Classes of meromorphic multivalent functions with Montel ’ s normalization

In the paper we define classes of meromorphic multivalent functions with Montel’s normalization. We investigate the coefficients estimates, distortion properties, the radius of starlikeness, subordination theorems and partial sums for the defined classes of functions. Some remarks depicting consequences of the main results are also mentioned.

One can obtain interesting results by applying normalization related to the Montel's normalization (cf.[7]) of the form where ρ is a fixed real number, −1 < ρ < 1.
We denote by M η ρ (p, k) the classes of functions f ∈ M (p, k) with Montel's normalization (3).
Also, by T η (p, k) (η ∈ R) we denote the class of functions f ∈ M (p, k) of the form ( 4) For η = 0 we obtain the class T 0 (p, k) of functions with positive coefficients.Finally, motivated by Silverman [10], we define the class ( 5) which is called the class of functions with varying argument of coefficients.Let α ∈ 0, p), r ∈ (0, 1 .A function f ∈ M (p, k) is said to be starlike of order α in D(r) if (6) Re zf (z) f (z) < −α (z ∈ D(r)).
We denote by MS * p (α) the class of all functions f ∈ M (p, p + 1) , which are starlike of order α in D.
It is easy to show that for a function f from the class T (p, k) the condition ( 6) is equivalent to the following (7) zf (z) Let B be a subclass of the class M (p, k).We define the radius of starlikeness of order α for the class B by Let functions f, F be analytic in U. We say that the function f is subordinate to the function F , and write f (z) ≺ F (z) (or simply f ≺ F ), if and only if there exists a function In particular, if F is univalent in U, we have the following equivalence For functions f, g ∈ M of the form by f * g we denote the Hadamard product (or convolution of f and g), defined by and ( 9) For the presented investigations we assume that ϕ, φ are functions of the form (10) ϕ(z where the sequences {α n }, {β n } are nonnegative real, and The family W (p, k; φ, ϕ; A, B) unifies a lot of new and also well-known classes of meromorphic functions.We list a few of them in the last section.
The object of the present paper is to investigate the coefficients estimates, distortion properties, the radius of starlikeness, subordination theorems and partial sums for the defined classes of functions.Some remarks depicting consequences of the main results are also mentioned.
Thus, it is sufficient to prove that , and using ( 12), we have Proof.In view of Theorem 1 we need only to show that each function f from the class T W η (p, k; φ, ϕ; A, B) satisfies the coefficient inequality (12).Let f ∈ T W η (p, k; φ, ϕ; A, B).Then by ( 13) and (1), we have Thus, putting z = re iη (0 ≤ r < 1), and applying (4), we obtain ( 14) It is clear that the denominator of the left hand said can not vanish for r ∈ 0, 1).Moreover, it is positive for r = 0, and in consequence for r ∈ 0, 1).Thus, by (14) we have which, upon letting r → 1 − , readily yields the assertion (12).
By applying Theorem 2 we can deduce the following result.
where {d n } is defined by (11).
Proof.For a function f of the form (1) with the normalization (3), we have Applying the equality (16) to (12), we obtain the assertions (15).
By applying Theorem 3 we obtain the following lemma.
Lemma 1.Let {d n } be defined by (11), −1 < ρ < 1, and let us assume that there exists an integer n 0 (n Then the function belongs to the class T W η ρ (p, k; φ, ϕ; A, B) for all positive real numbers a.Moreover, for all n (n ∈ N k ) such that the functions belong to the class T W η ρ (p, k; φ, ϕ; A, B).By Lemma 1 and Theorem 3, we have the following theorem.
Theorem 4. Let a function f of the form (4) belong to the class T W η ρ (p, k; φ, ϕ; A, B) and let {d n } be defined by (11).Then all of the coefficients a n for which are unbounded.Moreover, if there exists an integer n 0 (n 0 ∈ N k ) such that then all of the coefficients of the function f are unbounded.In the remaining cases The result is sharp, the functions f n of the form are the extremal functions.
By putting ρ = 0 in Theorem 3 and Corollary 4, we have the corollaries listed below.
Corollary 1.Let f ∈ T η (p, k) be a function of the form (2). Then f belongs to the class T W η 0 (p, k; φ, ϕ; A, B) if and only if where {d n } is defined by (11).

Corollary 2. If a function f of the form (2) belongs to the class
where d n is defined by (11).The result is sharp.The functions f n,η of the form are the extremal functions.

