Properties of the solutions of the Gauss equation

M. M. Sheremeta; Yu. S. Trukhan

It is proved that the Gauss equation $z(z-1)w''+((\alpha+\beta+1)z-\gamma)w'+\alpha\beta w=0$ apart from the hypergeometric function has a power solution with negative exponents. Its possible growth, close-to-convexity and $l$-index boundedness are investigated.

1. Golusin G.M., Geometric theory of functions of a complex variable. – American Math. Soc., 1966. – 676 p.

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

3. Kuznetsov D.S., Special functions. – M.: Vysshaya Shkola, 1965. – 423 p. (in Russian)

4. Sheremeta M.M. Properties of the hypergeometric function with positive parameters// Visnyk Lviv Univ., Ser. Mech. Math. – 2009. – V.70. – P. 183–190. (in Ukrainian)

5. 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.

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

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

	Properties of the solutions of the Gauss equation
Author	M. M. Sheremeta, Yu. S. Trukhan yurik93@mail.ru Ivan Franko National University of Lviv
Abstract	It is proved that the Gauss equation $z(z-1)w''+((\alpha+\beta+1)z-\gamma)w'+\alpha\beta w=0$ apart from the hypergeometric function has a power solution with negative exponents. Its possible growth, close-to-convexity and $l$-index boundedness are investigated.
Keywords	Gauss equation; hypergeometric function; close-to-convexity; $l$-index boundedness
Reference	1. Golusin G.M., Geometric theory of functions of a complex variable. – American Math. Soc., 1966. – 676 p. 2. Sheremeta M.M., Analytic functions of bounded index. – Lviv: VNTL Publishers, 1999. – 141 p. 3. Kuznetsov D.S., Special functions. – M.: Vysshaya Shkola, 1965. – 423 p. (in Russian) 4. Sheremeta M.M. Properties of the hypergeometric function with positive parameters// Visnyk Lviv Univ., Ser. Mech. Math. – 2009. – V.70. – P. 183–190. (in Ukrainian) 5. 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. 6. Mogra M.L. Hadamard product certain meromorphic univalent functions// J. Math. Anal. Appl. – 1991. – V.157. – P. 10–16. 7. Goodman A.W., Univalent function. – V.II, Mariner Publishing Co., 1983. – 158 p.
Pages	157-167
Volume	41
Issue	2
Year	2014
Journal	Matematychni Studii
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