Hadamard compositions of Gelfond-Leont’ev-Sǎlǎgean and Gelfond-Leont’ev-Ruscheweyh derivatives of functions analytic in the unit disk

Abstract

For analytic functions $$f(z)=z+\sum\limits_{k=2}^{\infty}f_kz^k \mbox{ and } g(z)=z+\sum\limits_{k=2}^{\infty}g_kz^k$$ in the unit disk properties of the Hadamard compositions $D^n_{l,[S]}f*D^n_{l,[S]}g$ and $D^n_{l,[R]}f*D^n_{l,[R]}g$ of their Gelfond-Leont'ev-S$\check{\text{a}}$l$\check{\text{a}}$gean derivatives $$D^n_{l,[S]}f(z)=z+\sum\limits_{k=2}^{\infty}\left(\frac{l_1l_{k-1}}{l_k}\right)^nf_kz^k$$ and Gelfond-Leont'ev-Ruscheweyh derivatives$$D^n_{l,[R]}f(z)=z+\sum\limits_{k=2}^{\infty}\frac{l_{k-1}l_n}{l_{n+k-1}}f_kz^k$$ are investigated. For study, generalized orders are used. A connection between the growth of the maximal term of the Hadamard composition of Gelfond-Leont'ev-S$\check{\text{a}}$l$\check{\text{a}}$gean derivatives or Gelfond-Leont'ev-Rusche\-weyh derivatives and the growth of the maximal term of these derivatives of Hadamard composition is established. Similar results are obtained in terms of the classical order and the lower order of the growth.