A Multiplier Transformation Defined by Convolution ... - CiteSeerX

International Mathematical Forum, 4, 2009, no. 37, 1823 - 1837

A Multiplier Transformation Defined by Convolution Involving nth Order Polylogarithm Functions K. Al-Shaqsi and M. Darus School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi 43600 Selangor D. Ehsan, Malaysia [email protected] [email protected] Abstract We define a multiplier transformation on the class A of analytic functions in the unit disk U = {z : |z| < 1} involving the nth order polylogarithm functions and introduce certain new subclass of strongly close-to-convex functions using this operator. Several interesting properties of these classes are obtained. Our results include several previous known results as special cases.

Mathematics Subject Classification: 30C45, 33C20 Keywords: nth order polylogarithm functions, Multiplier transformation, Subordination, Strongly close-to-convex

1

Introduction

Let A denote the class of functions of the form f (z) = z +

∞ X

ak z k ,

(1.1)

k=2

which are analytic in the unit disk U = {z : |z| < 1}. If f and g are analytic in U, we say that f is subordinate to g, written f ≺ g or f (z) ≺ g(z), if there exists Schwarz function w in U such that f (z) = g(w(z)). We denote by S ∗ (η) and C(η) the subclasses of A consisting of all analytic functions which are, respectively, starlike and convex of order η(0 ≤ η < 1) in

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K. Al-Shaqsi and M. Darus

U.(see, e.g., Srivastava and Owa [6]). If f ∈ A satisfies

¯ ¯ ¯ ³ zf 0 (z) ´¯ π ¯ ¯ −η ¯< β ¯ arg ¯ ¯ 2 f (z)

(z ∈ U),

for some η(0 ≤ η < 1) and β(0 < β ≤ 1), then f is said to be strongly starlike of order β and type η in U. If f ∈ A satisfies ¯ ¯ ¯ ³ ´¯ π 00 zf (z) ¯ ¯ −η ¯< β (z ∈ U), ¯ arg 1 + 0 ¯ ¯ 2 f (z) for some η(0 ≤ η < 1) and β(0 < β ≤ 1), then f is said to be strongly convex of order β and type η in U. We denote by S ∗ (β, η) and C(β, η), respectively, the subclasses of A consisting of all strongly starlike and strongly convex of order β and type η in U. It is obvious that f ∈ A belongs to C(β, η) if and only if zf 0 ∈ S ∗ (β, η). We also note that S ∗ (1, η) ≡ S ∗ (η) and C(1, η) ≡ C(η). In particular, the classes S ∗ (β, 0) and C(β, 0) have been extensively studied by Mocanu [18] and Nunokawa [14]. P k For functions f given by (1.1) and g(z) = z + ∞ k=2 bk z , let (f ∗ g)(z) denote the Hadamard product (or convolution) of f and g, defined by (f ∗ g)(z) = f (z) ∗ g(z) = z +

∞ X

ak bk z k .

k=2

Let f ∈ A. Denote by Dλ : A → A the operator defined by z Dλ = ∗ f (z) (λ > −1). (1 − z)λ+1 It is obvious that D0 f (z) = f (z), D1 f (z) = zf 0 (z) and z(z δ−1 f (z))(δ) , (δ ∈ N0 = N ∪ {0}). δ! ∞ ¡ ¢ P Note that Dδ f (z) = z + C(δ, k)ak z k where C(δ, k) = k+δ−1 and δ ∈ N0 . δ Dδ f (z) =

k=2

The operator Dδ f is called the Ruscheweyh derivative operator (see [25]). We recall here the definition of the well-known generalization of the Riemann Zeta and polylogarithm function, or simply the nth order polylogarithm function G(n; z) given by Φn (b; z) =

∞ X k=1

zk (k + b)n

(n, b ∈ C, z ∈ U).

(1.2)

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Multiplier transformation defined by convolution

Where any term with k + b = 0 is excluded (see Lerch [3] and also [[5], Section 1.10 and 1.12]). Using the definition of Gamma function [[5], p.27] a simply transformation produces the integral formula Z 1 1 tb Φn (b; z) = dt, z(log 1/t)n−1 ) Γ(n) 0 1 − tz Re b > −1 and Re n > 1, z we note that Φ−1 (0; z) = (1−z) 2 is Koebe function. For more about polylogarithm in the theory of univalent functions see [22] and [23].

