Inclusion relations for subclasses of multivalent functions defined by

Afr. Mat. https://doi.org/10.1007/s13370-018-0567-3

Inclusion relations for subclasses of multivalent functions defined by Srivastava–Saigo–Owa fractional differintegral operator A. O. Mostafa1 · M. K. Aouf1 · H. M. Zayed2

Received: 23 December 2016 / Accepted: 6 February 2018 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Abstract The purpose of this paper is to introduce subclasses of multivalent functions by using Srivastava–Saigo–Owa fractional differintegral operator and investigate various properties for these subclasses. Also, we investigate inclusion relations involving the operator F p,c . Keywords Starlike · Convex functions · Hadamard product (or convolution) · Generalized fractional derivative operator · Generalized fractional integral operator Mathematics Subject Classification 30C45 · 30C50

1 Introduction Let A( p) denote the class of functions f (z) of the form: f (z) = z p +

∞ 

ak z k ( p ∈ N = {1, 2, . . .}),

(1)

k= p+1

which are analytic and multivalent in the open unit disc U = {z : z ∈ C and |z| < 1}. Also, Let g(z) ∈ A( p), be given by

B

H. M. Zayed [email protected] A. O. Mostafa [email protected] M. K. Aouf [email protected]

1

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

2

Department of Mathematics, Faculty of Science, Menofia University, Shebin El Kom 32511, Egypt

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g(z) = z p +

∞ 

gk z k ( p ∈ N).

(2)

k= p+1

The Hadamard product (or convolution) of f (z) and g(z) is given by ( f ∗ g)(z) = z p +

∞ 

ak gk z k = (g ∗ f )(z).

(3)

k= p+1

Let Pk ( p, ρ) be the class of functions h(z) analytic in U and satisfying the properties h(0) = p and  2π    {h(z)} − ρ   dθ ≤ kπ,  (4)   p−ρ 0

where k ≥ 2 and 0 ≤ ρ < 1. This class was introduced by Aouf [2]. We note that, for p = 1, we obtain the class Pk (ρ) introduced by Padmanabhan and Parvatham [15] and for p = 1, ρ = 0, we have the well-known class Pk (1, 0) = Pk introduced by Pinchuk [16]. The case k = 2 gives the class P2 (ρ) = P (ρ) of functions with positive real part greater than ρ, and P2 (0) = P , gives the class P2 (0) = P of functions with positive real part. From (4), we have h(z) ∈ Pk (ρ) if and only if there exist p1 , p2 ∈ P (ρ) such that     k 1 1 k h(z) = + h 1 (z) − − h 2 (z) (z ∈ U). (5) 4 2 4 2 It is known that the class Pk (ρ) is a convex set (see [12]). We recall here the following generalized fractional integral and generalized fractional derivative operators due to Srivastava et al. [23] (see also [18]). Definition 1.1 For f (z) ∈ A( p), λ > 0, μ, η ∈ R and in terms of the function 2 F1 , the generalized fractional integral and derivative operators are defined by    ζ z −λ−μ z λ,μ,η I0,z f (z) = dζ, (6) (z − ζ )λ−1 f (ζ )2 F1 μ + λ, −η; λ; 1 − (λ) 0 z and λ,μ,η

J0,z

f (z) =

⎧ ⎪ ⎨ ⎪ ⎩

 d dz

z λ−μ

z

−λ 0 (z−ζ )

d j λ− j,μ,η J dz j 0,z

  f (ζ )2 F1 μ−λ,1−η;1−λ;1− ζz dζ (1−λ)

(0 ≤ λ < 1),

(7)

f (z) ( j ≤ λ < j + 1; j ∈ N),

where f (z) is an analytic function in a simply-connected region of the complex z-plane containing the origin with the order f (z) = O(|z|ε ), z → 0 when ε > max{0, μ − η} − 1 and the multiplicity of (z−ζ )λ−1 is removed by requiring log(z−ζ ) to be real when z−ζ > 0. We note that: λ,−λ,η

I0,z

λ,λ,η

f (z) = Dz−λ f (z) (λ > 0) and J0,z

f (z) = Dzλ f (z) (0 ≤ λ < 1).

