Differential Subordination Results for Analytic Functions in the Upper Half-Plane

and Applied Analysis 3 Using (16)–(18), and from (20), we obtain ψ (p (z) , p 󸀠 (z) , p 󸀠󸀠 (z) ; z) = φ( f 󸀠 (z) f (z) , f 󸀠󸀠 (z) f (z) − f 󸀠 (z) f (z) , f (z) [f 󸀠󸀠󸀠 (z) f 󸀠 (z) − (f 󸀠󸀠 (z)) 2 ] f (z) [f (z) f 󸀠󸀠 (z) − (f 󸀠 (z)) 2 ] − f 󸀠 (z) f (z) ; z) . (21)


Introduction
Let Δ denote the upper half-plane; that is, and let H[Δ] denote the class of functions  which are analytic in Δ and which satisfy the so-called hydrodynamic normalization (see [1][2][3]): Also let S[Δ] denote the class of all functions in H[Δ] which are univalent in Δ. Various basic properties concerning functions belonging to the class S[Δ] were developed in a series of articles (see, for details, [4][5][6]).
We denote by S * [Δ] the subclass of H[Δ] which consists of functions which are starlike in Δ.We first need to recall the notion of subordination in the upper half-plane.
Let  and  be members of H[Δ].The function  is subordinate to , written as  ≺  or () ≺ (), if there exists a function  ∈ H[Δ] with [Δ] ⊂ Δ such that () = (()).Furthermore, if the function  is univalent in Δ, then we have the following equivalence (cf.[7]): Using methods similar to those used in the unit disk, Rȃducanu and Pascu [7] have extended the theory of differential subordinations to the upper half-plane.In the following, we will list some definitions and theorems, which are required to prove our main results.
for  ∈ Δ, then In the present paper, by making use of the differential subordination results in the upper half-plane of Rǎducanu and Pascu [7] (which is a generalization of results in the unit disk obtained by Miller and Mocanu [8]), we determine certain appropriate classes of admissible functions and investigate some differential subordination properties of analytic functions in the upper half-plane.It should be remarked in passing that, in recent years, several authors obtained many interesting results associated with differential subordination and superordination in the unit disk; the interested reader may refer to, for example, [9][10][11][12][13][14][15][16][17][18].

The Main Subordination Results
We first define the following class of admissible functions that are required in proving our first result.
Proof.The proof of Theorem 7 is similar to that of [8,Theorem 2.3d,p.30]and so we choose to omit it.
and  is the best dominant.
Proof.Following the same arguments as in [8, Theorem 2.3e, p.31], we deduce that  is a dominant from Theorems 6 and 7.
Since  satisfies (27), it is also a solution of (24) and therefore  will be dominated by all dominants.Hence,  is the best dominant.

Also, we denote
by K[Δ] the subclass of H[Δ] which consists of functions which are convex in Δ.The classes S * [Δ] and K[Δ] were introduced by Stankiewicz [3].