Area functional and majorant series estimates in the class of bounded functions in the disk

In this article, the new inequalities for the weighted sums of coefficients in the class of bounded functions in the disk are obtained. We develop the methods of I.R.~Kayumov and S.~Ponnusamy, using E.~Reich's theorem on the majorization of subordinate functions. The sharp estimates for the area of the image of the disk of radius $r$ under the action of the function which is expanded into a lacunary series of standard form are obtained. Under significantly lower than in \cite{Khas} restrictions on the initial coefficient, the estimates for the Bohr--Bombieri function of the Hadamard convolution operator are proved. Using the example of the differentiation operator, it is shown that in some cases the new method for calculating the lower bound for the Bohr radius of the Hadamard operator with a fixed initial coefficient is more effective than the known one.