On certain subclass of Dirichlet series absolutely convergent in half-plane

  • M. M. Sheremeta Ivan Franko National University of Lviv, Lviv
Keywords: Dirichlet series; Hadamard composition; neighborhood of the function; differential equation.

Abstract

Denote by $\mathfrak{D}_0$ a class of absolutely convergent in half-plane $\Pi_0=\{s\colon \text{Re}\,s<0\}$ Dirichlet series
$F(s)=e^{sh}-\sum_{k=1}^{\infty}f_k\exp\{s(\lambda_k+h)\},\, s=\sigma+it$, where $h> 0$, $h<\lambda_k\uparrow+\infty$ and $f_k>0$.
For $0\le\alpha<h$ and $l\ge 0$ we say that $F$ belongs to the class $\mathfrak{DF}_h(l,\alpha)$ if and only if
$\text{Re}\{e^{-hs}((1-l)F(s)+\frac{l}{h}F'(s))\}>\frac{\alpha}{h}$,
and belongs to the class $\mathfrak{DG}_h(l,\alpha)$ if and only if
$\text{Re}\{e^{-hs}((1-l)F'(s)+\frac{l}{h}F''(s))\}>\alpha$ for all $s\in \Pi_0$. It is proved
that $F\in \mathfrak{DF}_h(l,\alpha)$ if and only if $ \sum_{k=1}^{\infty}(h+l\lambda_k)f_k\le h-\alpha$, and
$F\in \mathfrak{DG}_h(l,\alpha)$ if and only if $\sum_{k=1}^{\infty}(h+l\lambda_k)(\lambda_k+h)f_k\le h(h-\alpha)$.

If $F_j\in \mathfrak{DF}_h(l_j,\alpha_j)$, $j=1, 2$, where $l_j\ge0$ and $0\le \alpha_j<h$, then Hadamard composition
$(F_1*F_2)\in \mathfrak{D}F_h(l,\alpha)$, where $l=\min\{l_1,l_2\}$ and
$\alpha=h-\frac{(h-\alpha_1)(h-\alpha_2)}{h+l\lambda_1}$. Similar statement is correct for the class $F_j\in \mathfrak{DG}_h(l,\alpha)$.

For $j>0$ and $\delta>0$ the neighborhood of the function $F\in \mathfrak{D}_0$ is defined as follows
$O_{j,\delta}(F)=\{G(s)=e^{s}-\sum_{k=1}^{\infty}g_k\exp\{s\lambda_k\}\in \mathfrak{D}_0\colon
\sum_{k=1}^{\infty}\lambda^j_k|g_k-f_k|\le\delta\}$. It is described the neighborhoods of functions from classes $\mathfrak{DF}_h(l,\alpha)$ and $\mathfrak{DG}_h(l,\alpha)$.

Conditions on real parameters $\gamma_0,\,\gamma_1,\,\gamma_2,\,a_1$ and $a_2$ of the differential equation
$w''+(\gamma_0e^{2hs}+\gamma_1e^{hs}+\gamma_2) w=a_1e^{hs}+a_2e^{2hs}$ are found, under which this equation has a solution
either in $\mathfrak{DF}_h(l,\alpha)$ or in $\mathfrak{DG}_h(l,\alpha)$.

