Abstract
Let ${\mathcal U}(λ)$ denote the family of analytic functions $f(z)$, $f(0)=0=f'(0)-1$, in the unit disk $\ID$, which satisfy the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<λ$ for some $0<λ\leq 1$. The logarithmic coefficients $γ_n$ of $f$ are defined by the formula $\log(f(z)/z)=2\sum_{n=1}^\infty γ_nz^n$. In a recent paper, the present authors proposed a conjecture that if $f\in {\mathcal U}(λ)$ for some $0<λ\leq 1$, then $|a_n|\leq \sum_{k=0}^{n-1}λ^k$ for $n\geq 2$ and provided a n
Results & Lemmas (7)
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Theorem 1 · coeff
Theorem 1]. In this paper, we show that Conjecture 1 is true for n = 3, 4. and our proof includes an elegant proof of the case n = 2. The…
Theorem 1]. In this paper, we show that Conjecture 1 is true for n = 3, 4. and our proof includes an elegant proof of the case n = 2. The main results and their proofs are presented in Sections 2 and 3. 2. Logarithmic coefficients of functions in U(λ)
Theorem 1.
Theorem 1. For 0 < λ ≤1, the logarithmic coefficients of f ∈U(λ) satisfy the inequality (4) ∞ X n=1 |γn|2 ≤1 4 π2 6 + 2Li 2(λ) + Li 2(λ2) ,…
Theorem 1. For 0 < λ ≤1, the logarithmic coefficients of f ∈U(λ) satisfy the inequality (4) ∞ X n=1 |γn|2 ≤1 4 π2 6 + 2Li 2(λ) + Li 2(λ2) , where Li2 denotes the dilogarithm function given by Li2(z) = ∞
Corollary 1.
Corollary 1. The logarithmic coefficients of f ∈U satisfy the inequality (7) ∞ X n=1 |γn|2 ≤ ∞ X n=1 1 n2 = π2 6. We have equality in the…
Corollary 1. The logarithmic coefficients of f ∈U satisfy the inequality (7) ∞ X n=1 |γn|2 ≤ ∞ X n=1 1 n2 = π2 6 . We have equality in the last inequality for the Koebe function k(z) = z(1 −eiθz)−2. Further there exists a function f ∈U such that |γn| > 1/n for some n.
Theorem 1
Theorem 1, we conclude that ∞ X n=1 |γn(f1)|2 < π2 6,
Theorem 1, we conclude that ∞ X n=1 |γn(f1)|2 < π2 6 ,
Theorem 2.
Theorem 2. Let 0 < α ≤1 and G(α) be defined as above. Then the logarithmic coefficients γn of f ∈G(α) satisfy the inequalities (8) ∞ X n=1…
Theorem 2. Let 0 < α ≤1 and G(α) be defined as above. Then the logarithmic coefficients γn of f ∈G(α) satisfy the inequalities (8) ∞ X n=1 n2|γn|2 ≤ α 4(α + 2) and (9) ∞ X n=1 |γn|2 ≤α2
Corollary 2.
Corollary 2. The logarithmic coefficients γn of f ∈G:= G(1) satisfy the inequali- ties ∞ X n=1 n2|γn|2 ≤1 12 and ∞ X n=1 |γn|2 ≤1 4 Li 2 1 4…
Corollary 2. The logarithmic coefficients γn of f ∈G := G(1) satisfy the inequali- ties ∞ X n=1 n2|γn|2 ≤1 12 and ∞ X n=1 |γn|2 ≤1 4 Li 2 1 4
Theorem 3.
Theorem 3. Let f ∈U(λ) for 0 < λ ≤1 and let f(z) = z +a2z2 +a3z3 +· · ·. Then (15) |an| ≤1 −λn 1 −λ for 0 < λ < 1 and n = 2, 3, 4, and |an|…
Theorem 3. Let f ∈U(λ) for 0 < λ ≤1 and let f(z) = z +a2z2 +a3z3 +· · · . Then (15) |an| ≤1 −λn 1 −λ for 0 < λ < 1 and n = 2, 3, 4, and |an| ≤n for λ = 1 and n ≥2. The results are the best possible.
Function classes studied:
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