Abstract
Let $\es$ be the family of analytic and univalent functions $f$ in the unit disk $\D$ with the normalization $f(0)=f'(0)-1=0$, and let $γ_n(f)=γ_n$ denote the logarithmic coefficients of $f\in {\es}$. In this paper, we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families $\F(c)$ and $\G(δ)$ of functions $f\in {\es}$ defined by $$ {\rm Re} \left ( 1+\frac{zf''(z)}{f'(z)}\right )>1-\frac{c}{2}\, \mbox{ and } \, {\rm Re} \left
Results & Lemmas (15)
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Theorem 2.1.
Theorem 2.1. For −1 ≤B < A ≤1 and B ̸= 0, the logarithmic coefficients of f ∈S∗(A, B) satisfy the inequalities |γn| ≤A −B 2n for n ≥1, (3)…
Theorem 2.1. For −1 ≤B < A ≤1 and B ̸= 0, the logarithmic coefficients of f ∈S∗(A, B) satisfy the inequalities |γn| ≤A −B 2n for n ≥1, (3) and ∞ X n=1 |γn|2 ≤ A −B 2B 2 Li 2(B2),
Corollary 2.2.
Corollary 2.2. If f ∈S∗(A, −A) for 0 < A ≤1, then we have |γn| ≤A n for n ≥1, and ∞ X n=1 |γn|2 ≤Li2(A2). The first and second inequalities…
Corollary 2.2. If f ∈S∗(A, −A) for 0 < A ≤1, then we have |γn| ≤A n for n ≥1, and ∞ X n=1 |γn|2 ≤Li2(A2). The first and second inequalities are sharp for the functions kA;n(z) = z/(1− Azn)2/n and kA;1(z), respectively.
Theorem 2.1
Theorem 2.1 for the case B = 0 takes the following form.
Theorem 2.1 for the case B = 0 takes the following form.
Corollary 2.3.
Corollary 2.3. Let 0 < A ≤1 and f ∈A satisfy the inequality zf′(z) f(z) −1 < A, z ∈D, i.e., f ∈S∗(A, 0). Then the logarithmic coeffi- cients…
Corollary 2.3. Let 0 < A ≤1 and f ∈A satisfy the inequality zf′(z) f(z) −1 < A, z ∈D, i.e., f ∈S∗(A, 0). Then the logarithmic coeffi- cients of f satisfy the inequalities |γn| ≤A 2n for n ≥1, and ∞ X n=1 |γn|2 ≤A2 4 . Both inequalities are sharp for the functions kA;n(z) = zeAzn/n and kA;1(z), respectively. Our next result, which uses the method of proof of [1, Theorem 2.5]
Theorem 2.4.
Theorem 2.4. Let f ∈S∗(A, B) for −1 ≤B < A ≤1, and let t ≤2. Then we have ∞ X n=1 (n + 1)t|γn|2 ≤ A −B 2B 2 ∞ X n=1 (n + 1)t n2 |B|2n. 5
Theorem 2.4. Let f ∈S∗(A, B) for −1 ≤B < A ≤1, and let t ≤2. Then we have ∞ X n=1 (n + 1)t|γn|2 ≤ A −B 2B 2 ∞ X n=1 (n + 1)t n2 |B|2n. 5
Theorem 2.5.
Theorem 2.5. For |α| < π/2 and β ∈[0, 1), the logarithmic coefficients of f ∈Sα(β) satisfy the inequalities |γn| ≤(1 −β) n cos α for n ≥1,…
Theorem 2.5. For |α| < π/2 and β ∈[0, 1), the logarithmic coefficients of f ∈Sα(β) satisfy the inequalities |γn| ≤(1 −β) n cos α for n ≥1, (5) and ∞ X n=1 |γn|2 ≤π2 6 (1 −β)2 cos2 α. (6) Both inequalities are sharp for the function fα,β(z) = z/(1 −z)2(1−β) cos α. For the case of strongly starlike functions, we have the following.
Theorem 2.6.
Theorem 2.6. Let 0 < α ≤1 and An(α) = n X k=1 n −1 k −1 α k 2k. Then the logarithmic coefficients γn of f ∈SS∗ α satisfy the…
Theorem 2.6. Let 0 < α ≤1 and An(α) = n X k=1 n −1 k −1 α k 2k. Then the logarithmic coefficients γn of f ∈SS∗ α satisfy the inequalities |γn| ≤α n,
Theorem 2.7.