where Dz−λ f (z) denotes fractional integral operator and Dzλ f (z) denotes fractional derivative operator studied by Owa [13]. Goyal and Prajapat [9] (see also [17]) defined the operator ⎧ λ,μ,η ⎨ ( p+1−μ)( p+1−λ+η) z μ J0,z f (z) (0 ≤ λ < η + p + 1; z ∈ U), ( p+1)( p+1−μ+η) λ,μ,η, p (8) S0,z f (z) = ⎩ ( p+1−μ)( p+1−λ+η) z μ I −λ,μ,η f (z) (−∞ < λ < 0; z ∈ U). 0,z ( p+1)( p+1−μ+η)

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Inclusion relations for subclasses…

For a function f (z) ∈ A( p), of the form (1), we have λ,μ,η, p

S0,z

f (z) = z p 3 F2 (1, 1 + p, 1 + p + η − μ; 1 + p − μ, 1 + p + η − λ; z) ∗ f (z) = zp +

∞  n=1

( p + 1)n ( p + 1 − μ + η)n a p+n z p+n ( p + 1 − μ)n ( p + 1 − λ + η)n

( p ∈ N; μ, η ∈ R; μ < p + 1; −∞ < λ < η + p + 1),

(9)

where q Fs (q ≤ s + 1; q, s ∈ N0 , N ∪ {0}) is the well known generalized hypergeometric function (for more details, see [14] and [22]) and (υ)n is the Pochhammer symbol defined by  1 if n = 0, (υ)n = υ(υ + 1)(υ + 2) . . . (υ + n − 1) if n ∈ N. Let G λp,η,μ (z) = z p +

∞  n=1

( p + 1)n ( p + 1 − μ + η)n z p+n ( p + 1 − μ)n ( p + 1 − λ + η)n

( p ∈ N; μ, η ∈ R; μ < p + 1; −∞ < λ < η + p + 1).

(10)

We define a new function [G λp,η,μ ]−1 by means of the convolution  −1 zp G λp,η,μ (z) ∗ G λp,η,μ (z) = (δ > − p; z ∈ U). (1 − z)δ+ p λ,δ : A( p) → A( p) Tang et al. [24] (see also Aouf et al. [6] and [7]) defined the operator H p,η,μ by  −1 λ,δ H p,η,μ f (z) = G λp,η,μ (z) ∗ f (z).

For a function f (z) ∈ A( p), we have λ,δ H p,η,μ f (z) = z p +

∞  (δ + p)n ( p + 1 − μ)n ( p + 1 − λ + η)n a p+n z p+n . (1)n ( p + 1)n ( p + 1 − μ + η)n

(11)

n=1

m,δ,ζ

For m ∈ N0 , 0 < ζ ≤ 1, Aouf et al. [5] defined the operator N p,λ,μ,η : A( p) → A( p) as follows: λ,δ N p,λ,μ,η f (z) = H p,η,μ f (z), 0,δ,ζ

δ,ζ

1,δ,ζ

N p,λ,μ,η f (z) = N p,λ,μ,η f (z) λ,δ f (z) + ζ = (1 − ζ )H p,η,μ



z  λ,δ H p,η,μ f (z) p

 ∞   p + ζ n (δ + p)n ( p + 1 − μ)n ( p + 1 − λ + η)n =z + a p+n z p+n , p (1)n ( p + 1)n ( p + 1 − μ + η)n p

n=1

and (in general)

δ,ζ

m,δ,ζ

m−1,δ,ζ



N p,λ,μ,η f (z) = N p,λ,μ,η N p,λ,μ,η f (z)

 ∞   p + ζ n m (δ + p)n ( p + 1 − μ)n ( p + 1 − λ + η)n a p+n z p+n . =z + p (1)n ( p + 1)n ( p + 1 − μ + η)n p

n=1

(12)

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We note that: λ,δ f (z) (see [24]); (i) N p,λ,μ,η f (z) = H p,η,μ 0,δ,ζ

m,1,ζ

m f (z) (see [3]); (ii) N p, p, p,0 f (z) = Dζ, p

m,1,1 m (iii) N p, p, p,0 f (z) = D p f (z) (see [4] and [10]); m,1,ζ

(iv) N1,1,1,0 f (z) = Dζm f (z) (see [1]);

m,1,1 (v) N1,1,1,0 f (z) = D m f (z) (see [21]).

