) because ∞ k n {nq (p + 1) − (γ + p)} n=1
(1 − γ)
=1+
nk {nq (p + 1) − (−1)q (γ + p)} |an | + |bn | (1 − γ)
∞
|xn | +
n=2
∞
|yn | = 2.
n=1
In the following theorem, it is shown that the condition (2.2.1) is also necessary for functions fk = h + g k , where h and gk are of the form (1.1.7) and for q as odd positive integer. Theorem 2.2: Let fk = h + g k be given by (1.1.7). Then fk ∈ RH (k, γ, p, q) if and only if ∞ n=1
q
{n (p + 1) − (γ + p)}|an | +
∞
nk {nq (p + 1) − (−1)q (γ + p)}|bn | ≤ 2(1 − γ),
n=1
(2.2.4) where q is an odd positive integer.
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New subclass of harmonic univalent functions
Proof. Since RH (k, γ, p, q) ⊂ RH (k, γ, p, q) we only need to prove the “only if” part of the theorem. To this end, for functions fk of the form (1.1.7), we notice that the condition (1.1.6) is equivalent to ⎧ ⎫ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (1 − γ)z − nk [nq (1 + peiα ) − γ − peiα )]|an |z n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n=2 ⎪ ⎪ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k q iα q−1 iα n 2k+q−1 ⎪ ⎪ −(−1) n [n (1 + pe ) + (−1) (γ + pe )]|b n |z ⎪ ⎪ ⎨ ⎬ n=1 Re ∞ ∞ ⎪ ⎪ ⎪ ⎪ k n 2k n ⎪ ⎪ z − n |a |z + (−1) |b |z ⎪ ⎪ n n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n=2 n=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭
= Re
−
⎧ ∞ ⎪ ⎪ ⎪ nk [nq (1 + peiα ) − γ − peiα ]|an |z n−1 ⎪ ⎨ (1 − γ) − n=2
⎪ ⎪ ⎪ ⎪ ⎩
1−
∞
k
n |an |z
n=2
n−1
∞ z 2k + (−1) |bn |z n−1 z n=1
⎫ ∞ z ⎪ ⎪ (−1)2k+q−1 nk [nq (1 + peiα ) + (−1)q−1 (γ + peiα )]|bn |z n−1 ⎪ ⎪ ⎬ z n=1 1−
∞
k
n |an |z
n=2
n−1
∞ z 2k + (−1) nk |bn |z n−1 z n=1
⎪ ⎪ ⎪ ⎪ ⎭
≥ 0. The above condition must hold for all values of z on the positive real axis, where 0 ≤ z = r < 1, we must have ⎧ ∞ ∞ ⎪ k q n−1 2k+q−1 ⎪ ⎪ n (n − γ)|an |r − (−1) nk {nq + (−1)q−1 γ}|bn |r n−1 ⎪ ⎨ (1 − γ) − n=2 n=1 Re ∞ ∞ ⎪ ⎪ ⎪ 1− nk |an |r n−1 + (−1)2k nk |bn |r n−1 ⎪ ⎩ n=2
∞
n=1
nk (nq − 1)|an |r n−1 + (−1)2k+q−1
−peiα n=2 1−
∞ n=2
∞
⎫ ⎪ ⎪ [nq + (−1)q−1 ]|bn |r n−1 ⎪ ⎪ ⎬
n=1 ∞ 2k
nk |an |r n−1 + (−1)
n=1
nk |bn |r n−1
⎪ ⎪ ⎪ ⎪ ⎭
378
K. K. Dixit, A. L. Pathak, S. Porwal and R. Agarwal
≥ 0. Since Re(−eiα ) ≥ −|eiα | = −1, the above inequality reduces to (1 − γ) −
∞
nk {nq (p + 1) − γ − p}|an | −
n=2
1−
∞
∞
nk {nq (p + 1) − (−1)q (γ + p)}|bn |r n−1
n=1
nk |an |r n−1 +
n=2
∞
≥ 0. nk |bn |r n−1
n=1
(2.2.5) If the condition (2.2.4) does not hold then the numerator in (2.2.5) is negative for r sufficiently close to 1. Thus there exists a z0 = r0 in (0,1) for which the quotient in (2.2.5) is negative. This contradicts the condition for f ∈ RH (k, γ, p, q) and hence the result. Next we determine the extreme points of closed convex hulls of RH (k, γ, p, q). Theorem 2.3: Let fk be given by (1.1.7). Then fk ∈ RH (k, γ, p, q) if and only if ∞ fk (z) = (Xn hn (z) + Yn gkn (z))
(2.2.6)
n=1
