Certain Subclasses of Multivalent Functions ... - Semantic Scholar

Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 23, 1113 - 1123

Certain Subclasses of Multivalent Functions Associated with Fractional Calculus Operator N. Magesh Research and P.G. Studies in Mathematics Government Arts College (Men) Krishnagiri - 635001, Tamilnadu, India nmagi [email protected] S. Mayilvaganan and L. Mohanapriya Department of Mathematics Adhiyamaan College of Engineering (Autonomous) Hosur - 635109, Tamilnadu, India vaganan [email protected]

Abstract. By means of the generalized fractional calculus operator, we introduce and investigate a subclass of p−valent functions with negative coefficients. We obtain the coefficient estimates, distortion bounds for the class. Furthermore, we discuss inclusion relationship involving the neighborhood result of the same. Mathematics Subject Classification: 30C45 Keywords: Analytic functions, p-valent functions, Saigo hypergeometric fractional derivative operator, Saigo hypergeometric fractional integral operator, neighborhood properties, (n, δ)−neighborhood

1. Introduction Let Ap (n) denote the class of functions of the form (1.1.1)

p

f (z) = z +

∞ 

ak z k (p, n ∈ N = {1, 2, 3, . . . }),

k=n+p

which are analytic and p−valent in the open unit disk U = {z : z ∈ C and |z| < 1}.

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N. Magesh, S. Mayilvaganan and L. Mohanapriya

Let Tp (n) be the subclass of Ap (n), consisting of functions of the form ∞ 

f (z) = z p −

(1.1.2)

ak z k (p, n ∈ N; ak ≥ 0)

k=p+n

which are p−valent in U. The Hadamard product of two power series ∞ ∞   p k p ak z and g(z) = z + bk z k f (z) = z + k=n+p

k=n+p

is defined by (f ∗ g)(z) = z p +

(1.1.3)

∞ 

ak bk z k .

k=n+p

Let φp (a, c; z) be the incomplete beta function defined by (1.1.4)

∞  (a)k−p k z , a ∈ R, c ∈ R \ Z− φp (a, c; z) = z + 0 = {. . . , −2, −1, 0}, z ∈ U, (c)k−p p

k=p+n

where (a)k is the Pochhammer symbol defined by  Γ(a + k) 1, k=0 = (a)k = a(a + 1)(a + 2) . . . (a + k − 1), k ∈ N . Γ(a) Further, for f ∈ Ap (n) Lp (a, c)f (z) = φp (a, c; z) ∗ f (z), ∞  (a)k−p p = z + ak z k , z ∈ U, (c)k−p k=p+n where Lp (a, c) leads to Saitoh operator [22] for n = 1 and yields Carlson Shaffer operator [4] for n = 1 and p = 1. Definition 1.1. For real numbers μ > 0, γ and η, Saigo hypergeometric fracμ,γ,η is defined by tional integral operator I0,z (1.1.5)

z −μ−γ μ,γ,η f (z) = I0,z Γ(μ)

z (z − t)

μ−1

0

  t f (t)dt, 2 F1 μ + γ, −η; μ; 1 − z

where the function f (z) is analytic in a simply-connected region of the complex z−plane containing the origin, with the order f (z) = O(|z|ξ ) (z → 0; ξ > max{0, γ − η} − 1) and the multiplicity of (z − t)μ−1 is removed by requiring log(z − t) to be real when (z − t) > 0.

Subclasses of multivalent functions

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Definition 1.2. Under the hypotheses of Definition 1.1, Saigo hypergeometric μ,γ,η fractional derivative operator J0,z is defined by (1.1.6) μ,γ,η f (z) = J0,z

⎧ ⎨ dzd



z

z μ−γ (z−t)−μ 2 F1 (γ−μ,1−η;1−μ;1− zt )f (t)dt 0

⎩ dk

J μ−k,γ,η f (z), dz k 0,z

Γ(1−μ)

, 0 ≤ μ < 1, k ≤ μ < k + 1; k ∈ N,

where the multiplicity of (z − t)−μ is removed as in Definition 1.1. It may be remarked that μ,−μ,η f (z) = Dz−μ f (z) (μ > 0) I0,z

and μ,μ,η f (z) = Dzμ f (z) (0 ≤ μ < 1), J0,z

where Dz−μ denotes fractional integral operator and Dzμ denotes fractional derivative operator considered by Owa [15] and subsequently by Owa and Srivastava [16]. μ,γ,η : Ap (n) → Ap (n), be a generalized fractional differintegral operaLet U0,z tor defined by (1.1.7)

