Open Access

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

  • Neslihan Uyanik1,
  • Hitoshi Shiraishi2,
  • Shigeyoshi Owa2Email author and
  • Yasar Polatoglu3
Journal of Inequalities and Applications20112011:61

https://doi.org/10.1186/1029-242X-2011-61

Received: 10 April 2011

Accepted: 18 September 2011

Published: 18 September 2011

Abstract

For p-valently spirallike and p-valently Robertson functions in the open unit disk U , reciprocal classes S p ( α , β ) , and C p ( α , β ) are introduced. The object of the present paper is to discuss some interesting properties for functions f(z) belonging to the classes S p ( α , β ) and C p ( α , β ) .

2010 Mathematics Subject Classification

Primary 30C45

Keywords

Reciprocal classSubordinationSchwarz functionRobertson functionMiller and Mocanu lemma

1 Introduction

Let A p be the class of functions f(z) of the form
f ( z ) = z p + n = p + 1 a n z n
(1.1)

which are analytic in the open unit disk U = { z : z < 1 } .

For f ( z ) A p , we say that f(z) belongs to the class S p ( α , β ) if it satisfies
Re e i α z f ( z ) f ( z ) < β ( z U )
(1.2)

for some real α α < π 2 and β (β > p cos α).

When α = 0, the class S p ( 0 , β ) was studied by Polatoglu et al. [1], and the classes S 1 ( 0 , β ) and C 1 ( 0 , β ) were introduced by Owa and Nishiwaki [2].

Further, let C p ( α , β ) denote the subclass of A p consisting of functions f(z), which satisfy
Re e i α 1 + z f ( z ) f ( z ) < β ( z U )
(1.3)

for some real α α < π 2 and β (β > p cos α).

We note that f ( z ) C p ( α , β ) if and only if z f ( z ) p S p ( α , β ) , and that, f ( z ) S p ( α , β ) if and only if p 0 z f ( t ) t d t C p ( α , β ) .

Remark 1 If f ( z ) A p satisfies
Re e i α z f ( z ) f ( z ) > 0 ( z U ) ,
then we say that f(z) is p-valently spirallike in U (cf. [1]). Also, if f ( z ) A p satisfies
Re e i α 1 + z f ( z ) f ( z ) > 0 ( z U ) ,

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

Let P be the class of functions p(z) of the form
p ( z ) = 1 + n = 1 c n z n ( z U )
(1.4)
that are analytic in U and satisfy Re p ( z ) > 0 ( z U ) . A function p ( z ) P is called the Carathéodory function and satisfies
c n 2 ( n = 1 , 2 , 3 , )
(1.5)

with the equality for p ( z ) = 1 + z 1 - z (cf. [5]).

For analytic functions g(z) and h(z) in U , we say that g(z) is subordinate to h(z) if there exists an analytic function w(z) in U with w(0) = 0 and w ( z ) < 1 ( z U ) , and such that g(z) = h(w(z)). We denote this subordination by
g ( z ) h ( z ) ( z U ) .
(1.6)

If h(z) is univalent in U , then this subordination (1.6) is equivalent to g(0) = h(0) and g ( U ) h ( U ) (cf. [5]).

2 Subordinations for classes

We consider subordination properties of function f(z) in the classes S p ( α , β ) and C p ( α , β ) .

Theorem 1 A function f(z) belongs to the class S p ( α , β ) if and only if
e i α z f ( z ) f ( z ) 2 β - p e - i α + 2 ( p cos α - β ) 1 - z ( z U )
(2.1)

for some real α α < π 2 and β (β > p cos α).

The result is sharp for f(z) given by
f ( z ) = z p ( 1 - z ) 2 e - i α ( p cos α - β ) .
(2.2)
Proof. Let f ( z ) S p ( α , β ) . If we define the function w(z) by
β - e i α z f ( z ) f ( z ) + i p sin α β - p cos α = 1 + w ( z ) 1 - w ( z ) ( w ( z ) 1 ) ,
(2.3)
then we know that w(z) is analytic in U , w(0) = 0, and
Re 1 + w ( z ) 1 - w ( z ) > 0 ( z U ) .
(2.4)
Therefore, we have that w ( z ) < 1 ( z U ) . If follows from (2.3) that
e i α z f ( z ) f ( z ) = 2 β - p e - i α + 2 ( p cos α - β ) 1 - w ( z ) ( z U ) ,
(2.5)

which is equivalent to the subordination (2.1).

