Littlewood–Paley Conjecture for Certain Classes of Analytic Functions

The famous Littlewood–Paley conjecture is true for the starlike functions but it does not hold for close-to-convex functions. In fact, this conjecture does not hold for many well-defined subclasses of normalized univalent functions. The present work considers the classes of strongly α-logarithmic close-to-convex and logarithmic α-quasiconvex function of order β. For these classes, bounds on the initial coefficients and the Littlewood–Paley conjecture have been discussed. Applying these results, certain conditions are investigated under which the Littlewood–Paley conjecture holds for these classes for large values of the parameters involved therein. Relevant connections of our results with the existing ones are also pointed out.