Γ(1+p−γ)Γ(1+p+η−μ) μ,γ,η z γ J0,z f (z), 0 ≤ μ < η + p + 1, z ∈ U, μ,γ,η Γ(1+p)Γ(1+p+η−γ) U0,z f (z) = Γ(1+p−γ)Γ(1+p+η−μ) γ −μ,γ,η z I0,z f (z), −∞ < μ < 0, z ∈ U. Γ(1+p)Γ(1+p+η−γ) For n = 1 the generalized fractional differintegral operator was considered by Goyal and prajapat [7] (see also [18]). It is easily seen from (1.1.7) that for a function f of the form (1.1.1), we have ∞  (1 + p)k (1 + p + η − γ)k μ,γ,η U0,z f (z) = z p + ak z k (1 + p − γ) (1 + p + η − μ) k k k=n+p (1.1.8)

= z p 3 F1 (1, 1 + p, 1 + p + η − γ; 1 + p − γ, 1 + p + η − μ; z) ∗ f (z) (z ∈ U; p ∈ N; γ, η ∈ R; γ < p + 1; −∞ < μ < η + p + 1),

where ∗ is given by (1.1.3) and p Fq is well known generalized hypergeometric function. Note that zf  (z) zf  (z) + z 2 f  (z) 0,0,0 1,1,1 1,0,0 2,1,1 and U0,z f (z) = f (z), U0,z f (z) = U0,z f (z) = f (z) = . U0,z p p2 We also remark that μ,μ,η μ,γ,0 f (z) = U0,z f (z) = Ωμ,p U0,z z f (z),

is an extended fractional differintegral operator studied very rewhere Ωμ,p z cently by [17](see also [19]). On the other hand, if we set μ = −δ, γ = 0 and η = ν − 1,

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N. Magesh, S. Mayilvaganan and L. Mohanapriya

in (1.1.8) and using δ,0,ν−1 I0,z f (z)

1 = ν z Γ(δ)

z ν−1

t 0

 δ−1 t 1− f (t)dt, z

we obtain following p−valent generalization of multiplier transformation operator [9, 10]: Qδν,p f (z) (1.1.9)

 z   δ−1 t p+δ+ν −1 δ ν−1 = t f (t)dt 1− p+ν−1 zν z 0

∞  Γ(ν + k)Γ(p + δ + ν) p ak z k (ν > −p, δ + ν > −p) = z + Γ(δ + ν + k)Γ(p + ν) k=n+p

on the other hand, if we set μ = −1, γ = 0 and η = ν − 1, in (1.1.8), we obtain the generalized Berdardi-Libera integral operator Fν,p : Ap (n) → Ap (n) (ν > −p) defined by (1.1.10)

p+ν Fν,p f (z) = zν

z tν−1 f (t)dt 0

∞  p+ν = zp + ak z k (ν > −p). ν + k k=n+p

For the choice p = 1, n = 1 and ν ∈ N, the operator defined by (1.1.10) reduces to the well-known Bernardi integral operator. Also the operators Qδν,p and Fν,p were studied recently in [12]. Further, For f ∈ Ap (n) μ,γ,η μ,γ,η M0,z f (z) = φp (a, c, z) ∗ U0,z f (z) ∞  (a)k−p (1 + p)k−p (1 + p + η − γ)k−p p = z + ak z k (c)k−p (1 + p − γ)k−p (1 + p + η − μ)k−p k=p+n

(1.1.11)

p

= z +

∞ 

Φ(k)ak z k (z ∈ U),

k=p+n

where, (1.1.12)

Φ(k) =

(a)k−p (1 + p)k−p (1 + p + η − γ)k−p . (c)k−p (1 + p − γ)k−p (1 + p + η − μ)k−p

μ,γ,η The operator M0,z f (z) was studied by Kahairnar and More [8].