Conversely, we suppose that the subordination (2.1) holds true. Then, we have that
e i α z f ( z ) f ( z ) = 2 β - p e - i α + 2 ( p cos α - β ) 1 - w ( z ) ( z U ) ,
(2.6)
for some Shwarz function w(z), which is analytic in U , w(0) = 0, and w ( z ) < 1 ( z U ) . It is easy to see that the equality (2.6) is equivalent to the equality (2.3). Since
Re 1 + w ( z ) 1 - w ( z ) = Re β - e i α z f ( z ) f ( z ) + i p sin α β - p cos α > 0 ( z U ) ,
(2.7)
we conclude that
Re β - e i α z f ( z ) f ( z ) > 0 ( z U ) ,
(2.8)

which shows that f ( z ) S p ( α , β ) .

Finally, we consider the function f(z) given by (2.2). Then, f(z) satisfies
e i α z f ( z ) f ( z ) = 2 β - p e - i α + 2 ( p cos α - β ) 1 - z .
(2.9)

This completes the proof of the theorem.   □

Noting that f ( z ) C p ( α , β ) if and only if z f ( z ) p S p ( α , β ) , we also have

Corollary 1 A function f(z) belongs to the class C p ( α , β ) if and only if
e i α 1 + z f ( z ) f ( z ) 2 β - p e - i α + 2 ( p cos α - β ) 1 - z ( z U )
(2.10)

for some real α α < π 2 and β (β > p cos α).

The result is sharp for f(z) given by
f ( z ) = p z p - 1 ( 1 - z ) 2 e - i α ( p cos α - β ) .
(2.11)

3 Coefficient inequalities

Applying the properties for Carathéodory functions, we discuss the coefficient inequalities for f(z) in the classes S p ( α , β ) and C p ( α , β ) .

Theorem 2 If f(z) belongs to the class S p ( α , β ) , then
a p + k 1 k ! j = 0 k - 1 ( 2 ( β - p cos α ) + j ) ( k = 1 , 2 , 3 , ) .
(3.1)
The result is sharp for
f ( z ) = z p ( 1 - z ) 2 ( p - β )
(3.2)

for α = 0.

Proof. In view of Theorem 1, we can consider the function w(z) given by (2.3) for f ( z ) S p ( α , β ) . Since w(z) is the Schwarz function, the function q(z) defined by
q ( z ) = β - e i α z f ( z ) f ( z ) + i p sin α β - p cos α
(3.3)
is the Carathéodory function. If we write that
q ( z ) = 1 + n = 1 c n z n ,
(3.4)
then we see that
c n 2 ( n = 1 , 2 , 3 , )
and the equality holds true for q ( z ) = 1 + z 1 - z and its rotation. It is to be noted that the equation (3.3) is equivalent to
e i α z f ( z ) f ( z ) = β + i p sin α - ( β - p cos α ) q ( z ) .
(3.5)
This gives us that
e i α p z p + n = p + 1 n a n z n = p e i α - ( β - p cos α ) n = 1 c n z n z p + n = p + 1 a n z n ,
(3.6)
which implies that
e i α ( n - p ) a n = - ( β - p cos α ) ( c n - 1 + a 2 c n - 2 + + a n - 1 c 1 ) .
(3.7)
It follows from (3.7) that
a n 2 ( β - p cos α ) n - p ( 1 + a 2 + a 3 + + a n - 1 ) .
(3.8)
If n = p + 1, then we have that
a p + 1 2 ( β - p cos α ) .
(3.9)
If n = p + 2, then we also have that
a p + 2 2 ( β - p cos α ) 2 ( 1 + a 2 ) ( β - p cos α ) ( 1 + 2 ( β - p cos α ) ) .
(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
a p + k 2 ( β - p cos α ) k ( 1 + a 2 + a 3 + + a p + k - 1 ) 2 ( β - p cos α ) k 1 + 2 ( β - p cos α ) + 2 ( β - p cos α ) 2 ( 1 + 2 ( β - p cos α ) ) + 2 ( β - p cos α ) 3 ( 1 + 2 ( β - p cos α ) ) 1 + 2 ( β - p cos α ) 2 + + 1 ( k - 1 ) ! j = 0 k - 2 ( 2 ( β - p cos ) + j ) = 2 ( β - p cos α ) k ( 1 + 2 ( β - p cos α ) ) 1 + 2 ( β - p cos α ) 2 + 1 + 2 ( β - p cos α ) 2 2 ( β - p cos α ) 3 + + 2 ( β - p cos α ) ( k - 1 ) ! j = 1 k - 2 ( 2 ( β - p cos α ) + j ) = 1 k ! j = 0 k - 1 ( 2 ( β - p cos α ) + j ) .
(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
f ( z ) = z p j = 0 2 ( β - p ) j ( - z ) j = z p + 2 ( β - p ) z p + 1 + + 1 k ! j = 0 k - 1 ( 2 ( β - p ) + j ) z p + k + .
(3.12)

