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On the Taylor coefficients of a subclass of meromorphic univalent functions

Bappaditya Bhowmik, Firdoshi Parveen · 2017 📄 arXiv →

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Abstract

Let $\mathcal{V}_p(λ)$ be the collection of all functions $f$ defined in the unit disc $\ID$ having a simple pole at $z=p$ where $0<p<1$ and analytic in $\ID\setminus\{p\}$ with $f(0)=0=f'(0)-1$ and satisfying the differential inequality $|(z/f(z))^2 f'(z)-1|< λ$ for $z\in \ID$, $0<λ\leq 1$. Each $f\in\mathcal{V}_p(λ)$ has the following Taylor expansion: $$ f(z)=z+\sum_{n=2}^{\infty}a_n(f) z^n, \quad |z|<p. $$ In \cite{BF-3}, we conjectured that $$ |a_n(f)|\leq \frac{1-(λp^2)^n}{p^{n

Results & Lemmas (3)

Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.

Theorem 1. Theorem 1. Each f ∈Vp(λ) can be represented as f(z) = −pz (z −p)(1 −λpzw(z)), z ∈D, (2.1) where w ∈B. Also every f ∈Vp(λ) can be expressed…
Theorem 1. Each f ∈Vp(λ) can be represented as f(z) = −pz (z −p)(1 −λpzw(z)), z ∈D, (2.1) where w ∈B. Also every f ∈Vp(λ) can be expressed as f(z) = −pzu(z) (z −p)(1 −λp), z ∈D, (2.2) where u ∈B and u(0) = 1 −λp.
Theorem 2. Theorem 2. Let f ∈Vp(λ) have expansion of the form (1.1). Then the inequality (1.3) holds for n = 3, p ∈(0, 1/2]; for n = 4, p ∈(0, ( √ 3…
Theorem 2. Let f ∈Vp(λ) have expansion of the form (1.1). Then the inequality (1.3) holds for n = 3, p ∈(0, 1/2]; for n = 4, p ∈(0, ( √ 3 −1)/2] and for n = 5, p ∈ (0, ( √ 5 −1)/4]. Equality holds in the above inequality for the function (1.4).
Theorem 3. Theorem 3. Let f ∈Vp(λ) be of the form (1.1) in Dp. Then for n ≥3, |an(f)| ≤ 1 pn−1 + n−1 X k=1 λ2kp2k !1/2 n−1 X k=1 1 p2(n−k−1) !1/2,
Theorem 3. Let f ∈Vp(λ) be of the form (1.1) in Dp. Then for n ≥3, |an(f)| ≤ 1 pn−1 + n−1 X k=1 λ2kp2k !1/2 n−1 X k=1 1 p2(n−k−1) !1/2 ,

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