# Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series

• M. M. Sheremeta Ivan Franko National University of Lviv, Lviv
• O. B. Skaskiv Ivan Franko National University of Lviv, Lviv, Ukraine
Keywords: univalent function; psevdostarlike function; psevdoconvex function; Dirichlet series; several variables; directional derivative

### Abstract

The article introduces the concepts of pseudostarlikeness and pseudoconvexity in the direction of absolutely converges in $\Pi_0=\{s\in\mathbb{C}^p\colon \text{Re}\,s<0\}$, $p\in\mathbb{N},$ the multiple Dirichlet series of the form
$$F(s)=e^{(h,s)}+\sum\nolimits_{\|(n)\|=\|(n^0)\|}^{+\infty}f_{(n)}\exp\{(\lambda_{(n)},s)\}, \quad s=(s_1,...,s_p)\in {\mathbb C}^p,\quad p\geq 1,$$
where $\lambda_{(n^0)}>h$, $\text{Re}\,s<0\Longleftrightarrow (\text{Re}\,s_1<0,...,\text{Re}\,s_p<0)$,
$h=(h_1,...,h_p)\in {\mathbb R}^p_+$, $(n)=(n_1,...,n_p)\in {\mathbb N}^p$, $(n^0)=(n^0_1,...,n^0_p)\in {\mathbb N}^p$, $\|(n)\|=n_1+...+n_p$ and the sequences
$\lambda_{(n)}=(\lambda^{(1)}_{n_1},...,\lambda^{(p)}_{n_p})$ are such that $0<h_j<\lambda^{(j)}_1<\lambda^{(j)}_k<\lambda^{(j)}_{k+1}\uparrow+\infty$
as $k\to+\infty$ for every $j\in\{1,...,p\}$, and $(a,c)=a_1c_1+...+a_pc_p$ for $a=(a_1,...,a_p)$ and $c=(c_1,...,c_p)$. We say that $a>c$ if $a_j\ge c_j$ for all $1\le j\le p$ and there exists at least one $j$ such that $a_j> c_j$. Let ${\bf b}=(b_1,...,b_p)$ and $\partial_{{\bf b}}F( {s})=\sum\limits_{j=1}^p b_j\dfrac{\partial F( {s})}{\partial {s}_j}$ be the derivative of $F$ in the direction ${\bf b}$. In this paper, in particular, the following assertions were obtained:
1) If ${\bf b}>0$ and
$\sum\limits_{\|(n)\|=k_0}^{+\infty}(\lambda_{(n)},{\bf b})|f_{(n)}|\le (h,{\bf b})$
then $\partial_{{\bf b}}F( {s})\not=0$ in $\Pi_0:=\{s\colon \text{Re}\,s<0\}$, i.e. $F$ is conformal in $\Pi_0$ in the direction ${\bf b}$ (Proposition 1).
2) We say that function $F$ is pseudostarlike of the order $\alpha\in [0,\,(h,{\bf b}))$ and the type
$\beta >0$ in the direction ${\bf b}$ if
$\Big|\frac{\partial_{{\bf b}}F( {s})}{F(s)}-(h, {\bf b})\Big|<\beta\Big|\frac{\partial_{{\bf b}}F( {s})}{F(s)}- (2\alpha-(h, {\bf b}))\Big|,\quad s\in \Pi_0.$
Let $0\le \alpha<(h,{\bf b})$ and $\beta>0$. In order that the function $F$ is
pseudostarlike of the order $\alpha$ and the type $\beta$ in the direction ${\bf b}> 0$, it is sufficient and in the case, when all $f_{(n)}\le 0$, it is necessary that
$\sum\limits_{\|(n)\|=k_0}^{+\infty}\{((1+\beta)\lambda_{(n)}-(1-\beta)h,{\bf b})-2\beta\alpha\}|f_{(n)}|\le 2\beta ((h,{\bf b})-\alpha)$ (Theorem 1).

