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

M. M. Sheremeta; Yu. S. Trukhan

doi:10.15330/ms.48.1.14-23

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.

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), №2, 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), №2, 422–430.

4. Z.M. Sheremeta, The properties of entire solutions of one differential equation, Diff. uravnieniya, 36, (2000), №8, 1045–1050. Engl. transl.: Diff. Equat., 36 (2000), №8, 1155–1161.

5. Z.M. Sheremeta, Close-to-convexity of entire solutions of a differential equation, Mat. method. and fiz.- mech. polya, 42 (1999), №3, 31–35. (in Ukrainian)

6. Z.M. Sheremeta, On entire solutions of a differential equation, Mat. Stud., 14 (2000), №1, 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), №58, 54–56.

8. Z.M. Sheremeta, M.M. Sheremeta, Close-to-convexity for entire solutions of a differential equation, Diff. uravnieniya, 38 (2002), №4, 477–481. Engl. transl.: Diff. Equat., 38 (2002), №4, 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), №2, 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), №2, 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), №70, 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), №4, 62–74 (in Ukrainian). Engl. transl.: J. Math. Sci. 181 (2012), №3, 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), №1, 55–64.

	Starlike and convexity properties for p-valent solutions of the Shah differential equation
Author	M. M. Sheremeta, Yu. S. Trukhan m.m.sheremeta@gmail.com, yurkotrukhan@gmail.com 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), №2, 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), №2, 422–430. 4. Z.M. Sheremeta, The properties of entire solutions of one differential equation, Diff. uravnieniya, 36, (2000), №8, 1045–1050. Engl. transl.: Diff. Equat., 36 (2000), №8, 1155–1161. 5. Z.M. Sheremeta, Close-to-convexity of entire solutions of a differential equation, Mat. method. and fiz.- mech. polya, 42 (1999), №3, 31–35. (in Ukrainian) 6. Z.M. Sheremeta, On entire solutions of a differential equation, Mat. Stud., 14 (2000), №1, 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), №58, 54–56. 8. Z.M. Sheremeta, M.M. Sheremeta, Close-to-convexity for entire solutions of a differential equation, Diff. uravnieniya, 38 (2002), №4, 477–481. Engl. transl.: Diff. Equat., 38 (2002), №4, 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), №2, 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), №2, 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), №70, 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), №4, 62–74 (in Ukrainian). Engl. transl.: J. Math. Sci. 181 (2012), №3, 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), №1, 55–64.
Pages	14-23
Volume	48
Issue	1
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

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