Pseudostarlike and pseudoconvex Dirichlet series of the order α and the type β
Анотація
The concepts of the pseudostarlikeness of order α∈[0,1) and type β∈(0,1] and the pseudoconvexity of order α and type β are introduced for Dirichlet series with null abscissa of absolute convergence. In terms of coefficients, the pseudostarlikeness and the pseudoconvexity criteria of order α and type β are proved.
Let h≥1, Λ=(λk) be an increasing to +∞ sequence of positive numbers (λ1>h. We call a conformal function of the form F(s)=esh+∑∞k=1fkexp{sλk}, s=σ+it,
in Π0={s:Res<0} pseudostarlike of order α∈[0,1) and type
β∈(0,1] if
|F′(s)F(s)−h|<β|F′(s)F(s)−(2α−h)|,s∈Π0.
The main results of the article are contained in Theorems 1 and 2. Theorem 1 states: \textit{If α∈[0,1) and β∈(0,1] such that
∞∑k=1{(1+β)λk−2βα−h(1−β)}|fk|≤2β(h−α)
then the function F is pseudostarlike of order α and type β.}
The corresponding results for Hadamard compositions of such series are also established.
Посилання
Golusin G.M. Geometrical theory of functions of complex variables. – M.: Nauka, 1966. – 628 p. (in Russian); Engl. transl.: AMS: Translations of Mathematical monograph, 1969. – V.26. – 676 p.
Goodman A.W. Univalent functions and nonanalytic curves// Proc. Amer. Math. Soc. – 1957. – V.8, №3. – P. 597–601.
Sheremeta M.M. Geometric properties of analytic solutions of differential equations. – Lviv: Publisher I.E. Chyzhykov. – 2019. – 164 p.
Jack I.S. Functions starlike and convex of order // J. London Math. Soc. – 1971. – V.3. – P. 469–474.
Gupta V.P. Convex class of starlike functions// Yokohama Math. J. – 1984. – V.32. – P. 55–59.
Owa S. On certain classes of p-valent functions with negative coefficients// Simon Stevin. – 1985. – V.59. – P. 385–402.
El-Ashwah R.M., Aouf M.K., Moustava A.O. Starlike and convexity properties for p-valent hypergeometric functions// Acta Math. Univ. Comenianae. – 2010. – V.79, №1. – P. 55–64.
Juneja O.P., Reddy T.R. Meromorphic starlike and univalent functions with positive coefficients// Ann. Univ. Mariae Curie-Sklodowska. – 1985. – V.39. – P. 65–76.
Uralegaddi B.A. Meromorphic starlike functions with positive coefficients// Kyungpook. Math. J. – 1989. – V.29, №1. – P. 64–68.
Mogra M.L., Reddy T.R., Juneja O.P. Meromorphic univalent functions with positive coefficients// Bull. Austral. Math. Soc. – 1985. – V.32, №2. – P. 161–176.
Truhan Yu.S., Mulyava O.M. On meromorphically starlike functions of the order and the type , which satisfy Shah’s differential equations// Carpatian Math. Publ. – 2017. – V.9, №2. – P. 154–162.
Hadamard J. Th´eoremesur≤seriesentieres// Acta math. – 1899. – Bd.22. – S. 55–63.
Hadamard J. La s´erie de Taylor et son prolongement analitique // Scientia phys.-math. – 1901. – №12. – P. 43–62.
Bieberbach L. Analytische Fortzetzung. – Berlin, 1955.
Korobeinik Yu.F., Mavrodi N.N. Singular points of the Hadamard composition// Ukr. Math. Journ. – 1990. – V.42, №12. – P. 1711–1713. (in Russian); Engl. transl.: Ukr. Math. Journ. – 1990. – V.42, Issue 12. – P. 1545–1547.
Zalzman L. Hadamard product of shlicht functions// Proc. Amer. Math. Soc. – 1968. – V.19, №3. – P. 544–548.
Mogra M.L. Hadamard product of certain meromorphic univalent functions// J. Math. Anal. Appl. – 1991. – V.157. – P. 10–16.
Choi J.H., Kim Y.C., Owa S. Generalizations of Hadamard products of functions with negative coefficients// J. Math. Anal. Appl. – 1996. – V.199. – P. 495–501.
Aouf M.K., Silverman H. Generalizations of Hadamard products of meromorphic univalent functions with positive coefficients// Demonstratio Mathematica. – 2008. – V.51, №2. – P. 381–388.
Liu J., Srivastava P. Hadamard products of certain classes of p-valent starlike functions// RACSM. – 2019. – V.113. – P. 2001–205.
Holovata O.M., Mulyava O.M., Sheremeta M.M. Pseudostarlike, pseudoconvex and close-to-pseudoconvex Dirichlet series satisfying differential equations with exponential coefficients// Мath. methods and physicomech. fields. – 2018. – V.61, №1. – P. 57–70. (in Ukrainian)
Mulyava O.M., Sheremeta M.M. Properties of Hadamard compositions of derivatives of Dirichlet series// Visnyk of Lviv Univ. Ser Mech. Math. – 2012. – Issue 77. – P. 157–166. (in Ukrainian)
Авторське право (c) 2020 M.M. Sheremeta

Ця робота ліцензується відповідно до Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.