Distortion theorems.
From Theorem 2 we have the following lemma.
Lemma 2. Let a function f of the form (1) belong to the class T W η ρ (p, k; φ, ϕ; A, B).If the sequence {d n } defined by (11) satisfies the inequality Remark 1.The second part of Lemma 2 can be rewritten in terms of σ-neighborhood N σ defined by where Moreover, if (23) holds, then The result is sharp, with the extremal function f k,η of the form (21) and Proof.Suppose that the function f of the form (1) belongs to the class T W η ρ (p, k; φ, ϕ; A, B).By Lemma 2 we have If r ≤ ρ, then we obtain |f (z)| ≥ r −p .If r > ρ, then the sequence {(ρ n+p − r n+p )} is decreasing and negative.Thus, by (27), we obtain and we have the assertion (24).Making use of Lemma 2, in conjunction with (16), we readily obtain the assertion (26) of Theorem 5.

Corollary 4.
Let a function f belong to the class T W η 0 (p, k; φ, ϕ; A, B) and let the sequence {d n } be defined by (11).
Moreover, if The result is sharp, with the extremal function f k,η of the form (21).

The radius of starlikeness.
Theorem 6.The radius of starlikeness of order α for the class T W η (p, k; φ, ϕ; A, B) is given by , where d n is defined by (11).The functions f n,η of the form are the extremal functions.
Proof.A function f ∈ T η (p, k) of the form (1) is starlike of order α in D(r), 0 < r ≤ 1, if and only if it satisfies the condition (7).Since By Theorem 2, we have Thus, the condition (34) is true if The functions f n,η of the form (33) realize the equality in (35), and the radius r cannot be larger.Thus we have (32).
By Theorem 6 we have the following result.
Corollary 5. Let the sequence {d n + (B − A) ρ n+p }, where {d n } is defined by (11), be positive.The radius of starlikeness of order α for the class T W η ρ (p, k; φ, ϕ; A, B) is given by The functions f n,η of the form (21) are the extremal functions.

Subordination results.
Before stating and proving our subordination theorems for the class T W η (p, k; φ, ϕ; A, B) we need the following definition and lemma: Definition 1.A sequence {b n } of complex numbers is said to be a subordinating factor sequence if for each function f of the form (1) from the class S c we have
If p and k are odd, and η = 0, then the constant factor ε cannot be replaced by a larger number.
Proof.Let a function f of the form (1) belong to the class T W η (p, k; φ, ϕ; A, B) and suppose that a function g of the form belongs to the class S c .Then Thus, by Definition 1 the subordination result (39) holds true if {b n } is the subordinating factor sequence.By Lemma 2 we have Thus, by using Theorem 2 we obtain This evidently proves the inequality (38) and hence the subordination result (39).The inequality (40) follows from (39) by taking Next, we observe that the function f k,η of the form (33) belongs to the class T W η (p, k; φ, ϕ; A, B).If p and k are odd, and η = 0, then and the constant (41) cannot be replaced by any larger one.
hold true.If p and k are odd, and η = 0, then the constant factor ε in (42) cannot be replaced by a larger number.
6. Partial sums.Let f be a function of the form (1). Due to Silverman [9] and Silvia [11] we investigate the partial sums f m of the function f defined by In this section we consider partial sums of functions in the class T W η (p, k; φ, ϕ; A, B) and obtain sharp lower bounds for the ratios of real part of f to f m and f to f m .Theorem 8. Let the sequence {d n } defined by (11) be increasing and If a function f of the form (1) belongs to the class T W η (p, k; φ, ϕ; A, B) , then The bounds are sharp, with the extremal functions f n,η defined by (33).Thus, we have Re g(z) ≥ 0 (z ∈ U ) , which by (48) readily yields the assertion (45) of Theorem 8. Similarly, if we take

Proof. Since
and making use of (47), we can deduce that

2 .Theorem 1 .
Coefficients estimates.We first mention a sufficient condition for a function to belong to the class W (p, k; φ, ϕ; A, B).Let {d n } be defined by(11), −1 ≤ A < B ≤ 1.If a function f ∈ M (p,k) of the form (1) satisfies the condition (12) ∞ n=k d n |a n | ≤ (B − A) a −p , then f belongs to the class W (p, k; φ, ϕ; A, B).Proof.A function f of the form (1) belongs to the class W (p, k; φ, ϕ; A, B) if and only if there exists a function ω, |ω(z)| ≤ |z| (z ∈ D) , such that be a function of the form(4).Then f belongs to the class T W η (p, k; φ, ϕ; A, B) if and only if the condition(12) holds true.