- Research
- Open access
- Published:
New subclasses of analytic functions
Journal of Inequalities and Applications volume 2012, Article number: 24 (2012)
Abstract
For analytic functions f (z) in the open unit disk , subclasses , and are introduced. The object of the present article is to discuss some interesting properties of functions f (z) associated with classes , and .
Mathematics Subject Classification (2010): 30C45.
1. Introduction and Definitions
Let denotes the class of the normalized functions of the form
which are analytic in the open unit disk . Also, a function f (z) belonging to is said to be convex of order α if it satisfies
for some α(0 ≤ α < 1). We denote by the subclass of consisting of functions which are convex of order α in (see, [1, 2]). Further, a function f (z) belonging to is said to be in the class iff
for some α(0 ≤ α < 1).
For analytic functions f (z), Uyanik and Owa [3], obtained some interesting properties for analytic functions in the subclass defined by
associated with close-to-convex functions and starlike functions of order α.
In this article, we define the following subclass of analytic functions.
Definition 1.1. A function f (z) belonging to is said to be in the class , if it satisfies
for some complex numbers β1, β2, β3, and for some real λ > 0.
Example 1.2. Let us consider the function f γ (z), γ ∈ ℝ, given by
Then, we observe that
where
Therefore, if γ = 1, then
This implies that for λ ≥ 2 |β1|. If γ = 2, then
Therefore, for λ ≥ 10 |β1| + 6 |β2|. Further, if γ = 3; then we have
Thus, for λ ≥ 36 |β1| + 42 |β2| + 24 |β3|.
Now, let denotes the subclass of consisting of functions f (z) with
Also, we introduce the subclasses and of as follows:
2. Properties of the class
We first prove
Theorem 2.1. If satisfies
for some complex numbers β1, β2, β3 and for some real λ > 0, then .
Proof. We observe that
Therefore, if f (z) satisfies the inequality (2.1), then .
Next, we prove
Theorem 2.2. if with arg β1 = arg β2 = arg β3 = ϕ and a n = |a n |ei((n-1)θ-ϕ)(n = 2, 3,...), then we have
Proof. For , we see that
for all . Let us consider a point such that z = |z| e-iθ.
Then we have
Letting |z| → 1-, we obtain
Corollary 2.3. If with arg β1 = arg β2 = arg β3 = ϕ and a n = |a n | ei((n-1)θ-ϕ)(n = 2, 3,...), then we have
Example 2.4. Let us consider the function with arg β1 = arg β2 = arg β3 = ϕ and
Then, we see that
Corollary 2.5. If with arg β1 = arg β2 = arg β3 = ϕ and a n = |a n | ei((n-1)θ-ϕ)(n = 2, 3,...), then we have
with
and
with
Proof. In view of Theorem 2.1, we know that
Further, we note that
which is equivalent to
Thus, we have
and
Next, we observe that
which implies that
Therefore, we obtain that
and
3. Radius problem for the class
To obtain the radius problem for the class , we need the following lemma.
Lemma 3.1. If , then
Proof. Let . Then, we have
for all . Let us consider a point such that z = |z| e-iθ.
Then we have
Letting |z| → 1-, we obtain the inequality (3.1).
Corollary 3.2. If , then
Remark 3.3. By Lemma 3.1, we observe that if , then
Applying Theorem 2.1 and Lemma 3.1, we derive
Theorem 3.4. Let , and δ ∈ ℂ (0 < |δ| < 1). Then the function for (0 < |δ| ≤ |δ0(λ)|, where |δ0(λ)| is the smallest positive root of the equation
in 0 < |δ| < 1.
Proof. For , we see that
and
Thus, to show that , from Theorem 2.1, it is sufficient to prove that
Applying Cauchy-Schwarz inequality, we note that
We note that
thus, we have
Since
we see that
and thus, we obtain
Furthermore, we have
but
thus, we have
which yields
Therefore, from (3.3)-(3.6) with |δ|2 = x, we obtain
Now, let us consider the complex number δ (0 < |δ| < 1) such that
If we define the function h(|δ|) by
then we have h(0) = -λ < 0 and . This means that there exists some δ0 such that h(|δ0|) = 0 (0 < |δ0| < 1). This completes the proof of the theorem.
4. Radius problem for the class
For the class , we prove the following lemma.
Lemma 4.1. If , then
Proof. Let . Then, we have
for all . Let us consider a point . such that z = |z|e-iθ.
Then we have
Letting |z| → 1-, we obtain the inequality (4.1).
Corollary 4.2. If , then
Remark 4.3. If , then
Applying Theorem 2.1, Lemma 4.1 and using the same technique as in the proof of Theorem 3.4, we derive
Theorem 4.4. Let , and δ ∈ ℂ (0 < |δ| < 1). Then the function for (0 < |δ| ≤ |δ0(λ)|, where |δ0(λ)| is the smallest positive root of the equation
in 0 < |δ| < 1.
References
Duren PL: Univalent Functions. Springer-Verlag, Berlin; 1983.
Goodman AW: Univalent Functions. Volume 1–2. Mariner, Tampa; 1983.
Uyanik N, Owa S: New extensions for classes of analytic functions associated with close-to-convex and starlike of order α. Math Comput Model 2011, 54: 359–366. 10.1016/j.mcm.2011.02.020
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Frasin, B.A. New subclasses of analytic functions. J Inequal Appl 2012, 24 (2012). https://doi.org/10.1186/1029-242X-2012-24
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2012-24