Let ℳ denote the class of functions which are holomorphic in , where
By , where p, k are integer, , we denote the class of functions of the form
We note that for we have the class of functions which are meromorphic in , , and for we obtain the class of functions which are analytic in .
Let , , . A function is said to be convex of order α in if
A function is said to be starlike of order α in if
We denote by the class of all functions , which are convex of order α in and by we denote the class of all functions , which are starlike of order α in .
Let , . We define the radius of starlikeness of order α and the radius of convexity of order α for the class ℬ by
We say that a function is subordinate to a function , and write (or simply ), if there exists a function (, , ), such that
In particular, if F is univalent in , we have the following equivalence:
For functions of the form
by we denote the Hadamard product (or convolution) of f and g, defined by
For multivalent function , the normalization
is classical. One can obtain interesting results by applying Montel’s normalization (cf. ) of the form
where ρ is a fixed point from the unit disk . We see that for the normalization (4) is the classical normalization (3).
Let us denote by the class of functions with Montel’s normalization (4). It will be called the class of functions with two fixed points.
Also, by , , we denote the class of functions of the form
In particular, we obtain the class of functions with negative coefficients. Moreover, we define
The classes and are called the classes of functions with varying argument of coefficients. The class was introduced by Silverman  (see also ). It is easy to show that for , , the condition (2) is equivalent to the following:
Let A, B, δ be real parameters, , , , and let .
By we denote the class of functions such that
If , then the function
is univalent in and maps onto the disk , where
Thus, by definition of subordination the condition (9) is equivalent to the following:
If , then the function (10) maps the disc onto the half-plane . Thus, the condition (9) is equivalent to the following:
Now, we define the classes of functions with varying argument of coefficients related to the class . Let us denote
The class unifies various new and also well-known classes of analytic or meromorphic functions; see for example [1–36].
For the presented investigations we assume that φ, ϕ are the functions of the form
Moreover, let us put
The object of the present paper is to investigate the coefficients estimates, distortion properties, the radii of starlikeness and convexity, subordination theorems, partial sums and integral mean inequalities for the classes of functions with varying argument of coefficients. Some remarks depicting consequences of the main results are also mentioned.