On Hadamard composition of Gelfond-Leont'ev derivatives of entire and analytic functions in the unit disk

For an entire function and an analytic in the unit disk function the growth of the Hadamard composition of their Gelfond-Leont'ev derivatives is investigated in terms of generalized orders. A relationship between the behaviors of the maximal terms of Hadamard composition of Gelfond-Leont'ev derivatives and of the Gelfond-Leont'ev derivative of Hadamard composition is established.

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

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.

Hadamard compositions of Gelfond-Leont'ev derivatives of analytic functions

For analytic functions f and g, the growth of the Hadamard composition of their Gelfond-Leont'ev derivatives is investigated in terms of generalized orders. A relation between the behaviors of the maximal terms of the Hadamard composition of Gelfond-Leont'ev derivatives and those of the Gelfond-Leont'ev derivative of a Hadamard composition is established.

The Hadamard compositions of Dirichlet series absolutely converging in half-plane

For Dirichlet series with different finite abscissas of absolute convergence in terms of generalized orders the growth of the Hadamard composition of their derivatives is investigated.A relation between the behavior of the maximal terms of Hadamard composition of derivatives and of the derivative of Hadamard composition is established.