New subclasses of analytic functions

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.

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:

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|e^{i((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| e^{i((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| e^{i((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

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.

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.

The author declares that they have no competing interests.

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2. Goodman, AW: Univalent Functions. Mariner, Tampa (1983)

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