Now, for f ∈ A, n ∈ C, b ∈ C \ Z− and z ∈ U, we define the function G(n, b; z) by ∞ ³ X 1 + b ´n k n G(n, b; z) = (1 + b) Φn (b; z) = z . (1.3) k+b k=1 Also we introduce a function (G(n, b; z))(−1) given by z G(n, b; z) ∗ (G(n, b; z))(−1) = , (λ > −1, n ∈ C, b ∈ C \ Z− ),(1.4) (1 − z)λ+1 and obtain the following linear operator n Db,λ f (z) = (G(n, b; z))(−1) ∗ f (z).

(1.5)

Now we find the explicit form of the function (G(n, b; z))(−1) . It is well known that λ > −1 ∞

X (λ + 1)k z = z k+1 λ+1 (1 − z) k! k=0

(z ∈ U).

(1.6)

Putting (1.3) and (1.6) in (1.4), we get ∞ ³ X 1 + b ´n k=1

k+b

k

z ∗ (G(n, b; z))

(−1)

=

∞ X (k + λ − 1)! k=1

λ!(k − 1)!

zk .

Therefore the function (G(n, b; z))(−1) has the following form (G(n, b; z))(−1) =

∞ ³ X k + b ´n (k + λ − 1)! k=1

1+b

λ!(k − 1)!

zk

(z ∈ U).

Now we note that n Db,λ f (z)

=z+

∞ ³ X k + b ´n (k + λ − 1)! k=2

1+b

λ!(k − 1)!

ak z k

(n ∈ C, b ∈ C \ Z− , λ > −1; z ∈ U).

(1.7)

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Obviously, we observe that n+m n m Db,λ (Db,λ (z)) = Db,λ f (z) (n, m ∈ C, b ∈ C \ Z− , λ > −1; z ∈ U). n It is clear that Db,λ are multiplier transformations. For n ∈ Z, b = 1 and n λ = 0 the operators Db,λ were studied by Uralegaddi and Somanatha [1], and n are closely related to the multiplier transfor n ∈ Z, λ = 0 the operators Db,λ n is formations studied by Flett [26], also, for n = −1, λ = 0, the operators Db,λ the integral operator studied by Owa and Srivastava [21]. And for any negative n is the multiplier transforreal number n and b = 1, λ = 0 the operators Db,λ mation studied by Jung et al. [9], and for any nonnegative integer n and n b = λ = 0, the operators Db,λ is the differential operator defined by S˘al˘agean n is the [4]. Furthermore , for n = 0 and λ ∈ N0 = N ∪ {0}, the operators Db,λ δ differential operator D defined by Ruscheweyh [25]. For n, λ ∈ N0 and b = 0 the operators Dλn is the differential operator defined by authors [10]. Finally, for different choices of n, b and λ we obtain several operator investigated earlier by other author see, for example [16],[15] and [13]. Now we define new classes n of analytic functions by using the multiplier transformations Db,λ defined by (1.7) as follows: n For n ∈ C, b ∈ C \ Z− and λ > −1, let Kb,λ (γ, δ, η, A, B) be the class of functions f ∈ A satisfying the condition ¯ ¯ ¯³ z(Dn f (z))0 ´¯ π ¯ ¯ b,λ − γ ¯ < δ (0 ≤ γ < 1; 0 < δ ≤ 1; z ∈ U) ¯ n ¯ Db,λ g(z) ¯ 2 n for some g ∈ Sb,λ (η, A, B), where

( n Sb,λ (η, A, B) =

1 g∈A: 1−η

Ã

n z(Db,λ g(z))0 −η n Db,λ g(z)

!

1 + Az ≺ 1 + Bz

)

(n ∈ C, b ∈ C \ Z− , λ > −1, 0 ≤ η < 1; −1 ≤ B < A ≤ 1; z ∈ U). 0 1 0 (γ, 1, η, 1, −1) (γ, 1, η, 1, −1) and K0,0 (γ, 1, η, 1, −1) ≡ K0,0 We note that K0,1 are the classes of quasi-convex and close-to-convex functions of order γ and type η, respectively, introduced and studied by Noor and Alkhorasani [11] and 0 1 (0, δ, 0, 1, −1) is the class of (0, δ, 0, 1, −1) ≡ K0,1 Silverman [8]. Further, K0,0 strongly close-to- convex functions of order δ in the sense of Pommerenke [2]. n Finally, note that for integer n and λ ≥ 0, the class Kλ,0 (γ, δ, η, A, B) was studied by Cho and Kim [15].

