On the preservation of the boundedness of the $l$-index under the action of the Bernardi integral operator and the Ruscheweyh derivative - Dehnerys, Vitalii (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2026)
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.