Abstract
Let ${\mathcal H}$ denote the class of all normalized complex-valued harmonic functions $f=h+\bar{g}$ in the unit disk ${\mathbb D}$, and let $K=H+\bar{G}$ denote the harmonic Koebe function. Let $a_n,b_n, A_n, B_n$ denote the Maclaurin coefficients of $h,g,H,G$, and $${\mathcal F}=\{f=h+\bar{g}\in {\mathcal H}:\,|a_n|\leq A_n and |b_n|\leq B_n for n\geq 1}. $$ We show that the radius of univalence of the family ${\mathcal F}$ is $0.112903...$. We also show that this number is also the radius of
Results & Lemmas (13)
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 1.5.
Theorem 1.5. Let h and g have the form (1.1) and the coefficients of the series satisfy the conditions (1.4). Then f = h + g is…
Theorem 1.5. Let h and g have the form (1.1) and the coefficients of the series satisfy the conditions (1.4). Then f = h + g is close-to-convex (univalent), and starlike in the disk |z| < rS, where rS = 1 + √ 2 4 − r√ 2 + 1 8 ≈0.112903 is the root of the quadratic equation √ 2 r2 −(1 + 2 √ 2) r +
Corollary 1.6. · radius
Corollary 1.6. The radius of close-to-convexity and the radius of starlikeness for mappings in S∗0 H (resp. C0 H and TH) is at least…
Corollary 1.6. The radius of close-to-convexity and the radius of starlikeness for mappings in S∗0 H (resp. C0 H and TH) is at least 0.112903. Under the hypotheses of Theorem 1.5, all the partial sums of f are close-to-convex (univalent), and starlike in |z| < rS. Similar comments apply to the next two results. Another well-known result due to Clunie and Sheil-Small [6] states that the coef- ficients of the series of h and g of every convex function f = h + g ∈K0 H satisfy the inequalities (1.7)
Theorem 1.9.
Theorem 1.9. Let h and g have the form (1.1) and the coefficients of the series satisfy the conditions (1.7). Then f = h + g is…
Theorem 1.9. Let h and g have the form (1.1) and the coefficients of the series satisfy the conditions (1.7). Then f = h + g is close-to-convex (univalent), and starlike in the disk |z| < rS, where rS = 1 + 3p −18 + √ 330 62/3 − 1 3q 6(−18 + √ 330)
Theorem 1.9
Theorem 1.9 easily gives the following corollary although Theorem 1.9 is much more stronger.
Theorem 1.9 easily gives the following corollary although Theorem 1.9 is much more stronger.
Corollary 1.10. · radius
Corollary 1.10. The radius of close-to-convexity and the radius of starlikeness for convex mappings in S0 H is at least 0.164878.
Corollary 1.10. The radius of close-to-convexity and the radius of starlikeness for convex mappings in S0 H is at least 0.164878.
Theorem 1.11.
Theorem 1.11. Let h and g have the form (1.1) with |b1| = |g′(0)| < 1, and the coefficients of the series satisfy the conditions |an| + |bn|…
Theorem 1.11. Let h and g have the form (1.1) with |b1| = |g′(0)| < 1, and the coefficients of the series satisfy the conditions |an| + |bn| ≤c for all n ≥2. Then f = h + g is close-to-convex (univalent), and starlike in the disk |z| < rS, where rS = 1 − r c c + 1 −|b1|. The result is sharp.
Theorem 1.11
Theorem 1.11 helps to improve the Bloch-Landau’s theorem for bounded harmonic functions. Consider the class BM H of a harmonic mapping f of…
Theorem 1.11 helps to improve the Bloch-Landau’s theorem for bounded harmonic functions. Consider the class BM H of a harmonic mapping f of the unit disk D with f(0) = fz(0) = fz(0) −1 = 0, and |f(z)| < M for z ∈D. There are two important constants one is relative to the domain of the function while the other one, namely the Bloch constant, is defined relative to the range. In [3], authors proved that if f ∈BM H then f is univalent in |z| < ρ0 and f(|z| < ρ0) contains a disk |w| < R0, where ρ0 ≈
Theorem 1.12. · radius
Theorem 1.12. Let f ∈BM H. Then f = h + g is close-to-convex (univalent) in the disk |z| < r0, where rS = 1 − r 4M 4M + π and f(Dr0)…
Theorem 1.12. Let f ∈BM H . Then f = h + g is close-to-convex (univalent) in the disk |z| < r0, where rS = 1 − r 4M 4M + π and f(Dr0) contains a univalent disk of radius at least RS = rS −4M π r2 S 1 −rS .
Lemma 2.1.
Lemma 2.1. Let h and g have the form (1.1) with |b1| < 1, f = h + g, and satisfy the condition (2.2) ∞ X n=2 n|an| + ∞ X n=1 n|bn| ≤1. Then…
Lemma 2.1. Let h and g have the form (1.1) with |b1| < 1, f = h + g, and satisfy the condition (2.2) ∞ X n=2 n|an| + ∞ X n=1 n|bn| ≤1. Then f ∈C2 H, where C2 H = {f ∈SH : |fz(z) −1| < 1 −|fz(z)| in D}. The bound in (2.2) is sharp as the harmonic function
Lemma 2.3.
Lemma 2.3. Let h and g have the form (1.1) with |b1| < 1, f = h + g. Suppose f ∈C2 H. Then, we have the following (a) f is close-to-convex…
Lemma 2.3. Let h and g have the form (1.1) with |b1| < 1, f = h + g. Suppose f ∈C2 H. Then, we have the following (a) f is close-to-convex in D. (b) |an| −|bn| ≤1/n for n ≥2 whenever b1 = 0. The equality occurs, for example, for the function f(z) = z + eiθ n zn or f(z) = z + eiθ n zn for n ≥2 and θ real. (c) ∞ X n=2
Lemma 2.4.
Lemma 2.4. Let h and g have the form (1.1) with b1 = g′(0) = 0, f = h + g, and satisfy the condition (2.5) ∞ X n=2 n|an| + ∞ X n=2 n|bn|…
Lemma 2.4. Let h and g have the form (1.1) with b1 = g′(0) = 0, f = h + g, and satisfy the condition (2.5) ∞ X n=2 n|an| + ∞ X n=2 n|bn| ≤1. Then f ∈C2 H ∩S∗0 H . The following generalization of Lemma 2.1 is easy to obtain and so we omit its
Corollary 2.6.
Corollary 2.6. Let h and g have the form (1.1) with |b1| < 1−β for some β ∈[0, 1), and f = h + g. Then we have the following: (a) If the…
Corollary 2.6. Let h and g have the form (1.1) with |b1| < 1−β for some β ∈[0, 1), and f = h + g. Then we have the following: (a) If the coefficients of h and g satisfy the condition (2.7) ∞ X n=2 n|an| + ∞ X n=1 n|bn| ≤1 −β, then f ∈C2 H(β), where C2
Theorem 1.5
Theorem 1.5, it suffices to show that fr ∈C2 H ∩S∗0 H. According to Lemma 2.4,
Theorem 1.5, it suffices to show that fr ∈C2 H ∩S∗0 H . According to Lemma 2.4,
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