# Radius properties for analytic and p-valently starlike functions

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## Abstract

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

for some real λ > 0. The object of the present paper is to consider some radius properties for such that .

2010 Mathematics Subject Classification: Primary 30C45.

### Keywords

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

## 1 Introduction

Let be the class of functions f(z) of the form
(1.1)
which are analytic in the open unit disk and satisfy
(1.2)
For , we say that f(z) belongs to the class if it satisfies
(1.3)

for some real number λ > 0.

Let us consider a function f δ (z) given by
(1.4)
Then, we can write that
with
and
Thus, if δ = 2, then

This shows that for λ ≧ 1.

If δ = 3, then we have that

Which shows that for λ ≧ 5.

Further, if δ = 4, then

which shows that for λ ≧ 11.

If p = 1, then is defined by
(1.5)
for some real number λ > 0. Note that (1.5) is equivalent to

Therefore, this class was considered by Obradović and Ponnusamy [1]. Further-more, this class was extended as the class by Shimoda et al. [2].

Let denotes the subclass of consisting of f(z) which satisfy
(1.6)

for some real α (0 ≦ α < p).

A function is said to be p-valently starlike of order α in (cf. Robertson [3]).

## 2 Coefficient inequalities

For , we consider the sufficient condition for f(z) to be in the class .

Lemma 1 If satisfies
(2.1)

then .

Proof We note that
Therefore, if

then .

Example 1 If we consider a function given by
with
for np + 2, then we see that
Thus, this function f(z) satisfies the inequality (2.1). Also, we see that

Therefore, we say that .

Next, we discuss the necessary condition for the class .

Lemma 2 If satisfies
with b n = |b n | ei(n-p)θ(n = p + 1, p + 2, p + 3,...), then
Proof Let us define the function F(z) by
It follows that
for . Letting z = |z| e -iθ , we have that
If we take |z| → 1-, we obtain that
which implies that

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

(i)

and

(ii)

Our main result for the radius problem is contained in

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

This completes the proof of the theorem.

Corollary 1 Let with
and b n = |b n | ei(n-1)θ(n = 2, 3, 4,...). If δ ∈ ℂ (|δ| < 1), then belongs to the class for , where |δ0(λ)| is the smallest positive root of the equation
that is,
Remark 2 In view of (3.2), we define the function g(λ) by
Then, we have that

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

Remark 3 If we put in Theorem 1, then
Therefore, if we consider , then we see that
and if we make λ = 5, then we have that

## Notes

### References

1. 1.
Obradović M, Ponnusamy S: Radius properties for subclasses of univalent functions. Analysis 2005, 25: 183–188. 10.1524/anly.2005.25.3.183
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. 10.2307/1968451