Abstract
Keywords:
analytic; univalent functions; CauchySchwarz inequality1. 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 closetoconvex 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 β_{1}, β_{2}, β_{3}, 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 β_{1}. If γ = 2, then
Therefore, for λ ≥ 10 β_{1} + 6 β_{2}. Further, if γ = 3; then we have
Thus, for λ ≥ 36 β_{1} + 42 β_{2} + 24 β_{3}.
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
for some complex numbers β_{1}, β_{2}, β_{3 }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 β_{1 }= arg β_{2 }= arg β_{3 }= ϕ and a_{n }= a_{n}e^{i((n1)θϕ)}(n = 2, 3,...), then we have
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 β_{1 }= arg β_{2 }= arg β_{3 }= ϕ and a_{n }= a_{n} e^{i((n1)θϕ) }(n = 2, 3,...), then we have
Example 2.4. Let us consider the function with arg β_{1 }= arg β_{2 }= arg β_{3 }= ϕ and
Then, we see that
Corollary 2.5. If with arg β_{1 }= arg β_{2 }= arg β_{3 }= ϕ and a_{n }= a_{n} e^{i((n1)θϕ) }(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.
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).
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 ≤ δ_{0}(λ), where δ_{0}(λ) is the smallest positive root of the equation
in 0 < δ < 1.
and
Thus, to show that , from Theorem 2.1, it is sufficient to prove that
Applying CauchySchwarz 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 δ_{0 }such that h(δ_{0}) = 0 (0 < δ_{0} < 1). This completes the proof of the theorem.
4. Radius problem for the class
For the class , we prove the following lemma.
for all . Let us consider a point . such that z = ze^{iθ}.
Then we have
Letting z → 1^{}, we obtain the inequality (4.1).
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 ≤ δ_{0}(λ), where δ_{0}(λ) is the smallest positive root of the equation
in 0 < δ < 1.
Competing interests
The author declares that they have no competing interests.
References

Duren, PL: Univalent Functions. SpringerVerlag, Berlin (1983)

Uyanik, N, Owa, S: New extensions for classes of analytic functions associated with closetoconvex and starlike of order α. Math Comput Model. 54, 359–366 (2011). Publisher Full Text