Author Biography

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

Department of Mechanics and Mathematics, Professor

References

1. Lee S.K., Owa S., Srivastava H.M. Basic properties and characterization of a certain class of analytic functions with negative coefficients// Utilitas Math. – 1989. – V.36. – P. 121–128.
2. Aouf M.K., Darwish H.E. Basic properties and characterization of a certain class of analytic functions with negative coefficients, II// Utilitas Math. – 1994. – V.46. – P. 167–177.
3. Aouf M.K. A subclass of analytic p–valent functions with negative coefficients, I// Utilitas Math. – 1994. – V.46. – P. 219–231.
4. Altintas O. A subclass of analytic functions with negative coefficients// Hacettepe Bull. Natur. Sci. Engrg. – 1990.– V.19. – P. 15–24.
5. Altintas O. On a subclass of certain starlike functions with negative coefficients// Math. Japon. – 1991.– V.36. – P. 489–495.
6. Aouf M.K., Srivastava H.M. Certain families of analytic functions with negative coefficients// DMS-669-IR. – June 1994. – 49 p.
7. Aouf M.K., Hossen H.M., Srivastava H.M. A certain subclass of analytic p-valent functions with negative coefficients// Demonstratio Mathematica. – 1998. – V.51, No3. – P. 595–608.
8. Hadamard J. Theoreme sur le series entieres// Acta math. – 1899. – Bd.22. – S. 55–63.
9. Hadamard J. La serie de Taylor et son prolongement analitique // Scientia phys.– math. – 1901. – No12. – P. 43–62.
10. Bieberbach L. Analytische Fortzetzung – Berlin, 1955.
11. Korobeinik Yu.F., Mavrodi N.N. Singular points of the Hadamard composition// Ukr. Math. Journ. – 1990. – V.42, No12. – P. 1711–1713. (in Russian); Engl. transl.: Ukr. Math. Journ. – 1990. – V.42, No 12. P.1545-1547.
12. Zalzman L. Hadamard product of shlicht functions// Proc. Amer. Math. Soc. – 1968. – V.19, No3.P. 544–548.
13. Mogra M.L. Hadamard product of certain meromorphic univalent functions// J. Math. Anal. Appl.1991. – V.157. – P. 10–16.
14. 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.
15. Aouf M.K., Silverman H. Generalizations of Hadamard products of meromorphic univalent functions with positive coefficients// Demonstratio Mathematica. – 2008. – V.51, No2. – P. 381–388.
16. Liu J., Srivastava P. Hadamard products of certain classes of p–valent starlike functions// RACSM.2019. – V.113. – P. 2001–205.
17. Ruscheweyh S. Neighborhoods of univalent functions// Proc. Amer. Math. Soc. –1981. – V.81, No4.P. 521–527.
18. Sheremeta M.M. Pseudostarlike Dirichlet series of the order α and the type β// Mat. Stud. – 2020.V.54, No1. – P. 23–31.
19. Goodman A.W. Univalent functions and nonanalytic curves// Proc. Amer. Math. Soc. –1957. – V.8.P. 598–601.
20. Fournier R. A note on neighborhoods of univalent functions// Proc. Amer. Math. Soc. –1983. – V.87, No1. – P. 117–121.
21. Silverman H. Neighborhoods of a class of analytic functions// Far East J. Math. Sci. – 1995. – V.3, No2. – P. 165–169.
22. Altintas O., Neighborhoods of certain analytic functions with negative coefficients// Internat. J. Math. and Math. Sci. – 1996. – V.13, No4. – P. 210–219.
23. Altintas O., Ozkan O., Srivastava H.M. Neighborhoods of a class of analytic functions with negative coefficients// Applied Math. Lettr. – 2000. – V.13. – P. 63–67.
24. Frasin B.A., Daras M. Integral means and neighborhoods for analytic functions with negative coefficients// Soochow Journal Math. – 2004. – V.30, No2. – P. 217–223.
25. Murugusundaramoorthy G., Srivastava H.M. Neighborhoods of certain classes of analytic functions of complex order// J. Inequal. Pure and Appl. Math. – 2004. – V.5, No2. – Article 24.
26. Pascu M.N., Pascu N.R. Neighborhoods of univalent functions// Bull. Amer. Math. Soc. – 2011. – V.83. – P. 510–219.
27. Shah S.M. Univalence of a function f and its successive derivatives when f satisfies a differential equation, II// J. Math. anal. and appl. – 1989. – V.142. – P. 422–430.
28. Sheremeta Z.M. Close-to-convexity of entire solutions of a differential equation// Mat. methods and fiz.-mech. polya. – 1999. – V.42, No3. – P. 31–35. (in Ukrainian)
29. Sheremeta Z.M. On properties of entire solutions of a differential equation// Diff. uravnyeniya. – 2000. – V.36, No8. – P. 1–6. (in Russian)
30. Sheremeta Z.M. On entire solutions of a differential equation// Mat. Stud. – 2000. – V.14, No1. – P. 54–58.
31. Sheremeta Z.M., Sheremeta M.M. Close-to-convexity of entire solutions of a differential equation// Diff. uravnyeniya. – 2002. – V.38, No4. – P. 435–440. (in Russian)
32. Holovata O.M., Mulyava O.M., Sheremeta M.M. Pseudostarlike, pseudoconvex and close-to-pseudoconvex Dirichlet series satisfying differential equations with exponential coefficients// Mat. method. and fiz.- mech. polya. – 2018 – V.61, No1. – P. 57–70.
33. Sheremeta M.M. Geometric properties of analytic solution of differential equations – Publisher I. E. Chyzhykov. – 2019. – 164 p.
34. Sheremeta M.M. Entire Dirichlet series . – K.:ISDO, 1993. (in Ukrainian)
35. Sheremeta M.M. On the derivative of an entire Dirichlet series// Math. USSR Sbornik. – 1990. – V.65, No1. – P.133–139.
Published
2022-03-31
How to Cite
Sheremeta, M. M. (2022). On certain subclass of Dirichlet series absolutely convergent in half-plane. Matematychni Studii, 57(1), 32-44. https://doi.org/10.30970/ms.57.1.32-44
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Articles