Geodesic
ON THE COMPLEXITY OF FINDING A NECESSARY AND SUFFICIENT CONDITION FOR BLASCHKE-OSCILLATORY EQUATIONS
Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 543-554

Voir la notice de l'article provenant de la source Cambridge University Press

If A(z) belongs to the Bergman space , then the differential equation f′′+A(z)f=0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to . It is demonstrated that certain “nice” Blaschke sequences can be zero sequences of solutions in both cases when A ∈ or A ∉ . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ . 
HEITTOKANGAS, JANNE; REIJONEN, ATTE. ON THE COMPLEXITY OF FINDING A NECESSARY AND SUFFICIENT CONDITION FOR BLASCHKE-OSCILLATORY EQUATIONS. Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 543-554. doi: 10.1017/S0017089514000470
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[1] 1.Buckley, S., Koskela, P. and Vukotić, D., Fractional integration, differentiation, and weighted Bergman spaces, Math. Proc. Camb. Phil. Soc. 126 (2) (1999), 369–385. Google Scholar | DOI

[2] 2.Carleson, L., An interpolation problem for bounded analytic functions Am. J. Math. 80 (1958), 921–930. Google Scholar | DOI

[3] 3.Chuaqui, M., Gröhn, J., Heittokangas, J. and Rättyä, J., Zero separation results for solutions of second order linear differential equations, Adv. Math. 245 (2013), 382–422. Google Scholar | DOI

[4] 4.Duren, P., Theory of Hp spaces (Academic Press, New York, 1970). Google Scholar

[5] 5.Duren, P. and Schuster, A., Bergman Spaces, Mathematical Surveys and Monographs, vol. 100 (American Mathematical Soc., Providence, 2004). Google Scholar

[6] 6.Fricain, E. and Mashreghi, J., Integral means of the derivatives of Blaschke products, Glasg. Math. J. 50 (2) (2008), 233–249. Google Scholar | DOI

[7] 7.Gotoh, Y., On integral means of the derivatives of Blaschke products, Kodai Math. J. 30 (1) (2007), 147–155. Google Scholar | DOI

[8] 8.Gröhn, J. and Heittokangas, J., New findings on Bank-Sauer approach in oscillation theory Constr. Approx. 35 (2012), 345–361. Google Scholar | DOI

[9] 9.Hedenmalm, H., Korenblum, B. and Zhu, K., Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199 (Springer, New York, 2000). Google Scholar

[10] 10.Heittokangas, J., A survey on Blaschke-oscillatory differential equations, with updates, in Blaschke products and their applications, Fields Inst. Commun. vol. 65 (Fricain, E. and Mashreghi, J., Editors) (Springer, New York, 2013), 43–98. Google Scholar

[11] 11.Heittokangas, J., Solutions of f” + A(z)f = 0 in the unit disc having Blaschke sequences as the zeros, Comput. Methods Funct. Theory 5 (1) (2005), 49–63. Google Scholar | DOI

[12] 12.Heittokangas, J. and Laine, I., Solutions of f” + A(z)f = 0 with prescribed sequences of zeros, Acta Math. Univ. Comen. 124 (2) (2005), 287–307. Google Scholar

[13] 13.Heittokangas, J., Korhonen, R. and Rättyä, J., Linear differential equations with coefficients in weighted Bergman and Hardy spaces, Trans. Am. Math. Soc. 360 (2008), 1035–1055. Google Scholar | DOI

[14] 14.Kim, H., Derivatives of Blaschke products, Pac. J. Math. 114 (1) (1984), 175–190. Google Scholar | DOI

[15] 15.Linden, C., Hp-derivatives of Blaschke products, Mich. Math. J. 23 (1976), 43–51. Google Scholar | DOI

[16] 16.Nehari, Z., The Schwarzian derivative and Schlicht functions Bull. Am. Math. Soc. 55 (1949), 545–551. Google Scholar | DOI

[17] 17.Pommerenke, Ch., On the mean growth of the solutions of complex linear differential equations in the disk, Complex Var. Theory Appl. 1 (1) (1982), 23–38. Google Scholar

[18] 18.Protas, D., Blaschke products with derivative in Hp and Bp Mich. Math. J. 20 (1973), 393–396. Google Scholar

[19] 19.Šeda, V., On some properties of solutions of the differential equation y” + Ay = 0 with solutions having the prescribed zeros, Acta Fac. Nat. Univ. Comen. Math. 4 (1959), 223–253. (Slovak) Google Scholar

[20] 20.Yamashita, S., Gap series and α-Bloch functions, Yokohama Math. J. 28 (1980), 31–36. Google Scholar

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