The concepts of the pseudostarlikeness of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$ and the  pseudoconvexity of the order $\alpha$ and type $\beta$ are introduced for  Dirichlet series of the form $F(s)=e^{-sh}+\sum\limits_{j=1}^{n}a_j\exp\{-sh_j\}+\sum\limits_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$, where $h>h_n>\dots>h_1\ge 1$ and  $(\lambda_k)$ is an increasing to $+\infty$ sequence of positive numbers. Criteria for pseudostarlikeness and  pseudoconvexity in terms of coeffcients are proved. The obtained results and applied to the study of  meromorphic starlikeness and convexity of the Laurent series $f(s)=1/z^p+\sum_{j=1}^{p-1}a_j/z^j+\sum_{k=1}^{\infty}f_kz^k$. Conditions, under which the differential equation $w''+\gamma w'+(\delta e^{2sh}+\tau)w=0$ has a pseudostarlike or pseudoconvex solution of the order $\alpha$ and type $\beta=1$ are investigated.

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

[2] Goodman A. W. Univalent functions and nonanalytic curves. Proc. Amer. Math. Soc., 1957, 8 (3), 597–601.

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

[4] Jack I. S. Functions starlike and convex of order α. J. London Math. Soc., 1971, 3, 469 474.

[5] Gupta V. P. Convex class of starlike functions. Yokohama Math. J., 1984, 32, 55–59.

[6] Owa S. On certain classes of p-valent functions with negative coefficients. Simon Stevin, 1985, 59, 385– 402.

[7] El-Ashwah R. M., Aouf M. K., Moustava A. O. Starlike and convexity properties for p-valent hypergeometric functions. Acta Math. Univ. Comenianae, 2010, 79 (1), 55–64.

[8] Juneja O. P., Reddy T. R. Meromorphic starlike and univalent functions with positive coefficients. Ann. Univ. Mariae Curie-Sklodowska, 1985, (39), 65–76.

[9] Uralegaddi B. A. Meromorphic starlike functions with positive coefficients. Kyungpook. Math. J., 1989, 29 (1), 64–68.

[10] Mogra M. L., Reddy T. R., Juneja O. P. Meromorphic univalent functions with positive coefficients. Bull. Austral. Math. Soc., 1985, 32 (2), 161–176.

[11] Mulyava O. M., Truhan Yu. S. On meromorphically starlike functions of the order α and the type β, which satisfy Shah’s differential equations. Carpatian Math. Publ., 2017, 9 (2), 154–162.

[12] Royster W. C. Meromorphic starlike multivalent functions. Trans. Amer. Math. Soc., 1963, 107, 300–308.

[13] Faisal Imran, Darus M., Shah F. A. A critical studi of meromorphic starlike functions. TWMS J. Appl. Eng. Math., 2018, 8 (1), 238–242.

[14] 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, 61 (1), 57–70 (in Ukrainian).

[15] Sheremeta M. M. Pseudostarlike and pseudoconvex Dirichlet series of the order α and the type β. Mat. Stud., 2020, 54 (1), 23–31.