Abstract
Let $f=h+\overline{g}$ be a normalized harmonic mapping in the unit disk $\ID$. In this paper, we obtain the sharp radius of univalence, fully starlikeness and fully convexity of the harmonic linear differential operators $D_f^ε=zf_{z}-ε\overline{z}f_{\overline{z}}~(|ε|=1)$ and $F_λ(z)=(1-λ)f+λD_f^ε~(0\leqλ\leq 1)$ when the coefficients of $h$ and $g$ satisfy harmonic Bieberbach coefficients conjecture conditions. Similar problems are also solved when the coefficients of $h$ and $g$ satisfy the
Results & Lemmas (14)
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Lemma 1.1.
Lemma 1.1. Let f = h + g ∈H, where h and g are given by (1.1). Furthermore, let ∞ X n=2 n −α 1 −α |an| + ∞ X n=1 n + α 1 −α |bn| ≤1 and 0…
Lemma 1.1. Let f = h + g ∈H, where h and g are given by (1.1). Furthermore, let ∞ X n=2 n −α 1 −α |an| + ∞ X n=1 n + α 1 −α |bn| ≤1 and 0 ≤α < 1. Then f ∈FS∗ H(α).
Lemma 1.2.
Lemma 1.2. Let f = h + g ∈H, where h and g are given by (1.1). Furthermore, let ∞ X n=2 n(n −α) 1 −α |an| + ∞ X n=1 n(n + α) 1 −α |bn| ≤1…
Lemma 1.2. Let f = h + g ∈H, where h and g are given by (1.1). Furthermore, let ∞ X n=2 n(n −α) 1 −α |an| + ∞ X n=1 n(n + α) 1 −α |bn| ≤1 and 0 ≤α < 1. Then f ∈FCH(α). According to Rad´o-Kneser-Choquet theorem, a fully convex harmonic mapping is necessarily
Lemma 2.1.
Lemma 2.1. We have (a) ∞ X n=2 n rn−1 = r(2 −r) (1 −r)2, (b) ∞ X n=2 n2rn−1 = r 4 −3r + r2 (1 −r)3,
Lemma 2.1. We have (a) ∞ X n=2 n rn−1 = r(2 −r) (1 −r)2 , (b) ∞ X n=2 n2rn−1 = r 4 −3r + r2 (1 −r)3 ,
Theorem 2.2. · radius
Theorem 2.2. Let f = h + g, where h and g are given by (1.1), and the coefficients satisfy the conditions (1.2) for n ≥2. Then for Dǫ f = zfz…
Theorem 2.2. Let f = h + g, where h and g are given by (1.1), and the coefficients satisfy the conditions (1.2) for n ≥2. Then for Dǫ f = zfz −ǫ zfz (|ǫ| = 1), (1) the radius of fully starlikeness of order α is rs(α), where rs(α) is the unique root of the equation pα(r) = 0 in the interval (0, 1), where pα(r) = 1 −α −(17 −9α)r + (13 −21α)r2 −21(1 −α)r3 + 10(1 −α)r4 −2(1 −α)r5, (2.2) (2) the radius of fully starlikeness is ru ≈0.0614313, where ru is the unique root of the equation 1 −17r + 13r2 −21
Theorem 2.3.
Theorem 2.3. Under the hypothesis of Theorem 2.2, for Dǫ f = zfz −ǫ zfz (|ǫ| = 1) is fully convex of order α in |z| < rc(α), where rc(α) is…
Theorem 2.3. Under the hypothesis of Theorem 2.2, for Dǫ f = zfz −ǫ zfz (|ǫ| = 1) is fully convex of order α in |z| < rc(α), where rc(α) is the unique root of the equation 1 −α −2(15 −7α)r −12(1 + 3α)r2 −2(29 −21α)r3 + 29(1 −α)r4 −12(1 −α)r5 + 2(1 −α)r6 = 0 in the interval (0, 1). In particular, Dǫ f is fully convex in |z| < rc ≈0.0328348, where rc is the unique root of the equation 1 −30r −12r2 −58r3 + 29r4 −12r5 + 2r6 = 0 in the interval (0, 1). All results are sharp.
Theorem 2.4.
