Research

# Reciprocal classes of p-valently spirallike and p-valently Robertson functions

Neslihan Uyanik1, Hitoshi Shiraishi2, Shigeyoshi Owa2* and Yasar Polatoglu3

Author Affiliations

1 Department of Mathematics, Kazim Karabekir Faculty of Education, Ataturk University, Erzuram 25240, Turkey

2 Department of Mathematics, Kinki University, Higashi-Osaka, 577-8502 Osaka, Japan

3 Department of Mathematics and Computer Sciences, Faculty of Science and Letters, Istanbul Kultur University, Istanbul, Turkey

For all author emails, please log on.

Journal of Inequalities and Applications 2011, 2011:61  doi:10.1186/1029-242X-2011-61

 Received: 10 April 2011 Accepted: 18 September 2011 Published: 18 September 2011

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

For p-valently spirallike and p-valently 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 lemma

### 1 Introduction

Let be the class of functions f(z) of the form

(1.1)

which are analytic in the open unit disk .

For , we say that f(z) belongs to the class if it satisfies

(1.2)

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

(1.3)

for some real and β (β > p cos α).

We note that if and only if , and that, if and only if .

Remark 1 If satisfies

then we say that f(z) is p-valently spirallike in (cf. [1]). Also, if satisfies

then f(z) is said to be p-valently Robertson function in (cf. [3,4]). Therefore, defined by (1.2) is the reciprocal class of p-valently spirallike functions in , and defined by (1.3) is the reciprocal class of p-valently Robertson functions in .

Let be the class of functions p(z) of the form

(1.4)

that are analytic in and satisfy . A function is called the Carathéodory function and satisfies

(1.5)

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

(1.6)

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

(2.1)

for some real and β (β > p cos α).

The result is sharp for f(z) given by

(2.2)

Proof. Let . If we define the function w(z) by

(2.3)

then we know that w(z) is analytic in , w(0) = 0, and

(2.4)

Therefore, we have that . If follows from (2.3) that

(2.5)

which is equivalent to the subordination (2.1).

Conversely, we suppose that the subordination (2.1) holds true. Then, we have that

(2.6)

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

(2.7)

we conclude that

(2.8)

which shows that .

Finally, we consider the function f(z) given by (2.2). Then, f(z) satisfies

(2.9)

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

(2.10)

for some real and β (β > p cos α).

The result is sharp for f(z) given by

(2.11)

### 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

(3.1)

The result is sharp for

(3.2)

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

(3.3)

is the Carathéodory function. If we write that

(3.4)

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

(3.5)

This gives us that

(3.6)

which implies that

(3.7)

It follows from (3.7) that

(3.8)

If n = p + 1, then we have that

(3.9)

If n = p + 2, then we also have that

(3.10)

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

(3.11)

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

(3.12)

Thus, this function f(z) satisfies the equality in (3.1).   □

Corollary 2 If f(z) belongs to the class , then

(3.13)

The result is sharp for f(z) defined by

(3.14)

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

(3.15)

and

(3.16)

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 .

Theorem 3 If , then we have

(4.1)

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

(4.2)

Letting z = re(0 ≦ r < 1), we see that

(4.3)

Let us define

(4.4)

Then, we know that h'(t) ≧ 0. This implies that

(4.5)

which is equivalent to

(4.6)

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

Corollary 3 If , then

(4.7)

for | z | = r < 1. The equalities hold true for

(4.8)

Corollary 4 If , then we have

(4.9)

for | z | = r < 1. The equalities hold true for f(z) defined by (2.11).

Corollary 5 If , then

(4.10)

for | z | = r < 1. The equalities hold true for f(z) defined by

(4.11)

### 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

(5.1)

then we can write

(5.2)

where k is real and k ≧ 1.

Applying Lemma 1, we derive

Theorem 4 If satisfies

(5.3)

for some real β > p, then .

Proof. Let us define the function w(z) by

(5.4)

Then we see that w(z) is analytic in and w(0) = 0.

It follows from (5.4) that

(5.5)

We suppose that there exists a point such that

Then, Lemma 1 gives us that w(z0) = eand z0w'(z0) = ke. For such a point z0, we have that

(5.6)

This contradicts our condition (5.3). Therefore, there is no such that |w (z0) | = 1. This implies that , that is, that

(5.7)

Thus, we observe that .   □

Further, we derive

Theorem 5 If for some real , then

(5.8)

Proof. We consider the function w(z) such that

(5.9)

for and for .

Then, we know that

(5.10)

for .

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(z0) = eand z0w'(z0) = ke(k ≧ 1). This gives us that

(5.11)

which contradicts the inequality (5.10). Thus, there is no point such that |w (z0) | = 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

(5.12)

Finally, we consider the coefficient estimates for functions f(z) to be in the classes and .

Theorem 6 If satisfies

(5.13)

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

(5.14)

Then . It follows that

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

Corollary 7 If satisfies

for some real , then .

Further, we have

Theorem 7 If satisfies

for some real , β (β > p cos α) and k (0 ≦ k p cos α), then

Corollary 8 If satisfies

for some real , then .

### Acknowledgements

This paper was completed when the first author was visiting Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan, between February 17 and February 26, 2011.

### References

1. Polatoglu, Y, Blocal, M, Sen, A, Yavuz, E: An investigation on a subclass of p-valently starlike functions in the unit disc. Turk J Math. 31, 221–228 (2007)

2. Owa, S, Nishiwaki, J: Coefficient estimates for certain classes of analytic functions. J Inequal Pure Appl Math. 3, 1–5 (2002)

3. Aouf, MK, Al-Oboudi, FM, Haidan, MM: On some results for λ-spirallike and λ-Robertson functions of complex order. Publ Inst Math. 75, 93–98 (2005)

4. Robertson, MS: Univalent functions f(z) for which zf'(z) is spirallike. Michigan Math J. 16, 315–324 (1969)

5. Duren, PL: Univalent Functions. Springer, New York (1983)

6. Miller, SS, Mocanu, PT: Second order differential inequalities in the complex plane. J Math Anal Appl. 65, 289–305 (1978). Publisher Full Text

7. Jack, IS: Functions starlike and convex of order α. J Lond Math Soc. 3, 469–474 (1971). Publisher Full Text