Let denote the class of functions:
which are analytic in the unit disk. Denote by the class of normalized univalent functions in U.
Let. We say that is subordinate to (written as) if there is a function w analytic in U, with, for all. If g is univalent, then if and only if and  .
Definition 1 (  ). Let and. The operator is defined by
Remark 1. If and, then .
Remark 2. For in (2), we obtain the Salagean differential operator.
From (2), the following relations holds:
and from which, we get
Definition 2 (  ). Let, and. Then
This operator is a particular case of the operator defined in  and it is easy to see that for any,.
Next, we define the new subclasses of.
Definition 3. A function belongs to the class if and only if
Remark 4. if and only if.
Definition 4. Let, and, the set of functions satisfying:
i) is continuous in a domain of,
iii) when and for.
Several examples of members of the set have been mentioned in   and (  , p. 27).
2. Preliminary Lemmas
Let P denote the class of functions which are analytic in U and satisfy.
Lemma 1 (   ) Let with corresponding domain. If is defined as the set of functions given as which are regular in U and satisfy:
ii) when.Then in U.
More general concepts were discussed in  -  .
Lemma 2 (  ). Let and be complex constants and a convex univalent function in U satisfying, and. Suppose satisfies the differential subordination:
If the differential subordination:
has univalent solution in U. Then and is the best dominant in (6).
The formal solution of (6) is given as
see   .
Lemma 3 (  ). Let and be complex constants and regular in U with, then the solution of (7) given by (8) is univalent in U if (i) Re
, (ii) (iii).
3 Main Results
Theorem 1. Let and a convex univalent function in U satisfying
, and,. Let. If, then.
Proof. From (4), we have
If we suppose, we need to show that. Using the above equation and (4) and Remark 4, it suffices to show that if, then.
By (2) and (3) we have
Applying Lemma 2 with and, the proof is complete.W
Theorem 2. Let and a convex univalent function in U satisfying, and. Let. If, then
is the best dominant.
Proof. Let, then by Remark 4,
By (9), we have
To show that, by Remark 4, it suffices to show that
Now, considering the differential equation
whose solution is obtained from (8). If we proof that is univalent in U, our re-
sult follows trivially from Lemma 2. Setting and in Lemma 3, we have
where, so that by logarithmic differentiation, we have
Hence, is univalent in U since it satisfies all the conditions of Lemma 3. This completes the proof.W
Proof. Let. By Remark 4
From (9), let with for. Conditions (i) and (ii) of Lemma 1 are clearly satisfied by. Next, Then if. Hence, Using Remark 4, which complete the proof.W
Corollary 1. All functions in are starlike univalent in U.
Proof. The proof follows directly from Theorem 3 and Remark 4.W
Corollary 2. The class “clone” the analytic representation of convex functions.
Proof. The proof is obvious from the above corollary and Definition 4.W
The functions and are examples of functions in.
Theorem 4. The class is preserve under the Bernardi integral transformation:
Proof. let, then by Remark 4. From (10) we get
Applying on (10) and noting from Remark 1 that, we have
Let and noting that, we get
Let for. Then satisfies all the conditions of Lemma 1 and so Þ By Remark 4.W
Theorem 5. Let. Then f has integral representation:
Proof. Let. Then by Remark 4, and so for some
But, so that
Applying the operator in Definition 2, we have the result.W
With, we have the extremal function for this new subclass of which is
Theorem 6. Let. Then
The function given by (13) shows that the result is sharp.
Proof. Let, then by Remark 4,. Since it is well known that for any, , then from Remark 1 we get the result.W
Theorem 7. Let. Then
Proof. Let. Then by Theorem 6, we have
Also, upon differentiating, we get
for. This complete the proof.W
The authors appreciates the immense role of Dr. K.O. Babalola (a senior lecturer at University of Ilorin, Ilorin, Nigeria) in their academic development.
 Al-Oboudi, F.M. (2004) On Univalent Functions Defined by a Generalized Salagean Operator. International Journal of Mathematics and Mathematical Sciences, 27, 1429-1436.
 Babalola, K.O. and Opoola, T.O. (2006) Iterated Integral Transforms of Caratheodory Functions and Their Applications to Analytic and Univalent Functions. Tamkang Journal of Mathematics, 37, 355-366.
 Miller, S.S. and Mocanu, P.T. (1978) Second Order Differential Inequalities in the Complex Plane. Journal of Mathematical Analysis and Applications, 65, 289-305.
 Babalola, K.O. and Opoola, T.O. (2008) On the Coefficients of Certain Analytic and Univalent Functions. In: Dragomir, S.S. and Sofo, A., Eds., Advances in Inequalities for Series, Nova Science Publishers, 5-17.