Also, we note that: m,1,ζ

m,ζ

(i) N p,λ,μ,η f (z) = N p,λ,μ,η f (z) = z p + a p+n z p+n ; m,δ,ζ

m,δ,ζ

m,1,ζ

m,δ,ζ

(ii) N p,λ,λ,η f (z) = N p,λ

(iii) N p,μ,μ,η f (z) = N p,μ

m ( p + 1 − μ) ( p + 1 − λ + η) ∞  n n p+ζ n p (1) ( p + 1 − μ + η) n n n=1

m (δ + p) ( p + 1 − λ) ∞  n n p+ζ n a p+n z p+n ; p (1) ( p + 1) n n n=1 m (δ + p) ( p + 1 − μ) ∞  n n p+ζ n f (z) = z p + a p+n z p+n . p (1) ( p + 1) n n n=1

f (z) = z p +

It is easy to verify from (12) that 

m,δ,ζ m+1,δ,ζ m,δ,ζ ζ z N p,λ,μ,η f (z) = p N p,λ,μ,η f (z) − p(1 − ζ )N p,λ,μ,η f (z) (ζ > 0), 

m,δ,ζ m,δ,ζ m,δ,ζ z N p,λ+1,μ,η f (z) = ( p + η − λ)N p,λ,μ,η f (z) − (η − λ) N p,λ+1,μ,η f (z), and



m,δ,ζ m,δ+1,ζ m,δ,ζ z N p,λ,μ,η f (z) = (δ + p)N p,λ,μ,η f (z) − δ N p,λ,μ,η f (z).

(13) (14)

(15) m,δ,ζ

We now define the following subclasses of the class A( p) by means of the operator N p,λ,μ,η as follows: p

Definition 1.2 Let f (z) ∈ A( p), z ∈ U. Then f (z) ∈ Rk (m, δ, ζ, λ, μ, η; ρ) if and only if



m,δ,ζ z N p,λ,μ,η f (z) (16) ∈ Pk ( p, ρ) . m,δ,ζ N p,λ,μ,η f (z) p

Also, f (z) ∈ Vk (m, δ, ζ, λ, μ, η; ρ) if and only if z f (z) p ∈ Rk (m, δ, ζ, λ, μ, η; ρ) . p

(17)

2 Main results Unless otherwise mentioned, we assume throughout this paper that p ∈ N, μ, η ∈ R, μ < p + 1, −∞ < λ < η + p + 1, m ∈ N0 , 0 < ζ ≤ 1, k ≥ 2 and 0 ≤ ρ < 1. To establish our results, we need the following lemma due to Miller and Mocanu [11]. Lemma 2.1 [11]. Let φ(u, v) be a complex valued function φ : D → C, D ⊂ C2 and let u = u 1 + iu 2 , v = v1 + iv2 . Suppose that the function φ(u, v) satisfies (i) φ(u, v) is continuous in D;

123

Inclusion relations for subclasses…

(ii) (1, 0) ∈ D and  {φ(1, 0)} > 0; (iii) for all (iu 2 , v1 ) ∈ D such that v1 ≤ − 21 (1 + u 22 ),  {φ(iu 2 , v1 )} ≤ 0. 2

Let p(z)  = 1 + p1 z + p2 z + · · · be regular in U such that ( p(z), zp (z)) ∈ D for all z ∈ U. If  φ( p(z), zp (z)) > 0 for all z ∈ U, then  { p(z)} > 0.

