On the preservation of the boundedness of the $l$-index under the action of the Bernardi integral operator and the Ruscheweyh derivative, Про збереження обмеженості $l$-індексу під дією інтегрального оператора Бернарді та похідної Рушевея

<jats:p>We study the action of the Bernardi integral operator and the Ruscheweyh derivative on analytic functions of bounded $l$-index in a disc.  Sufficient conditions are given for functions of bounded $l$-index under which their images under the action of the Bernardi integral operator, as well as the Rusheweyh derivative operator, are also functions of bounded $l$-index with the same function $l$.  The proof is based on the theorem of M.M. Sheremeta and Z.M. Sheremeta (1999), which contains a sufficient condition for the boundedness of the l-index of an analytic function in a disc. It is formulated in the form of a restriction on the Taylor coefficients of a given analytic function of bounded $l$-index.</jats:p>