Definition 2.1. Let denote the subclass of consisting of a function which satisfies the inequality

for , and for all .

Several well-known subclasses of functions are special cases of for suitable choices of the parameters, and are listed below.

Theorem 2.2. Let satisfies

for some , then where

Proof. Suppose that

it is true for some , then it is suficient to prove that

For , we define the function by

which gives,

and

From (7) and (8) we get,

We know that

and

Therefore,

and , then the class analogues the class was studied by Thirucheran and Stalin [18].

Corollary 2.3. Let satisfies

for some , then , where

Let , then the class reduces to

which analogue the class introduced by Sevtap Sumer Eker and Ozlem Guney[20].

Corollary 2.4. Let satisfies the inequality

for some , then ,

where

If we put and , then the class reduces to

which analogues the class introduced by Sevtap Sumer Eker and Owa [19].

Corollary 2.5.Let satisfies

for some , then , where

If , and , then the class , which reduces to the form

and analogues the class introduced by Kadioglu [21].

Corollary 2.6. Let satisfies the coefficient inequality

for some , then .

Theorem 2.7.If for , then

Where

and

If replace = 0 and in (15), then the class , reduces to the class , which is studied by Thirucheran and Stalin [18].

Corollary 2.8.If for , then

where and .

If , we get the results of subclass introduced by Sevtap Sumer Eker and Ozlem Guney [20].

Corollary 2.9. If for , then

j_1}^{k-1} \sum_{j_1=2}^{k-2} \frac{(1 + (j_1 - 1)\delta)(1 + (j_2 - 1)\delta)^n}{|v_{j_1} v_{j_2}|} \\&+ \dots + \beta^{k-2} \prod_{j=2}^{k-1} \frac{(1 + (j - 1)\delta)^n}{|v_j|} \Biggr\}\end{align*} \end{document} ]]>

where and .

If and , we get the result of the class introduced by Sevtap Sumer Eker and Owa [19].

Corollary 2.10.If for , then

where and .

Now we define the subclass , which consists of the function

Now determine the extreme points for the subclass .

Theorem 2.11. Let

then if and only if it can be expressed in the form , where 0 \end{document} ]]> and

Proof. Suppose that

which shows that satisfies the condition (20). Hence, . Conversely, suppose that . Since,

Let

and

then we obtain

and , then the class reduces and analogues the class studied by Thirucheran and Stalin [18].

Corollary 2.12. Let

then if and only if it can be expressed in the form , where 0 \end{document} ]]> and

Let , we get the result of the class introduced by Sevtap Sumer Eker and Ozlem Guney [20].

Corollary 2.13. Let and

then if and only if it can be expressed , where ~ } 0 , \textstyle \sum _ { j = 1 } ^ { \infty } \eta _ { j } = 1 \end{document} ]]> .

Theorem 2.14. (Littlewood [23]) If and are analytic in with , then

for 0 \end{document} ]]> and .

Theorem 2.15. Let and suppose that is defined by

If there exists an analytic function given by

then

Proof. We must show that

By the help of Littlewood subordination theorem, it is suficient to show that

Let

Which readily yields . Further, we prove that the analytic function satisfies using Schwarz lemma. We know that

Let and , then the class reduces to the class ，which is investigated by Thirucheran and Stalin [18].

Corollary 2.16. Let and suppose that is defined by

If there exists an analytic function given by

then

Definition 2.17. Let denote the subclass of consisting of function which satisfies the inequality

For some and all .

Several well known subclasses of functions are the special cases of for suitable choices of the parameters.

Remark 2.18.

Theorem 2.19. Let satisfies

For some and all , then , where

Proof. Suppose that

is true. For some and all , then it is suficient to prove that

For , then define the function by

Then

Put and in , then this class reduces as , which was studied by Thirucheran and Stalin [24].

Corollary 2.20. Let satisfies

For some , and all , then , where

If and , then the class reduces to

which analogs to the class introduced by Sevtap Sumer Eker and Ozlem Guney[20].

Corollary 2.21. Let satisfies the inequality

for some , then ， where

If , and , then the class given the class ， which is discussed by Sevtap Sumer Eker and Owa [19]

Corollary 2.22. Let satisfies

for some , then , where

We define the subclass , which consists of the function

Now we determine the extreme points of the subclass .

Theorem 2.23.Let and

then if and only if it can be expressed in the form

and

Let and , then the class reduces and analogs the class, which is examined by Thirucheran and Stalin [24].

Corollary 2.24. Let and

then if and only if it can be expressed in the form

and

If and , we get the result of the class introduced by Sevtap Sumer Eker and Ozlem Guney [20].

Corollary 2.25. Let and

then if and only if it can be expressed

Theorem 2.26.Let and suppose that is defined by

If there exists an analytic function given by

then

and , then the class reduces and analogs to the the class studied by Thirucheran and Stalin [24].

Corollary 2.27. Let and suppose that is defined by

If there exists an analytic function given by

then

In 1933, Fekete-Szego [10] obtained the maximum value of as a function of the real parameter , for the function of class . Since then, the various authors were investigated and obtained the Fekete-Szego inequalities for diferent subclasses of the class $[25], [10], [26], [27], [28], [29], [30], [31] .In this article, we introduced two new subclasses of univalent functions which are defined by using Raducanu-Ohran diferential operator in the open unit disc. For these subclasses, we obtain the Fekete-Szego inequality .

If replacing special values for the subclass, we obtained Several well known subclasses.

Remark 2.28.

Remark 2.29.

Lemma 2.30. If is a function with positive real part in and is a complex number, then

The result is sharp for the function is given by and .

Lemma 2.31. If . is a function with positive real part in , then

when 1 \end{document} ]]>, the equality holds if and only if or one of its rotations.

If , then the equality holds if and only if or one of its rotations.

If , the equality holds if and only if or one of its rotation.

If , the equality holds if and only if is the reciprocal of one of the function such that equality holds in case of .

Also the above upper bound is sharp and it can be improved as follows: when

Theorem 2.32. Let , with . satisfies the inequality

then

where and

Then, the result is sharp.

Proof. If , then there is a Schwarz function , analytic in with and in such that

Define by

Since is a Schwarz function, it is clear that 0 \end{document} ]]> and .

Using Lemma (5.2.2), we get

and

where

Hence

Therefore, the result (40) is sharp for the function defined by

Theorem 2.33. Let with . If satisfies the inequality

then

where

and

Proof. Let , then

Let , then

Let , then

To show that the bounds are sharp, for the functions well defined by

well defined by

and

It is clear that

belongs to the class

.

If \end{document} ]]>, then equality holds if and only if is or one of its rotations. If , then equality holds if and only if is or one of its rotations. , then equality holds if and only if is or one of its rotations. If , then equality holds if and only if is or one of its rotations.

Theorem 2.34. let , with . If satisfies the inequality

then

where

and

Then, the result is sharp.