On differential sandwich theorems of p -valent analytic functions deﬁned by the integral 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 an integral operator. Relevant connection of the results, which are presented in this paper with various known results are also considered.

the Hadamard product (or convolution) of f (z) and g(z) is defined by (1. 3) For f, g ∈ H (U ), we say that the function f is subordinate to g, if there exists a Schwartz 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 [5,6]).
Supposing that h and k are two analytic functions in U , let ϕ(r, s, t; z) : C 3 × U → C.
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 k(z) ≺ ϕ(h(z), zh (z), z 2 h (z); z), (1.4) then h is 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 the best subordinant.
Recently, Miller and Mocanu [7] obtained sufficient conditions on the functions k, q and ϕ for which the following implication holds: Using these results, Bulboaca [3] considered certain classes of first-order differential superordinations, as well as superordination-preserving integral operators [4]. Ali et al. [1], using the results from [3], obtained sufficient conditions for certain normalized analytic functions to satisfy where q 1 and q 2 are given univalent normalized functions in U . Very recently, Shanmugam et al. [14][15][16][17] obtained the sandwich results for certain classes of analytic functions. Further subordination results can be found in [8,9,13,[18][19][20].
For p ∈ N, n ∈ N 0 = N ∪ {0}, λ > 0 and f ∈ A( p), we consider the integral operator defined as follows: a k z k , We note that:

and (in general)
Also we note that I n p,1 f (z) = I n p f (z), where I n p is p-valent Salagean integral operator In this paper, we will derive several subordination results, superordination results and sandwich results involving the operator I n p,λ .

Preliminaries
In order to prove our subordination and superordination results, we make use of the following known definition and results.

Definition 2.1 [7] Denote by Q the set of all functions f (z) that are analytic and injective on
where and are such that f (ζ ) = 0 for ζ ∈ ∂U \E( f ).

Lemma 2.2 [6] Let the function q(z) be univalent in the unit disc U and let θ and ϕ be analytic in a domain
If p is analytic with p(0) = q(0), p(U ) ⊆ D and and q(z) is the best dominant.

Lemma 2.3 [16] Let q be a convex univalent function in U and let
If p(z) is analytic in U with p(0) = q(0) and and q is the best dominant.

Lemma 2.4 [5] 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
and q(z) is the best subordinant.
Lemma 2.5 [7] Let q be convex univalent in U and γ ∈ C. Further assume that and q is the best subordinant.
The last lemma gives us a necessary and sufficient condition for the univalence of a special function which will be used in some particular case. Lemma 2.6 [11] The function q(z) = (1 − z) −2ab is univalent in the unit disc U if and only if |2ab − 1| ≤ 1 or |2ab + 1| ≤ 1.

Subordination results
Unless otherwise mentioned, we assume throughout this paper that λ > 0, p, n ∈ N and the powers understood as principal values. Theorem 3.1 Let q be univalent in U , with q(0) = 1, and suppose that

3)
and q is the best dominant of (3.2).
Proof If we consider the analytic function From (3.5), by using the identity (1.6), a simple computation shows that hence the subordination (3.2) is equivalent to An application of Lemma 2.3, with ψ = 1 and γ = λμ p 2 , leads to (3.3).
Taking q(z) It is easy to check that the function ϕ(ζ ) = 1−ζ 1+ζ , |ζ | < |B|, is convex in U , and since ϕ(ζ ) = ϕ(ζ ) for all |ζ | < |B|, it follows that the image φ(U ) is a convex domain symmetric with respect to the real axis, hence Then, the inequality (3.7) is equivalent to hence we obtain the following result.

If f ∈ A( p) satisfies the following subordination condition
and 1+Az 1+Bz is the best dominant of (3.9).
For p = A = −B = 1, the above corollary reduces as follows: If f ∈ A satisfies the following subordination condition (3.10) then and 1+z 1−z is the best dominant of (3.10). Theorem 3.4 Let q(z) be univalent in U, with q(0) = 1 and q(z) = 0 for all z ∈ U. Let γ, μ ∈ C * and ν, η ∈ C * with ν + η = 0. Let f ∈ A( p) and suppose that f and q satisfy the next conditions: and q is the best dominant of (3.13).

Proof Let
(3.14) According to (3.11) the function h(z) is analytic in U , and differentiating (3.14) logarithmically with respect to z, we obtain In order to prove our result we will use Lemma 2.2. In this lemma consider θ(w) = 1 and ϕ(w) = γ w , then θ is analytic in C and ϕ(w) = 0 is analytic in C * . Also, if we let and g(z) = θ(q(z)) + Q(z) = 1 + γ zq (z) q(z) from (3.12), we see that Q(z) is a starlike function in U . From (3.12), we also have and then, by using Lemma 2.2 we deduce that the subordination (3.13) implies h(z) ≺ q(z), and the function q is the best dominant of (3.13).

Corollary 3.5
Let μ ∈ C * . Let f ∈ A( p) and suppose that and 1+Az 1+Bz is the best dominant of (3.15). Putting in Theorem 3.4, then combining this with Lemma 2.6 we obtain the next result due to Obradovic et al. [8,Theorem 1]. (3.16) and (1 − z) −2ab is the best dominant of (3.16).

Remark 3.7
For a = 1, Corollary 3.6 reduces to the recent result of Srivastava and Lashin [19].
Let f ∈ A such that f (z) z = 0 for all z ∈ U , and let μ ∈ C * . If
From the assumption (3.20) we see that Q is starlike in U and that thus, by applying Lemma 2.2, the proof of Theorem 3.10 is completed.

Remark 4.6
Putting λ = 1 in the above results, we obtain the corresponding results for the p-valent Salagean integral operator I n p .