Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series

Yıl 2023, Cilt: 6 Sayı: 4, 130 - 139, 18.12.2023

Myroslav Sheremeta

https://doi.org/10.32323/ujma.1359248

Öz

Let $p\in {\Bbb N}$, $s=(s_1,\ldots,s_p)\in {\Bbb C}^p$, $h=(h_1,\ldots,h_p)\in {\Bbb R}^p_+$, $(n)=(n_1,\ldots,n_p)\in {\Bbb N}^p$ and the sequences $\lambda_{(n)}=(\lambda^{(1)}_{n_1},\ldots,\lambda^{(p)}_{n_p})$ are such that $0<\lambda^{(j)}_1<\lambda^{(j)}_k<\lambda^{(j)}_{k+1}\uparrow+\infty$
as $k\to\infty$ for every $j=1,\ldots,p$. For $a=(a_1,\ldots,a_p)$ and $c=(c_1,\ldots,c_p)$ let $(a,c)=a_1c_1+\ldots+a_pc_p$, and we say that $a>c$ if $a_j> c_j$ for all $1\le j\le p$. For a multiple Dirichlet series \begin{align*}F(s)=e^{(s,h)}+\sum\limits_{\lambda_{(n)}>h}f_{(n)}\exp\{(\lambda_{(n)},s)\}\end{align*} absolutely converges in $\Pi^p_0=\{s:\text{Re}\,s<0\}$, concepts of pseudostarlikeness and pseudoconvexity are introduced and criteria for pseudostarlikeness and the pseudoconvexity are proved. Using the obtained results, we investigated neighborhoods of multiple Dirichlet series, Hadamard compositions, and properties of solutions of some differential equations.

Anahtar Kelimeler

Differential equation, Hadamard composition, Multiple Dirichlet series, Neighborhood, Pseudostarlikeness, Pseudoconvexity

Kaynakça

• [1] G. M. Golusin, Geometrical theory of functions of complex variables, M. Nauka, 1966. (in Russian); Engl. Transl.: AMS: Translations of Mathematical Monograph, 26 (1969).
• [2] A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8(3) (1957), 597–601.
• [3] M. M.Sheremeta, Geometric Properties of Analytic Solutions of Differential Equations, Lviv: Publisher I. E. Chyzhykov, 2019.
• [4] I. S. Jack, Functions starlike and convex of order a, 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(4) (1981), 521–527.
• [8] R. Fournier, A note on neighborhoods of univalent functions, Proc. Amer. Math. Soc., 87(1) (1983), 117–121.
• [9] H. Silverman, Neighborhoods of a class of analytic functions, Far East J. Math. Sci., 3(2) (1995), 165–169.
• [10] O. Altıntas¸, Neighborhoods of certain analytic functions with negative coefficients, Int. J. Math. and Math. Sci. 13(4) (1996), 210–219.
• [11] O. Altıntaş, O. 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(2) (2004), 217–223.
• [13] G. Murugusundaramoorthy, H. M. Srivastava, Neighborhoods of certain classes of analytic functions of complex order, J. Inequal. Pure and Appl. Math., 5(2) (2004), 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. J., 42(12) (1990), 1711–1713. (in Russian); Engl. Transl.: Ukr. Math. J., 42(12) (1990), 1545–1547.
• [19] L. Zalzman, Hadamard product of shlicht functions, Proc. Amer. Math. Soc., 19(3) (1968), 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(2) (2008), 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, Mat. Methods and Fiz.-Mech. Polya., 61(1) (2018), 57–70.
• [25] M. M. Sheremeta, Pseudostarlike and pseudoconvex Dirichlet series of the order a and the type b, Mat. Stud. 54(1) (2020), 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(1) (2000), 54–58.
• [28] Z. M. Sheremeta, M. M. Sheremeta, Convexity of entire solutions of a differential equation, Mat. Methods and Fiz.-Mech. Polya., 47(2) (2004), 181–185.
• [29] M. M. Sheremeta, Pseudostarlike and pseudoconvex solutions of a differential equation with exponential coefficients, Mat. Stud., 56(1) (2021), 39–47.
• [30] M. M. Sheremeta, On certain subclass of Dirichlet series absolutely convergent in half-plane, Mat. Stud., 57(1)(2022), 32–44.
• [31] M. M. Sheremeta, O. B. Skaskiv, Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series, Mat. Stud., 58(2) (2023), 182–200.