Abstract
Let $\ID$ denote the open unit disk and $f:\,\ID\TO\BAR\IC$ be meromorphic and univalent in $\ID$ with the simple pole at $p\in (0,1)$ and satisfying the standard normalization $f(0)=f'(0)-1=0$. Also, let $f$ have the expansion $$f(z)=\sum_{n=-1}^{\infty}a_n(z-p)^n,\quad |z-p|<1-p, $$ such that $f$ maps $\ID$ onto a domain whose complement with respect to $\BAR{\IC}$ is a convex set (starlike set with respect to a point $w_0\in \IC, w_0\neq 0$ resp.). We call these functions as concave (meromorp
Results & Lemmas (9)
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Theorem 1.1.
Theorem 1.1. Let p ∈(0, √ 5−1 2 ] and f ∈Co(p) have the expansion (1.2). Then (1.6) p −(1 −p2) a0 a−1 ≤ p |a−1|, i.e. a−1 −1 −p2 p
Theorem 1.1. Let p ∈(0, √ 5−1 2 ] and f ∈Co(p) have the expansion (1.2). Then (1.6) p −(1 −p2) a0 a−1 ≤ p |a−1|, i.e. a−1 −1 −p2 p
Theorem 1.2.
Theorem 1.2. If f ∈Co(p) with p ∈(0, 1) and has the expansion (1.2), then we have for (n ≥3) an−2 −(1 −p2)an−1 p
Theorem 1.2. If f ∈Co(p) with p ∈(0, 1) and has the expansion (1.2), then we have for (n ≥3) an−2 −(1 −p2)an−1 p
Theorem 3.1.
Theorem 3.1. For 0 < p < 1, let f ∈Σs(p, w0). Then there exists a function ω holomorphic in D such that ω(D) ⊂D, ω(0) = −1 2( 1 w0 + p + 1…
Theorem 3.1. For 0 < p < 1, let f ∈Σs(p, w0). Then there exists a function ω holomorphic in D such that ω(D) ⊂D, ω(0) = −1 2( 1 w0 + p + 1 p) and (3.2) f(z) = w0 + pw0(1 + zω(z))2 (z −p)(1 −zp) , z ∈D.
Corollary 2
Corollary 2] in which they use the notation σ∗(p, w0) in place of Σs(p, w0). By this corollary we get if f ∈Σs(p, w0), then
Corollary 2] in which they use the notation σ∗(p, w0) in place of Σs(p, w0). By this corollary we get if f ∈Σs(p, w0), then
Corollary 3.1.
Corollary 3.1. For 0 < p < 1, let f ∈Σs(p, w0). Then, we have (3.3) w0 + p(1 + p2) (1 −p2)2 ≤ 2p2 (1 −p2)2. In particular, one has p (1 +…
Corollary 3.1. For 0 < p < 1, let f ∈Σs(p, w0). Then, we have (3.3) w0 + p(1 + p2) (1 −p2)2 ≤ 2p2 (1 −p2)2. In particular, one has p (1 + p)2 ≤|w0| ≤ p (1 −p)2.
Theorem 3.2.
Theorem 3.2. Let f ∈Σs(p, w0) have the Laurent expansion (1.2). Then (i) a−1 − pw0 1 −p2 ≤p|w0| 1 −p2 | p w0 + p2 + 1| + 2p2 2 + | p w0 +…
Theorem 3.2. Let f ∈Σs(p, w0) have the Laurent expansion (1.2). Then (i) a−1 − pw0 1 −p2 ≤p|w0| 1 −p2 | p w0 + p2 + 1| + 2p2 2 + | p w0 + p2 + 1|
Theorem 3.2
Theorem 3.2 turns out to be a−1 − pw0 1 −p2 ≤ p2 1 −p2(p + 2)|w0|, p ∈(0, 1). Applying the triangle inequality in the above inequality and…
Theorem 3.2 turns out to be a−1 − pw0 1 −p2 ≤ p2 1 −p2(p + 2)|w0|, p ∈(0, 1). Applying the triangle inequality in the above inequality and the inequality (ii) in Theorem 3.2 we get (3.6) |a−1| ≤p(1 + p) 1 −p |w0|, p ∈(0, 1) and
Theorem 3.2.
Theorem 3.2. □ In view of the estimate (3.6) we observe that there was a minor error in one of the results of Livingston, namely Theorem 9…
Theorem 3.2. □ In view of the estimate (3.6) we observe that there was a minor error in one of the results of Livingston, namely Theorem 9 in [6]. Indeed a counterexample is given by the function g(z) = −zp (z −p)(1 −pz) ∈Σs p, −p 1 + p2 .
Theorem 3.3.
Theorem 3.3. If f ∈Σs(p, w0) and has the Laurent expansion (1.2), then we have |a−1| ≥p(1 −p) 1 + p |w0|. The inequality is sharp for the…
Theorem 3.3. If f ∈Σs(p, w0) and has the Laurent expansion (1.2), then we have |a−1| ≥p(1 −p) 1 + p |w0|. The inequality is sharp for the function g(z) = −zp (z −p)(1 −pz) = w0 + pw0 (z −p)(1 −pz)(1 −z)2 ∈Σs(p, w0) where w0 = −p (1−p)2.
Function classes studied:
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