Theorem 2.7. Let f ∈F(c) for c ∈(0, 3]. Then the logarithmic coeffi- cients γn of f for n = 1, 2,..., 5, satisfy the inequalities …
Theorem 2.7. Let f ∈F(c) for c ∈(0, 3]. Then the logarithmic coeffi- cients γn of f for n = 1, 2, . . . , 5, satisfy the inequalities |γ1| ≤c 4,
Corollary 2.8.
Corollary 2.8. Let F(3), i.e. Re (1 + (zf ′′(z)/f ′(z))) > −1/2 in D. Then we have |γ1| ≤3 4, |γ2| ≤7 16, |γ3| ≤5 16, |γ4| ≤1 40
Corollary 2.8. Let F(3), i.e. Re (1 + (zf ′′(z)/f ′(z))) > −1/2 in D. Then we have |γ1| ≤3 4, |γ2| ≤7 16, |γ3| ≤5 16, |γ4| ≤1 40
Theorem 2.10.
Theorem 2.10. Let f ∈G(c) for c ∈(0, 1]. Then the logarithmic coef- ficients γn of f satisfy the inequalities |γ1| ≤c 4, |γ2| ≤c 12, and…
Theorem 2.10. Let f ∈G(c) for c ∈(0, 1]. Then the logarithmic coef- ficients γn of f satisfy the inequalities |γ1| ≤c 4, |γ2| ≤c 12, and |γ3| ≤c 24. The inequalities are attained for f ′(z) = (1 −zn)c/n, n = 1, 2, 3. 8
Lemma 3.1.
Lemma 3.1. [13, Corollary 3.1d.1, p. 76] Let h be starlike in D, with h(0) = 0 and a ̸= 0. If an analytic function p(z) = a+anzn+an+1zn+1+·…
Lemma 3.1. [13, Corollary 3.1d.1, p. 76] Let h be starlike in D, with h(0) = 0 and a ̸= 0. If an analytic function p(z) = a+anzn+an+1zn+1+· · · satisfies the subordination relation zp′(z) p(z) ≺h(z), then p(z) ≺q(z) = a exp n−1 Z z 0 h(t) t dt
Lemma 3.2.
Lemma 3.2. [4, Theorem 6.3, p. 192] (see also [19, Rogosinski’s Theo- rem II (i)]) Let f(z) = P∞ n=1 anzn and g(z) = P∞ n=1 bnzn be…
Lemma 3.2. [4, Theorem 6.3, p. 192] (see also [19, Rogosinski’s Theo- rem II (i)]) Let f(z) = P∞ n=1 anzn and g(z) = P∞ n=1 bnzn be analytic in D, and suppose that f ≺g, where g is univalent in D. Then n X k=1 |ak|2 ≤ n X k=1 |bk|2, n = 1, 2, . . . .
Lemma 3.3.
Lemma 3.3. [4, Theorem 6.4 (i), p. 195] (see also [19, Rogosinski’s Theorem X]) Let f(z) = P∞ n=1 anzn and g(z) = P∞ n=1 bnzn be analytic…
Lemma 3.3. [4, Theorem 6.4 (i), p. 195] (see also [19, Rogosinski’s Theorem X]) Let f(z) = P∞ n=1 anzn and g(z) = P∞ n=1 bnzn be analytic in D. Suppose that f ≺g, where g is univalent and convex in D. Then |an| ≤|g′(0)| = |b1|, n = 1, 2, . . . . Our next lemma due to Prokhorov and Szynal [18] is crucial in the investigation of fourth and fifth logarithmic coefficients bound for F(c) and G(c).
Lemma 3.4.
Lemma 3.4. [18, Lemma 2] Let φ(z) = P∞ k=1 ckzk ∈B be a Schwarz function and Ψ(φ) = |c3 + µc1c2 + υc3 1|. 9
Lemma 3.4. [18, Lemma 2] Let φ(z) = P∞ k=1 ckzk ∈B be a Schwarz function and Ψ(φ) = |c3 + µc1c2 + υc3 1|. 9
Theorem 2.7
Theorem 2.7, f ∈G(c) if and only if 1 + zf ′′(z) f ′(z) ≺1 − cz 1 −z. Hence, we get β1 = −cc1, β2 = −c c2 + c2 1 , and β3 = −c c3 +…
Theorem 2.7, f ∈G(c) if and only if 1 + zf ′′(z) f ′(z) ≺1 − cz 1 −z . Hence, we get β1 = −cc1, β2 = −c c2 + c2 1 , and β3 = −c c3 + 2c1c2 + c3
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