Abstract
Let $\mathcal{A}$ be the family of analytic and normalized functions in the open unit disc $|z|<1$. In this article we consider the following classes \begin{equation*}
\mathcal{R}(α,β):=\left\{ f\in \mathcal{A}: {\rm Re}\left\{f'(z)+\frac{1+e^{iα}}{2}zf''(z)\right\}>β,\, |z|<1\right\} \end{equation*} and \begin{equation*}
\mathcal{L}_α(b):=\left\{f\in\mathcal{A}:\left|f'(z)
+\frac{1+e^{iα}}{2}zf''(z)-b\right|< b,\, |z|<1 \right\}, \end{equation*} where $-π<α\leq π$, $0\leq β<1$ and $b>1/2$
Results & Lemmas (14)
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Lemma 1.1.
Lemma 1.1. (see [19]) A necessary and sufficient condition for f to be in the class Lα(b) is f ′(z) + 1 + eiα 2 zf ′′(z) ≺φb(z) (z ∈∆), where…
Lemma 1.1. (see [19]) A necessary and sufficient condition for f to be in the class Lα(b) is f ′(z) + 1 + eiα 2 zf ′′(z) ≺φb(z) (z ∈∆), where φb is given by (1.1). 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -1.0
Lemma 1.2.
Lemma 1.2. A function f ∈A belongs to the class R(α, β) if, and only if, f ′(z) + 1 + eiα 2 zf ′′(z) ≺1 + (1 −2β)z 1 −z (z ∈∆, 0 ≤β < 1, −π…
Lemma 1.2. A function f ∈A belongs to the class R(α, β) if, and only if, f ′(z) + 1 + eiα 2 zf ′′(z) ≺1 + (1 −2β)z 1 −z (z ∈∆, 0 ≤β < 1, −π < α ≤π). To prove of our main results we need the following lemma.
Lemma 1.3.
Lemma 1.3. [10, p. 35] Let Ξ be a simply connected domain in the complex plane C and let t be a complex number such that Re t > 0. Suppose…
Lemma 1.3. [10, p. 35] Let Ξ be a simply connected domain in the complex plane C and let t be a complex number such that Re{t} > 0. Suppose that a function ψ : C2 × ∆→C satisfies the condition ψ(iρ, σ; z) ̸∈Ξ for all real ρ, σ ≤−| t −iρ |2 /(2Re t) and all z ∈∆. If the function p(z) defined by p(z) = t + t1z + t2z2 + · · · is analytic in ∆and if ψ(p(z), zp′(z); z) ∈Ξ, then Re{p(z)} > 0 in ∆. This paper is organized as follows. In Section 2 some properties of the classes R(α, β) and Lα(b) are studi
Theorem 2.3
Theorem 2.3, we employ the same technique as in [5, Theorem 2.1].
Theorem 2.3, we employ the same technique as in [5, Theorem 2.1].
Theorem 2.1.
Theorem 2.1. Let β ∈[0, 1) and α ∈(−π, π]. If f ∈A belongs to the class R(α, β), then Re f ′(z) > β (0 ≤β < 1). This means that R(α, β)…
Theorem 2.1. Let β ∈[0, 1) and α ∈(−π, π]. If f ∈A belongs to the class R(α, β), then Re{f ′(z)} > β (0 ≤β < 1). This means that R(α, β) ⊂C(β).
Corollary 2.1.
Corollary 2.1. If f ∈Lα, then Re f ′(z) > 0 (z ∈∆) and thus f is univalent.
Corollary 2.1. If f ∈Lα, then Re{f ′(z)} > 0 (z ∈∆) and thus f is univalent.
Theorem 2.2.
Theorem 2.2. Let β ∈[0, 1) and α ∈(−π, π]. If f ∈A belongs to the class R(α, β), then we have Re f(z) z > β (0 ≤β < 1).
Theorem 2.2. Let β ∈[0, 1) and α ∈(−π, π]. If f ∈A belongs to the class R(α, β), then we have Re f(z) z > β (0 ≤β < 1).
Corollary 2.2.
Corollary 2.2. If f ∈Lα, then Re f(z)/z > 0 in the open unit disc ∆. We shall require the following lemma in order to prove of the next…
Corollary 2.2. If f ∈Lα, then Re{f(z)/z} > 0 in the open unit disc ∆. We shall require the following lemma in order to prove of the next result.
Lemma 2.1.
Lemma 2.1. Let φb(z) be defined by (1.1) for b > 1/2. Then φb(∆) = Ωb where Ωb:= w ∈C: 0 < Re w < 2b.
Lemma 2.1. Let φb(z) be defined by (1.1) for b > 1/2. Then φb(∆) = Ωb where Ωb := {w ∈C : 0 < Re{w} < 2b}.
Theorem 2.3.
Theorem 2.3. Let f ∈A be a member of the class Lα(b) where b > 1/2 and α ∈(−π, π]. Then 0 < Re f ′(z) < 2b (z ∈∆).
Theorem 2.3. Let f ∈A be a member of the class Lα(b) where b > 1/2 and α ∈(−π, π]. Then 0 < Re{f ′(z)} < 2b (z ∈∆).
Theorem 2.4.
Theorem 2.4. Assume that b > 1/2, α ∈(−π, π] and f ∈Lα(b). Then for each |z| = r < 1 we have (2.8) 1 − (2b −1)r b + (b −1)r ≤Re f ′(z) +…
Theorem 2.4. Assume that b > 1/2, α ∈(−π, π] and f ∈Lα(b). Then for each |z| = r < 1 we have (2.8) 1 − (2b −1)r b + (b −1)r ≤Re f ′(z) + 1 + eiα 2 zf ′′(z) ≤1 + (2b −1)r b + (b −1)r.
Corollary 2.3.
Corollary 2.3. Let f ∈Lα(1). Then we have 1 −r < Re f ′(z) + 1 + eiα 2 zf ′′(z) < 1 + r (|z| = r < 1).
Corollary 2.3. Let f ∈Lα(1). Then we have 1 −r < Re f ′(z) + 1 + eiα 2 zf ′′(z) < 1 + r (|z| = r < 1).
Corollary 2.4.
Corollary 2.4. By a simple geometric observation and applying (2.9) and (2.10), we have arg f ′(z) + 1 + eiα 2 zf ′′(z) < arcsin (2b…
Corollary 2.4. By a simple geometric observation and applying (2.9) and (2.10), we have arg f ′(z) + 1 + eiα 2 zf ′′(z) < arcsin (2b −1)r b + (b −1)r (|z| = r < 1, b > 1/2). 3. The radius of univalence of 2-th section sum of f ∈R(α, β) In this section, we obtain the radius of univalence of 2-th section sum of f ∈ R(α, β). We recall that the Taylor polynomial sk(z) = sk(f)(z) of f defined by sk(z) = sk(f)(z) = z + a2z2 + · · · + akzk,
Theorem 3.1.
Theorem 3.1. The 2-th section sum of f ∈R(α, β) is univalent in the disc |z| < √10 + 6 cos α 4(1 −β) (−π < α ≤π, 0 ≤β < 1). The number…
Theorem 3.1. The 2-th section sum of f ∈R(α, β) is univalent in the disc |z| < √10 + 6 cos α 4(1 −β) (−π < α ≤π, 0 ≤β < 1). The number √10+6 cos α 4(1−β) cannot be replaced by a greater one.
Function classes studied:
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