Properties of analytic solutions of a differential equation

M. M. Sheremeta; Yu. S. Trukhan

doi:10.30970/ms.52.2.138-143

For an analytic in the unit disk $\{z\colon \,|z|\le1\}$ solution of the differential equation $z(z-1) w''+\beta_1 z w'+\gamma_2w=0$ the close-to-convexity, possible growth and the $l$-index boundedness are investigated.

1. Goluzin G.M. Geometric theory of functions of a complex variable - M.: Nauka, 1966 (in Russian), Engl. transl.: AMS: Translations of Mathematical monograph, 26 (1969), 676p.

2. Kaplan W. Close-to-convex schlicht functions, Michigan Math. J., 1 (1952), №2, 169–185.

3. Alexander J.W. Functions which map the interior of the unit circle upon simple regions, Ann. of Math., 17 (1915/1916), 12–22.

4. Goodman A.W. Univalent function, V. II., Mariner Publishing Co., 1983, 158 p.

5. Shah S.M. Univalence of a function f and its successive derivatives when f satisfies a differential equation, II, J. Math. anal. and appl., 142 (1989), 422–430.

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

7. Sheremeta Z.M. On properties of entire solutions of a differential equation, Diff. uravnieniya, 36 (2000), №8, 1–6 (in Russian), Engl. transl.: Diff. Equat., 36 (2000), №8, 1155–1161.

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

9. Sheremeta Z.M. On close-to-convexsity of entire solutions of a differential equation, Visnyk of the Lviv Univ. Series Mech-Math., (2000), №57, 88–91. (in Ukrainian)

10. Sheremeta Z.M., Sheremeta M.M. Close-to-convexsity of entire solutions of a differential equation, Diff. uravnieniya, 38 (2002), №4, 435–440. (in Russian)

11. Mahola Ya.S., Sheremeta M.M. 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.

12. Mahola Ya.S., Sheremeta M.M. On properties of entire solutions of linear differential equations with polynomial coefficients, Mat. method. and fiz.-mech. polya, 53 (2010), №4, 162–174. (in Ukrainian)

13. Mahola Ya.S., Sheremeta M.M. On entire solutions with two-member recurrent formula for Taylor coefficients of linear differential equation, Mat. Stud., 36 (2011), №2, 133–141.

14. Mahola Ya.S., Sheremeta M.M. Close-to-convexity of an entire solution of linear differential equation of the third order with polynomial coefficients of the third degree, Visnyk of the Chernivtsi Univ., Series Math., 528 (2010), 88–92. (in Ukrainian)

15. Sheremeta M.M. Analytic functions of bounded index. – Lviv: VNTL Publishers, 1999, 141 p.

16. Sheremeta Z.M. Index boundedness of an entire solution of a differential equation, Mat. Stud., 19 (2003), №2, 208–212.

17. Sheremeta Z.M. On l-index boundedness of an entire solution of a differential equation, Visnyk of the Lviv Univ. Series Mech-Math., 63 (2004), 148–151. (in Ukrainian)

18. Zabolotsky M.V., Sheremeta Z.M. On index boundedness of entire solution to one differential equation, Mat. method. and fiz.-mech. polya, 47 (2004), №2, 181–185. (in Ukrainian)

19. Sheremeta M.M., Sheremeta Z.M. Properties of entire solutions of differential equations, Ukr. Math Zh., 58 (2006), №12, 1693–1703. (in Ukrainian); Engl. transl.: Ukrainian Math. J., 56 (2006), №12, 1924–1934.

20. Sheremeta Z.M., Sheremeta M.M. Boundedness of l-index of analytic functions represented by power series. Visnyk of the Lviv Univ. Series Mech-Math., 66 (2006), 208–213. (in Ukrainian)

21. Sheremeta Z.M., Sheremeta M.M. On the l-index boundedness of entire solutions of a differential equation, Dopovidi NAN Ukraine, 56 (2007), №2, 31–36. (in Ukrainian)

