Subordination and superordination results of p − valent analytic functions involving a linear operator

In this paper we derive some subordination and superordination results for certain p−valent analytic functions in the open unit disc, which are acted upon by a class of a linear operator. Some of our results improve and generalize previously known results.

Also, let A(p) be the subclass of the functions f ∈ H(U ) of the form: and set A ≡ A(1).For functions f (z) ∈ A(p), given by (1.1), and g(z) given by the Hadamard product (or convolution) of f (z) and g(z) is defined by .3)2000 Mathematics Subject Classification: 30C45.

T. M. Seoudy
For f, g ∈ H(U ), we say that the function f is subordinate to g, if there exists a Schwarz function w, i.e, w ∈ H(U ) with w(0) = 0 and |w(z)| < 1, z ∈ U, such that f (z) = g(w(z)) for all z ∈ U.This subordination is usually denoted by f (z) ≺ g(z).It is well-known that, if the function g is univalent in U , then f (z) ≺ g(z) is equivalent to f (0) = g(0) and f (U ) ⊂ g(U ) (see [6] and [11]).
Supposing that h and k are two analytic functions in U , let φ(r, s, t; z) : If h and ϕ(h(z), zh ′ (z), z 2 h ′′ (z); z) are univalent functions in U and if h satisfies the second-order superordination then h is called to be a solution of the differential superordination (1.4).A function q ∈ H(U ) is called a subordinant of (1.4), if q(z) ≺ h(z) for all the functions h satisfying (1.4).A univalent subordinant q that satisfies q(z) ≺ q(z) for all of the subordinants q of (1.4), is said to be the best subordinant.
Recently, Miller and Mocanu [12] obtained sufficient conditions on the functions k, q and ϕ for which the following implication holds: Using these results, Bulboaca [4] considered certain classes of first-order differential superordinations, as well as superordination-preserving integral operators [5].Ali et al. [1], using the results from [4], obtained sufficient conditions for certain normalized analytic functions to satisfy where q 1 and q 2 are given univalent normalized functions in U .

Preliminaries
In order to prove our subordination and superordination results, we make use of the following known definition and results.Definition 2.1.[12] Denote by Q the set of all functions f (z) that are analytic and injective on U \E(f ), where and are such that f Lemma 2.2.[11] Let the function q(z) be univalent in the unit disc U and let θ and ϕ be analytic in a domain D containing q(U ) with ϕ(w) = 0 when w ∈ q(U ).
If p is analytic with p(0) = q(0), p(U ) ⊆ D and then p(z) ≺ q(z) and q(z) is the best dominant.
Lemma 2.3.[6] Let q(z) be convex univalent in the unit disc U and let θ and ϕ be analytic in a domain D containing q(U ).Suppose that then q(z) ≺ p(z) and q(z) is the best subordinant.
The following lemma gives us a necessary and sufficient condition for the univalence of a special function which will be used in some particular case.Lemma 2.4.[15] The function q

Main Results
Unless otherwise mentioned, we assume throughout this paper that p ∈ N, m ∈ N 0 , ℓ ≥ 0, λ ≥ 0 and the power understood as principal values.
Theorem 3.1.Let q(z) be univalent in U such that q(0) = 1, q(z) = 0 and zq is starlike in U. Let f ∈ A(p) and suppose that f and q satisfy the next conditions: and where then I m,ℓ p,q,s,λ (α 1 )f (z) and q is the best dominant of (3.3).
Proof: Let According to (3.1) the function h(z) is analytic in U , and differentiating (3.5) logarithmically with respect to z, we obtain By using the identity (1.10), we obtain

T. M. Seoudy
In order to prove our result we will use Lemma 2.2.In this lemma consider θ(w) = χ + ζw + δw 2 and ϕ(w) = γ w , then θ is analytic in C and ϕ(w) = 0 is analytic in C * .Also, if we let q(z) .
We see that Q(z) is starlike function in U .From (3.2), we also have and then, by using Lemma 2.2 we deduce that the subordination (3.3) implies h(z) ≺ q(z), and the function q is the best dominant of (3.3).
1, we obtain the following result which improves the corresponding work of Shammugam et al. [17,Theorem 3].✷ Corollary 3.2.Let q(z) be univalent in U such that q(0) = 1, q(z) = 0 and zq and suppose that q satisfies (3.2).If where and q is the best dominant of (3.7).
in Theorem 3.10, and using Lemma 2.3 we obtain the next result.
Theorem 3.10.Let q be convex in U such that q (0) = 1 and zq ′ (z) q(z) is starlike in U. Further assume that

.16)
If Ψ (z) given by (3.4) is univalent in U and satisfies the following superordination condition then q(z) ≺ I m,ℓ p,q,s,λ (α 1 )f (z) z p µ , and q is the best subordinant of (3.17).Corollary 3.11.Let q be convex in U such that q (0) = 1 and zq If Λ (z) given by (3.8) is univalent in U and satisfies the following superordination condition and q is the best subordinant of (3.20).
Let H(U ) denotes the class of analytic functions in the open unit disc U = {z ∈ C : |z| < 1} and let H[a, p] denotes the subclass of the functions f ∈ H(U ) of the form: f (z) = a + a p z p + a p+1 z p+1 + ...(a ∈ C; p ∈ N = {1, 2, ..}).
and q(z) = (1 − z) −2ab in Theorem 3.1, then combining this together with Lemma 2.4 we obtain the next result due to Obradovic et al. [14, Theorem 1].