where h1 (z) = z, (1 − γ) z n , (n = 2, 3, ...), + 1) − (γ + p)} (1 − γ) z n , (n = 1, 2, ...), gkn (z) = z + (−1)k k q n {n (p + 1) − (−1)q (γ + p)} ∞ (Xn + Yn ) = 1, hn (z) = z −
nk {nq (p
n=1
Xn ≥ 0, Yn ≥ 0 and q is an odd positive integer. In particular, the extreme points of RH (k, γ, p, q) are {hn } and {gkn }. Proof. For functions fk of the form (2.2.6), we have fk (z) = =
∞ n=1 ∞ n=1
(Xn hn (z) + Yn gkn (z)) (Xn + Yn )z −
∞ n=2
(1 − γ) Xn z n nk {nq (p + 1) − (γ + p)}
379
New subclass of harmonic univalent functions
+ (−1)k
∞ n=1
(1 − γ) Yn z n . nk {nq (p + 1) − (−1)q (γ + p)}
Then ∞ nk {nq (p + 1) − (γ + p)} n=2
(1 − γ)
|an | + =
∞ nk {nq (p + 1) − (−1)q (γ + p)} n=1 ∞
(1 − γ) Xn +
n=2
∞
|bn |
Y n = 1 − X1 ≤ 1
n=1
and so fk ∈ RH (k, γ, p, q). Conversely, suppose that fk ∈ RH (k, γ, p, q). Setting nk {nq (p + 1) − (γ + p)} an , (n = 2, 3, ...), (1 − γ) nk {nq (p + 1) − (−1)q (γ + p)} bn , (n = 1, 2, ...), Yn = (1 − γ) ∞ where (Xn + Yn ) = 1, we obtain Xn =
fk (z) =
n=1 ∞
(Xn hn (z) + Yn gkn (z))
n=1
as required. The following theorem gives the distortion bounds for functions in RH (k, γ, p, q) which yields a covering result for this class. Theorem 2.4: Let fk ∈ RH (k, γ, p, q). Then for |z| = r < 1, we have (1 − γ) {(p + 1) − (−1)q (γ + p)} 1 − |b1 | r 2 |fk (z)| ≤ (1 + |b1 |)r + k 2 {2q (p + 1) − (γ + p)} {2q (p + 1) − (γ + p)} and
{(p + 1) − (−1)q (γ + p)} 1 (1 − γ) − |b1 | r 2 , |fk (z)| ≥ (1 − |b1 |)r − k q q 2 {2 (p + 1) − (γ + p)} {2 (p + 1) − (γ + p)} where q is an odd positive integer. Proof. We only prove the right hand inequality. The proof for the left hand inequality is similar and will be omitted. Let fk ∈ RH (k, γ, p, q). Taking the absolute value of fk , we obtain |fk (z)| ≤ (1 + |b1 |)r +
∞ n=2
(|an | + |bn |)r n
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K. K. Dixit, A. L. Pathak, S. Porwal and R. Agarwal
≤ (1 + |b1 |)r +
∞
(|an | + |bn |)r 2
n=2
2k {2q (p + 1) − (γ + p)} (1 − γ) (|an | + |bn |)r 2 ≤ (1 + |b1 |)r + k q 2 {2 (p + 1) − (γ + p)} n=2 (1 − γ) ∞ k n {nq (p + 1) − (γ + p)} (1 − γ) ≤ (1 + |b1 |)r + k q |an | 2 {2 (p + 1) − (γ + p)} n=2 (1 − γ) nk {nq (p + 1) − (−1)q (γ + p)} |bn | r 2 + (1 − γ) (1 − γ) {(p + 1) − (−1)q (γ + p)} ≤ (1 + |b1 |)r + k q 1− |b1 | r 2 2 {2 (p + 1) − (γ + p)} (1 − γ) (1 − γ) 1 {(p + 1) − (−1)q (γ + p)} ≤ (1 + |b1 |)r + k − |b1 | r 2 . 2 {2q (p + 1) − (γ + p)} {2q (p + 1) − (γ + p)} ∞
The following covering result follows from the left hand inequality in Theorem 2.4. Corollary 2.5: Let fk of the form (1.1.7) be so that fk ∈ RH (k, γ, p, q). Then 2k [2q (p + 1) − p] − 1 − (2k − 1)γ w : |w| < 2k [2q (p + 1) − (γ + p)] 2k [2q (p + 1) − p] − [(p + 1) − (−1)q p] − (2k + 1)γ|b1 | ⊂ fk (U). − 2k [2q (p + 1) − (γ + p)] For our next theorem, we need to define the convolution of two harmonic functions of the form fk (z) = z −
∞
|an |z n + (−1)k
n=2
∞
|bn |z n
n=1
and Fk (z) = z −
∞
n
k
|An |z + (−1)
n=2
∞
|Bn |z n .
