Subordination and superordination of certain linear operator on meromorphic functions

Using the methods of differential subordination and superordination, sufficient conditions are determined on the differential linear operator of meromorphic functions in the punctured unit disk to obtain, respectively, the best dominant and the best subordinant. New sandwich-type results are also obtained.


Introduction.
Let H(U ) be the class of functions analytic in U = {z : z ∈ C and |z| < 1} and H[a, n] be the subclass of H(U ) consisting of functions of the form f (z) = a + a n z n + a n+1 z n+1 + . . ., with H = H [1,1]. Let f and F be members of H(U ).The function f is said to be subordinate to F , or F is said to be superordinate to f , if there exists a function ω analytic in U with ω(0) = 0 and |ω(z)| < 1 (z ∈ U ), such that f (z) = F (ω(z)).In such a case we write f (z) ≺ F (z).If F is univalent, then f (z) ≺ F (z) if and only if f (0) = F (0) and f (U ) ⊂ F (U ) (see [5] and [6]).
Denote by Q the set of all functions q(z) that are analytic and injective on Ū \E(q) where E(q) = ζ ∈ ∂U : lim z→ζ q(z) = ∞ , and are such that q (ζ) = 0 for ζ ∈ ∂U \E(q).Further let the subclass of Q for which q(0) = a be denoted by Q(a) and Q(1) ≡ Q 1 .
In particular, if then q(U ) = U M = {w : |w| < M }, q(0) = a, E(q) = ∅ and q ∈ Q (a).In this case, we set Ψ n [Ω, M, a] = Ψ n [Ω, q], and in the special case when the set Ω = U M , the class is simply denoted by Ψ n [M, a].
Let p denote the class of all p-valent functions of the form: For two functions f given by (1.1) and g given by the Hadamard product (or convolution) of f and g is defined by For a function f in the class p given by (1.1), we define a linear operator D n λ,p : p → p as follows: and (in general) (1.4) We note that: (i) The operator D n 1,p = D n p was introduced and studied by Aouf and Hossen [2], Liu and Owa [3], Liu and Srivastava [4], Srivastava and Patel [7]; (ii) The operator D n 1,1 = D n was considered and studied by Uralegaddi and Somanatha [8].
In the present paper, by making use of the differential subordination and superordination results of Miller and Mocanu [5, Theorem 2.3b, p. 28] and [6, Theorem 1, p. 818], certain classes of admissible functions are determined so that subordination as well as superordination implications of functions associated with the linear operator D n λ,p hold.Ali et al. [1] have considered a similar problem for Liu-Srivastava linear operator on meromorphic functions.Additionally, several differential sandwich-type results are obtained.

Subordination results involving the linear operator D n
λ,p .Unless otherwise mentioned, we assume throughout this paper that λ > 0, p ∈ N and n ∈ N 0 .The following class of admissible functions is required in our first result.

Definition 3.
Let Ω be a set in C and q(z) ∈ Q 1 ∩H.The class of admissible functions Φ D [Ω, q] consists of those functions φ : C 3 × U → C which satisfy the admissibility condition where z ∈ U , ζ ∈ ∂U \E (q) and k ≥ 1.
The proof shall make use of Lemma 1.Using equations (2.2)-(2.4),and from (2.6), we obtain (2.7) The proof is completed, if it can be shown that the admissibility condition for φ ∈ Φ D [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 1.Note that If Ω = C is a simply connected domain, then Ω = h (U ) for some conformal mapping h(z) of U onto Ω.In this case the class The following result is an immediate consequence of Theorem 1.
. Our next result is an extension of Theorem 1 to the case where the behavior of q(z) on ∂U is not known.
The next theorem yields the best dominant of the differential subordination (2.8).
If f (z) ∈ p satisfies (2.8), then z p D n λ,p f (z) ≺ q (z) , and q(z) is the best dominant.
Proof.Following the same arguments in [4, Theorem 2.3e, p. 31], we deduce that q(z) is a dominant from Theorems 2 and 3. Since q(z) satisfies (2.9), it is also a solution of (2.8) and therefore q(z) will be dominated by all dominants.Hence q(z) is the best dominant.
In the particular case q(z) = 1 + M z, M > 0, and in view of Definition 3, the class of admissible functions Φ D [Ω, q], denoted by Φ D [Ω, M ], is described below.

