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Radius properties for analytic and p-valently starlike functions

  • Neslihan Uyanik
  • Shigeyoshi Owa
Open Access
Research
  • 1.3k Downloads

Abstract

Let A p Open image in new window be the class of functions f(z) which are analytic in the open unit disk U Open image in new window and satisfy z p f ( z ) 0 ( z U ) Open image in new window. Also, let S p * ( α ) Open image in new window denotes the subclass of A p Open image in new window consisting of f(z) which are p-valently starlike of order α(0 ≦ α < p). A new subclass U p ( λ ) Open image in new window of A p Open image in new window is introduced by

z 2 z p - 1 f ( z ) - 1 z λ ( z U ) Open image in new window

for some real λ > 0. The object of the present paper is to consider some radius properties for f ( z ) S p * ( α ) Open image in new window such that δ - p f ( δ z ) U p ( λ ) Open image in new window.

2010 Mathematics Subject Classification: Primary 30C45.

Keywords

Analytic radius problem Cauchy-Schwarz inequality p-valently starlike of order α. 

1 Introduction

Let A p Open image in new window be the class of functions f(z) of the form
f ( z ) = z p + n = p + 1 a n z n ( p = 1 , 2 , 3 , ) Open image in new window
(1.1)
which are analytic in the open unit disk U = { z : | z | < 1 } Open image in new window and satisfy
z p f ( z ) = 1 + n = p + 1 b n z n - p 0 ( z U ) . Open image in new window
(1.2)
For f ( z ) A p Open image in new window, we say that f(z) belongs to the class U p ( λ ) Open image in new window if it satisfies
z 2 z p - 1 f ( z ) - 1 z λ ( z U ) Open image in new window
(1.3)

for some real number λ > 0.

Let us consider a function f δ (z) given by
f δ ( z ) = z p ( 1 - z ) δ ( δ ) . Open image in new window
(1.4)
Then, we can write that
f δ ( z ) = z p 1 + n = 1 a n z n Open image in new window
with
a n = ( - 1 ) n δ n Open image in new window
and
z 2 z p - 1 f δ ( z ) - 1 z = n = 1 ( n - 1 ) a n z n < n = 1 ( n - 1 ) | a n | . Open image in new window
Thus, if δ = 2, then
z 2 z p - 1 f 2 ( z ) - 1 z < 1 . Open image in new window

This shows that f 2 ( z ) U p ( λ ) Open image in new window for λ ≧ 1.

If δ = 3, then we have that
z 2 z p - 1 f 3 ( z ) - 1 z < 5 Open image in new window

Which shows that f ( z ) U p ( λ ) Open image in new window for λ ≧ 5.

Further, if δ = 4, then
z 2 z p - 1 f 4 ( z ) - 1 z < 1 1 Open image in new window

which shows that f ( z ) U p ( λ ) Open image in new window for λ ≧ 11.

If p = 1, then f ( z ) U 1 ( λ ) Open image in new window is defined by
z 2 1 f ( z ) - 1 z λ ( z U ) Open image in new window
(1.5)
for some real number λ > 0. Note that (1.5) is equivalent to
f ( z ) z f ( z ) 2 - 1 λ ( z U ) . Open image in new window

Therefore, this class U 1 ( λ ) Open image in new window was considered by Obradović and Ponnusamy [1]. Further-more, this class was extended as the class U ( β 1 , β 2 ; λ ) Open image in new window by Shimoda et al. [2].

Let S p * ( α ) Open image in new window denotes the subclass of A p Open image in new window consisting of f(z) which satisfy
R e z f ( z ) f ( z ) > α ( z U ) Open image in new window
(1.6)

for some real α (0 ≦ α < p).

A function f ( z ) S p * ( α ) Open image in new window is said to be p-valently starlike of order α in U Open image in new window (cf. Robertson [3]).

2 Coefficient inequalities

For f ( z ) A p Open image in new window, we consider the sufficient condition for f(z) to be in the class U p ( λ ) Open image in new window.

Lemma 1 If f ( z ) A p Open image in new window satisfies
n = p + 2 ( n - p - 1 ) | b n | λ , Open image in new window
(2.1)

then f ( z ) U 1 ( λ ) Open image in new window.