Lemma 2.2 [20]. Let p be analytic function in U with p(0) = 1 and  { p(z)} > 0 with z ∈ U. Then for s > 0 and μ = −1 (complex number),   szp (z)  p(z) + > 0 (|z| < r0 ) , p(z) + μ where r0 = 

  |μ + 1| , A = 2(s + 1)2 + μ2  − 1,    A + A2 − μ2 − 1

and this result is the best possible. Lemma 2.3 [19]. Let ψ be convex and let g be functions in U. Then for F analytic in U with ψ ∗ Fg F(0) = 1, is contained in the convex hull of F(U). ψ∗g p

p

p

Theorem 2.1 Rk (m + 1, δ, ζ, λ, μ, η; ρ) ⊂ Rk (m, δ, ζ, λ, μ, η; ρ) ⊂ Rk (m, δ, ζ, λ +1, μ, η; ρ) Proof We first show that p

p

Rk (m + 1, δ, ζ, λ, μ, η; ρ) ⊂ Rk (m, δ, ζ, λ, μ, η; ρ) .

(18)

p Rk

Let f (z) ∈ (m + 1, δ, ζ, λ, μ, η; ρ) and let



m,δ,ζ z N p,λ,μ,η f (z) = H (z) = ( p − ρ)h(z) + ρ m,δ,ζ N p,λ,μ,η f (z)     = k4 + 21 [(1 − ρ)h 1 (z) + ρ] − k4 − 21 [(1 − ρ)h 2 (z) + ρ] , (19) where h i (z) is analytic in U with h i (0) = 1 for i = 1, 2. Using identity (13) in (19), we get m+1,δ,ζ

p N p,λ,μ,η f (z) p(1 − ζ ) + ( p − ρ)h(z) + ρ. = m,δ,ζ ζ N ζ p,λ,μ,η f (z)

(20)

Differentiating (20) logarithmically with respect to z, we obtain 

m+1,δ,ζ z N p,λ,μ,η f (z) ( p − ρ)zh (z) . = ( p − ρ)h(z) + ρ + m+1,δ,ζ p(1−ζ ) + ( p − ρ)h(z) + ρ N p,λ,μ,η f (z) ζ

(21)

p

Since f (z) ∈ Rk (m + 1, δ, ζ, λ, μ, η; ρ), then h i (z) +

p(1−ζ ) ζ

zh i (z) + ( p − ρ)h i (z) + ρ

∈ P ( p) (i = 1, 2).

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Defining the function (u, v) = u +

v p(1−ζ ) ζ

+ ( p − ρ)u + ρ

,

(22)

where u = h i (z) = u 1 + iu 2 , v = zh i (z) = v1 + iv2 , we have   p(1−ζ ) +ρ ζ (i) (u, v) is continuous in D = C − × C (0 < ζ ≤ 1); ρ−p (ii) (1, 0) ∈ D and  {(1, 0)} = 1 > 0; (iii) for all (iu 2 , v1 ) ∈ D such that v1 ≤ − 21 (1 + u 22 ),   v1  {(iu 2 , v1 )} =  iu 2 + p(1−ζ ) + ( p − ρ) (iu 2 ) + ρ ζ v1 = 2 p(1−ζ ) + ρ + ( p − ρ)2 u 22 ζ (1 + u 22 ) ≤− < 0. 2 ) 2u2 2 p(1−ζ + ρ + 2( p − ρ) 2 ζ Since (u, v) satisfies the hypotheses of Lemma 2.1, we obtain the inclusion relation (18). By using similar arguments as in the first inclusion together with (14) instead of (13), we obtain p p (23) Rk (m, δ, ζ, λ, μ, η; ρ) ⊂ Rk (m, δ, ζ, λ + 1, μ, η; ρ) . Combining the inclusion relationships (18) and (23), we complete the proof of Theorem 2.1. 

p

p

p

Theorem 2.2 Vk (m + 1, δ, ζ, λ, μ, η; ρ) ⊂ Vk (m, δ, ζ, λ, μ, η; ρ) ⊂ Vk (m, δ, ζ, λ +1, μ, η; ρ). Proof Applying (17) and Theorem 2.1, we obtain p

f (z) ∈ Vk (m + 1, δ, ζ, λ, μ, η; ρ)

z f (z) p ∈ Rk (m + 1, δ, ζ, λ, μ, η; ρ) p z f (z) p ⇒ ∈ Rk (m, δ, ζ, λ, μ, η; ρ) p p ⇔ f (z) ∈ Vk (m, δ, ζ, λ, μ, η; ρ) .