Definition 4.
Let Ω be a set in C and M > 0. The class of admissible functions Φ D [Ω, M ] consists of those functions φ : Corollary 2 can now be written the following form: To use Corollary 2, we need to show that φ ∈ Φ D [Ω, M ], that is, the admissibility condition (2.10) is satisfied.This follows from The required result now follows from Corollary 2.
Theorem 4 shows that the result is sharp.The differential equation Next, let us note that D 0 1,p f (z) = f (z), and By taking n = 0 and λ = 1, Corollary 5 shows that for f ∈ p , if z p zf (z) + pf (z) ≺ M z, then z p f (z) ≺ 1 + M z.

Definition 5.
Let Ω be a set in C and q(z) ∈ Q 1 ∩ H.The class of admissible functions Φ D,1 [Ω, q] consists of those functions φ : C 3 × U → C which satisfy the admissibility condition: where z ∈ U , ζ ∈ ∂U \E (q) and k ≥ 1.
Proof.Define an analytic function g(z) in U by (2.12) .
By making use of (1.5) in (2.13), we get .
Define the transformations from C 3 to C by (2.16)
In the case Ω = C is a simply connected domain with Ω = h (U ) for some conformal mapping h(z) of U onto Ω, the class Φ D,1 [h (U ) , q] is written as The following result is an immediate consequence of Theorem 5.
In the particular case q(z) = 1 + M z, M > 0, the class of admissible functions kM for all real θ, λ > 0 and k ≥ 1.
In the special case and Corollary 6 takes the following form: Proof.This follows from Corollary 6 by taking φ (u, v, w; z) = v − u and Ω = h (U ), where h (z) = λM 1+M z, M > 0. To use Corollary 6 we need to show that φ ∈ Φ D,1 [M ], i.e., the admissibility condition (2.19) is satisfied.This follows from Hence the result is easily deduced from Corollary 6.

Superordination results of the linear operator D n
λ,p .In this section we obtain differential superordination for functions associated with the linear operator D n λ,p .For this purpose the class of admissible functions is given in the following definition.

Definition 7.
Let Ω be a set in C and q(z) ∈ H with zq (z) = 0.The class of admissible functions Φ D [Ω, q] consists of those functions φ : C 3 × Ū → C which satisfy the admissibility condition: where z ∈ U, ζ ∈ ∂U , and m ≥ 1.
From (2.6), we see that the admissibility condition for φ ∈ Φ D [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 2. Hence ψ ∈ Ψ [Ω, q], and by Lemma 2, Proceeding similarly as in the previous section, we establish the following result as an immediate consequence of Theorem 7. The following theorem proves the existence of the best subordinant of (3.2) for an appropriate φ.
Theorem 9. Let h(z) be analytic in U and φ : z) ≺ z p D n λ,p f (z) and q(z) is the best subordinant.
Proof.The proof is similar to the proof of Theorem 4 and is therefore omitted.
Combining Theorems 2 and 8, we obtain the following sandwich-type theorem.
Corollary 9. Let h 1 (z) and q 1 (z) be analytic functions in Definition 8. Let Ω be a set in C and q(z) ∈ H with zq (z) = 0.The class of admissible functions Φ D,1 [Ω, q] consists of those functions φ : C 3 × Ū → C which satisfy the admissibility condition: where z ∈ U, ζ ∈ ∂U and m ≥ 1.Now we will give the dual result of Theorem 5 for differential superordination. .
If Ω = C is a simply connected domain and Ω = h(U ) for some conformal mapping h(z) of U onto Ω, then the class Φ D,1 [h (U ) , q] is written as Proceeding similarly as in the previous section, we establish the following result as an immediate consequence of Theorem 10. .
Combining Theorems 6 and 11, we obtain the following sandwich-type theorem.
Remark.Putting λ = 1 in the above results, we obtain the similar results associated with the operator D n p .
Theorems 7 and 8 can only be used to obtain subordinants of differential superordination of the form (3.1) or (3.2).