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

Corollary 2 If f(z) belongs to the class C p ( α , β ) , then
a p + k 1 ( k - 1 ) ! j = 0 k - 1 ( 2 ( β - p cos α ) + j ) ( k = 1 , 2 , 3 , ) .
(3.13)
The result is sharp for f(z) defined by
f ( z ) = p z p - 1 ( 1 - z ) 2 ( p - β )
(3.14)

for α = 0.

Remark 2 We know that the extremal functions for f ( z ) S p ( α , β ) is f(z) given by (2.2) and for f ( z ) C p ( α , β ) is f(z) given by (2.11). But, we see that
a p + k 1 k ! j = 0 k - 1 2 e - i α ( β - p cos α ) + j
(3.15)
and
a p + k 1 ( k - 1 ) ! j = 0 k - 1 2 e - i α ( β - p cos α ) + j
(3.16)

for such functions.

Therefore, the extremal functions for f ( z ) S p ( α , β ) and f ( z ) C p ( α , β ) 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 z f ( z ) f ( z ) .

Theorem 3 If f ( z ) S p ( α , β ) , then we have
p cos α - ( 2 β - p cos α ) r 1 - r Re e i α z f ( z ) f ( z ) p cos α + ( 2 β - p cos α ) r 1 + r
(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
g ( z ) = 2 β - p e - i α + 2 ( p cos α - β ) 1 - z ( z U ) .
(4.2)
Letting z = r e (0 r < 1), we see that
Re g ( z ) = 2 β - p cos α + 2 ( p cos α - β ) ( 1 - r cos θ ) 1 + r 2 - 2 r cos θ .
(4.3)
Let us define
h ( t ) = 1 - r t 1 + r 2 - 2 r t ( t = cos θ ) .
(4.4)
Then, we know that h'(t) 0. This implies that
2 β - p cos α + 2 ( p cos α - β ) 1 - r Re g ( z ) 2 β - p cos α + 2 ( p cos α - β ) 1 + r ,
(4.5)
which is equivalent to
p cos α - ( 2 β - p cos α ) r 1 - r Re g ( z ) p cos α + ( 2 β - p cos α ) r 1 + r .
(4.6)

Noting that e i α z f ( z ) f ( z ) g ( z ) ( z U ) by Theorem 1 and g(z) is univalent in U , 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 f ( z ) S p ( 0 , β ) , then
p - ( 2 β - p ) r 1 - r Re z f ( z ) f ( z ) p + ( 2 β - p ) r 1 + r
(4.7)
for | z | = r < 1. The equalities hold true for
f ( z ) = z p ( 1 - z ) 2 ( p - β ) .
(4.8)
Corollary 4 If f ( z ) C p ( α , β ) , then we have
p cos α - ( 2 β - p cos α ) r 1 - r Re e i α 1 + z f ( z ) f ( z ) p cos α + ( 2 β - p cos α ) r 1 + r
(4.9)

