Let denote the class of all normalized functions of the form

which are analytic in the open unit disk E = {z : |z| < 1}.

Let , , , (β) and (β) denote the subclasses of consisting of functions that are univalent, convex, starlike, convex of order β and starlike of order β in E respectively, see [10]. For the functions

the convolution (Hadamard product) is defined as

Let f and g be analytic functions in E. Then f is said to be subordinate to g written as f ≺ g and f(z) ≺ g(z), z ∈ E, if there exits a Schwarz function ω analytic in E, with ω (0) = 0 and |ω(z)| < 1 for z ∈ E, such that f(z) = g (ω(z)), z ∈ E.

Recently, the theory of fractional calculus has found interesting applications in the theory of analytic functions. The classical definitions of fractional operators and their generalizations have fruitfully been applied in obtaining, for example, the characterization properties, coefficient estimates, distortion inequalities and convolution structures for various subclasses of analytic functions.

The fractional derivative of order α, 0 ≤ α < 1 is defined in [23] as follows

where the function f(z) is analytic in a simply connected domain in the complex plane containing the origin and the multiplicity of (z−t)^−α is removed by requiring log(z − t) ∈ ℝ, whenever (z − t) > 0. The gamma function Γ is defined as

Note that

Owa and Srivastava [24, 30] introduced the operator Ω^α : → as follows

where the incomplete beta function φ(a, c; z) is defined as follows

where a, c are complex numbers different from 0, −1, −2, …. It can be seen that

Mishra and Gochhayat [19] have studied some properties of the operator Ω^α and introduced new subclass of k-uniformly convex functions. In a recent paper Ibrahim and Darus [12] introduced the fractional differential subordination based on the operator Ω^α. Srivastava and Mishra [29] studied the applications of fractional calculus to the parabolic starlike and uniformly convex functions using the operator Ω^α.

Al-Oboudi [1, 2] defined the linear multiplier fractional differential operator of order 1 as follows

where 0 ≤ λ ≤ 1, 0 ≤ α < 1 and

Obviously

where Ω^α was defined by (1.4) and

Bulut [5, 6, 7, 8] used the Al-Oboudi fractional differential operator of order n to define some new integral operators and obtain interesting results. Recently, Noor et al [22] used fractional derivative to define some new subclasses of analytic functions in the conic regions. Now for a > 0, let

Ozkan [25] defined the convolution of f given by (1.1) and k_a(z) such that

Now extending the concept of [4] and [25, 30], we define the Srivastava-Owa-Ruscheweyh La,λα:A→A as follows

which is called the Srivastava-Owa-Ruscheweyh fractional derivative operator.

It can easily be seen that

• For α = 0, λ = 0. La,00f(z)=(ka*f) (z),

  see [25].

• For a = n + 1 > 0, α = 0, λ = 0. Ln+1,00f(z)=z(1-z)n+1*f(z),

  see [28].

• For a = n + 1 > 0, α = 0. Ln+1,00f(z)=z(1-z)n+1*Dλ0f(z),

  see [26].

From the definition of La,λα, we can establish the following identity as well

We assume that h is analytic, convex, univalent in E with h(0) = 1 and ℜ h(z) > 0, z ∈ E. Using the operator La,λα, we define the following

Definition 1.1

Let Pa,λα(h,δ) be the class of functions f ∈ satisfying

for a > 0, 0 ≤ δ ≤ 1, 0 ≤ λ ≤ 1, 0 ≤ α < 1 and for all z ∈ E.

We note that the class Pa,00(h,δ)=Pa(h,δ), was studied by Ozkan in [25] and the class Pa,00(h,0)=Sa(h), was studied by Padmanabhan and Parvatham in [26].

Obviously, for the special choices of function h and parameters α, a, λ, we have the following relationship.

Definition 1.2

Let Ja,λα(h,γ) denote the subclass of consisting of the functions f which satisfies the following condition

for some a > 0, 0 ≤ λ ≤ 1, 0 ≤ α < 1, 0 ≤ γ ≤ 1 and for all z ∈ E. Note that

see [25].

Definition 1.3

Let Ma,λα(h,γ) be the class of functions f ∈ which satisfies the following condition

for some a > 0, 0 ≤ λ ≤ 1, 0 ≤ α < 1, 0 ≤ γ ≤ 1 and for all z ∈ E.

To prove our main results, we need the following Lemmas.

Lemma 2.1.([18])

Let h be analytic, univalent, convex in E, with h(0) = 1 and ℜ [β₁h(z)+γ₁] > 0, β₁, γ₁ ∈ ℂ, z ∈ E. If p is analytic in E, with p(0) = h(0), then

implies

Lemma 2.2.([11])

Let h be analytic, univalent, convex in E, with h(0) = 1. Let p be analytic in E, with p(0) = h(0). If γ₂ ≠ 0, ℜ γ₂ ≥ 0 and

then

where

Lemma 2.3.([9])

If ψ ∈ , g ∈ and F is analytic function with ℜ (F(z)) > 0 for z ∈ E, then

Lemma 2.4.([21])

Let p(z) and q(z) be analytic in E, with p(0) = q(0) = 1 and ℜe(q(z))>12 for |z| < ρ, (0 < ρ ≤ 1). Then the image of E_ρ = {z : |z| < ρ} under p * q is a subset of closed convex hull of p(E).