### Author Biography

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

Department of Mechanics and Mathematics, Professor

### References

1. G.M. Golusin, Geometrical theory of functions of complex variables, M.: Nauka, 1966, 628 p.(in Russian); Engl. transl.: AMS: Transl. Math. Monograph., 26 (1969), 676 p.
2. A.W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8 (1957), No3, 597–601.
3. M.M. Sheremeta, Geometric properties of analytic solutions of differential equations, Lviv: Publ. I.E. Chyzhykov, 2019, 164 p.
4. I.S. Jack, Functions starlike and convex of order α, J. London Math. Soc., 3 (1971), 469–474.
5. V.P. Gupta, Convex class of starlike functions, Yokohama Math. J., 32 (1984), 55–59.
6. A.W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8 (1957), 598–601.
7. S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 81 (1981), No4, 521–527.
8. R. Fournier, A note on neighborhoods of univalent functions, Proc. Amer. Math. Soc., 87 (1983), No1, 117–121.
9. H. Silverman, Neighborhoods of a class of analytic functions, Far East J. Math. Sci., 3 (1995), No2200
10. O. Altintas, Neighborhoods of certain analytic functions with negative coefficients, Internat. J. Math. and Math. Sci., 13 (1996), No4, 210–219.
11. O. Altintas, Ö. Özkan, H.M. Srivastava, Neighborhoods of a class of analytic functions with negative coefficients, Applied Math. Lettr., 13 (2000), 63–67.
12. B.A. Frasin, M. Daras, Integral means and neighborhoods for analytic functions with negative coefficients, Soochow Journal Math., 30 (2004), No2, 217–223.
13. G. Murugusundaramoorthy, H.M. Srivastava, Neighborhoods of certain classes of analytic functions of complex order, J. Inequal. Pure Appl. Math., 5 (2004), No2, Article 24.
14. M.N. Pascu, N.R. Pascu, Neighborhoods of univalent functions, Bull. Amer. Math. Soc., 83 (2011), 510–219.
15. J. Hadamard, Theoreme sur le series entieres, Acta math., 22 (1899), 55–63.
16. J. Hadamard, La serie de Taylor et son prolongement analitique, Scientia phys.-math., 12 (1901), 43–62.
17. L. Bieberbach, Analytische Fortzetzung, Berlin, 1955.
18. Yu.F. Korobeinik, N.N. Mavrodi, Singular points of the Hadamard composition, Ukr. Math. Journ., 42 (1990), No12, 1711–1713. (in Russian); Engl. transl.: Ukr. Math. Journ., 42 (1990), No12, 1545–1547.
19. L. Zalzman, Hadamard product of shlicht functions, Proc. Amer. Math. Soc., 19 (1968), No3, 544–548.
20. M.L. Mogra, Hadamard product of certain meromorphic univalent functions, J. Math. Anal. Appl., 157 (1991), 10–16.
21. J.H. Choi, Y.C. Kim, S. Owa, Generalizations of Hadamard products of functions with negative coefficients, J. Math. Anal. Appl., 199 (1996), 495–501.
22. M.K. Aouf, H. Silverman, Generalizations of Hadamard products of meromorphic univalent functions with positive coefficients, Demonstr. Math., 51 (2008), No2, 381–388.
23. J. Liu, P. Srivastava, Hadamard products of certain classes of p-valent starlike functions, RACSM, 113 (2019), 2001–205.
24. O.M. Holovata, O.M. Mulyava, M.M. Sheremeta, Pseudostarlike, pseudoconvex and close-to-pseudoconvex Dirichlet series satisfying differential equations with exponential coefficients, Math. methods and phys-mech. fields., 61 (2018), No1, 57–70 (in Ukrainian).
25. M.M. Sheremeta, Pseudostarlike and pseudoconvex Dirichlet series of the order α and the type β, Mat. Stud., 54 (2020), No1, 23–31.
26. S.M. Shah, Univalence of a function f and its successive derivatives when f satisfies a differential equation, II, J. Math. Anal. and Appl., 142 (1989), 422–430.
27. Z.M. Sheremeta, On entire solutions of a differential equation, Mat. Stud., 14 (2000), No1, 54–58.
28. Z.M. Sheremeta, M.M. Sheremeta, Convexity of entire solutions of a differential equation, Mat. methods and fiz.-mech. polya, 47 (2004), No2, 181–185 (in Ukrainian).
29. A. Bandura, O. Skaskiv, Analog of Hayman’s theorem and its application to some system of linear partial differential equations, J. Math. Phys., Anal., Geom., 15 (2019), No2, 170–191. doi: 10.15407/mag15.02.170
30. A. Bandura, O. Skaskiv, Linear directional differential equations in the unit ball: solutions of bounded L-index, Math. Slovaca., 69 (2019), No5, 1089–1098. doi: 10.1515/ms-2017-0292
31. A. Bandura, O. Skaskiv, L. Smolovyk, Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction, Demonstr. Math., 53 (2019), No1, 482–489. doi: 10.1515/dema-2019-0043
32. A. Bandura, O. Skaskiv, Entire functions of bounded L-index: Its zeros and behavior of partial logarithmic derivatives, J. Complex Analysis, 2017 (2017), 1–10. Article ID 3253095. doi: 10.1155/2017/3253095
33. A. Bandura, O. Skaskiv, Analytic functions in the unit ball of bounded L-index in joint variables and of bounded L-index in direction: a connection between these classes, Demonstr. Math., 52 (2019), No1, 82–87. doi: 10.1515/dema-2019-0008
34. A. Bandura, O. Skaskiv, Functions analytic in the unit ball having bounded L-index in a direction, Rocky Mountain J. Math., 49 (4) (2019), 1063–1092. doi: 10.1216/RMJ-2019-49-4-1063
Published
2023-01-23
How to Cite
Sheremeta, M. M., & Skaskiv, O. B. (2023). Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series. Matematychni Studii, 58(2), 182-200. https://doi.org/10.30970/ms.58.2.182-200
Issue
Section
Articles