Skip to main content
SpringerLink
Log in

Pseudostarlike, Pseudoconvex, and Close-to-Pseudoconvex Dirichlet Series Satisfying Differential Equations with Exponential Coefficients

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

The concepts of pseudostarlikeness, pseudoconvexity, and closeness to pseudoconvexity are introduced for the Dirichlet series with the null abscissa of absolute convergence. The obtained results are used to study the properties of solutions of the differential equations with exponential coefficients.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Amer. Math. Soc., Providence, RI (1969).

  2. A. F. Leont’ev, Series of Exponential Functions [in Russian], Nauka, Moscow (1976).

  3. Ya. S. Mahola and M. M. Sheremeta, “On properties of entire solutions of linear differential equations with polynomial coefficients,” Mat. Met. Fiz.-Mekh. Polya, 53, No. 4, 62–74 (2010); English translation: J. Math. Sci., 181, No. 3, 366–382 (2012), https://doi.org/10.1007/s10958-012-0691-9.

  4. Ya. Mahola and M. Sheremeta, “Closeness to convexity of the entire solution of a linear differential equation with polynomial coefficients,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Issue 70, 122–127 (2009).

  5. Z. M. Sheremeta, “Closeness to convexity for the entire solution of a differential equation,” Mat. Met. Fiz.-Mekh. Polya, 42, No. 3, 31–35 (1999).

    MATH  Google Scholar 

  6. Z. M. Sheremeta, “The properties of entire solutions of one differential equation,” Differents. Uravn., 36, No. 8, 1045–1050 (2000); English translation: Differents. Equat., 36, No. 8, 1155–1161 (2000), https://doi.org/10.1007/BF02754183.

  7. Z. M. Sheremeta and M. M. Sheremeta, “Convexity of the entire solutions of one differential equation,” Mat. Met. Fiz.-Mekh. Polya, 47, No. 2, 186–191 (2004).

    MATH  Google Scholar 

  8. Z. M. Sheremeta and M. N. Sheremeta, “Closeness to convexity for entire solutions of a differential equation,” Differents. Uravn., 38, No. 4, 477–481 (2002); English translation: Differents. Equat., 38, No. 4, 496–501 (2002), https://doi.org/10.1023/A:1016355531151.

  9. Z. Sheremeta, “On closeness to convexity for the entire solutions of a differential equation,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Issue 58, 54–56 (2000).

  10. M. N. Sheremeta, “Full equivalence of the logarithms of the maximum modulus and the maximal term of an entire Dirichlet series,” Mat. Zametki, 47, No. 6, 119–123 (1990); English translation: Math. Notes, 47, No. 6, 608–611 (1990), https://doi.org/10.1007/BF01170894.

  11. M. N. Sheremeta, “On the derivative of an entire Dirichlet series,” Mat. Sb., 137(179). No. 1(9), 128–139 (1988); English translation: Math. USSR-Sb., 65, No. 1, 133–145 (1990).

  12. M. N. Sheremeta, “A relation between the maximal term and maximum of the modulus of the entire Dirichlet series,” Mat. Zametki, 51, No. 5, 141–148 (1992); English translation: Math. Notes, 51, No. 5, 522–526 (1992), https://doi.org/10.1007/BF01262189.

  13. J. W. Alexander, “Functions which map the interior of the unit circle upon simple regions,” Ann. Math., 17, No. 1, 12–22 (1915).

    Article  MathSciNet  Google Scholar 

  14. N. E. Cho, O. S. Kwon, and V. Ravichandran, “Coefficient, distortion and growth inequalities for certain close-to-convex functions,” J. Inequal. Appl., 2011, 100–106 (2011), https://doi.org/10.1186/1029-242X-2011-100.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. W. Goodman, Univalent Functions, Vol. II, Mariner Publ., Tampa (1983), https://zbmath.org/1041.30501.

  16. A. W. Goodman, “Univalent functions and nonanalytic curves,” Proc. Amer. Math. Soc., 8, 598–601 (1957), https://doi.org/10.1090/S0002-9939-1957-0086879-9.

    Article  MathSciNet  MATH  Google Scholar 

  17. O. P. Juneja and T. R. Reddy, “Meromorphic starlike univalent functions with positive coefficients,” Ann. Univ. M. Curie-Sklodowska. Sec. A: Math., 39, 65–76 (1985).

