A Note on Meromorphic Functions Associated With Beseel Function Defined by Hilbert Sapce Operator

: In this paper,we introduce and study a new subclass of meromorphic functions associated with a certain differential operator on Hilbert space. For this class, we obtain several properties like the coefficient inequality, growth and distortion theorem, radius of close-to-convexity, starlikeness and meromorphically convexity and integral transforms. Further, it is shown that this class is closed under convex linear combina-tions.

We say that u is in the class Σ * (α) of such functions. Similarly, a function u ∈ Σ given by (1) is said to be meromorphically convex of order α if if satis es the following: We say that u is in the class Σ k (α) of such functions. For a function u ∈ Σ is given by (1) is said to be meromorphically close to convex of order β and α if there exists a function v ∈ Σ * (α) such that We say that u is in the class K(β, α). The class Σ * (α) and various other subclasses of Σ having been studied rather extensively by Clunie [3],Miller [8],Pommerenke [11],Royster [12], Akgul [1,2], Venkateswarlu [13], Sakar and Guney [9,10] et al. In recent years, many authors investigated the subclass of meromorphic functions with positive coe cient Juneja and Reddy [7] class Σp functions of the form Which are regular and univalent in U, the function in this class are said to be meromorphic functions with positive coe cients. For functions u ∈ Σp given by (1) and v ∈ Σp given by v(z) = z + ∞ η= bη z η , (bη ≥ ).
We de ne the Hadamard product(or convolution) of u and v by We recall here the generalized Bessel function of the rst kind of order γ(see [5]), denote by (Where Γ stands for the Gamma Euler function) which is the particular solution of the second-order linear homogeneous di erential equation (see, for details, [14]) We introdue the function ψ de ned, in terms of the generalized Bessel function w by By using the well-known Pochhammer symbol (x)µ de ned for x ∈ C and in terms of the Euler gamma function by We obtain the following series representation for the function ψ(z) Corresponding to the function ψ de ned the Bessel operator S c λ by the following Hadamard product Let Where I is the identity operator on H and C is positively oriented simple closed Recti able closed contour containing the spectrum σ(T) in the interior domain [4]. The operator u(T) can also be de ned by the following series This converges in the norm topology. The class of all functions u ∈ Σ, aη ≥ is de ned by Σp. The object of the present paper is to investigate the following subclass Σp associated with the di erential operator S c λ u(z).
The main object of the paper is to study usual properties of the geometric function theory such as coecients bounds, growth and distortion properties, a radius of convexity, convex linear combination, convolution properties, integral operators andδ-neighborhoods for the class σp(α, β, λ, c, T).

Coe cient Bounds
We rst give a characterization of the class σp(α, β, λ, c, T) by nding necessary and su cient condition for a functions in the class. This characterization implies coe cient estimates.

Theorem 2.1. A function u ∈ Σp given by (2) is in the class σp(α, β, λ, c, T) for all contraction T with T ≠ θ if and only if
The result is sharp for the function Proof. Suppose that (4) is true for ≤ β < and ≤ α < . Then and so u ∈ Σp is in the class σp(α, β, λ, c, T). Conversely suppose that u ∈ σp(α, β, λ, c, T) satis es the coe cients inequality (4 ). Since u ∈ σp(α, β, λ, c, T) then Letting r → in the above inequality, we obtain the assertion (4). This completes the proof of the theorem.

Corollary 2.2. If a function u(z) ∈ Σp given by
The result is sharp for the function u of the form (5).

Distortion Bounds
In this section we obtain growth and the distortion bounds for the class σp(α, β, λ, c, T).
Since T < , the above inequality becomes Using (9), we get the result.
The result is sharp for

Extreme points
In this section we obtain extreme bounds for the class σp(α, β, λ, c, T).
This completes the proof of the theorem.

Hence completes the proof
Radii of close-to-convexity, starlikeness and convexity

c, T). Then u is meromorphically close-to-convex of order
The result is sharp for the extremal function given by (5 ).
By Theorem 4, we have So the inequality Then (13) holds if This yields the close-to-convexity of the function and completes the proof. ∈ σp(α, β, λ, c, T). Then u is meromorphically starlike of order γ ( ≤ γ < ) in the disc |z| < r , where

Theorem 5.2. Let u
The result is sharp for the extremal function given by (5 ).

Theorem 5.3. Let u ∈ σp(α, β, λ, c, T). Then u is meromorphically convex of order
The result is sharp for the extremal function given by (5 ).
Proof. By using the technique employed in the proof of the Theorem 5.1, we can show that for |z| < r and prove that the assertion of the theorem is true.
Proof. We need to nd the largest ρ such that We want only to show that On the other hand, from (22), we have Therefore in view of (23) and (24), it is enough to nd the largest ρ that (η, λ, c) . Proof. Since u, v ∈ σp(α, β, λ, c, T), we have Combining the last two inequalities, we get But we need to nd the largest ρ such that The inequity (28) would hold if Then we have

Integral operators
In this section, we consider integral transforms of functions in the class σp(α, β, λ, c, T) of the type considered by Goel and Sohi [11]. is an increasing function of η, η ≥ . We obtained the desired result.

Con ict of interest:
Authors state no con ict of interest.