Classes of multivalent analytic and meromorphic functions with two fixed points
© Dziok; licensee Springer 2013
Received: 20 December 2012
Accepted: 18 March 2013
Published: 5 April 2013
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.
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 .
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 .
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.
Let A, B, δ be real parameters, , , , and let .
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 .
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.
which, upon letting , readily yields the assertion (15). □
By applying Theorem 2, we can deduce following result.
Then the conditions (15) and (16) are equivalent. □
From Theorem 3, we obtain the following lemma.
By Lemma 1 and Theorem 3, we have following two corollaries.
is the extremal function.
then all of the coefficients of the class are unbounded.
By putting in Theorem 3 and Corollary 1, we have the corollaries listed below.
are the extremal functions.
3 Distortion theorems
From Theorem 2, we have the following lemma.
as the following corollary.
The result is sharp, with the extremal function of the form (21) and .
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.
The result is sharp, with the extremal function of the form (24).
4 The radii of convexity and starlikeness
are the extremal functions.
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.
The functions of the form (33) are the extremal functions.
Thus, by Theorems 5 and 6 we have the following corollary.
5 Subordination results
Before stating and proving our subordination theorems for the class , we need the following definition and lemma.
Lemma 3 
If p and are odd, and , then the constant factor ε cannot be replaced by a larger number.
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 , we obtain integral means inequalities for the functions from the class .
Lemma 4 
Applying Lemma 4 and Theorem 2, we prove the following result.
where is defined by (33).
Thus, by definition of subordination we have (44) and this completes the proof. □
By using (17) in Theorem 8 we have the following corollary.
where is defined by (21).
7 Partial sums
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 .
The bounds are sharp, with the extremal functions defined by (21).
This completes the proof. □
The bounds are sharp, with the extremal functions defined by (21).
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
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 , and Wisniowska et al.  and , respectively (see also ).
was introduced and studied by Raina and Bansal .
defined by Srivastava et al. .
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].
Dedicated to Professor Hari M Srivastava.
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