Theorem 2.4. Let f = h + g, where h and g are given by (1.1), and the coefficients satisfy the conditions (1.4) for n ≥2. Then for Dǫ f = zfz…
Theorem 2.4. Let f = h + g, where h and g are given by (1.1), and the coefficients satisfy the conditions (1.4) for n ≥2. Then for Dǫ f = zfz −ǫ zfz (|ǫ| = 1),
Theorem 2.4
Theorem 2.4, it suffices to show that Dǫ,r f ∈FS∗ H(α), where Dǫ,r f (z) is defined by (2.1). Accordingly, we consider S2 = ∞ X n=2 n −α 1 −α…
Theorem 2.4, it suffices to show that Dǫ,r f ∈FS∗ H(α), where Dǫ,r f (z) is defined by (2.1). Accordingly, we consider S2 = ∞ X n=2 n −α 1 −α |nan|rn−1 + ∞ X n=2
Theorem 2.5. · radius
Theorem 2.5. Let f = h + g, where h and g are given by (1.1), and the coefficients satisfy the conditions (1.4) for n ≥2. Then Dǫ f = zfz −ǫ…
Theorem 2.5. Let f = h + g, where h and g are given by (1.1), and the coefficients satisfy the conditions (1.4) for n ≥2. Then Dǫ f = zfz −ǫ zfz (|ǫ| = 1) is fully convex of order α in |z| < rc(α), where rc(α) is the unique root of the equation 1 −α −(23 −10α)r + (19 −30α)r2 −(41 −42α)r3 + (30 −31α)r4 −12(1 −α)r5 + 2(1 −α)r6 = 0 in the interval (0, 1). In particularly, Dǫ f is fully convex in |z| < rc ≈0.0449935, where rc is the unique root of the equation 1 −23r + 19r2 −41r3 + 30r4 −12r5 + 2r6 =
Theorem 3.1.
Theorem 3.1. Let f = h + g, where h and g have the form (1.1) with Jf(0) = 1 −|b1|2 > 0. If ∞ X n=2 (1 −λ + λn)n (|an| + |bn|) ≤1 −|b1| (λ…
Theorem 3.1. Let f = h + g, where h and g have the form (1.1) with Jf(0) = 1 −|b1|2 > 0. If ∞ X n=2 (1 −λ + λn)n (|an| + |bn|) ≤1 −|b1| (λ ≥0). (3.1) holds, then f ∈K2 H(λ).
Theorem 3.2.
Theorem 3.2. Let h and g have the form (1.1) and the coefficients of the series satisfy the condi- tions (1.2). Then, f = h + g satisfies the…
Theorem 3.2. Let h and g have the form (1.1) and the coefficients of the series satisfy the condi- tions (1.2). Then, f = h + g satisfies the inequality |h′ λ(z) −1| < 1 −|g′ λ(z)| (λ ≥0) in the disk |z| < rs and is fully starlike in |z| < rs, where rs is the root of the equation 1 −(11 + 6λ)r + (21 −8λ)r2 −(19 + 2λ)r3 + 10r4 −2r5 = 0 (3.2) in the interval (0, 1). The result is sharp.
Corollary 3.3. · radius
Corollary 3.3. Let f = h + g, where h and g are given by (1.1), and the coefficients satisfy the conditions (1.2) for n ≥2. Then the radius…
Corollary 3.3. Let f = h + g, where h and g are given by (1.1), and the coefficients satisfy the conditions (1.2) for n ≥2. Then the radius of fully starlikeness for F(z) = (1 −λ)f + λDǫ f (0 ≤ λ ≤1) is at least rs, where rs is the root of the equation (3.2) in the interval (0, 1).
Theorem 3.4.
Theorem 3.4. Let h and g have the form (1.1) and the coefficients of the series satisfy the condi- tions (1.4). Then, f = h + g ∈K2 H(λ) (λ…
Theorem 3.4. Let h and g have the form (1.1) and the coefficients of the series satisfy the condi- tions (1.4). Then, f = h + g ∈K2 H(λ) (λ ≥0) in the disk |z| < rc and is fully starlike in |z| < rc, where rc is the root of the equation 1 −4(2 + λ)r + (13 −2λ)r2 −8r3 + 2r4 = 0 (3.4) in the interval (0, 1). The result is sharp.
Corollary 3.5. · radius
Corollary 3.5. Let f = h + g, where h and g are given by (1.1), and the coefficients satisfy the conditions (1.4) for n ≥2. Then the radius…
Corollary 3.5. Let f = h + g, where h and g are given by (1.1), and the coefficients satisfy the conditions (1.4) for n ≥2. Then the radius of fully starlikeness for F(z) = (1 −λ)f + λDǫ f is at least rc. where rc is the root of the equation (3.4) in the interval (0, 1). The result is sharp.
Theorem 3.6.
Theorem 3.6. Let h and g have the form (1.1) with |b1| = |g′(0)| < 1, and the coefficients satisfy the conditions |an| + |bn| ≤c for n ≥2.…
Theorem 3.6. Let h and g have the form (1.1) with |b1| = |g′(0)| < 1, and the coefficients satisfy the conditions |an| + |bn| ≤c for n ≥2. Then, f = h + g ∈K2 H(λ) and is fully starlike in |z| < rv, where rv is the root of the equation Φc,|b1|,λ(r) = 0 in the interval (0, 1), where Φc,|b1|,λ(r) = (1 + c −|b1|)(1 −r)3 −c [1 + (2λ −1)r] . (3.5) The result is sharp.
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