On the third logarithmic coefficient in some subclasses of close-to-convex functions

Abstract

For analytic functions f in the unit disk $${\mathbb {D}}$$D normalized by $$f(0)=0$$f(0)=0 and $$f'(0)=1$$f′(0)=1 satisfying in $${\mathbb {D}}$$D respectively the conditions $${{\,\mathrm{Re}\,}}\{ (1-z)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z+z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z)^2f'(z) \} > 0,$$Re{(1-z)f′(z)}>0,Re{(1-z2)f′(z)}>0,Re{(1-z+z2)f′(z)}>0,Re{(1-z)2f′(z)}>0, the sharp upper bound of the third logarithmic coefficient in case when $$f''(0)$$f′′(0) is real was computed.

The third logarithmic coefficient for the class S

“…It is a classical result that for the class S we have |a 3 − a2 2 | ≤ 1 (see[9, p.5]), which is by (2.4) equivalent with If we put x 1 = 1 and x 3 = 0 in (2.5), then we get|ω 11 | 2 + 3|ω 13 | 2 ≤ 1,…”

Two applications of Grunsky coefficients in the theory of univalent functions

“…Let A be the class of functions f which are analytic in the open unit disc D = {z : |z| < 1} of the form (1) f (z) = z + a 2 z 2 + a 3 z 3 + • • • , and let S be the subclass of A consisting of functions that are univalent in D.…”

Second Hankel Determinant of Logarithmic Coefficients of Convex and Starlike Functions of Order Alpha

“…and are unknown for n ≥ 3. Logarithmic coefficients is one of the topic recently being of the research interest by various authors (e.g., [1,2,6,9,13,20]).…”