New Subclasses concerning Some Analytic and Univalent Functions
Considering a function which is analytic and starlike in the open unit disc and a function which is analytic and convex in we introduce two new classes and concerning . The object of the present paper is to discuss some interesting properties for functions in the classes and
1. Introduction and Preliminaries
Let be the class of functions which are analytic in the open unit disk with and .
Let denote the subclass of consisting of functions which are univalent in . Also, let be the subclass of consisting of which are starlike of order in . Further, we say that if satisfies . A function is said to be convex of order in (cf. [1–3]).
With the above definitions for classes , , , and , it is known thatand if and only if
The function given byis in the class and the function given byis in the class .
With the function given by (4), we introduce a class of analytic functions with series expansion in such thatfor some real , where we take the principal value for If satisfiesfor some real , then we say that
Also, if satisfiesfor some real , then we say that
With the above definitions for the classes and , we have that if and only if and that if and only if .
2. Some Properties
In this section, we consider some properties of functions with series expansion given by (4).
Theorem 1. If is given by (4), then for and for .
Proof. For given by (4), we see that for andfor This shows that for Further, we have that for andfor Lettingwe have thatThus, we see thatfor . This completes the proof of the theorem.
Corollary 2. A functionbelongs to the class and
Next, we discuss some properties of functions for
Theorem 3. If given by (5) satisfiesfor some , then
The equality holds true for given by
Further, we obtain the following.
Theorem 5. Let be given by (5) with Then, if and only iffor some The equality holds true for
Proof. Theorem 3 implies that if satisfies (20), then Next, we suppose that Then,If we consider , then we have thatThen, we obtain thatThis gives us that is, Thus, if and only if the coefficient inequality (20) holds true.
Further, for the class , we have the following.
Theorem 6. Let be given by (5) with Then, if and only iffor some The equality holds true for
3. Radius Problems
In this section, we considerfor some real . Then, we say that and for any real
Now, we derive the following.
Theorem 7. If is given by (29) with , then
Proof. For given by (29), we have thatfor . This gives us Lettingwe see that This gives us
Corollary 8. If is given by (29) with , thenfor
Proof. If we considerthen
Remark 9. If in (35), thenand if , then
4. Partial Sums
Finally, we consider the partial sums of given by (5). In view of (5), we writefor some real Recently, Darus and Ibrahim  and Hayami et al.  have shown some interesting results for some partial sums of analytic functions.
Now, we derive the following.
Theorem 10. Let be given by (40) with Then,
Corollary 11. Let be given by (40) with Then,
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The work here is supported by MOHE Grant FRGS/1/2016/STG06/UKM/01/1.
P. L. Duren, Univalent Functions, vol. 259, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983.
A. W. Goodman, Geometric Theory of Functions, vol. I and II, Mariner, Tampa, Fla, USA, 1983.View at: MathSciNet