Proof We note that
z 2 z p - 1 f ( z ) - 1 z = n = p + 1 ( n - p - 1 ) b n z n - p < n = p + 1 ( n - p - 1 ) | b n | . Open image in new window
Therefore, if
n = p + 1 ( n - p - 1 ) | b n | = n = p + 2 ( n - p - 1 ) | b n | λ , Open image in new window

then f ( z ) U p ( λ ) Open image in new window.

Example 1 If we consider a function f ( z ) A p Open image in new window given by
z p f ( z ) = 1 + b p + 1 z + n = p + 2 λ e i φ ( n - p ) ( n - p - 1 ) 2 z n - p 0 ( z U ) Open image in new window
with
b n = λ e i φ ( n - p ) ( n - p - 1 ) 2 ( λ > 0 , φ ) Open image in new window
for np + 2, then we see that
n = p + 2 ( n - p - 1 ) | b n | = n = p + 2 λ e i φ ( n - p ) ( n - p - 1 ) < λ n = p + 2 1 n - p - 1 - 1 n - p = λ . Open image in new window
Thus, this function f(z) satisfies the inequality (2.1). Also, we see that
| z 2 ( z p 1 f ( x ) 1 z ) | = | n = p + 2 λ e i φ n p 1 ) ( n p ) z n p | < λ n = p + 2 ( 1 n p 1 1 n p ) = λ . Open image in new window

Therefore, we say that f ( z ) U p ( λ ) Open image in new window.

Next, we discuss the necessary condition for the class S p * ( α ) Open image in new window.

Lemma 2 If f ( z ) S p * ( α ) Open image in new window satisfies
z p f ( z ) = 1 + n = p + 1 b n z n - p 0 ( z U ) Open image in new window
with b n = |b n | ei(n-p)θ(n = p + 1, p + 2, p + 3,...), then
n = p + 1 ( n + α - 2 p ) | b n | p - α . Open image in new window
Proof Let us define the function F(z) by
F ( z ) = z p f ( z ) = 1 + n = p + 1 b n z n - p . Open image in new window
It follows that
R e z f ( z ) f ( z ) = R e p - z F ( z ) F ( z ) = R e p - n = p + 1 ( n - 2 p ) b n z n - p 1 + n = p + 1 b n z n - p = R e p - n = p + 1 ( n - 2 p ) | b n | e i ( n - p ) θ z n - p 1 + n = p + 1 | b n | e i ( n - p ) θ z n - p > α Open image in new window
for z U Open image in new window. Letting z = |z| e -iθ , we have that
p - n = p + 1 ( n - 2 p ) | b n | | z | n - p 1 + n = p + 1 | b n | | z | n - p > α ( z U ) . Open image in new window
If we take |z| → 1-, we obtain that
p - n = p + 1 ( n - 2 p ) | b n | 1 + n = p + 1 | b n | α Open image in new window
which implies that
n = p + 1 ( n + α - 2 p ) | b n | p - α . Open image in new window

Remark 1 If we take p = 1 in Lemmas 1 and 2, then we have that

(i) f ( z ) A 1 , n = 2 ( n - 2 ) | b n | λ f ( z ) U 1 ( λ ) Open image in new window

and

(ii) f ( z ) S * ( α ) , | b n | = | b n | e i ( n - 1 ) θ n = 2 ( n + α - 2 ) | b n | 1 - α . Open image in new window