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Multiplier transformation defined by convolution

In the present paper, we give some argument properties of analytic functions belonging to A which contain the basic inclusion relationships among n the classes Kb,λ (γ, δ, η, A, B). The integral preserving properties in connection n with the operator Db,λ defined by (1.7) are also considered. Furthermore, we obtain the previous results by Bernardi [20], Libera [19], Noor [12], Noor and Alkhorasani [11] and Cho and Kim [15] as special cases.

2

Main results

To derive our results we need the following lemmas: Lemma 2.1 [17]. Let β, ν be complex numbers. Let φ ∈ P be convex univalent in U with φ(0) = 1 and < [βφ(z) + ν] > 0, z ∈ U. If p(z) = 1 + p1 z + p2 z 2 + · · · is analytic in U with p(0) = 1, then p(z) +

zp0 (z) ≺ φ(z) ⇒ p(z) ≺ φ(z), βp(z) + ν

(z ∈ U).

Lemma 2.2 [24]. Let φ ∈ P be convex univalent in U and w be analytic in U with < w(z) ≥ 0, z ∈ U If p is analytic in U with p(0) = φ(0), then p(z) + w(z)zp0 (z) ≺ φ(z) ⇒ p(z) ≺ φ(z),

(z ∈ U).

Lemma 2.3 [14]. Let p be analytic in U with p(0) = 1 and p(z) 6= 0 in U. Suppose that there exists a point z0 ∈ U such that ¯ ¯ π ¯ ¯ (2.1) ¯ arg p(z)¯ < α for |z| < |z0 | 2 and ¯ ¯ π ¯ ¯ (2.2) ¯ arg p(z0 )¯ < α for (0 < α ≤ 1). 2 Then we have z0 p0 (z0 ) = ikα, p(z0 ) where

(2.3)

1´ 1³ a+ k≥ 2 a

when

arg p(z0 ) =

1³ 1´ a+ 2 a

when

π arg p(z0 ) = − α 2

k≤−

π α 2

(2.4) (2.5)

and 1

p(z0 ) α = ±ia (a > 0).

(2.6)

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K. Al-Shaqsi and M. Darus

At first, with the help of Lemma 2.1, we obtain the following Theorem 2.4 Let h be convex univalent in U with h(0) = 1 and Re h(z) > 0. If a function f ∈ A satisfies the condition ! Ã n+1 z(Db,λ f (z))0 1 − η ≺ h(z), n+1 1−η Db,λ f (z) then 1 1−η

Ã

n z(Db,λ f (z))0 −η n f (z) Db,λ

! ≺ h(z),

for 0 ≤ η < 1, n ∈ C, b ∈ C \ Z− , λ > −1 and z ∈ U. Proof. Let 1 p(z) = 1−η

Ã

! n z(Db,λ f (z))0 −η , n Db,λ f (z)

where p is analytic function with p(0) = 1. By using the equation n+1 n n z(Db,λ f (z))0 = (b + 1)Db,λ f (z) − bDb,λ f (z),

(2.7)

n+1 Db,λ f (z) b + η + (1 − η)p(z) = (b + 1) n . Db,λ f (z)

(2.8)

we get

Taking logarithmic derivatives in both sides of (2.8) and multiplying by z, we have à ! n z(Db,λ+1 f (z))0 1 zp0 (z) − η = p(z) + (z ∈ U). n 1−η Db,λ+1 f (z) (1 − η)p(z) + η + b Applying Lemma 2.1, it follows that p ≺ φ, that is ! à n z(Db,λ f (z))0 1 − η ≺ h(z) (z ∈ U). n 1−η Db,λ f (z) Theorem 2.5 Let h be convex univalent in U with h(0) = 1 and Re h(z) > 0. If a function f ∈ A satisfies the condition à ! n z(Db,λ+1 f (z))0 1 − η ≺ h(z), n 1−η Db,λ+1 f (z)

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Multiplier transformation defined by convolution

then 1 1−η

Ã

n z(Db,λ f (z))0 −η n Db,λ f (z)

! ≺ h(z),

for 0 ≤ η < 1, n ∈ C, b ∈ C \ Z− , λ > −1 and z ∈ U. Proof. Let 1 p(z) = 1−η

Ã

! n z(Db,λ f (z))0 −η , n Db,λ f (z)

where p is analytic function with p(0) = 1. By using the equation n n n z(Db,λ f (z))0 = (λ + 1)Db,λ+1 f (z) − λDb,λ f (z).

