Logarithmic Coefficients of the Inverse of Univalent Functions

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 coefficients based on the proof by de Branges. In this paper, following Roth's ideas, we will show more general sharp inequalities with convex sequences as weight functions and then establish several consequences of them. We also consider the inequality with the help of de Branges system of linear ODE for non-convex sequences where the proof is partly assisted by computer. Also, we apply some of those inequalities to improve previously known results.2010 Mathematics Subject Classification. Primary 30C50; Secondary 30C75.

Logarithmic Coefficients for Univalent Functions Defined by Subordination

“…2. For A = e iα (e iα − 2β cos α), where β ∈ [0, 1) and α ∈ (−π/2, π/2) in the above expression, then we get the results obtained by Ponnusamy et al [7] (Theorem 2.5).…”

“…and it concludes the inequality (7). Finally, we suppose that ϕ is starlike with respect to 1 in U, which implies φ(z) is starlike, and thus by Lemma 4(ii), we obtain 2n|γ…”

“…All inequalities in (5), (7), and (8) are sharp such that for any n ∈ N, there is the function f n given by z f n (z) f n (z) = ϕ(z n ) and the function f given by z f (z) f (z) = ϕ(z), respectively.…”

“…Thus, every convex function is starlike but not conversely; however, each starlike function is convex in the disk of radius 2 − √ 3. Lately, several researchers have subsequently investigated similar problems in the direction of the logarithmic coefficients, the coefficient problems, and differential subordination [3][4][5][6][7][8][9][10][11], to mention a few. For example, the rotation of Koebe function k(z) = z(1 − e iθ ) −2 for each θ has logarithmic coefficients γ n = e iθn /n, n ≥ 1.…”

Bohr Phenomenon for Locally Univalent Functions and Logarithmic Power Series

“…(compare [21]). We also compute the Bohr radius for log (f −1 (w)/w) with respect to the inequality (1.4), where r = |w| and f ∈ S(or S * ).…”