Certain New Classes of Analytic Functions with Varying Arguments

We introduce certain new classes and , which represent the κ uniformly starlike functions of order α and type β with varying arguments and the κ uniformly convex functions of order α and type β with varying arguments, respectively. Moreover, we give coefficients estimates, distortion theorems, and extreme points of these classes.


Introduction
Let  denote the class of functions of the following form: that are analytic and univalent in the open unit disc  =   ℂ ∶ ||  .
e class   ST(  denote the class of  uniformly starlike functions of order  and type  and the class   UCV(  denotes the class of  uniformly convex functions of order  and type .
Also we note that which are the uniformly starlike functions of order  and type  and uniformly convex functions of order  and type , respectively.
�e�nition � (see [11]).A function () of the form ( 1) is said to be in the class (  ) if  ∈  and arg(  ) =   for all   .If furthermore there exist a real number  such that   + ( − 1)   ( ), then () is said to be in the class (  , ).e union of (  , ) taken over all possible sequences {  } and all possible real numbers  is denoted by .
In this paper we obtain coefficient bounds for functions in the classes  − ST(, ) and  − UCV(, ), respectively, further we obtain distortion bounds and the extreme points for functions in these classes.
We shall need the following lemmas.
By using ( 4) and ( 6) we can obtain the following lemma.In the following theorems, we show that the conditions ( 6) and ( 16) are also necessary for functions ()    ST( ) and   UCV( ), respectively.eorem 6.Let () be of the form (1) (32) Similarly, we get is completes the proof of eorem 10.Finally the result is sharp for the following function: at   || − 2 .
Corollary 11.Under the hypotheses of eorem 8, () is included in a disc with center at the origin and radius  1 given by eorem 12. Let the function () de�ned by (1) be in the class  − ST(, ).en e result is sharp.
Proof.Similarly Φ() is an increasing function of  ( ≥ 2), in view of eorem 6, we have that is, us we have Similarly Finally, we can see that the assertions of eorem 12 are sharp for the function () de�ned �y (34).is completes the proof of eorem 12.
Corollary 13.Under the hypotheses of eorem 12,  ′ () is included in a disc with center at the origin and radius  2 given by Using the same technique used in eorems 10 and 12, we get the following theorems.( e result is sharp for the function given by (43).