On Dirichlet series similar to Hadamard compositions in half-plane


  • A.I. Bandura Ivano-Frankivsk National Technical University of Oil and Gas, 15 Karpatska str., 76019, Ivano-Frankivsk, Ukraine
  • O.M. Mulyava National University of Food Technologies, 68 Volodymyrska str., 01601, Kyiv, Ukraine
  • M.M. Sheremeta Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine


Dirichlet series, Hadamard composition, generalized order, generalized type, generalized convergence class, pseudostarlikeness, pseudoconvexity
Published online: 2023-06-29


Let $F(s)=\sum\limits_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ and $F_j(s)=\sum\limits_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\},$ $j=\overline{1,p},$ be Dirichlet series with exponents $0\le\lambda_n\uparrow+\infty,$ $n\to\infty,$ and the abscissas of absolutely convergence equal to $0$. The function $F$ is called Hadamard composition of the genus $m\ge 1$ of the functions $F_j$ if $a_n=P(a_{n,1},\dots ,a_{n,p})$, where $$P(x_1,\dots ,x_p)=\sum\limits_{k_1+\dots+k_p=m}c_{k_1\dots\, k_p}x_1^{k_1}\cdots x_p^{k_p}$$ is a homogeneous polynomial of degree $m$. In terms of generalized orders and convergence classes the connection between the growth of the functions $F_j$ and the growth of the Hadamard composition $F$ of the genus $m\ge 1$ of $F_j$ is investigated. The pseudostarlikeness and pseudoconvexity of the Hadamard composition of the genus $m\ge 1$ are studied.

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How to Cite
Bandura, A.; Mulyava, O.; Sheremeta, M. On Dirichlet Series Similar to Hadamard Compositions in Half-Plane. Carpathian Math. Publ. 2023, 15, 180-195.