22. Sheremeta M.M., Trukhan Yu.S. Properties of the solutions of the Gauss equation, Mat. Stud., 41 (2014), №2, 157–167.

	Properties of analytic solutions of a differential equation
Author	M. M. Sheremeta¹, Yu. S. Trukhan² m_m_sheremeta@gmail.com¹, yurkotrukhan@gmail.com² Lviv Ivan Franko National University, Lviv, Ukraine
Abstract	For an analytic in the unit disk $\{z\colon \,\|z\|\le1\}$ solution of the differential equation $z(z-1) w''+\beta_1 z w'+\gamma_2w=0$ the close-to-convexity, possible growth and the $l$-index boundedness are investigated.
Keywords	differential equation; close-to-convexity; l-index boundedness
DOI	doi:10.30970/ms.52.2.138-143
Reference	1. Goluzin G.M. Geometric theory of functions of a complex variable - M.: Nauka, 1966 (in Russian), Engl. transl.: AMS: Translations of Mathematical monograph, 26 (1969), 676p. 2. Kaplan W. Close-to-convex schlicht functions, Michigan Math. J., 1 (1952), №2, 169–185. 3. Alexander J.W. Functions which map the interior of the unit circle upon simple regions, Ann. of Math., 17 (1915/1916), 12–22. 4. Goodman A.W. Univalent function, V. II., Mariner Publishing Co., 1983, 158 p. 5. Shah S.M. Univalence of a function f and its successive derivatives when f satisfies a differential equation, II, J. Math. anal. and appl., 142 (1989), 422–430. 6. Sheremeta Z.M. Close-to-convexsity of entire solutions of a differential equation, Mat. method. and fiz.-mech. polya, 42 (1999), №3, 31–35. (in Ukrainian) 7. Sheremeta Z.M. On properties of entire solutions of a differential equation, Diff. uravnieniya, 36 (2000), №8, 1–6 (in Russian), Engl. transl.: Diff. Equat., 36 (2000), №8, 1155–1161. 8. Sheremeta Z.M. On entire solutions of a differential equation, Mat. Stud., 14 (2000), №1, 54–58. 9. Sheremeta Z.M. On close-to-convexsity of entire solutions of a differential equation, Visnyk of the Lviv Univ. Series Mech-Math., (2000), №57, 88–91. (in Ukrainian) 10. Sheremeta Z.M., Sheremeta M.M. Close-to-convexsity of entire solutions of a differential equation, Diff. uravnieniya, 38 (2002), №4, 435–440. (in Russian) 11. Mahola Ya.S., Sheremeta M.M. 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. 12. Mahola Ya.S., Sheremeta M.M. On properties of entire solutions of linear differential equations with polynomial coefficients, Mat. method. and fiz.-mech. polya, 53 (2010), №4, 162–174. (in Ukrainian) 13. Mahola Ya.S., Sheremeta M.M. On entire solutions with two-member recurrent formula for Taylor coefficients of linear differential equation, Mat. Stud., 36 (2011), №2, 133–141. 14. Mahola Ya.S., Sheremeta M.M. Close-to-convexity of an entire solution of linear differential equation of the third order with polynomial coefficients of the third degree, Visnyk of the Chernivtsi Univ., Series Math., 528 (2010), 88–92. (in Ukrainian) 15. Sheremeta M.M. Analytic functions of bounded index. – Lviv: VNTL Publishers, 1999, 141 p. 16. Sheremeta Z.M. Index boundedness of an entire solution of a differential equation, Mat. Stud., 19 (2003), №2, 208–212. 17. Sheremeta Z.M. On l-index boundedness of an entire solution of a differential equation, Visnyk of the Lviv Univ. Series Mech-Math., 63 (2004), 148–151. (in Ukrainian) 18. Zabolotsky M.V., Sheremeta Z.M. On index boundedness of entire solution to one differential equation, Mat. method. and fiz.-mech. polya, 47 (2004), №2, 181–185. (in Ukrainian) 19. Sheremeta M.M., Sheremeta Z.M. Properties of entire solutions of differential equations, Ukr. Math Zh., 58 (2006), №12, 1693–1703. (in Ukrainian); Engl. transl.: Ukrainian Math. J., 56 (2006), №12, 1924–1934. 20. Sheremeta Z.M., Sheremeta M.M. Boundedness of l-index of analytic functions represented by power series. Visnyk of the Lviv Univ. Series Mech-Math., 66 (2006), 208–213. (in Ukrainian) 21. Sheremeta Z.M., Sheremeta M.M. On the l-index boundedness of entire solutions of a differential equation, Dopovidi NAN Ukraine, 56 (2007), №2, 31–36. (in Ukrainian) 22. Sheremeta M.M., Trukhan Yu.S. Properties of the solutions of the Gauss equation, Mat. Stud., 41 (2014), №2, 157–167.
Pages	138-143
Volume	52
Issue	2
Year	2019
Journal	Matematychni Studii
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