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Classes of multivalent analytic and meromorphic functions with two fixed points
Fixed Point Theory and Applications volume 2013, Article number: 86 (2013)
Abstract
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 classes of functions with two fixed points. Some remarks depicting consequences of the main results are also mentioned.
MSC:30C45, 30C50, 30C55.
1 Introduction
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
respectively.
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. [1]) 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 [2] (see also [3]). 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
and
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.
2 Coefficients estimates
We first mention a sufficient condition for the function to belong to the class .
Theorem 1 Let and . If and
then .
Proof If , then we have
Thus, by (15), we obtain (11) and consequently . Let now . Then simply calculations give
Thus, by (15) we obtain (12). Hence and the proof is complete. □
Theorem 2 Let . Then if and only if the condition (15) holds true.
Proof Let . In view of Theorem 1, we need only show that f satisfies the coefficient inequality (15). Putting in the conditions (11) and (12) we obtain
By (8), it is clear that the denominator of the left hand side cannot vanish for . Moreover, it is positive for , and in consequence for . Thus, we have
which, upon letting , readily yields the assertion (15). □
By applying Theorem 2, we can deduce following result.
Theorem 3 Let . Then if and only if it satisfies (4) and
Proof For a function with the normalization (4), we have
Then the conditions (15) and (16) are equivalent. □
From Theorem 3, we obtain the following lemma.
Lemma 1 Let there exist an integer such that
Then the function
belongs to the class for all positive real numbers a. Moreover, for all such that
the functions
belongs to the class for all positive real numbers a and
By Lemma 1 and Theorem 3, we have following two corollaries.
Corollary 1 Let
If
then the nth coefficient of the class satisfies the following inequality:
The estimation (20) is sharp, the function of the form
is the extremal function.
Corollary 2 If
then the nth coefficient of the class is unbounded. Moreover, if there exists such that
then all of the coefficients of the class are unbounded.
By putting in Theorem 3 and Corollary 1, we have the corollaries listed below.
Corollary 3 Let . Then if and only if
Corollary 4 If , then
The result is sharp. The functions of the form
are the extremal functions.
3 Distortion theorems
From Theorem 2, we have the following lemma.
Lemma 2 Let . If the sequence satisfies the inequality
then
Moreover, if
then
The second part of Lemma 2 may be formulated in terms of σ-neighborhood defined by
as the following corollary.
Corollary 5 If the sequence satisfies (26), then , where
Theorem 4 Let , . If the sequence satisfies (25), then
where
Moreover, if (26) holds, then
The result is sharp, with the extremal function of the form (21) and .
Proof Suppose that the function f of the form (1) belongs to the class . By Lemma 2 we have
and
If , then we obtain . If , then the sequence is decreasing and negative. Thus, by (30), we obtain
and we have the assertion (27). Making use of Lemma 2, in conjunction with (17), we readily obtain the assertion (29) of Theorem 4. □
Putting in Theorem 4 we have the following corollary.
Corollary 6 Let , . If (), then
Moreover, if (), then
The result is sharp, with the extremal function of the form (24).
4 The radii of convexity and starlikeness
Theorem 5 If , then
The functions of the form
are the extremal functions.
Proof A function of the form (1) is starlike of order α in if and only if it satisfies the condition (7). Since
the condition (7) is true if
By Theorem 2, we have
Thus, the condition (34) is true if
that is, if
It follows that each function is starlike of order α in , where
The functions of the form (33) realize equality in (35), and the radius r cannot be larger. Thus we have (32). □
The following result may be proved in much the same way as Theorem 5.
Theorem 6 If , then
The functions of the form (33) are the extremal functions.
It is clear that for
the extremal function of the form (33) belongs to the class . Moreover, we have
Thus, by Theorems 5 and 6 we have the following corollary.
Corollary 7 Let the sequence be positive, . Then
5 Subordination results
Before stating and proving our subordination theorems for the class , we need the following definition and lemma.
Definition 1 A sequence of complex numbers is said to be a subordinating factor sequence if for each function we have
Lemma 3 [36]
A sequence is a subordinating factor sequence if and only if
Theorem 7 Let the sequence satisfy the inequality (25). If and , then
and
where
If p and are odd, and , 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 and suppose that a function g of the form
belongs to the class . Then
where
Thus, by Definition 1, the subordination result (39) holds true if is the subordinating factor sequence. By (25), 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 of the form (33) belongs to the class . If p and are odd, and , then
and the constant (41) cannot be replaced by any larger one. □
Remark 1 By using (17) in Theorem 7, we obtain the result related to the class . Moreover, by putting , we have the following corollary.
