Solution of logarithmic coefficients conjectures for some classes of convex functions

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 gamma(n) of the functions f belonging to some well-known classes like C(1 + alpha z) for alpha is an element of (0, 1], and CVhpl(1/2). For example, it is conjectured that if the function f is an element of C(1 + alpha z), then the logarithmic coefficients of f satisfy the inequalities|gamma(n)| <= alpha /2n(n+ 1), n is an element of N.Equality is attained for the function L-alpha,L-n, that is,log L-alpha,L-n(z)/z =2 sigma(infinity)(n=1) gamma(n)(L-alpha,L-n)z(n )= alpha/n(n + 1) z(n)+ ..., z is an element of U.The aim of this paper is to confirm that these conjectures hold for the coefficient gamma(n0-1) whenever the function f has the form f(z) = z + sigma(infinity)(k=n0) a(k)z(k), z is an element of U for some n(0) is an element of N, n(0) >= 2.(c) 2023 Mathematical Institute Slovak Academy of Sciences