    MathSciNet  MATH  Google Scholar 

  18. W. Kaplan, “Close-to-convex schlicht functions,” Michigan Math. J., 1, No. 2, 169–185 (1952), https://doi.org/10.1307/mmj/1028988895.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Kowalczyk and E. Leś-Bomba, “On a subclass of close-to-convex functions,” Appl. Math. Lett., 23, No. 10, 1147–1151 (2010), https://doi.org/10.1016/j.aml.2010.03.004.

    Article  MathSciNet  MATH  Google Scholar 

  20. Ya. S. Mahola and M. M. Sheremeta, “Properties of entire solutions of a linear differential equation of n th order with polynomial coefficients of n th degree,” Mat. Studii, 30, No. 2, 153–162 (2008).

  21. M. L. Mogra, “Hadamard product of certain meromorphic univalent functions,” J. Math. Anal. Appl., 157, No. 1, 10–16 (1991), https://doi.org/10.1016/0022-247X(91)90133-K.

    Article  MathSciNet  Google Scholar 

  22. M. L. Mogra, T. R. Reddy, and O. P. Juneja, “Meromorphic univalent functions with positive coefficients,” Bull. Austral. Math. Soc., 32, No. 2, 161–176 (1985), https://doi.org/10.1017/S0004972700009874.

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Owa and N. N. Pascu, “Coefficient inequalities for certain classes of meromorphically starlike and meromorphically convex functions,” J. Inequal. Pure Appl. Math., 4, No. 1, Art. 17 (2003), http://jipam.vu.edu.au.

  24. B. Şeker, “On certain new subclass of close-to-convex functions,” Appl. Math. Comput., 218, No. 3, 1041–1045 (2011).

    MathSciNet  MATH  Google Scholar 

  25. S. M. Shah, “Univalence of a function f and its successive derivatives when f satisfies a differential equation. II,” J. Math. Anal. Appl., 142, No. 2, 422–430 (1989), https://doi.org/10.1016/0022-247X(89)90011-5.

    Article  MathSciNet  Google Scholar 

  26. Z. M. Sheremeta, “On entire solutions of a differential equation,” Mat. Studii, 14, No. 1, 54–58 (2000), https://doi.org/10.1016/j.amc.2011.03.018.

    Article  MathSciNet  MATH  Google Scholar 

  27. H. Tang, G.-T. Deng, S.-H. Li, “On a certain new subclass of meromorphic close-to-convex functions,” J. Inequal. Appl., 2013, 164–169 (2013), https://doi.org/10.1186/1029-242X-2013-164.

    Article  MathSciNet  MATH  Google Scholar 

  28. B. A. Uralegaddi, “Meromorphically starlike functions with positive and fixed second coefficients,” Kyungpook Math. J., 29, No. 1, 64–68 (1989).

    MathSciNet  MATH  Google Scholar 

  29. Z. Wang, C. Gao, and S. Yuan, “On certain subclass of close-to-convex functions,” Acta Math. Acad. Paedagog. Nyházi, 22, No. 2, 171–177 (2006).

    MathSciNet  MATH  Google Scholar 

  30. Z.-G. Wang, Y. Sun, and N. Xu, “Some properties of certain meromorphic close-to-convex functions,” Appl. Math. Lett., 25, No. 3, 454–460 (2012), https://doi.org/10.1016/j.aml.2011.09.035.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. I. Franko Lviv National University, Lviv, Ukraine

    O. M. Holovata & M. M. Sheremeta

  2. Kiev National University of Food Technologies, Kiev, Ukraine

    O. M. Mulyava

Authors
  1. O. M. Holovata
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. M. Mulyava
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. M. Sheremeta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. M. Holovata.

Additional information

Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 61, No. 1, pp. 57–70, January–March, 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Holovata, O.M., Mulyava, O.M. & Sheremeta, M.M. Pseudostarlike, Pseudoconvex, and Close-to-Pseudoconvex Dirichlet Series Satisfying Differential Equations with Exponential Coefficients. J Math Sci 249, 369–388 (2020). https://doi.org/10.1007/s10958-020-04948-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-020-04948-1

Keywords

Search

Navigation