On close-to-pseudoconvex Dirichlet series

  • O. M. Mulyava Kyiv National University of Food Technologies
  • M. M. Sheremeta Ivan Franko National University of Lviv, Lviv, Ukraine
  • M.G. Medvediev V.I. Vernadsky Taurida National University
Keywords: Dirichlet series, lose-to-pseudoconvexity, differential equation


For a Dirichlet series of form $F(s)=\exp\{s\lambda_1\}+\sum\nolimits_{k=2}^{+\infty}f_k\exp\{s\lambda_k\}$ absolutely convergent in the half-plane $\Pi_0=\{s\colon \mathop{\rm Re}s<0\}$ new sufficient conditions
for the close-to-pseudoconvexity are found and the obtained result is applied to studying of solutions linear differential equations of second order with exponential coefficients. In particular, are proved the following statements:

1) Let $\lambda_k=\lambda_{k-1}+\lambda_1$ and $f_k>0$ for all $k\ge 2$. If $1\le\lambda_2f_2/\lambda_1\le 2$ and $\lambda_kf_k-\lambda_{k+1}f_{k+1}\searrow q\ge 0$ as $k\to+\infty$ then function of form {\bf(1)} is close-to-pseudoconvex in $\Pi_0$ (Theorem 3). This theorem complements Alexander's criterion obtained for power series.
2) If either $-h^2\le\gamma\le0$ or $\gamma=h^2$ then differential equation $(1-e^{hs})^2w''-h(1-e^{2hs})w'+\gamma e^{2hs}=0$ $(h>0, \gamma\in{\mathbb R})$ has a solution $w=F$ of form {\bf(1)} with the exponents $\lambda_k=kh$ and the the abscissa of absolute convergence $\sigma_a=0$ that is close-to-pseudoconvex in $\Pi_0$ (Theorem 4).

Author Biographies

O. M. Mulyava, Kyiv National University of Food Technologies

Kyiv National University of Food Technologies

Kyiv, Ukraine

M. M. Sheremeta, Ivan Franko National University of Lviv, Lviv, Ukraine

Department of Mechanics and Mathematics, Professor

Lviv, Ukraine

M.G. Medvediev, V.I. Vernadsky Taurida National University

V.I. Vernadsky Taurida National University

Kyiv, Ukraine


G.M. Golusin, Geometrical theory of functions of complex variables, M.: Nauka, 1966. (in Russian); Engl. transl.: AMS: Translations of Mathematical monograph, V.26, 1969.

W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J., 1 (1952), №2, 169–185.

J.M. Alexander, Functions which map the interior of the unit circle upon simple regions, Annals Math., (1915), 12–22.

S.M. Shah, Univalence of a function f and its successive derivatives when f satisfies a differential equation, II, J. Math. Anal. Appl., 142 (1989), 422–430.

Z.M. Sheremeta, On entire solutions of a differential equation, Mat. Stud., 14 (2000), №1, 54–58.

Ya.S. Mahola, M.M. Sheremeta, Properties of entire solutions of a linear differential equation of n − th order with polynomial coefficients of n − th degree, Mat. Stud., 30 (2008), №2, 153–162.

K.I. Dosyn, M.M. Sheremeta, On the existence of meromorphically starlike and meromorphically convex solutions of Shah’s differential equation, Mat. Stud., 42 (2014), №1, 44–53.

O.M. Mulyava, Yu.S. Trukhan, On meromorphically starlike functions of the order α and the type β, which satisfy Shah’s differential equations, Carpatian Math. Publ., 9 (2017), №2, 154–162. doi:10.15330/cmp.9.2.154-162.

O.M. Holovata, O.M. Mulyava, M.M. Sheremeta, Pseudostarlike, pseudoconvex and close-topseudoconvex Dirichlet series satisfying differential equations with exponential coefficients, Мath. methods and phys-mech. fields, 61 (2018), №1, 57–70. (in Ukrainian)

M.M. Sheremeta, Geometric properties of analytic solutions of differential equations, Lviv: Publisher I.E. Chyzhykov, 2019.

S. Mandelbrojt, Dirichlet series: Principles and methods. Springer, Netherlands, 1972.

How to Cite
Mulyava, O. M., Sheremeta, M. M., & Medvediev, M. (2024). On close-to-pseudoconvex Dirichlet series. Matematychni Studii, 61(2), 214-218. https://doi.org/10.30970/ms.61.2.214-218
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