n=1
The convolution of fk and Fk is defined as (fk ∗ Fk )(z) = fk (z) ∗ Fk (z) ∞ ∞ n k |an ||An |z + (−1) |bn ||Bn |z n . =z− n=2
n=1
(2.2.7)
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New subclass of harmonic univalent functions
Theorem 2.6: For 0 ≤ β ≤ γ < 1, let fk ∈ RH (k, γ, p, q) and Fk ∈ RH (k, β, p, q). Then the convolution fk ∗ Fk ∈ RH (k, γ, p, q) ⊂ RH (k, β, p, q). Proof. The convolution fk ∗ Fk is given by (2.2.7). We wish to show that the coefficients of fk ∗ Fk satisfy the required condition given in Theorem 2.2. For Fk ∈ RH (k, β, p, q), we note that |An | ≤ 1 and |Bn | ≤ 1. Now, for the convolution function fk ∗ Fk , we obtain ∞ nk {nq (p + 1) − (β + p)}
≤
n=2 ∞ n=2
≤
(1 − β)
|an ||An | +
∞ nk {nq (p + 1) − (−1)q (β + p)}
(1 − β)
n=1
|bn ||Bn |
∞ nk {nq (p + 1) − (β + p)} nk {nq (p + 1) − (−1)q (β + p)} |an | + |bn | (1 − β) (1 − β) n=1
∞ nk {nq (p + 1) − (γ + p)}
(1 − γ)
n=2
|an | +
∞ nk {nq (p + 1) − (−1)q (γ + p)}
(1 − γ)
n=1
|bn |
≤1, since 0 ≤ β ≤ γ < 1 and fk ∈ RH (k, γ, p, q). Therefore fk ∗ Fk ∈ RH (k, γ, p, q) ⊂ RH (k, β, p, q). Next we discuss the convex combinations of the class RH (k, γ, p, q). Theorem 2.7: The family RH (k, γ, p, q) is closed under convex combination. Proof. For i = 1, 2, ..., suppose that fki ∈ RH (k, γ, p, q), where fki (z) = z −
∞
n
k
|ain |z + (−1)
n=2
∞
|bin |z n .
n=1
Then by Theorem 2.2, ∞ nk {nq (p + 1) − (γ + p)}
(1 − γ)
n=2
|ain | +
∞ nk {nq (p + 1) − (−1)q (γ + p)}
(1 − γ)
n=1
|bin | ≤ 1. (2.2.8)
For
∞
ti = 1, 0 ≤ ti ≤ 1, the convex combination of fki may be written as
i=1 ∞
ti fki (z) = z −
i=1
∞ ∞ n=2
ti |ain | z n + (−1)k
∞ ∞ n=1
i=1
ti |bin | z n .
i=1
Then by (2.2.8) ∞ nk {nq (p + 1) − (γ + p)} (1 − γ) n=2
∞ i=1
ti |ain |
∞ nk {nq (p + 1) − (−1)q (γ + p)} + (1 − γ) n=1
∞ i=1
ti |bin |
382
= ≤
∞ i=1 ∞
K. K. Dixit, A. L. Pathak, S. Porwal and R. Agarwal ti
∞ ∞ nk {nq (p + 1) − (γ + p)} nk {nq (p + 1) − (−1)q (γ + p)} |ain | + |bin | (1 − γ) (1 − γ) n=2 n=1
ti = 1
i=1
and therefore
∞
ti fki (z) ∈ RH (k, γ, p, q).
i=1
1
Acknowledgement: The present research of the author is supported by
the University Grant Commission, Government of India under Grant No. F. 6-1(49)/2007 (MRP/Sc/NRCB).
References [1] O.P. AHUJA, Planar harmonic univalent and related mappings, J. Ineq. Pure Appl. Math., 6 (4) (2005), Art. 122. [2] J. CLUNIE AND T. SHEIL-SMALL, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. AI Math., 9 (1984), 3-25. [3] P. DUREN, Harnomic Mappings in the plane, Cambridge University Press, (2004), ISBN 05216412173. [4] J.M. JAHANGIRI, G. MURUGUSUNDARAMOORTHY AND K. VIJAYA, Salagean-type harmonic univalent functions, South. J. Pure and Appl. Math., Issue 2(2002), 77-82. [5] J.M. JAHANGIRI, CHAN KIM YONG AND 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. [6] T. ROSY, B. ADOLPH STEPHEN AND K.G. SUBRAMANIAN, Goodman-Ronning type harmonic univalent functions, Kyungpook Math. J., 41 (2001), 45-54. [7] G.S. SALAGEAN, Subclass of univalent functions, Complex AnalysisFifth Romanian Finish Seminar, Bucharest, 1 (1983), pp. 362-372. [8] S. YALCIN, A new class of Salagean-type harmonic univalent functions, Appl. Math. Letters, 18 (2005), 191-198. ¨ URK ¨ [9] S. YALCIN, M. OZT AND M. YAMANKARADENIZ, On the subclass of Salagean-type harmonic univalent functions, J. Ineq. Pure Appl. Math., 8 (2) (2007), Art. 54.
New subclass of harmonic univalent functions
Received: July, 2008
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