THE SHARP BOUND OF THE SECOND HANKEL DETERMINANT OF LOGARITHMIC COEFFICIENTS FOR STARLIKE AND CONVEX FUNCTIONS

2024/11/06 by VASUDEVARAO ALLU, Vasudevarao Allu, AMAL SHAJI +1
Mathematics · #Analytic and geometric function theory #Holomorphic and Operator Theory #Differential Equations and Boundary Problems

paper · pdf · doi:10.1017/s0004972724000947

Abstract

Abstract Let \mathcal S denote the class of univalent functions in the open unit disc \mathbb D:=\z∈ \mathbb C: |z|<1\ with the form f(z)= z+∑ _n=2^∞ a_n zⁿ . The logarithmic coefficients γ _n of f∈ \mathcal S are defined by F_f(z):= log (f(z)/z)=2∑ _n=1^∞ γ _nzⁿ . The second Hankel determinant for logarithmic coefficients is defined by \beginalign* H_2,2(F_f/2) = \beginvmatrix γ₂ amp; γ₃
γ₃ amp; γ₄ \endvmatrix =γ₂γ₄ -γ₃². \endalign* We obtain sharp upper bounds of the second Hankel determinant of logarithmic coefficients for starlike and convex functions.

Citations

• A proof of the Bieberbach conjecture
• SECOND HANKEL DETERMINANT FOR LOGARITHMIC INVERSE COEFFICIENTS OF CONVEX AND STARLIKE FUNCTIONS

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