(2.9)

Then, by using the arguments similar to Theorem 4.5.2. Taking h(z) = (1 + Az)/(1 + Bz) (−1 ≤ B < A ≤ 1) in Theorem 2.1 and Theorem 2.2, we have n+1 n n Corollary 2.6 The inclusion relations, Sb,λ (η, A, B) ⊂ Sb,λ (η, A, B) and Sb,λ+1 (η, A, B) ⊂ n Sb,λ (η, A, B), holds for n ∈ C, b ∈ C \ Z− , λ > −1.

Letting n = b = 0, λ = 1 and h(z) = ((1 + z)/(1 − z))β (0 < β ≤ 1) in above theorems, we have the following inclusion relation. Corollary 2.7 C(β, η) ⊂ S ∗ (β, η). Theorem 2.8 Let h be convex univalent in U with h(0) = 1 and and Re h(z) > 0. If a function f ∈ A satisfies the condition à ! n z(Db,λ f (z))0 1 − η ≺ h(z), n 1−η Db,λ f (z) then 1 1−η

Ã

n z(Db,λ F (z))0 −η n Db,λ F (z)

! ≺ h(z),

(0 ≤ η < 1, n ∈ C, b ∈ C \ Z− , λ > −1; z ∈ U), where F be the integral operator defined by Z c + 1 z c−1 t f (t)dt F (z) = zc 0

(c > −1).

(2.10)

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K. Al-Shaqsi and M. Darus

Proof. From (2.10), we have n n n z(Db,λ F (z))0 = (c + 1)Db,λ f (z) − cDb,λ F (z).

(2.11)

Let 1 p(z) = 1−η

Ã

n z(Db,λ F (z))0 −η n F (z) Db,λ

!

where p is analytic function with p(0) = 1. Then, by using (2.11), we get n f (z) Db,λ c + η + (1 − η)p(z) = (c + 1) n . Db,λ F (z)

(2.12)

Taking logarithmic derivatives in both sides of (2.12) and multiplying by z, we have à ! n z(Db,λ f (z))0 1 zp0 (z) − η = p(z) + (z ∈ U). n 1−η Db,λ f (z) (1 − η)p(z) + η + c Therefore, by Lemma 2.1, we have à ! n z(Db,λ F (z))0 1 − η ≺ h(z) n 1−η Db,λ F (z)

(z ∈ U).

Letting h(z) = (1 + Az)/(1 + Bz) (−1 ≤ B < A ≤ 1) in Theorem 2.8, we have immediately n n Corollary 2.9 If f ∈ Sb,λ (η, A, B), then F ∈ Sb,λ (η, A, B), where F is the integral operator defined by (2.10).

Now, we derive Theorem 2.10 Let f ∈ A and 0 < δ ≤ 1, 0 ≤ γ < 1. If ¯ Ã !¯ n+1 0 ¯ π ¯ f (z)) z(D ¯ ¯ b,λ arg − γ ¯< δ ¯ n+1 ¯ 2 ¯ Db,λ g(z) n+1 (η, A, B), then for some g ∈ Sb.λ ¯ Ã !¯ ¯ ¯ π n z(Db,λ f (z))0 ¯ ¯ − γ ¯ < α, ¯ arg n ¯ ¯ 2 Db,λ g(z)

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Multiplier transformation defined by convolution

where α(0 < α ≤ 1) is the solution of the equation ´ ³ ( α cos π t α + π2 tan−1 (1−η)(1+A) 2 1 f or B 6= −1, +η+b+α sin π2 t1 δ= 1+B f or B = −1, α and t1 =

2 sin−1 π

Ã

!

(1 − η)(A − B) . (1 − η)(1 − AB) + (η + b)(1 − B 2 )

Proof. Let 1 p(z) = 1−γ

(2.13)

Ã

(2.14)

! n z(Db,λ f (z))0 −γ . n Db,λ g(z)

Using (2.7) and simplifying, we have n+1 n n [(1 − γ)p(z) + γ]Db,λ g(z) = (b + 1)Db,λ f (z) − bDb,λ f (z).

