# Starlike and convexity properties for p-valent solutions of the Shah differential equation

Author
Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract
The starlikeness and the convexity in the unit disc and the growth of an entire function $f(z)=z^p+\sum\nolimits_{n=p+1}^{\infty}f_n z^n$, $p\in{\Bbb N}$, satisfying the differential equation $z^2 w''+(\beta_0z^2+\beta_1z)w'+(\gamma_0 z^2+\gamma_1 z+\gamma_2) w=0$ ($\beta_0,\,\beta_1,\,\gamma_0,\,\gamma_1,\,\gamma_2$ are complex parameters) are studied.
Keywords
starlikeness; convexity; entire function; differential equation
DOI
doi:10.15330/ms.48.1.14-23
Reference
1. G.M. Goluzin, Geometric theory of functions of a complex variable, M.: Nauka, 1966 (in Russian). Engl. transl.: AMS: Translations of Mathematical monograph, 26 (1969), 676p.

2. W. Kaplan, Close-to-convex schlicht functions, Michigan Math.J., 1 (1952), ¹2, 169–185.

3. S.M. Shah, Univalence of a function $f$ and its successive derivatives when $f$ satisfies a differential equation II, J. Math. Anal. Appl., 142 (1989), ¹2, 422–430.

4. Z.M. Sheremeta, The properties of entire solutions of one differential equation, Diff. uravnieniya, 36, (2000), ¹8, 1045–1050. Engl. transl.: Diff. Equat., 36 (2000), ¹8, 1155–1161.

5. Z.M. Sheremeta, Close-to-convexity of entire solutions of a differential equation, Mat. method. and fiz.- mech. polya, 42 (1999), ¹3, 31–35. (in Ukrainian)

6. Z.M. Sheremeta, On entire solutions of a differential equation, Mat. Stud., 14 (2000), ¹1, 54–58.

7. Z.M. Sheremeta, On the close-to-convexity of entire solutions of a differential equation, Visn. Lviv Un-ty. Mech.Math., (2000), ¹58, 54–56.

8. Z.M. Sheremeta, M.M. Sheremeta, Close-to-convexity for entire solutions of a differential equation, Diff. uravnieniya, 38 (2002), ¹4, 477–481. Engl. transl.: Diff. Equat., 38 (2002), ¹4, 496–501.

9. Z.M. Sheremeta, M.M. Sheremeta, Convexity of entire solutions of a differential equation, Mat. method. and fiz.-mech. polya, 47 (2004), ¹2, 186–191. (in Ukrainian)

10. Ya.S. Mahola, M.M. Sheremeta, Properties of entire solutions of a linear differential equation of n-th order with polynomial coefficients of n-th degree, Mat. Stud., 30 (2008), ¹2, 153–162.

11. Ya.S. Mahola, M.M. Sheremeta, Close-to-convexsity an entire solution of a linear differential equation with polynomial coefficients, Visnyk of the Lviv Univ. Series Mech-Math., (2009), ¹70, 122–127. (in Ukrainian)

12. Ya.S. Mahola, M.M. Sheremeta, On the properties of entire solutions of linear differential equations with polynomial coefficients, Mat. metody and fiz.-mech. polya, 53 (2010), ¹4, 62–74 (in Ukrainian). Engl. transl.: J. Math. Sci. 181 (2012), ¹3, 366–382.

13. S. Owa, On certain classes of p-valent functions with negative coefficients, Simon Stevin, 59 (1985), 385–402.

14. R.M. El-Ashwah, M.K. Aouf, A.O. Moustafa, Starlike and convexity properties for $p$-valent hypergeometric functions, Acta Math. Univ. Comenianae, 79 (2010), ¹1, 55–64.

Pages
14-23
Volume
48
Issue
1
Year
2017
Journal
Matematychni Studii
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