⇔

Thus

p

p

Vk (m + 1, δ, ζ, λ, μ, η; ρ) ⊂ Vk (m, δ, ζ, λ, μ, η; ρ) .

(24)

The right part of Theorem 2.2 can be proved by similar arguments. This completes the proof of Theorem 2.2. 

p

p

Theorem 2.3 If f (z) ∈ Rk (m, δ, ζ, λ, μ, η; ρ). Then f (z) ∈ Rk (m, δ + 1, ζ, λ, μ, η; ρ) for   |μ + 1| |z| < r0 =  (25) , A = 2(s + 1)2 + μ2  − 1,    2 2   A+ A − μ −1

123

Inclusion relations for subclasses…

with

1 δ+ρ , μ= . p−ρ p−ρ

s=

(26)

p

Proof Let f (z) ∈ Rk (m, δ, ζ, λ, μ, η; ρ) and let



m,δ,ζ z N p,λ,μ,η f (z) m,δ,ζ

N p,λ,μ,η f (z)

= ( p − ρ)g(z) + ρ =

k 4

+

1 2



[(1 − ρ)g1 (z) + ρ] −

k 4

−

1 2



[(1 − ρ)g2 (z) + ρ] ,(27)

where gi (z) is analytic in U with gi (0) = 1 for i = 1, 2. Using identity (15) in (27), we get m+1,δ,ζ

(δ + p)

N p,λ,μ,η f (z) m,δ,ζ

N p,λ,μ,η f (z)

= ( p − ρ)g(z) + δ + ρ.

(28)

Differentiating (28) logarithmically with respect to z, we obtain ⎡ ⎤ 

1 m+1,δ,ζ zg (z) z N f (z) p,λ,μ,η 1 ⎢ p−ρ ⎥ . − ρ ⎦ = g(z) + ⎣ m+1,δ,ζ δ+ρ p−ρ N p,λ,μ,η f (z) g(z) + p−ρ where  {gi (z)} > 0 (i = 1, 2). Applying Lemma 2.2 with s = ⎧ ⎪ ⎪ ⎨

1 zg (z) p−ρ i  gi (z) + δ+ρ ⎪ ⎪ ⎩ gi (z) + p−ρ

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

1 p−ρ ,

μ=

δ+ρ p−ρ ,

(29)

we get

> 0 (|z| < r0 ) ,

where r0 is given by (25). This completes the proof of Theorem 2.3.



p

Theorem 2.4 Let ψ be convex and let f (z) ∈ R2 (m, δ, ζ, λ, μ, η; ρ). Then ψ ∗ f ∈ p R2 (m, δ, ζ, λ, μ, η; ρ). Proof Let G = ψ ∗ f , then 

m,δ,ζ z N p,λ,μ,η G(z) m,δ,ζ

p N p,λ,μ,η G(z)

=

=



m,δ,ζ z N p,λ,μ,η (ψ ∗ f ) (z) m,δ,ζ

p N p,λ,μ,η (ψ ∗ f ) (z)



m,δ,ζ  z N p,λ,μ,η f (z) m,δ,ζ ψ∗ N p,λ,μ,η f (z) m,δ,ζ p N p,λ,μ,η f (z) m,δ,ζ

ψ ∗ N p,λ,μ,η f (z) 

m,δ,ζ ψ ∗ F(z) N p,λ,μ,η f (z) , = m,δ,ζ ψ ∗ N p,λ,μ,η f (z)

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where F(z) =



m,δ,ζ z N p,λ,μ,η f (z) m,δ,ζ

p N p,λ,μ,η f (z)

p

. Since f (z) ∈ R2 (m, δ, ζ, λ, μ, η; ρ), it follows that

N p,λ,μ,η f (z) ∈ S ∗ (ρ) ⊆ S ∗ . From Lemma 2.3, we see that m,δ,ζ



m,δ,ζ z N p,λ,μ,η f (z) m,δ,ζ

p N p,λ,μ,η f (z) p in the convex hull of F(U) and consequently ψ ∗ f ∈ R2 (m, δ, ζ, λ, μ, η; ρ).