(2.15)

Differentiating (2.15) and multiplying by z, we obtain n n (1 − γ)zp0 (z)Db,λ g(z) + [(1 − γ)p(z) + γ]z(Db,λ g(z))0 n+1 n = (b + 1)z(Db,λ f (z))0 − bz(Db,λ f (z))0

(2.16)

n+1 n Since g ∈ Sb.λ (η, A, B), by Corollary 2.6, we know that g ∈ Sb.λ (η, A, B). Let à ! n z(Db,λ g(z))0 1 q(z) = −η . n 1−η Db,λ g(z)

Then, using (2.7) once again, we have (1 − η)q(z) + η + b = (b + 1)

n+1 Db,λ g(z) . n Db,λ g(z)

(2.17)

¿From (2.16) and (2.17), we obtain à ! n+1 f (z))0 z(Db,λ zp0 (z) 1 − η = p(z) + . n+1 1−γ (1 − η)q(z) + η + b g(z) Db,λ While, by using the result of Silverman and Silvia [7], we have ¯ ¯ ¯ 1 − AB ¯¯ A − B ¯ (z ∈ U; B 6= −1) ¯< ¯q(z) − ¯ 1 − B2 ¯ 1 − B2

(2.18)

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K. Al-Shaqsi and M. Darus

and 1−A (z ∈ U; B 6= −1). 2 Then, from (2.18) and (2.19), we obtain Re {q(z)} >

(2.19)

πφ

(1 − η)q(z) + η + b = ρei 2 , where

½

(1−η)(1−A) 1−B

+ η + b < ρ < (1−η)(1+A) +η+b 1+B −t1 < φ < t1 for B 6= −1,

when t1 is given by (2.14), and ½ (1−η)(1−A)

+η+b 0). Then we obtain à ! z0 p0 (z0 ) arg p(z0 ) + (1 − η)q(z0 ) + η + b à ! π i πφ −1 = α + arg 1 + iαk(ρe 2 ) 2 à ! αk sin π2 (1 − φ) π −1 ≥ α + tan 2 ρ + αk cos π2 (1 − φ) à ! α cos π2 t1 π −1 ≥ α + tan (1−η)(1+A) 2 + η + b + α sin π2 t1 1+B π = δ, 2 where δ and t1 are given by (2.13) and (2.14), respectively. Similarly, for the case B = −1, we have ! à π z0 p0 (z0 ) ≥ α. arg p(z0 ) + (1 − η)q(z0 ) + η + b 2

Multiplier transformation defined by convolution

1833

These evidently contradict the assumption of Theorem 2.10. 1 Next, suppose that p(z0 ) α = −ia(a > 0). Applying the same method as the above, we have ! à z0 p0 (z0 ) arg p(z0 ) + (1 − η)q(z0 ) + η + b ! à π α cos t π 2 1 ≤ − α − tan−1 (1−η)(1+A) 2 + η + b + α sin π2 t1 1+B π = − δ, 2 where δ and t1 are given by (2.13) and (2.14), respectively. Similarly, for the case B = −1, we have à ! z0 p0 (z0 ) π arg p(z0 ) + ≤ α. (1 − η)q(z0 ) + η + b 2 These also are contradiction to the assumption of Theorem 2.10. Therefore we complete the proof of Theorem 2.10. Theorem 2.11 Let f ∈ A and 0 < δ ≤ 1, 0 ≤ γ < 1. If ¯ !¯ à ¯ π ¯ n z(Db,λ+1 f (z))0 ¯ ¯ − γ ¯< δ ¯ arg n ¯ 2 ¯ Db,λ+1 g(z) n for some g ∈ Sb.λ+1 (η, A, B), then ¯ à !¯ ¯ ¯ π n z(Db,λ f (z))0 ¯ ¯ − γ ¯ < α, ¯ arg n ¯ ¯ 2 Db,λ g(z)

where α(0 < α ≤ 1) is the solution of the equation, δ and t1 are given by (2.13) and (2.14), respectively. Proof. By using the equation (2.9) and the same methods to prove Theorem 2.10. From Theorem 2.10 and Theorem 2.11, we see easily the following: n+1 n (γ, δ, η, A, B) ⊂ Kb,λ (γ, δ, η, A, B) Corollary 2.12 The inclusion relations, Kb,λ n n − and Kb,λ+1 (γ, δ, η, A, B) ⊂ Kb,λ (γ, δ, η, A, B) , n ∈ C, b ∈ C \ Z , λ > −1.

