Hankel and Toeplitz determinants of logarithmic coefficients of Inverse functions for certain classes of univalent functions

2023/08/03 by Mandal, Sanju, Roy, Partha Pratim, Ahamed, Molla Basir
#30C35 #30H05 #Complex Variables (math.CV) #FOS: Mathematics #Primary 30A10 #Secondary 30C45

paper · doi:10.48550/arxiv.2308.01548

Abstract

The Hankel and Toeplitz determinants H_2,1(F_f^-1/2) and T_2,1(F_f^-1/2) are defined as: H_2,1(F_f^-1/2):= \beginvmatrix Γ₁ amp; Γ₂ Γ₂ amp; Γ₃ \endvmatrix and T_2,1(F_f^-1/2):= \beginvmatrix Γ₁ amp; Γ₂ Γ₂ amp; Γ₁ \endvmatrix where Γ₁, Γ₂, and Γ₃ are the first, second and third logarithmic coefficients of inverse functions belonging to the class S of normalized univalent functions. In this article, we establish sharp inequalities |H_2,1(F_f^-1/2)|≤ 1/4, |H_2,1(F_f^-1/2)| ≤ 1/36, |T_2,1(F_f^-1/2)|≤ 5/16 and |T_2,1(F_f^-1/2)|≤ 145/2304 for the logarithmic coefficients of inverse functions for the classes starlike functions and convex functions with respect to symmetric points. In addition, our findings are substantiated further through the incorporation of illustrative examples, which support the strict inequality and lend credence to our conclusions.

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