Let ℋ denote the class of analytic functions in the open unit disk , and be the subclass of ℋ consisting of normalized analytic functions of the form

For the analytic functions f and g, we say that f is subordinate to g and write f ≺ g, if there exists an analytic function w in such that w(0) = 0 and f(z) = g(w(z)). In particular, if g is univalent in , then we have the following equivalence:

Further, for functions , given by fj(z)=z+∑n=2∞an,j zn (j = 1, 2), the Hadamard product (or Convolution) of f₁ and f₂ is defined by

Let ℛ(β) denote the class of functions such that , and let ℘(β) denote the class of functions f ∈ ℋ such that f(0) = 1 and . For β = 0, we denote ℛ(β) and ℘(β) simply by ℛ and ℘, respectively.

Let ℋ^∞ denote the space of all bounded analytic functions in . This is Banach algebra with respect to the norm . For f ∈ ℋ, we define

The function f ∈ ℋ belongs to the Hardy space ℋ^p (0 < p ≤ ∞), if M_p(r, f) is bounded for all r ∈ [0, 1). Clearly, we have

(see [3, p. 2]). For 1 ≤ p ≤ ∞, ℋ^p is a Banach space with the norm defined by

(see [3, p. 23]). Following are two widely known results (see [8]) for the Hardy space ℋ^p:

In [10], Ponnusamy studied the Hardy space of hypergeometric functions. Further, Baricz [1] obtained the conditions for the generalized Bessel functions such that it belongs to Hardy space and Yagmur and Orhan [16] studied the same problem for the generalized Struve functions.

In [6] (see also [7]), Ibrahim studied the following generalized fractional integral operator in the complex plane ℂ:

where the function f(z) is analytic in a simply connected region of ℂ containing the origin, and the multiplicity of (z^μ+1 −ξ^μ+1)^α−1 is removed by requiring log(z^μ+1 − ξ^μ+1) to be real when (z^μ+1−ξ^μ+1) > 0. We observe that, if we take μ = 0 in (1.4), we arrive at the Srivastva-Owa fractional integral operator [13] (see also [11]) and when μ→−1⁺ in (1.4), we obtain the Hadamard fractional integral operator [5].

For α > 0 and μ > −1, we define a fractional integral operator Ωzα,μ:A→A by

Note that Ωz0,0f(z),Ωz1,0f(z)=ℒf(z) and Ωzα,0f(z)=L(2,α+2)f(z), where ℒ and L(a, c) denotes the Libera integral operator [9] and Carlson-Shaffer operator [2], respectively. Also, we observe that the operator Ωzα,μ satisfies the following recurrence relation

Corresponding to fractional integral operator Ωzα,μf(z), a fractional differential operator Φ^α,μf(z) was studied in [6, 7] to obtain its boundedness in Bergman space and certain geometric properties were also discussed.

In order to derive our main results, we recall here the following lemmas:

Lemma 2.1.([15])

Let F and G be analytic functions inand F(0) = G(0). If H(z) = zG′(z) is starlike function inand zF′ (z) ≺ zG′ (z), then

Lemma 2.2.([14])

For α < 1, β < 1, we have ℘(α) * ℘(β) ⊂ ℘(δ), where δ = 1 − 2(1 − α)(1 − β). The value of δ is best possible.

Our first main result is given by Theorem 2.3 below.

Theorem 2.3

Let α > 1 and μ > −1. If f(z) ∈ ℛ, thenΩzα,μf(z)∈ℛ∩ℋ∞.

Remark 2.4

If we consider Ψ(z)=-z-2 log(1-z)=z+2∑n=2∞znn, then from the geometrical descriptions of image domains of functions in the class ℛ, we can easily verify that Ψ ∈ ℛ and not belongs to ℋ^∞, but as per the Theorem 2.3, function Ωzα,μΨ(z)∈ℋ∞∩ℛ, where

Theorem 2.5

Let, α > 1 and μ > −1. If

then Ωzα,μf(z)z∈℘.

Theorem 2.6

Let, α > 1 and μ > −1. If g ∈ ℛ(1/2) andΩzα,μfsatisfy the inequality (2.6), thenΩzα,μf(z)*g(z)∈ℛ.

The authors wish to thank the referee for her/his valuable suggestions, which have improved the paper. The work was supported by the University Grant Commission, India (Project Number: MRP-MAJOR-MATH-2013-19114).