Differential Sandwich Theorems for Analytic Functions Defined by an Extended Multiplier Transformation

In this investigation, we obtain some applications of first order differential subordination and superordination results involving an extended multiplier transformation and other linear operators for certain normalized analytic functions. Some of our results improve previous results.


 
H U be the class of functions analytic in the open unit disk consisting of functions of the form: For simplicity, let

  
,1 H a H  a .Also, let A be the subclass of , consisting of functions of the form: (1.2) we say that is subordinate to if there exists Schwarz function  , w z 0 which (by definition) is analytic in with and Furthermore, if the function g z , is univalent in U then we have the following equivalence , (cf., e.g.[1,2]; see also [3]): We denote this subordination by then is a solution of the differential superordination (1.3).Note that if p f is subordinate to g , then g is superordinate to f .An analytic function is called a subordinant if for all satisfying (1.3).A univalent subordinant that satisfies q for all subordinants of (1.3) is called the best subordinant.Recently Miller and Mocanu [4] obtained conditions on the functions and  for which the following implication holds: Using the results of Miller and Mocanu [4], Bulboaca [5] considered certain classes of first-order differential superordinations as well as superordination-preserving integral operators [6].Ali et al. [7] have used the results of Bulboaca [5] and obtained sufficient conditions for normalized analytic functions f to satisfy: where 1 and 2 are given univalent functions in .Also, Tuneski [8] obtained a sufficient condition for star- Recently Shanmugam et al. [9] obtained sufficient conditions for a normalized analytic functions to satisfy Many essentially equivalent definitions of multiplier transformation have been given in literature (see [10][11][12].

Also if f z A
 , then we can write In this paper, we obtain sufficient conditions for the normalized analytic function f defined by using an extended multiplier transformation to satisfy: and and are given univalent functions in .

Definitions and Preliminaries
In order to prove our results, we shall make use of the following known results.

Definition 2.1. [4]
Denote by the set of all functions f that are analytic and injective on Let the function be univalent in the open unit disc and  and  be analytic in a domain containing then and is the best dominant.Taking is analytic in , and then and is the best dominant.Lemma 2.

[5]
Let the function be univalent in the open unit disc and  and  be analytic in a domain containing then and is the best subordinant.
Taking   w w , and is univalent in and then and q is the best subordinant.

Applications to an Extended Multiplier Transformation and Sandwich Theorems
Then the function is analytic in and U   Therefore, differentiating (3.3) logarithmically with respect to and using the identity (1.6) in the resulting equation, we have and therefore, the theorem follows by applying Lemma 2.2.Putting in Theorem 3.1, we have the following corollary.
, and is the best dominant.Now, by appealing to Lemma 2.4 it can be easily prove the following theorem.
is univalent in , and and is the best subordinant.
and is the best subordinant.
in Theorem 3.2, we have Corollary 3.8.

Let be convex univalent in
is univalent in , and and is the best subordinant.Taking   in Theorem 3.2, we have is univalent inU , and in Theorem 3.2, we have Corollary 3.10.

Let be convex univalent in
and is the best subordinant.Combining Theorems 3.1 and 3.2, we get the following sandwich theorem.
is univalent in , and  and 1 and 2 are respectively, the best subordinant and the best dominant.
and q is the best dominant.
Proof.Define the function by in Theorem 3.4, we have Corollary 3.12.Let be convex univalent in , .Further, assume that (3.1) holds.
and is the best dominant.Theorem 3.5.

Let be convex univalent in
, , and is the best subordinant.Proof.The proof follows by applying Lemma 3.4.Combining Theorems 3.4 and 3.5, we get the following sandwich theorem.
2 are respectively the best subordinant and the best dominant.