In this section, we will prove our main results.

Theorem 3.1

Assume that a ≥ 1, 0 ≤ δ ≤ 1, 0 ≤ λ ≤ 1, 0 ≤ α < 1 and

Then

Corollary 3.2.([25])

If (3.1) is satisfied for α = 0 and λ = 0, then Pa+1,00(h,δ)⊂Pa,00(h,δ).

Corollary 3.3

If (3.1) is satisfied for δ = 0, α = 0, λ = 0, a = 1 and h(z)=1+z1-z,P2,00(1+z1-z,0)⊂P1,00(1+z1-z,0). That is ⊂ , z ∈ E.

Theorem 3.4

Assume that

If f∈Pa,λα(h,δ) for a ≥ 1, then Fc∈Pa,λα(h,δ), where F_c is defined as

Theorem 3.5

For a > 0, 0 ≤ γ ≤ 1, 0 ≤ λ ≤ 1, 0 ≤ α < 1, we have

Theorem 3.6

If f∈Ja,λα(h,γ), then Fc∈Ja,λα(h,γ), where F_c is defined by (3.3).

Corollary 3.7

If f∈Ja,00(h,γ)=Ja(h,γ), then F_c ∈ (h, γ), where F_c is defined by (3.3).

Theorem 3.8

For a > 0, 0 ≤ γ ≤ 1, 0 ≤ λ ≤ 1, 0 ≤ α < 1, we have

Corollary 3.9.([25])

For α = 0, λ = 0, we have

Theorem 3.10

• f∈Ma,λα(h,γ)⇔zf′∈Ja,λα(h,γ),

• Ma,λα(h,γ)⊂Ja,λα(h,γ).

Corollary 3.11

• f∈Ma,00(h,γ)⇔zf′∈Ja,00(h,γ),

• Ma,00(h,γ)⊂Ja,00(h,γ).

Theorem 3.12

For γ > η ≥ 0, 0 ≤ λ ≤ 1, 0 ≤ α < 1, we have

• Ja,λα(h,γ)⊂Ja,λα(h,η),

• Ma,λα(h,γ)⊂Ma,λα(h,η).

Theorem 3.13

If Ψ ∈ and f∈Pa,λα(h,δ), then f∈Pa,λα(h,δ), for a ≥ 1.

Theorem 3.14

For δ ≥ 0, 0 ≤ λ ≤ 1, 0 ≤ α < 1, we have

For different choices of analytic function h, we have some applications of the main results. For k ∈ [0,∞), Kanas [13, 14, 15] defined the conic domain Ω_k as follows

For k = 0, this domain represents the whole right half plane. Also we obtain the right branch of hyperbola for 0 < k < 1, the parabola for k = 1 and the ellipse for k > 1. The following functions p_k(z) are univalent with p_k (0) = 1, pk′(0)>0 and plays the role of extremal functions mapping E onto the conic domains Ω_k

Here u(z)=z-t1-tz,t∈(0,1), t ∈ (0, 1), z ∈ E and z is chosen such that k=cosh (πR′(t)4R(t)), R(t) is Legender’s elliptic integral of the first kind and R′ (t) is the complementary integral of R(t). For details, we refer to [13], [14], [15] and [20]. In [16] some linear operators associated with k-uniformly convex functions were considered. Now by choosing h(z) = p_k(z) in Theorem 3.14, we can easily prove the following

Corollary 4.1

For δ ≥ 0, 0 ≤ λ ≤ 1, 0 ≤ α < 1, we have

where

Some of the special cases are given as follows.

• Let k = 0. Then f∈Pa,λα(1+z1-z,δ)⇒f∈Pa,λα(11-z,0). That is ℜe(z(La,λαf(z))′La,λαf(z))>12 in E.

• For k = 1, we have Pa,λα((1+2π2[log(1+z1-z)]2,δ))⊂Pa,λα(q1,0)

  and ℜe(z(La,λαf(z))′La,λαf(z))>q1(-1)=12.

• Let k > 1 and f∈Pa,λα(pk,δ)⇒f∈Pa,λα(z(z-k) log(1-zk),0). That is ℜe(z(La,λαf(z))′La,λαf(z))>1(k+1) log(1+1k).

• For the case k = 2, we note that Pa,λα(p2,δ)⊂Pa,λα(q2,0). This gives us ℜe(z(La,λαf(z))′La,λαf(z))>q2(-1)13 log32≈0.813.

Now we prove a radius result for the class Ma,λα(h,σ).

Theorem 4.2

Let f∈Ma,λα(h,0). Then f∈Ma,λα(h,σ), for |z| < r_σ, where