Abstract
I. M. Milin proposed, in his 1971 paper, a system of inequalities for the logarithmic coefficients of normalized univalent functions on the unit disk of the complex plane. This is known as the Lebedev-Milin conjecture and implies the Robertson conjecture which in turn implies the Bieberbach conjecture. In 1984, Louis de Branges settled the long-standing Bieberbach conjecture by showing the Lebedev-Milin conjecture. Recently, O.~Roth proved an interesting sharp inequality for the logarithmic coef
Results & Lemmas (11)
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Theorem 1.1.
Theorem 1.1. Let pn, n = 1, 2, 3,..., be a convex sequence of non-negative numbers with p1 > 0 such that P∞ n=1(pn/n) < +∞. For f ∈S with…
Theorem 1.1. Let pn, n = 1, 2, 3, . . ., be a convex sequence of non-negative numbers with p1 > 0 such that P∞ n=1(pn/n) < +∞. For f ∈S with expansion (1.1), the inequality (1.3) ∞ X n=1 npn|γn|2 ≤ ∞ X n=1 pn n holds. Moreover, the inequality is strict unless f(z) has the form z/(1 −eiθz)2 for some θ ∈R.
Theorem 1.1
Theorem 1.1 if and only if r < 1. Then we obtain the sharp inequality for the logarithmic area ∞ X n=1 n|γn|2r2n ≤ ∞ X n=1 r2n n = log 1 1…
Theorem 1.1 if and only if r < 1. Then we obtain the sharp inequality for the logarithmic area ∞ X n=1 n|γn|2r2n ≤ ∞ X n=1 r2n n = log 1 1 −r2, which is known as the Bazilevi˘c conjecture and proved by Milin and Grinshpan [10] (see also [9]). The next fundamental example is pn = n−α for a constant α > 0. Since
Lemma 2.1.
Lemma 2.1. For each choice of the following, the sequence pn (n = 1, 2, 3,...) is positive and convex. (1) pn = 1 n + α and α > −1, (2) pn…
Lemma 2.1. For each choice of the following, the sequence pn (n = 1, 2, 3, . . .) is positive and convex. (1) pn = 1 n + α and α > −1, (2) pn = n n2 + an + b for a, b ∈R with a + b + 1 > 0, a + 3 ≥0 and (6 + a)b ≤6. (3) pn = n (n + α)(n + β) for α > −1, β > −1 with (α + β + 6)αβ ≤6 and αβ ≤6. (4) pn = 1 n2 + an + b for a, b ∈R with a + b + 1 > 0, a + 2 ≥0 and b ≤a2 + 6a + 11. (5) pn =
Corollary 2.2.
Corollary 2.2. For the logarithmic coefficients γn of f ∈S, the following inequalities hold. Each of them is strict unless f is not a…
Corollary 2.2. For the logarithmic coefficients γn of f ∈S, the following inequalities hold. Each of them is strict unless f is not a rotation of the Koebe function z/(1 −z)2.
Corollary 2.3.
Corollary 2.3. (1) ∞ X n=1 n2 (n + 1)3|γn|2 ≤ζ(3) −1, (2) ∞ X n=1 n (n + 1)3|γn|2 ≤1 6 18 −π2 −6ζ(3)
Corollary 2.3. (1) ∞ X n=1 n2 (n + 1)3|γn|2 ≤ζ(3) −1, (2) ∞ X n=1 n (n + 1)3|γn|2 ≤1 6 18 −π2 −6ζ(3)
Theorem 3.1.
Theorem 3.1. Let pn, n = 1, 2, 3,..., be a sequence of non-negative numbers and set qn = pn −pn+1 and λn = qn −qn+1. Suppose that there…
Theorem 3.1. Let pn, n = 1, 2, 3, . . ., be a sequence of non-negative numbers and set qn = pn −pn+1 and λn = qn −qn+1. Suppose that there exists a number N ≥1 satisfying the following three conditions: (0) pN+1 > 0, (i) λn ≥0 for n > N, (ii) Qk(x) = N−k X j=0 νj+kP (2k,0) j (x) > 0 for −1 < x < 1 and k = 1, 2, . . . , N, where νm = qm −qN+1. Then the inequality ∞
Proposition 3.2.
Proposition 3.2. Under the hypothesis of Theorem 3.1, a necessary condition for (3.1) is vk = vk,N = [(N−k)/2] X j=0 λk+2j = λk + λk+2 +…
Proposition 3.2. Under the hypothesis of Theorem 3.1, a necessary condition for (3.1) is vk = vk,N = [(N−k)/2] X j=0 λk+2j = λk + λk+2 + λk+4 + · · · + λN′ ≥0, k = 1, 2, . . . , N, where N′ = N if N −k is even and N′ = N −1 if N −k is odd.
Theorem 3.3.
Theorem 3.3. For the logarithmic coefficients γn of a function f ∈S, the inequality ∞ X n=1 n2 n2 + 4/3|γn|2 ≤B2/ √ 3 = (2π/ √ 3) coth(2π/ √…
Theorem 3.3. For the logarithmic coefficients γn of a function f ∈S, the inequality ∞ X n=1 n2 n2 + 4/3|γn|2 ≤B2/ √ 3 = (2π/ √ 3) coth(2π/ √ 3) −1 8/3 = 0.98727 · · · holds, where the inequality is strict unless f is a rotation of the Koebe function.
Theorem 3.4.
Theorem 3.4. Let β = 1/20. For the logarithmic coefficients γn of a function f ∈S, the sharp inequality ∞ X n=1 n3|γn|2 (n + 1)2(n + β) ≤E1,β…
Theorem 3.4. Let β = 1/20. For the logarithmic coefficients γn of a function f ∈S, the sharp inequality ∞ X n=1 n3|γn|2 (n + 1)2(n + β) ≤E1,β = 20 192 1 −γ −ψ 21 20 −20 19
Theorem 4.1.
Theorem 4.1. Let f ∈S, b = |f ′′(0)|/2!, and let H be defined by (4.2). Then H ∈U in the disk |z| < r2(b), where r2(b) ≥r2(0) ≈0.558509 is…
Theorem 4.1. Let f ∈S, b = |f ′′(0)|/2!, and let H be defined by (4.2). Then H ∈U in the disk |z| < r2(b), where r2(b) ≥r2(0) ≈0.558509 is the solution of the equation log 1 1 −r2 + −r2 + 23r4 + 18r6 (1 −r2)3 = 20 4E1,1/20 −5b2/84 in 0 < r < 1 for b ∈[0, 2] and E1,1/20 is the constant given in Theorem 3.4. The method of the proof is along the line of [13] but based on Theorem 3.4 instead of the Roth inequality.
Theorem 4.2.
Theorem 4.2. Let f ∈S and b = |f ′′(0)|/2!. Then Pf ∈U on the disk |z| < r4(b). Here, r = r4(b) is the solution of the equation 4B2/ √ 3…
Theorem 4.2. Let f ∈S and b = |f ′′(0)|/2!. Then Pf ∈U on the disk |z| < r4(b). Here, r = r4(b) is the solution of the equation 4B2/ √ 3 −3b2 7 r4(r6 + 2r4 + 11r2 + 4) = 3 4(1 −r2)5 in 0 < r < 1 for b ∈[0, 2] and B2/ √ 3 is the constant given in Theorem 3.3. The function r4(b) is increasing in 0 ≤b ≤2 and r4(b) ≥r4(0) ≈0.362012.
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