3 Radius problems

Our main result for the radius problem is contained in

Theorem 1 Let f ( z ) S p * ( α ) Open image in new window (p - 1 ≦ α < p) with
z p f ( z ) = 1 + n = p + 1 b n z n - p 0 ( z U ) . Open image in new window
and b n = | b n | ei(n-p)θ(n = p + 1, p + 2, p + 3, ...). If δ ( | δ | < 1 ) Open image in new window, then 1 δ p f ( δ z ) Open image in new window belongs to the class U p ( λ ) Open image in new window for 0 < | δ | | δ 0 ( λ ) | Open image in new window, where |δ0(λ)| is the smallest positive root of the equation
| δ | 2 1 - α - ( 1 - | δ | 2 ) λ = 0 , Open image in new window
(3.1)
that is,
| δ 0 ( λ ) | = λ λ + 1 - α . Open image in new window
(3.2)
Proof Since
f ( δ z ) = δ p z p + n = p + 1 a n δ n z n , Open image in new window
we have that
z p 1 δ p f ( δ z ) = 1 + n = p + 1 b n δ n - p z n - p . Open image in new window
In view of Lemma 1, we have to show that
n = p + 2 ( n - p - 1 ) | b n | | δ | n - p λ . Open image in new window
Note that f ( z ) S p * ( α ) Open image in new window satisfies
| b n | p - α n + α - 2 p < 1 ( p - 1 α < p ) . Open image in new window
Applying Cauchy-Schwarz inequality, we obtain that
n = p + 2 ( n - p - 1 ) | b n | | δ | n - p n = p + 2 ( n - p - 1 ) | b n | 2 1 2 n = p + 2 ( n - p - 1 ) | δ | 2 ( n - p ) 1 2 n = p + 2 ( n - p - 1 ) | δ | 2 ( n - p ) 1 2 p - α . Open image in new window
Let |δ|2 = x. Then, we have that
n = p + 2 ( n - p - 1 ) x n - p = x 2 n = p + 2 ( n - p - 1 ) x n - p - 2 = x 2 n = p + 2 x n - p - 1 = x 2 n = 1 x n - 1 = x 2 ( 1 - x ) 2 . Open image in new window
This gives us that
n = p + 2 ( n - p - 1 ) | b n | | δ | n - p | δ | 2 p - α 1 - | δ | 2 . Open image in new window
Let us define the function h(|δ|) by
h ( | δ | ) = | δ | 2 p - α - ( 1 - | δ | 2 ) λ . Open image in new window
Then, h (|δ|) satisfies h (0) = < 0 and h ( 1 ) = p - α > 0 Open image in new window. Indeed, we have that h (|δ0(λ) |) = 0 for
0 < | δ 0 ( λ ) | = λ λ + p - α < 1 . Open image in new window

This completes the proof of the theorem.

Corollary 1 Let f ( z ) S 1 * ( α ) ( 0 α < 1 ) Open image in new window with
z f ( z ) = 1 + n = 2 b n z n - 1 0 ( z U ) Open image in new window
and b n = |b n | ei(n-1)θ(n = 2, 3, 4,...). If δ ∈ ℂ (|δ| < 1), then 1 δ f ( δ z ) Open image in new window belongs to the class U 1 ( λ ) Open image in new window for 0 < | δ | | δ 0 ( λ ) | Open image in new window, where |δ0(λ)| is the smallest positive root of the equation
| δ | 2 1 - α - ( 1 - | δ | 2 ) λ = 0 , Open image in new window
that is,
| δ 0 ( λ ) | = λ λ + 1 - α . Open image in new window
Remark 2 In view of (3.2), we define the function g(λ) by
g ( λ ) = | δ 0 ( λ ) | = λ λ + p - α . Open image in new window
Then, we have that
g ( λ ) = 1 2 p - α λ ( λ + p - α ) 3 > 0 Open image in new window

for λ > 0. Therefore, |δ0(λ)| given by (3.2) is increasing for λ > 0.

Remark 3 If we put α = p - 1 2 Open image in new window in Theorem 1, then
| δ 0 ( λ ) | = 2 λ 2 λ + 2 . Open image in new window
Therefore, if we consider λ = 1 2 Open image in new window, then we see that
δ 0 1 2 = 1 1 + 2 = 0 . 6 4 3 5 9 Open image in new window
and if we make λ = 5, then we have that
| δ 0 ( 5 ) | = 1 0 1 0 + 2 = 0 . 9 3 6 0 0 Open image in new window

Notes

References

  1. 1.
    Obradović M, Ponnusamy S: Radius properties for subclasses of univalent functions. Analysis 2005, 25: 183–188. 10.1524/anly.2005.25.3.183MATHGoogle Scholar
  2. 2.
    Shimoda Y, Hayami T, Owa S: Notes on radius properties of certain univalent functions. Acta Univ Apul 2009, 377–383. (Special Issue)Google Scholar
  3. 3.
    Robertson MS: On the theory of univalent functions. Ann Math 1936, 37: 374–408. 10.2307/1968451CrossRefGoogle Scholar

Copyright information

© Uyanik and Owa; 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.

Authors and Affiliations

  1. 1.Department of Mathematics, Kazim Karabekir Faculty of EducationAtatürk UniversityErzurumTurkey
  2. 2.Department of MathematicsKinki UniversityHigashi-Osaka, OsakaJapan

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