Corollary 8 Let the sequence satisfy the inequality (25). If and , then conditions (39) and (40) hold true. If p and are odd, and , then the constant factor cannot be replaced by a larger number.
6 Integral means inequalities
Due to Littlewood [22], we obtain integral means inequalities for the functions from the class .
Lemma 4 [22]
Let f, g be functions analytic in . If , then
Applying Lemma 4 and Theorem 2, we prove the following result.
Theorem 8 Let the sequence satisfy (25), . If , then
where is defined by (33).
Proof For function f of the form (1), the inequality (43) is equivalent to the following:
By Lemma 4, it suffices to show that
Setting
and using (25) and Theorem 2 we obtain
and
Thus, by definition of subordination we have (44) and this completes the proof. □
By using (17) in Theorem 8 we have the following corollary.
Corollary 9 Let the sequence satisfy (25), . If , then
where is defined by (21).
7 Partial sums
Let f be a function of the form (1). Due to Silvia [27], we investigate the partial sums of the function f defined by
In this section, we consider partial sums of functions in the class and obtain sharp lower bounds for the ratios of real part of f to and to .
Theorem 9 Let the sequence be increasing and . If , then
and
The bounds are sharp, with the extremal functions defined by (21).
Proof
Since
by Theorem 1, we have
Let
Applying (48), we find that
Thus, we have () and by (49) we have the assertion (46) of Theorem 9. Similarly, if we take
and making use of (48), we can deduce that
which leads us immediately to the assertion (47) of Theorem 9. In order to see that the function of the form (21) gives the results sharp, we observe that
This completes the proof. □
Theorem 10 Let the sequence be increasing and . If , then
The bounds are sharp, with the extremal functions defined by (21).
Proof
By setting
and
the proof is analogous to that of Theorem 9, and we omit the details. □
Remark 2 By using (17) in Theorems 9 and 10, we obtain the results related to the class .
8 Concluding remarks
We conclude this paper by observing that, in view of the subordination relation (9), choosing the functions ϕ and φ, we can consider new and also well-known classes of functions. Let , , and
The class generalize well-known classes, which were investigated in earlier works; see, for example, [5, 23, 28, 30]. In particular, the class contains functions , which satisfies the condition
It is related to the class of starlike functions with respect to n-symmetric points. Moreover, putting , we obtain the class defined by the following condition:
The class is related to the class of starlike functions. In particular, we have
Analogously, the class
contains functions , which satisfy the condition
It is related to the class of δ-uniformly convex functions of order γ with respect to n-symmetric points. Moreover, putting , we obtain the class defined by the following condition:
The class is related to the class of δ-uniformly convex functions of order γ. The classes
are the well-known classes of δ-starlike functions of order γ and δ-uniformly convex functions of order γ, respectively. In particular, the classes , were introduced by Goodman [18], and Wisniowska et al. [29] and [19], respectively (see also [20]).
We note that the class
was introduced and studied by Raina and Bansal [24].
If we set
where is the generalized hypergeometric function, then we obtain the class
defined by Srivastava et al. [26].
Let λ be a convex parameter. A function belongs to the class
if it satisfies the following condition:
Moreover, a function belongs to the class
if it satisfies the following condition:
The considered classes are defined by using the convolution or equivalently by the linear operator
By choosing the function φ, we can obtain a lot of important linear operators, and in consequence new and also well-known classes of functions. We can listed here some of these linear operators as the Salagean operator, the Cho-Kim-Srivastava operator, the Dziok-Raina operator, the Hohlov operator, the Dziok-Srivastava operator, the Carlson-Shaffer operator, the Ruscheweyh derivative operator, the generalized Bernardi-Libera-Livingston operator, the fractional derivative operator and so on (see, for the precise relationships [14, 17]).
If we apply the results presented in the paper to the classes discussed above, we can lead to several results. Some of these were obtained in earlier works; see, for example, [3–17, 21, 23–26, 30–35].
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
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Dziok, J. Classes of multivalent analytic and meromorphic functions with two fixed points. Fixed Point Theory Appl 2013, 86 (2013). https://doi.org/10.1186/1687-1812-2013-86
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DOI: https://doi.org/10.1186/1687-1812-2013-86
Keywords
- analytic functions
- varying arguments
- fixed points
- Montel’s normalization
- subordination
- Hadamard product