Abstract
New sufficient conditions, concerned with the coefficients of harmonic functions $f(z)=h(z)+\bar{g(z)}$ in the open unit disk $\mathbb{U}$ normalized by $f(0)=h(0)=h'(0)-1=0$, for $f(z)$ to be harmonic close-to-convex functions are discussed. Furthermore, several illustrative examples and the image domains of harmonic close-to-convex functions satisfying the obtained conditions are enumerated.
Results & Lemmas (12)
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 1.1.
Theorem 1.1. If f(z) = h(z) + g(z) ∈H satisfies g′(z) = zh′(z) and Re 1 + zh′′(z) h′(z) > −1 2 for all z ∈U, then f(z) ∈C0 H ⊂S0 H. A…
Theorem 1.1. If f(z) = h(z) + g(z) ∈H satisfies g′(z) = zh′(z) and Re 1 + zh′′(z) h′(z) > −1 2 for all z ∈U, then f(z) ∈C0 H ⊂S0 H. A simple and interesting example is below. Example 1.1. The function
Theorem 1.2.
Theorem 1.2. If f(z) ∈H satisfies ∞ X n=2 n|an| + ∞ X n=1 n|bn| ≦1, then f(z) ∈CH. Example 1.2. The function f(z) = z + 1 5z5 belongs to the…
Theorem 1.2. If f(z) ∈H satisfies ∞ X n=2 n|an| + ∞ X n=1 n|bn| ≦1, then f(z) ∈CH. Example 1.2. The function f(z) = z + 1 5z5 belongs to the class C0 H ⊂CH and satisfies the condition of Theorem 1.2. Indeed, f(z) maps U onto the
Lemma 1.1.
Lemma 1.1. If h(z) and g(z) are analytic in U with |h′(0)| > |g′(0)| and h(z) + εg(z) is close-to-convex for each ε (|ε| = 1), then f(z) =…
Lemma 1.1. If h(z) and g(z) are analytic in U with |h′(0)| > |g′(0)| and h(z) + εg(z) is close-to-convex for each ε (|ε| = 1), then f(z) = h(z) + g(z) is harmonic close-to-convex.
Lemma 1.2.
Lemma 1.2. If f(z) = h(z) + g(z) is locally univalent in U and h(z) + εg(z) is convex for some ε (|ε| ≦1), then f(z) is univalent…
Lemma 1.2. If f(z) = h(z) + g(z) is locally univalent in U and h(z) + εg(z) is convex for some ε (|ε| ≦1), then f(z) is univalent close-to-convex.
Lemma 1.3.
Lemma 1.3. If a function F(z) = z + ∞ P n=2 Anzn is analytic in U and satisfies ∞ X n=2
Lemma 1.3. If a function F(z) = z + ∞ P n=2 Anzn is analytic in U and satisfies ∞ X n=2
Theorem 2.1.
Theorem 2.1. If f(z) ∈H satisfies the following condition ∞ X n=2 nan −eiϕ(n −1)an−1 + ∞ X n=1 nbn −eiϕ(n −1)bn−1 ≦1 for some real number ϕ…
Theorem 2.1. If f(z) ∈H satisfies the following condition ∞ X n=2 nan −eiϕ(n −1)an−1 + ∞ X n=1 nbn −eiϕ(n −1)bn−1 ≦1 for some real number ϕ (0 ≦ϕ < 2π), then f(z) ∈CH.
Theorem 2.2.
Theorem 2.2. If f(z) ∈H is locally univalent in U and satisfies ∞ X n=2
Theorem 2.2. If f(z) ∈H is locally univalent in U and satisfies ∞ X n=2
Theorem 2.3.
Theorem 2.3. If f(z) ∈H is locally univalent in U with ∞ X n=2 n2|an| ≦1, then f(z) ∈CH. Furthermore, taking α = 1 and β = 0 in the…
Theorem 2.3. If f(z) ∈H is locally univalent in U with ∞ X n=2 n2|an| ≦1, then f(z) ∈CH. Furthermore, taking α = 1 and β = 0 in the theorem, we have
Corollary 2.1.
Corollary 2.1. If f(z) ∈H is locally univalent in U and satisfies ∞ X n=2 n |(n + 1)an −(n −1)an−1| + (n −1) |nan −(n −2)an−1| ≦2, then f(z)…
Corollary 2.1. If f(z) ∈H is locally univalent in U and satisfies ∞ X n=2 {n |(n + 1)an −(n −1)an−1| + (n −1) |nan −(n −2)an−1|} ≦2, then f(z) ∈CH. Example 2.2. The function f(z) = − Z z 0 log(1 −t) t dt + z + (1 −z) log(1 −z)
Lemma 3.1.
Lemma 3.1. Let cn ∞ k=0 be a convex null sequence. Then, the function p(z) = c0 2 + ∞ X n=1 cnzn is analytic and satisfies Re(p(z)) > 0 in…
Lemma 3.1. Let {cn}∞ k=0 be a convex null sequence. Then, the function p(z) = c0 2 + ∞ X n=1 cnzn is analytic and satisfies Re(p(z)) > 0 in U. Applying the above lemma, we deduce
Theorem 3.1.
Theorem 3.1. For some b (|b| < 1) and some convex null sequence cn ∞ n=0 with c0 = 2, the function f(z) = h(z) + g(z) = z + ∞ X n=2 cn−1 n…
Theorem 3.1. For some b (|b| < 1) and some convex null sequence {cn}∞ n=0 with c0 = 2, the function f(z) = h(z) + g(z) = z + ∞ X n=2 cn−1 n zn + b
Theorem 3.2.
Theorem 3.2. For some b (|b| < 1) and some convex null sequence cn ∞ n=0 with c0 = 2, the function f(z) = h(z) + g(z) = z + ∞ X n=2 1 n …
Theorem 3.2. For some b (|b| < 1) and some convex null sequence {cn}∞ n=0 with c0 = 2, the function f(z) = h(z) + g(z) = z + ∞ X n=2 1 n 1 + n−1 X j=1 cj
Function classes studied:
Related Papers