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Definition 1.3. A function f ∈ Tp (n) is said to be in the class S(b, λ, β; p) if it satisfies the following inequality 

   μ,γ,η μ,γ,η 1  z(M0,z f (z)) + λz 2 (M0,z f (z))   (1.1.13)   < β (z ∈ U), μ,γ,η μ,γ,η  − p  b (1 − λ)M0,z f (z) + λz(M0,z f (z))  where b ∈ C∗ = C\{0}, p ∈ N, 0 ≤ λ ≤ 1, 0 < β ≤ 1. Our definition of the function class S(b, λ, β; p) is motivated essentially by the earlier investigations of Aouf [3], Darwich and Aouf [5], Mahzoon and Latha [11], Orhan and Kamali [14], Raina and Srivastava [20], in each of which further details and references to other closely related subfamilies of the normalized p-valently analytic function class Tp (n) can be found. The object of the present paper is to investigate the various properties and characteristics of analytic p-valent functions belonging to the subclass S(b, λ, β; p) which we have defined here. Apart from deriving coefficient bounds for the function class, we establish distortion bounds, inclusion relationship involving the (n, δ)−neighborhoods of analytic p−valent functions belonging to this subclass. 2. Coefficient Estimates In this section, we determine the coefficient inequality for functions to be in the subclass S(b, λ, β; p). Theorem 2.1. A function f (z) defined by (1.1.2) is in S(b, λ, β; p) if and only if (2.2.1)

∞ 

[1 + λ(k − 1)](k − p + β|b|)Φ(k)ak ≤ β|b|[1 + λ(p − 1)],

k=p+n

where 0 ≤ λ ≤ 1, 0 < β ≤ 1 and Φ(k) is given by (1.1.12). Proof. Let f (z) ∈ S(b, λ, β; p). Assume that inequality (2.2.1) holds true. Then we find that    z(M μ,γ,η f (z)) + λz 2 (M μ,γ,η f (z))    0,z 0,z (2.2.2)  μ,γ,η μ,γ,η  − p  (1 − λ)M0,z f (z) + λz(M0,z f (z))     ∞ k     k=p+n [1 + λ(k − 1)](k − p)Φ(k)ak z ∞ = p  k  z [1 + λ(p − 1)] + k=p+n [1 + λ(k − 1)]Φ(k)ak z  ∞ k−p k=p+n [1 + λ(k − 1)](k − p)Φ(k)ak |z|  ≤ k−p [1 + λ(p − 1)] + ∞ k=p+n [1 + λ(k − 1)]Φ(k)ak |z| < β|b|.

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Choosing values of z on real axis and letting z → 1− , we have ∞  [1 + λ(k − 1)](k − p + β|b|)Φ(k)ak ≤ β|b|[1 + λ(p − 1)]. k=p+n

Conversely, assume that f (z) ∈ S(b, λ, β; p), then in the view of (1.1.11) and (1.1.13), we get the following inequality

   μ,γ,η μ,γ,η z(M0,z f (z)) + λz 2 (M0,z f (z)) Re − p > −β|b| μ,γ,η μ,γ,η (1 − λ)M0,z f (z) + λz(M0,z f (z)) 

 k pz p [1 + λ(p − 1)] + ∞ k[1 + λ(k − 1)]Φ(k)a z k k=p+n − p + β|b| > 0 Re k z p [1 + λ(p − 1)] + ∞ k=p+n [1 + λ(k − 1)]Φ(k)ak z

  k β|b|[1 + λ(p − 1)]z p + ∞ [1 + λ(k − 1)](k − p + β|b|)Φ(k)a z k k=p+n  Re > 0. k [1 + λ(p − 1)]z p + ∞ k=p+n [1 + λ(k − 1)]Φ(k)ak z Since Re(−eiθ ) ≥ −|eiθ | = −1, the above inequality reduces to  k β|b|[1 + λ(p − 1)]r p − ∞ k=p+n [1 + λ(k − 1)](k − p + β|b|)Φ(k)ak r  > 0. k [1 + λ(p − 1)]r p − ∞ k=p+n [1 + λ(k − 1)]Φ(k)ak r Letting r → 1− and by the mean value theorem we have desired inequality (2.2.1). This completes the proof of the Theorem 2.1. The result is sharp for the function f (z) given by f (z) = z p +

β|b|[1 + λ(p − 1)] zk . [1 + λ(k − 1)](k − p + β|b|)Φ(k)

Corollary 2.1. Let the function f defined by (1.1.2) be in the class S(b, λ, β; p). Then β|b|[1 + λ(p − 1)] ak ≤ (k ≥ p + n, n ∈ N), [1 + λ(k − 1)](k − p + β|b|)Φ(k) where Φ(k)is given by (1.1.12). 3. Distortion Bounds In this section, we obtain distortion bounds for functions in the class S(b, λ, β; p). Theorem 3.1. Let the function f defined by (1.1.2)be in the class S(b, λ, β; p). Then for |z| = r we have rp −