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K. Al-Shaqsi and M. Darus

Taking n = b = 0 and λ = 1 in Theorem 2.10, we have Corollary 2.13 Let f ∈ A. ¯ Ã !¯ ¯ ¯ π (zf 0 (z))0 ¯ ¯ − γ ¯ arg ¯< δ 0 ¯ ¯ 2 g (z)

(0 ≤ γ < 1; 0 < δ ≤ 1),

1 for some g ∈ S0,1 (η, A, B), then ¯ Ã !¯ ¯ ¯ π 0 zf (z) ¯ ¯ − γ ¯ < α, ¯ arg ¯ ¯ 2 g(z)

where α(0 < α ≤ 1) is the solution of the equation given by (2.13). Remark 2.14 If we put A = 1, B = −1 and δ = 1 in Corollary 2.13, then we see that every quasi-convex function of order γ and type η is close-to-convex function of order γ and type η, which reduces the result obtained by Noor [12]. Theorem 2.15 Let f ∈ A and 0 < δ ≤ 1, 0 ≤ γ < 1. If ¯ !¯ Ã ¯ π ¯ n z(Db,λ f (z))0 ¯ ¯ −γ ¯< δ ¯ arg n ¯ 2 ¯ Db,λ g(z) n for some g ∈ Sb.λ (η, A, B), then ¯ !¯ Ã ¯ ¯ π n 0 z(D F (z)) ¯ ¯ b,λ − γ arg ¯ ¯ < α, n ¯ ¯ 2 Db,λ G(z)

where F is integral operator defined by (1.10) and α(0 < α ≤ 1) is the solution of the equation (2.13). Proof. Let 1 p(z) = 1−γ

Ã

! n z(Db,λ F (z))0 −γ . n Db,λ G(z)

n n Since g ∈ Sb.λ (η, A, B), we have from Theorem 2.8 that G ∈ Sb.λ (η, A, B). Using (2.11) we have n n n F (z) f (z) − cDb,λ g(z) = (c + 1)Db,λ [(1 − γ)p(z) + γ]Db,λ

Then, by a simple calculation, we get (1 − γ)zp0 (z) + [(1 − γ)p(z) + γ][(1 − η)q(z) + c + η] = (c + 1)

n z(Db,λ f (z))0 , n Db,λ G(z)

Multiplier transformation defined by convolution

where 1 q(z) = 1−γ Hence we have 1 1−γ

Ã

Ã

n f (z))0 z(Db,λ −η n g(z) Db,λ

1835

! n z(Db,λ G(z))0 −γ . n Db,λ G(z)

! = p(z) +

zp0 (z) . (1 − η)q(z) + η + b

The remaining part of the proof in Theorem 2.15 is similar to that of Theorem 2.10 and so we omit it. From Theorem 2.15, we see easily the following: Corollary 2.16 If f ∈ Kn b, λ(γ, δ, η, A, B), then F ∈ Kn b, λ(γ, δ, η, A, B), where F is the integral operator defined by (2.10). Remark 2.17 If we take n = b = 0, λ = 1 and n = b = λ = 0 with δ = 1, A = 1 and B = −1 in Corollary 2.16, respectively, then we have the corresponding results obtained by Noor and Alkhorasani [11]. Furthermore, taking n = b = λ = γ = 0, A = 1, B = −1 and δ = 1 in Corollary 2.16, we obtain the classical result by Bernardi [20], which implies the result studied by Libera [19]. AcknowledgementThe work presented here was supported by Fundamental Research Grant Scheme: UKM-ST-01-FRGS0055-2006, Malaysia.

References [1] B. A. Uralegaddi and C. Somanatha, Certain classes of univalent functions, In Current Topics in Analytic Function Theory, (Edited by H .M. Srivastava and S. Owa), pp. 371-374, World Scientific, Singapore, (1992). [2] Ch. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, Gttingen. (1975). P∞ 2kπix /(w + k)s , Acta [3] E. Lerch, Note sur la fonction R(w, x, s) = 0 e Math. (Stockholm) 11, (1887), 19-24. [4] G. S¸. S˘al˘agean, Subclasses of univalent functions, Lecture Note in Math.(Springer-Verlag), 1013, (1983), 362-372. [5] H. Bateman, Higher transcendental functions, Vol I (A. Erdelyi, W. Mangnus, F. Oberhettinger and F. G. Tricomi, eds.) Mc Graw-Hill, New York, (1953).

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