Solution of logarithmic coefficients conjectures for some classes of convex functions

Abstract

In [Logarithmic coefficient bounds and coefficient conjectures for classes associated with convex functions, J. Funct. Spaces 2021 (2021), Art. ID 6690027], Alimohammadi et al. presented a few conjectures for the logarithmic coefficients γ n of the functions f belonging to some well-known classes like C ( 1 + α z ) $ \mathcal{C}(1+\alpha z) $ for α ∈ (0, 1], and C V h p l ( 1 / 2 ) $ \mathcal{CV}_{hpl}(1/2) $ . For example, it is conjectured that if the function f ∈ C ( 1 + α z ) $ f\in\mathcal{C}(1+\alpha z) $ , then the logarithmic coefficients of f satisfy the inequalities | γ n | ≤ α 2 n ( n + 1 ) , n ∈ N . $$ |\gamma_n|\le\dfrac{\alpha}{2n(n+1)},\quad n\in\mathbb{N}. $$ Equality is attained for the function L α, n , that is, log L α , n ( z ) z = 2 ∑ n = 1 ∞ γ n ( L α , n ) z n = α n ( n + 1 ) z n + … , z ∈ U . $$ \log\dfrac{L_{\alpha,n}(z)}{z}=2\sum\limits_{n=1}^{\infty}{\gamma_n(L_{\alpha,n})z^n} =\frac{\alpha}{n(n+1)}z^n+\dots,\quad z\in\mathbb{U}. $$ The aim of this paper is to confirm that these conjectures hold for the coefficient γ n 0−1 whenever the function f has the form f ( z ) = z + ∑ k = n 0 ∞ a k z k $ f(z)=z+\sum\limits_{k=n_{0}}^{\infty}{a_kz^k} $ , z ∈ U $ z\in\mathbb{U} $ for some n 0 ∈ N $ n_0\in\mathbb{N} $ , n 0⩾2.

Logarithmic Coefficients Inequality for the Family of Functions Convex in One Direction

“…In [17], the researchers obtained the bounds of the logarithmic coefficients for particular subfamilies of univalent functions and found the sharp upper bound for γ n when n = 1, 2, 3 if f belongs to the family G(γ) (see also [18]). Logarithmic coefficients problem was also considered for the another well-known classes, for example see [12,[19][20][21][22][23][24][25].…”

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