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

Corollary 5 If f ( z ) C p ( 0 , β ) , then
p - ( 2 β - p ) r 1 - r Re 1 + z f ( z ) f ( z ) p + ( 2 β - p ) r 1 + r
(4.10)
for | z | = r < 1. The equalities hold true for f(z) defined by
f ( z ) = p z p - 1 ( 1 - z ) 2 ( p - β ) .
(4.11)

5 Sufficient conditions

We consider some sufficient conditions for f(z) to be in the classes S p ( 0 , β ) and C p ( 0 , β ) .

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 U with w(0) = 0. If there exists a point z 0 U such that
max z z 0 w ( z ) = w ( z 0 ) ,
(5.1)
then we can write
z 0 w ( z 0 ) = k w ( z 0 ) ,
(5.2)

where k is real and k 1.

Applying Lemma 1, we derive

Theorem 4 If f ( z ) A p satisfies
Re z f ( z ) f ( z ) - z f ( z ) f ( z ) > p + β 2 β ( z U )
(5.3)

for some real β > p, then f ( z ) S p ( 0 , β ) .

Proof. Let us define the function w(z) by
z f ( z ) f ( z ) = p + ( p - 2 β ) w ( z ) 1 - w ( z ) ( w ( z ) 1 ) .
(5.4)

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

It follows from (5.4) that
Re z f ( z ) f ( z ) - z f ( z ) f ( z ) = Re 1 - ( p - 2 β ) z w ( z ) p + ( p - 2 β ) w ( z ) - z w ( z ) 1 - w ( z ) > p + β 2 β ( z U ) .
(5.5)
We suppose that there exists a point z 0 U such that
max z z 0 w ( z ) = w ( z 0 ) = 1 .
Then, Lemma 1 gives us that w(z0) = e and z0w'(z0) = k e . For such a point z0, we have that
Re z 0 f ( z 0 ) f ( z 0 ) - z 0 f ( z 0 ) f ( z 0 ) = Re 1 - ( p - 2 β ) k e i θ p + ( p - 2 β ) e i θ - k e i θ 1 - e i θ = 1 + ( 2 β - p ) k ( p cos θ + p - 2 β ) p 2 + ( p - 2 β ) 2 + 2 p ( p - 2 β ) c o s θ + k 2 1 - ( 2 β - p ) k 2 β + k 2 = 1 - ( β - p ) k 2 β p + β 2 β .
(5.6)
This contradicts our condition (5.3). Therefore, there is no z 0 U such that |w (z0) | = 1. This implies that w ( z ) < 1 ( z U ) , that is, that
z f ( z ) f ( z ) - p z f ( z ) f ( z ) + ( p - 2 β ) < 1 ( z U ) .
(5.7)

Thus, we observe that f ( z ) S p ( 0 , β ) .   □

Further, we derive

Theorem 5 If f ( z ) S p ( 0 , β ) for some real β p + 1 2 , then
Re z p f ( z ) > 1 2 β - 2 p + 1 ( z U ) .
(5.8)
Proof. We consider the function w(z) such that
z p f ( z ) = 1 + ( 1 - 2 γ ) w ( z ) 1 - w ( z ) ( w ( z ) 1 )
(5.9)

for γ = 1 2 β - 2 p + 1 and for f ( z ) S p ( 0 , β ) .

Then, we know that
Re z f ( z ) f ( z ) = Re p - ( 1 - 2 γ ) z w ( z ) 1 + ( 1 - 2 γ ) w ( z ) - z w ( z ) 1 - w ( z ) < β
(5.10)

for z U .

