Logarithmic coefficients and a coefficient conjecture for univalent functions

Abstract

Abstract. Let U(λ) denote the family of analytic functions f (z), f (0) = 0 = f (0) − 1, in the unit disk D, which satisfy the condition z/f (z) 2 f (z) − 1 < λ for some 0 < λ ≤ 1. The logarithmic coefficients γ n of f are defined by the formula log(f (z)/z) = 2 ∞ n=1 γ n z n . In a recent paper, the present authors proposed a conjecture that if f ∈ U(λ) for some 0 < λ ≤ 1, then |a n | ≤ n−1 k=0 λ k for n ≥ 2 and provided a new proof for the case n = 2. One of the aims of this article is to present a proof of this conjecture for n = 3, 4 and an elegant proof of the inequality for n = 2, with equality for f (z) = z/[(1 + z)(1 + λz)]. In addition, the authors prove the following sharp inequality for f ∈ U(λ):where Li 2 denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of S.

Logarithmic Coefficients Problems in Families Related to Starlike and Convex Functions

“…In [15], the authors considered the logarithmic coefficients of the functions f in the class G(c) for some c ∈ (0, 1] and they got the estimate |γ n | ≤ c 2(c + 1)n , n ∈ N.…”

Inequalities for weighted sums of Mertens functions

“…Let µ(n) be the Möbius function of the positive integer n, that is, (a) µ(1) = 1, (b) µ(n) = 0, if a square number is a divisor of n, (c) µ(n) = (−1) r , if n is the product of r pairwise disjoint prime numbers. During their efforts to prove a coefficient conjecture (see [4,Conjecture 1]) for some classes of univalent functions, the second and the third authors of the present paper considered an inequality that concerned the Mertens function. See also [5].…”

Bohr Phenomenon for Locally Univalent Functions and Logarithmic Power Series

“…We know that proving inequalities concerning |γ n |'s is considered to be a challenging problem till date (see f.i. [17,22] and references therein) due to the unavailability of the sharp bounds on |γ n |'s for n ≥ 3, where f ∈ S. Inspired by this fact, in this article we have considered the problem of establishing Bohr inequalities similar to the inequality (1.1) for log(f (z)/z). More precisely, we will say that log(f (z)/z) has Bohr radius r 0 ∈ (0, 1] if…”