β|b|[1 + λ(p − 1)] r p+n ≤ |f (z)| [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

≤ rp +

β|b|[1 + λ(p − 1)] r p+n [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

Subclasses of multivalent functions

1119

and the result are sharp for f given by f (z) = z p +

β|b|[1 + λ(p − 1)] z p+n . [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

Proof. Given that f (z) ∈ S(b, λ, β; p), from the equation (2.2.1), we have [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

∞ 

ak

k=p+n

≤

∞ 

[1 + λ(k − 1)](k − p + β|b|)Φ(k)ak ≤ β|b|[1 + λ(p − 1)],

k=p+n

which is equivalent to, ∞ 

(3.3.1)

ak ≤

k=p+n

β|b|[1 + λ(p − 1)] . [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

Using (1.1.2) and (3.3.1), we obtain ∞ 

p

f (z) = z +

ak z k

k=p+n ∞ 

|f (z)| ≤ |z|p + ≤ rp +

ak |z|k

k=p+n ∞ 

ak r k

k=p+n

≤ r p + r p+n

∞ 

ak

k=p+n

≤ rp +

β|b|[1 + λ(p − 1)] r p+n . [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

Similarly, f (z) ≥ r p − r p+n

β|b|[1 + λ(p − 1)] . [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

This completes the proof of Theorem 3.1. Theorem 3.2. Let the function f defined by (1.1.2)be in the class S(b, λ, β; p). Then for |z| = r we have (p + n)β|b|[1 + λ(p − 1)] r p+n−1 [1 + λ(p + n − 1)][n + β|b|]Φ(p + n) (p + n)β|b|[1 + λ(p − 1)]  ≤ |f (z)| ≤ pr p−1 + r p+n−1 [1 + λ(p + n − 1)][n + β|b|]Φ(p + n) pr p−1 −

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N. Magesh, S. Mayilvaganan and L. Mohanapriya

and the result are sharp for f given by β|b|[1 + λ(p − 1)] f (z) = z p + z p+n . [1 + λ(p + n − 1)][n + β|b|]Φ(p + n) Proof. From (1.1.2) and (3.3.1) ∞   p−1 + kak z k−1 f (z) = pz k=p+n ∞ 



|f (z)| ≤ p|z|p−1 + ≤ pr p−1 +

kak |z|k−1

k=p+n ∞ 

kak r k−1

k=p+n

≤ pr p−1 +

(p + n)β|b|[1 + λ(p − 1)] r p+n−1 . [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

Similarly, (p + n)β|b|[1 + λ(p − 1)] r p+n−1 . [1 + λ(p + n − 1)][n + β|b|]Φ(p + n) This completes the proof of Theorem 3.2. 

f (z) ≥ pr p−1 −

4. Inclusion and Neighborhood Results In this section, we prove certain inclusion relationship for functions belonging to the class S(b, λ, β; p) and also, we determine the neighborhood properties of functions belonging to the subclass S α (b, λ, β; p). Following the works of Goodman [6], Ruschewehy [21] and Altintas et al. [1, 2] (see also [13, 14, 20]) we defined the (n, δ)- neighborhood of a function f ∈ Tp (n) by (4.4.1) Nn,δ (f ) =

g ∈ Tp (n) : g(z) = z p −

∞ 

bk z k and

k=p+n

In particular, for the function e(z) = z (4.4.2) Nn,δ (e) =

g ∈ Tp (n), g(z) = z p −

p

∞ 

 k|ak − bk | ≤ δ

.

k=p+n

(p ∈ N)

∞  k=p+n

bk z k and

∞ 

 k|bk | ≤ δ

.

k=p+n

A function f ∈ Tp (n) defined by (1.1.2) is said to be in the class S α (b, λ, β; p) if there exists a function h ∈ S(b, λ, β; p), such that     f (z)  < p − α (z ∈ U, 0 ≤ α < p).  − 1 (4.4.3)   h(z)

Subclasses of multivalent functions

Theorem 4.1. If Φ(k) ≥ Φ(p + n) for k ≥ p + n, (4.4.4)

δ=

1121

n ∈ N and

(p + n)β|b|[1 + λ(p − 1)] [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

then S(b, λ, β; p) ⊆ Nn,δ (e).

