On the growth of solutions of 𝑓”+𝑔𝑓’+ℎ𝑓=0

Hellerstein, Simon; Miles, Joseph; Rossi, John

doi:10.1090/S0002-9947-1991-1005080-X

On the growth of solutions of

Authors: Simon Hellerstein, Joseph Miles and John Rossi
Journal: Trans. Amer. Math. Soc. 324 (1991), 693-706
MSC: Primary 30D20; Secondary 34A20
DOI: https://doi.org/10.1090/S0002-9947-1991-1005080-X
MathSciNet review: 1005080
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Abstract: Suppose and are entire functions with the order of less than the order of . If the order of does not exceed $\tfrac{1} {2}$ , it is shown that every (necessarily entire) nonconstant solution of the differential equation has infinite order. This result extends previous work of Ozawa and Gundersen.

[1] P. D. Barry, On a theorem of Besicovitch, Quart. J. Math. Oxford Ser. (2) 14 (1963), 293–302. MR 0156993, https://doi.org/10.1093/qmath/14.1.293
[2] P. D. Barry, Some theorems related to the 𝑐𝑜𝑠𝜋𝜌 theorem, Proc. London Math. Soc. (3) 21 (1970), 334–360. MR 0283223, https://doi.org/10.1112/plms/s3-21.2.334
[3] David Drasin and Daniel F. Shea, Pólya peaks and the oscillation of positive functions, Proc. Amer. Math. Soc. 34 (1972), 403–411. MR 0294580, https://doi.org/10.1090/S0002-9939-1972-0294580-X
[4] David Drasin and Daniel F. Shea, Convolution inequalities, regular variation and exceptional sets, J. Analyse Math. 29 (1976), 232–293. MR 0477619, https://doi.org/10.1007/BF02789980
[5] Gary G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), no. 1, 415–429. MR 920167, https://doi.org/10.1090/S0002-9947-1988-0920167-5
[6] W. K. Hayman, An inequality for real positive functions, Proc. Cambridge Philos. Soc. 48 (1952), 93–105. MR 0045775
[7] W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
[8] W. K. Hayman and Joseph Miles, On the growth of a meromorphic function and its derivatives, Complex Variables Theory Appl. 12 (1989), no. 1-4, 245–260. MR 1040924, https://doi.org/10.1080/17476938908814369
[9] W. K. Hayman and J. F. Rossi, Characteristic, maximum modulus and value distribution, Trans. Amer. Math. Soc. 284 (1984), no. 2, 651–664. MR 743737, https://doi.org/10.1090/S0002-9947-1984-0743737-2
[10] W. K. Hayman and F. M. Stewart, Real inequalities with applications to function theory, Proc. Cambridge Philos. Soc. 50 (1954), 250–260. MR 0061638
[11] E. L. Ince, Ordinary differential equations, Longmans, Green and Co., London, 1927.
[12] Bo Kjellberg, On the minimum modulus of entire functions of lower order less than one, Math. Scand. 8 (1960), 189–197. MR 0125967, https://doi.org/10.7146/math.scand.a-10608
[13] Joseph Miles, Bounds on the ratio 𝑛(𝑟,𝑎)/𝑆(𝑟) for meromorphic functions, Trans. Amer. Math. Soc. 162 (1971), 383–393. MR 0285711, https://doi.org/10.1090/S0002-9947-1971-0285711-X
[14] Mitsuru Ozawa, On a solution of 𝑤′′+𝑒^{-𝑧}𝑤′+(𝑎𝑧+𝑏)𝑤=0, Kodai Math. J. 3 (1980), no. 2, 295–309. MR 588459
[15] John Rossi, Second order differential equations with transcendental coefficients, Proc. Amer. Math. Soc. 97 (1986), no. 1, 61–66. MR 831388, https://doi.org/10.1090/S0002-9939-1986-0831388-8
[16] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959. MR 0114894