On the Moduli Difference of Initial Logarithmic and Inverse Logarithmic Coefficients of Ma-Minda Convex Functions
Abstract
Let 𝒞(Φ) denote a subclass of class $ \mathcal{C}$ of convex functions in the unit disc 𝔻={z:|z|<1, z∈ ℂ} defined by Re {1+ ( zf′ ′(z))/f′(z) } ≺ Φ(z), where Φ represents the Ma--Minda function in 𝔻. For f∈𝒞(Φ), we find the sharp estimates of ||γ2|−|γ1||, where γ1 and γ2 are the first two logarithmic coefficients of f. The bound is also obtained for the inverse logarithmic coefficients Γ1 and Γ2. We then show that, under suitable conditions, the bounds for ||γ2|−|γ1|| and ||Γ2|−|Γ1|| are identical, which gives the invariance result between these functionals. We also present bounds for the differences for various classes of univalent functions by choosing particular function Φ. Thus our results unify several results studied in the literature.
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BibTeX
@misc{ali2026moduli,
title = {{On the Moduli Difference of Initial Logarithmic and Inverse Logarithmic Coefficients of Ma-Minda Convex Functions}},
author = {Rashid Ali and Mohsan Raza and Umar Raza},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20645},
eprint = {2610.20645},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20645},
}
RIS
TY - UNPB
TI - On the Moduli Difference of Initial Logarithmic and Inverse Logarithmic Coefficients of Ma-Minda Convex Functions
AU - Rashid Ali
AU - Mohsan Raza
AU - Umar Raza
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.20645
UR - https://arxiv.org/abs/2610.20645
AB - Let 𝒞(Φ) denote a subclass of class $ \mathcal{C}$ of convex functions in the unit disc 𝔻={z:|z|<1, z∈ ℂ} defined by Re {1+ ( zf^′ ′(z))/f^′(z) } ≺ Φ(z), where Φ represents the Ma--Minda function in 𝔻. For f∈𝒞(Φ), we find the sharp estimates of ||γ_2|−|γ_1||, where γ_1 and γ_2 are the first two logarithmic coefficients of f. The bound is also obtained for the inverse logarithmic coefficients Γ_1 and Γ_2. We then show that, under suitable conditions, the bounds for ||γ_2|−|γ_1|| and ||Γ_2|−|Γ_1|| are identical, which gives the invariance result between these functionals. We also present bounds for the differences for various classes of univalent functions by choosing particular function Φ. Thus our results unify several results studied in the literature.
ER -