(4.4.5)

Proof. Let f ∈ S(b, λ, β; p). Then in view of assertion (2.2.1) of Theorem 2.1 and the condition Φ(k) ≥ Φ(p + n) for k ≥ p + n, n ∈ N, we get [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

∞ 

ak ≤

k=p+n

∞ 

[1 + λ(k − 1)](k − p + β|b|)Φ(k)ak

k=p+n

≤ β|b|[1 + λ(p − 1)],

(4.4.6) which implies (4.4.7)

∞ 

ak ≤

k=p+n

β|b|[1 + λ(p − 1)] . [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

Applying assertion (2.2.1) of Theorem (2.1) in conjunction with (4.4.7), we obtain ∞  ak ≤ β|b|[1 + λ(p − 1)] [1 + λ(p + n − 1)][n + β|b|]Φ(p + n) (p + n)[1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

k=p+n ∞ 

ak ≤ (p + n)β|b|[1 + λ(p − 1)]

k=p+n ∞  k=p+n

kak ≤

(p + n)β|b|[1 + λ(p − 1)] =δ [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

which by virtue of (4.4.1) establishes the inclusion relation(4.4.5). Theorem 4.2. If h ∈ S(b, λ, β; p) and (4.4.8)

  δ [1 + λ(p + n − 1)][n + β|b|]Φ(p + n) α=p− , (p + n) [1 + λ(p + n − 1)][n + β|b|]Φ(p + n) − β|b|[1 + λ(p − 1)]

then Nn,δ (h) ⊂ S α (b, λ, β; p). Proof. Suppose that f ∈ Nn,δ (h), we then find from (4.4.1) that ∞  k=p+n

k|ak − bk | ≤ δ,

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N. Magesh, S. Mayilvaganan and L. Mohanapriya

which readily implies the following co-efficient inequality ∞ 

(4.4.9)

|ak − bk | ≤

k=p+n

δ p+n

(n ∈ N).

Next, since h ∈ S(b, λ, β; p) in the view of (4.4.7), we have ∞ 

(4.4.10)

k=p+n

bk ≤

β|b|[1 + λ(p − 1)] . [1 + λ(p + n − 1)][n + β|b|]Φ(p + n)

Using (4.4.9)and (4.4.10), we get ∞     f (z) k=p+n |ak − bk |  ≤   − 1   k(z) 1− ∞ k=p+n bk δ   β|b|[1+λ(p−1)] (p + n) 1 − [1+λ(p+n−1)][n+β|b|]Φ(p+n)   δ [1 + λ(p + n − 1)][n + β|b|]Φ(p + n) ≤ (p + n) [1 + λ(p + n − 1)][n + β|b|]Φ(p + n) − β|b|[1 + λ(p − 1)] (4.4.11)= p − α, ≤

provided that α is given by (4.4.8), thus by condition (4.4.3), f ∈ S α (b, λ, β; p), where α is given by (4.4.8). References ¨ Ozkan ¨ [1] O. Altinta¸s, O. and H. M. Srivastava, Neighborhoods of a class of analytic functions with negative coefficients, Appl. Math. Lett. 13 (2000), no. 3, 63–67. ¨ Ozkan ¨ [2] O. Altinta¸s, O. and H. M. Srivastava, Neighborhoods of a certain family of multivalent functions with negative coefficients, Comput. Math. Appl. 47 (2004), no. 10-11, 1667–1672. [3] M. K. Aouf, On certain subclasses of multivalent functions with negative coefficients defined by using a differential operator, Bull. Inst. Math. Acad. Sin. (N.S.) 5 (2010), no. 2, 181–200. [4] B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984), no. 4, 737–745. [5] H. E. Darwish and M. K. Aouf, Neighborhoods of certain subclasses of analytic functions with negative coefficients, Bull. Korean Math. Soc. 48 (2011), no. 4, 689–695. [6] A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc. 8 (1957), 598–601. [7] S. P. Goyal and J. K. Prajapat, A new class of analytic p-valent functions with negative coefficients and fractional calculus operators, Tamsui Oxf. J. Math. Sci. 20 (2004), no. 2, 175–186. [8] S. M. Khairnar and M. More, On a subclass of multivalent β-uniformly starlike and convex functions defined by a linear operator, IAENG Int. J. Appl. Math. 39 (2009), no. 3, 175–183. [9] I. B. Jung, Y. C. Kim and H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl. 176 (1993), no. 1, 138–147.

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Received: November, 2011