Since w(z) is analytic in U and w(0) = 0, we suppose that there exists a point z 0 U such that
max z z 0 w ( z ) = w ( z 0 ) = 1 .
Then, applying Lemma 1, we can write that w(z0) = e and z0w'(z0) = k e (k 1). This gives us that
Re z 0 f ( z 0 ) f ( z 0 ) = Re p - ( 1 - 2 γ ) k e i θ 1 + ( 1 - 2 γ ) e i θ - k e i θ 1 - e i θ = p + ( 1 - 2 γ ) k 2 γ + k 2 p + 1 - γ 2 γ = β ,
(5.11)
which contradicts the inequality (5.10). Thus, there is no point z 0 U such that |w (z0) | = 1. This means that w ( z ) < 1 ( z U ) , and that,
Re z p f ( z ) > 1 2 β - 2 p + 1 ( z U ) .

This completes the proof of the theorem.   □

Letting z f ( z ) p instead of f(z) in Theorem 5, we have

Corollary 6 If f ( z ) C p ( α , β ) for some β p + 1 2 , Then
Re p z p - 1 f ( z ) > 1 2 β - 2 p + 1 ( z U ) .
(5.12)

Finally, we consider the coefficient estimates for functions f(z) to be in the classes S p ( α , β ) and C p ( α , β ) .

Theorem 6 If f ( z ) A p satisfies
n = p + 1 n e i α - k + n e i α - ( 2 β - k ) a n p e i α - ( 2 β - k ) - p e i α - k
(5.13)

for some real α α < π 2 , β (β > p cos α), and k (0 k p cos α), then f ( z ) S p ( α , β )

Proof. It is to be noted that if f ( z ) A p satisfies
e i α z f ( z ) f ( z ) - k e i α z f ( z ) f ( z ) - ( 2 β - k ) < 1 ( z U ) ,
(5.14)
Then f ( z ) S p ( α , β ) . It follows that
e i α z f ( z ) f ( z ) - k e i α z f ( z ) f ( z ) - ( 2 β - k ) = e i α z f ( z ) - k f ( z ) e i α - ( 2 β - k ) f ( z ) < p e i α - k + n = p + 1 n e i α - k a n p e i α - ( 2 β - k ) - n = p + 1 n e i α - ( 2 β - k ) a n .

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 f ( z ) A p satisfies
n = p + 1 ( n - β ) a n ( β - p )

for some real β p < β < p + 1 2 , then f ( z ) S p ( 0 , β ) .

Further, we have

Theorem 7 If f ( z ) A p satisfies
n = p + 1 n n e i α - k + n e i α - ( 2 β - k ) a n p p e i α - ( 2 β - k ) - p e i α - k

for some real α α < π 2 , β (β > p cos α) and k (0 k p cos α), then f ( z ) C p ( α , β )

Corollary 8 If f ( z ) A p satisfies
n = p + 1 n ( n - β ) a n p ( β - p )

for some real β p < β < p + 1 2 , then f ( z ) C p ( α , β ) .

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Kazim Karabekir Faculty of Education, Ataturk University
(2)
Department of Mathematics, Kinki University
(3)
Department of Mathematics and Computer Sciences, Faculty of Science and Letters, Istanbul Kultur University

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 2007, 31: 221–228.Google Scholar
  2. Owa S, Nishiwaki J: Coefficient estimates for certain classes of analytic functions. J Inequal Pure Appl Math 2002, 3: 1–5.MathSciNetGoogle Scholar
  3. Aouf MK, Al-Oboudi FM, Haidan MM: On some results for λ -spirallike and λ -Robertson functions of complex order. Publ Inst Math 2005, 75: 93–98.MathSciNetView ArticleGoogle Scholar
  4. Robertson MS: Univalent functions f ( z ) for which zf' ( z ) is spirallike. Michigan Math J 1969, 16: 315–324.MathSciNetView ArticleGoogle Scholar
  5. Duren PL: Univalent Functions. Springer, New York; 1983.Google Scholar
  6. Miller SS, Mocanu PT: Second order differential inequalities in the complex plane. J Math Anal Appl 1978, 65: 289–305. 10.1016/0022-247X(78)90181-6MathSciNetView ArticleGoogle Scholar
  7. Jack IS: Functions starlike and convex of order α . J Lond Math Soc 1971, 3: 469–474. 10.1112/jlms/s2-3.3.469MathSciNetView ArticleGoogle Scholar

Copyright

© Uyanik et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. 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.