Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions

It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If S denotes the class of functions f(z) = z + Sigma(infinity)(n=2)a(n)z(n) analytic and univalent in the open unit disk U, then the logarithmic coefficients gamma(n)(f) of the function f is an element of S are defined by log (f(z)/z) = 2 Sigma(infinity)(n =1)gamma(n)(f)z(n). In the current paper, the bounds for the logarithmic coefficients gamma(n) for some well-known classes like C(1+alpha z) for alpha is an element of (0, 1] and CVhpl (1/2) were estimated. Further, conjectures for the logarithmic coefficients.n for functions f belonging to these classes are stated. For example, it is forecasted that if the function f is an element of C(1 + alpha z), then the logarithmic coefficients of f satisfy the inequalities vertical bar gamma(n)vertical bar <= 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.