On the existence of meromorphically starlike and meromorphically
convex solutions of Shah’s differential equation

K. I. Dosyn; M. M. Sheremeta

We establish conditions under which the differential equation of S.~Shah

$z^2w''+(\beta_{0}z^2+\beta_1{z}) w'+(\gamma_{0}z^2+\gamma_{1}z+\gamma_{2})w=0$ has meromorphically starlike and meromorphically convex solutions of order

$\alpha \in [0,1)$ .

1. Goluzin G.M., Geometrical theory of functions of complex variables. - M.: Nauka, 1966. – 628 p. (in Russian)

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

3. Shah S.M. Univalence of a function f and its successive derivatives when f satisfies a differential equation, II// J. Math. Anal. and Appl. – 1989. – V.142, №2. – P. 422–430.

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

5. Sheremeta Z.M. The properties of entire solutions of a differential equation// Diff. Equat. – 2000. – V.36, №8. – P. 1045–1050. (in Russian)

6. Sheremeta Z.M. On entire solutions of a differential equation// Mat. Stud. – 2000. – V.14, №1. – P. 54–58.

7. Sheremeta Z.M. On the close-to-convexity of entire solutions of a differential equation// Visnyk of Lviv Un-tu, ser. mech.-mat. – 2000. – V.57. – P. 88–91. (in Ukrainian)

8. Sheremeta Z.M., Sheremeta M.M. Closeness to convexity for entire solutions of a differential equation// Diff. Equat. – 2002. – V.38, №4. – P. 496–501. Translated from Differential’nye Uravneniya. – 2002. – V.38, №4. – P. 477–481.

9. Sheremeta Z.M., Sheremeta M.M. Convexity of entire solutions of a differential equation// Mat. methods and fiz.-mech. polya. – 2004. – V.47, №2. – P. 186–191. (in Ukrainian)

10. Juneja O.P., Reddy T.R. Meromorphic starlike and univalent functions with positive coefficients// Ann. Univ. Mariae Curie-Sklodowska. – 1985. – V.39. – P. 65–76.

11. Mogra M.L. Hadamard product certain meromorphic univalent functions// J. Math. Anal. Appl. – 1991. – V.157. – P. 10–16.

12. Owa Sh., Pascu N.N. Coefficient inequalities for certain classes of meromorphically starlike and meromorphically convex functions// J. of Inequalities in Pure and Appl. Math. – 2003. – V.4, №1. – P. 1–6.

	On the existence of meromorphically starlike and meromorphically convex solutions of Shah’s differential equation
Author	K. I. Dosyn, M. M. Sheremeta kseniya_dosyn@gmail.com, m_m_sheremeta@list.ru Ivan Franko National University of Lviv
Abstract	We establish conditions under which the differential equation of S.~Shah $z^2w''+(\beta_{0}z^2+\beta_1{z}) w'+(\gamma_{0}z^2+\gamma_{1}z+\gamma_{2})w=0$ has meromorphically starlike and meromorphically convex solutions of order $\alpha \in [0,1)$ .
Keywords	starlike function; convex function; differential equation
Reference	1. Goluzin G.M., Geometrical theory of functions of complex variables. - M.: Nauka, 1966. – 628 p. (in Russian) 2. Kaplan W. Close-to-convex schlicht functions// Michigan Math. J. – 1952. – V.1, №2. – P. 169–185. 3. Shah S.M. Univalence of a function f and its successive derivatives when f satisfies a differential equation, II// J. Math. Anal. and Appl. – 1989. – V.142, №2. – P. 422–430. 4. Sheremeta Z.M. Close-to-convexity of entire solutions of a differential equation// Mat. methods and fiz.-mech. polya. – 1999. – V.42, №3. – P. 31–35. (in Ukrainian) 5. Sheremeta Z.M. The properties of entire solutions of a differential equation// Diff. Equat. – 2000. – V.36, №8. – P. 1045–1050. (in Russian) 6. Sheremeta Z.M. On entire solutions of a differential equation// Mat. Stud. – 2000. – V.14, №1. – P. 54–58. 7. Sheremeta Z.M. On the close-to-convexity of entire solutions of a differential equation// Visnyk of Lviv Un-tu, ser. mech.-mat. – 2000. – V.57. – P. 88–91. (in Ukrainian) 8. Sheremeta Z.M., Sheremeta M.M. Closeness to convexity for entire solutions of a differential equation// Diff. Equat. – 2002. – V.38, №4. – P. 496–501. Translated from Differential’nye Uravneniya. – 2002. – V.38, №4. – P. 477–481. 9. Sheremeta Z.M., Sheremeta M.M. Convexity of entire solutions of a differential equation// Mat. methods and fiz.-mech. polya. – 2004. – V.47, №2. – P. 186–191. (in Ukrainian) 10. Juneja O.P., Reddy T.R. Meromorphic starlike and univalent functions with positive coefficients// Ann. Univ. Mariae Curie-Sklodowska. – 1985. – V.39. – P. 65–76. 11. Mogra M.L. Hadamard product certain meromorphic univalent functions// J. Math. Anal. Appl. – 1991. – V.157. – P. 10–16. 12. Owa Sh., Pascu N.N. Coefficient inequalities for certain classes of meromorphically starlike and meromorphically convex functions// J. of Inequalities in Pure and Appl. Math. – 2003. – V.4, №1. – P. 1–6.
Pages	44-53
Volume	42
Issue	1
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On the existence of meromorphically starlike and meromorphically convex solutions of Shah’s differential equation