Abstract
For pvalently spirallike and pvalently Robertson functions in the open unit disk , reciprocal classes , and are introduced. The object of the present paper is to discuss some interesting properties for functions f(z) belonging to the classes and .
2010 Mathematics Subject Classification
Primary 30C45
Keywords:
Reciprocal class; Subordination; Schwarz function; Robertson function; Miller and Mocanu lemma1 Introduction
Let be the class of functions f(z) of the form
which are analytic in the open unit disk .
For , we say that f(z) belongs to the class if it satisfies
for some real and β (β > p cos α).
When α = 0, the class was studied by Polatoglu et al. [1], and the classes and were introduced by Owa and Nishiwaki [2].
Further, let denote the subclass of consisting of functions f(z), which satisfy
for some real and β (β > p cos α).
We note that if and only if , and that, if and only if .
then we say that f(z) is pvalently spirallike in (cf. [1]). Also, if satisfies
then f(z) is said to be pvalently Robertson function in (cf. [3,4]). Therefore, defined by (1.2) is the reciprocal class of pvalently spirallike functions in , and defined by (1.3) is the reciprocal class of pvalently Robertson functions in .
Let be the class of functions p(z) of the form
that are analytic in and satisfy . A function is called the Carathéodory function and satisfies
with the equality for (cf. [5]).
For analytic functions g(z) and h(z) in , we say that g(z) is subordinate to h(z) if there exists an analytic function w(z) in with w(0) = 0 and , and such that g(z) = h(w(z)). We denote this subordination by
If h(z) is univalent in , then this subordination (1.6) is equivalent to g(0) = h(0) and (cf. [5]).
2 Subordinations for classes
We consider subordination properties of function f(z) in the classes and .
Theorem 1 A function f(z) belongs to the class if and only if
for some real and β (β > p cos α).
The result is sharp for f(z) given by
Proof. Let . If we define the function w(z) by
then we know that w(z) is analytic in , w(0) = 0, and
Therefore, we have that . If follows from (2.3) that
which is equivalent to the subordination (2.1).
Conversely, we suppose that the subordination (2.1) holds true. Then, we have that
for some Shwarz function w(z), which is analytic in , w(0) = 0, and . It is easy to see that the equality (2.6) is equivalent to the equality (2.3). Since
we conclude that
Finally, we consider the function f(z) given by (2.2). Then, f(z) satisfies
This completes the proof of the theorem. □
Noting that if and only if , we also have
Corollary 1 A function f(z) belongs to the class if and only if
for some real and β (β > p cos α).
The result is sharp for f(z) given by
3 Coefficient inequalities
Applying the properties for Carathéodory functions, we discuss the coefficient inequalities for f(z) in the classes and .
Theorem 2 If f(z) belongs to the class , then
The result is sharp for
for α = 0.
Proof. In view of Theorem 1, we can consider the function w(z) given by (2.3) for . Since w(z) is the Schwarz function, the function q(z) defined by
is the Carathéodory function. If we write that
then we see that
and the equality holds true for and its rotation. It is to be noted that the equation (3.3) is equivalent to
This gives us that
which implies that
It follows from (3.7) that
If n = p + 1, then we have that
If n = p + 2, then we also have that
Thus, the coefficient inequality (3.1) is true for n = p + 1 and n = p + 2. Next, we suppose that (3.1) holds true for n = p + 1, p + 2, p + 3, ..., p + k  1. Then
This means that the inequality (3.1) holds true for n = p + k. Therefore, by the mathematical induction, we prove the coefficient inequality (3.1).
Finally, let us consider the function f(z) given by (3.2). Then, f(z) can be written by
Thus, this function f(z) satisfies the equality in (3.1). □
Corollary 2 If f(z) belongs to the class , then
The result is sharp for f(z) defined by
for α = 0.
Remark 2 We know that the extremal functions for is f(z) given by (2.2) and for is f(z) given by (2.11). But, we see that
and
for such functions.
Therefore, the extremal functions for and do not satisfy the equalities in (3.1) and (3.13), respectively.
Furthermore, if we consider α = 0 in Theorem 2, then we obtain the corresponding result due to Polatoglu et al. [1].
4 Inequalities for the real parts
We discuss some problems of inequalities for the real parts of .
for  z  = r < 1. The equalities hold true for f(z) given by (2.2).
Proof. By virtue of Theorem 1, we consider the function g(z) defined by
Letting z = re^{iθ }(0 ≦ r < 1), we see that
Let us define
Then, we know that h'(t) ≧ 0. This implies that
which is equivalent to
Noting that by Theorem 1 and g(z) is univalent in , we prove the inequality (4.1). Since the subordination (2.1) is sharp for f(z) given by (2.2), we say that the equalities in (4.1) are attained by the function f(z) given by (2.2). □
Taking α = 0 in Theorem 3, we have
for  z  = r < 1. The equalities hold true for
for  z  = r < 1. The equalities hold true for f(z) defined by (2.11).
for  z  = r < 1. The equalities hold true for f(z) defined by
5 Sufficient conditions
We consider some sufficient conditions for f(z) to be in the classes and .
To discuss our sufficient conditions, we have to recall here the following lemma by Miller and Mocanu [6] (also due to Jack [7]).
Lemma 1 Let w(z) be analytic in with w(0) = 0. If there exists a point such that
then we can write
where k is real and k ≧ 1.
Applying Lemma 1, we derive
Proof. Let us define the function w(z) by
Then we see that w(z) is analytic in and w(0) = 0.
It follows from (5.4) that
We suppose that there exists a point such that
Then, Lemma 1 gives us that w(z_{0}) = e^{iθ }and z_{0}w'(z_{0}) = ke^{iθ}. For such a point z_{0}, we have that
This contradicts our condition (5.3). Therefore, there is no such that w (z_{0})  = 1. This implies that , that is, that
Further, we derive
Theorem 5 If for some real , then
Proof. We consider the function w(z) such that
Then, we know that
Since w(z) is analytic in and w(0) = 0, we suppose that there exists a point such that
Then, applying Lemma 1, we can write that w(z_{0}) = e^{iθ }and z_{0}w'(z_{0}) = ke^{iθ }(k ≧ 1). This gives us that
which contradicts the inequality (5.10). Thus, there is no point such that w (z_{0})  = 1. This means that , and that,
This completes the proof of the theorem. □
Letting instead of f(z) in Theorem 5, we have
Corollary 6 If for some , Then
Finally, we consider the coefficient estimates for functions f(z) to be in the classes and .
for some real , β (β > p cos α), and k (0 ≦ k ≦ p cos α), then
Proof. It is to be noted that if satisfies
Therefore, if f(z) satisfies the coefficient estimate (5.13), then we know that f(z) satisfies the inequality (5.14). This completes the proof of the theorem. □
Letting α = 0 and k = p in Theorem 6, we have
Further, we have
for some real , β (β > p cos α) and k (0 ≦ k ≦ p cos α), then
Acknowledgements
This paper was completed when the first author was visiting Department of Mathematics, Kinki University, HigashiOsaka, Osaka 5778502, Japan, between February 17 and February 26, 2011.
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