is contained 

3 Inclusion relationships involving the operator F p,c Using the operator F p,c defined by Cho et al. [8] for a function f (z) of the form (1) by 



c+ p F p,c f (z) = zc

z t c−1 f (t)dt (c > − p),

(30)

0

we find that

 

 m,δ,ζ  m,δ,ζ m,δ,ζ  z N p,λ,μ,η F p,c f (z) = (c + p)N p,λ,μ,η f (z) − cN p,λ,μ,η F p,c f (z) (c > − p). (31) In this section, we derive inclusion relations involving the operator F p,c . p

Theorem 3.1 If f (z) ∈ Rk (m, δ, ζ, λ, μ, η; ρ), then for c > − p, p Rk (m, δ, ζ, λ, μ, η; ρ) .



 F p,c f (z) ∈

p

Proof Let f (z) ∈ Rk (m, δ, ζ, λ, μ, η; ρ) and let

 

m,δ,ζ  z N p,λ,μ,η F p,c f (z) = ( p − ρ)q(z) + ρ  m,δ,ζ  N p,λ,μ,η F p,c f (z)     = k4 + 21 [(1 − ρ)q1 (z) + ρ] − k4 − 21 [(1 − ρ)q2 (z) + ρ] , (32) where qi (z) is analytic in U with qi (0) = 1 for i = 1, 2. Using identity (31) in (32), we get m,δ,ζ

(c + p)

N p,λ,μ,η f (z) m,δ,ζ





N p,λ,μ,η F p,c f (z)

= c + (1 − ρ)q(z) + ρ.

Differentiating (33) logarithmically with respect to z, we obtain 

m,δ,ζ z N p,λ,μ,η f (z) ( p − ρ)zq (z) . = ( p − ρ)q(z) + ρ + m,δ,ζ c + ( p − ρ)q(z) + ρ N f (z) p,λ,μ,η

p

Since f (z) ∈ Rk (m, δ, ζ, λ, μ, η; ρ), then ( p − ρ)q(z) + ρ +

123

( p − ρ)zq (z) ∈ Pk ( p, ρ), c + ( p − ρ)q(z) + ρ

(33)

(34)

Inclusion relations for subclasses…

or, equivalently, qi (z) +

zqi (z) ∈ P ( p) (i = 1, 2). c + ( p − ρ)qi (z) + ρ

Defining the function (u, v) = u +

v , c + ( p − ρ)u + ρ

(35)

where u = m i (z) = u 1 + iu 2 , v = zm i (z) = v1 + iv2 , we have   c+ρ × C (c > − p); (i) (u, v) is continuous in D = C − ρ−p (ii) (1, 0) ∈ D and  {(1, 0)} = 1 > 0; (iii) for all (iu 2 , v1 ) ∈ D such that v1 ≤ − 21 (1 + u 22 ),   v1  {(iu 2 , v1 )} =  iu 2 + c + ( p − ρ) (iu 2 ) + ρ (c + ρ) v1 = (c + ρ)2 + ( p − ρ)2 u 22 ≤−

(c + ρ) (1 + u 22 ) 2 (c + ρ)2 + 2( p − ρ)2 u 22

< 0.

Since (u, v) satisfies the hypotheses of Lemma 2.1, we obtain the required result. This completes the proof of Theorem 3.1. 

  p Theorem 3.2 If f (z) ∈ Vk (m, δ, ζ, λ, μ, η; ρ), then for c > − p, F p,c f (z) ∈ p Vk (m, δ, ζ, λ, μ, η; ρ). Proof Applying (17) and Theorem 3.1, we obtain p

f (z) ∈ Vk (m, δ, ζ, λ, μ, η; ρ)

z f (z) p ∈ Rk (m, δ, ζ, λ, μ, η; ρ) p  

z F p,c ( f ) p ⇒ ∈ Rk (m, δ, ζ, λ, μ, η; ρ) p p ⇔ F p,c ( f ) ∈ Vk (m, δ, ζ, λ, μ, η; ρ) . ⇔

Thus, the proof of Theorem 3.2 is completed.



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