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 $f”+gf’+hf=0$
HTML articles powered by AMS MathViewer

by Simon Hellerstein, Joseph Miles and John Rossi

Trans. Amer. Math. Soc. 324 (1991), 693-706

DOI: https://doi.org/10.1090/S0002-9947-1991-1005080-X

PDF | Request permission

Abstract:

Suppose $g$ and $h$ are entire functions with the order of $h$ less than the order of $g$. If the order of $g$ does not exceed $\tfrac {1} {2}$, it is shown that every (necessarily entire) nonconstant solution $f$ of the differential equation $f'' + gf’ + hf = 0$ has infinite order. This result extends previous work of Ozawa and Gundersen.

References

P. D. Barry, On a theorem of Besicovitch, Quart. J. Math. Oxford Ser. (2) 14 (1963), 293–302. MR 156993, DOI 10.1093/qmath/14.1.293
P. D. Barry, Some theorems related to the $\textrm {cos}\,\pi \rho$ theorem, Proc. London Math. Soc. (3) 21 (1970), 334–360. MR 283223, DOI 10.1112/plms/s3-21.2.334
David Drasin and Daniel F. Shea, Pólya peaks and the oscillation of positive functions, Proc. Amer. Math. Soc. 34 (1972), 403–411. MR 294580, DOI 10.1090/S0002-9939-1972-0294580-X
David Drasin and Daniel F. Shea, Convolution inequalities, regular variation and exceptional sets, J. Analyse Math. 29 (1976), 232–293. MR 477619, DOI 10.1007/BF02789980
Gary G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), no. 1, 415–429. MR 920167, DOI 10.1090/S0002-9947-1988-0920167-5
W. K. Hayman, An inequality for real positive functions, Proc. Cambridge Philos. Soc. 48 (1952), 93–105. MR 45775, DOI 10.1017/s0305004100027407
W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
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, DOI 10.1080/17476938908814369
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, DOI 10.1090/S0002-9947-1984-0743737-2
W. K. Hayman and F. M. Stewart, Real inequalities with applications to function theory, Proc. Cambridge Philos. Soc. 50 (1954), 250–260. MR 61638, DOI 10.1017/s0305004100029297

Ordinary differential equations

Bo Kjellberg, On the minimum modulus of entire functions of lower order less than one, Math. Scand. 8 (1960), 189–197. MR 125967, DOI 10.7146/math.scand.a-10608
Joseph Miles, Bounds on the ratio $n(r,\,a)/S(r)$ for meromorphic functions, Trans. Amer. Math. Soc. 162 (1971), 383–393. MR 285711, DOI 10.1090/S0002-9947-1971-0285711-X
Mitsuru Ozawa, On a solution of $w^{\prime \prime }+e^{-z}w^{\prime } +(az+b)w=0$, Kodai Math. J. 3 (1980), no. 2, 295–309. MR 588459
John Rossi, Second order differential equations with transcendental coefficients, Proc. Amer. Math. Soc. 97 (1986), no. 1, 61–66. MR 831388, DOI 10.1090/S0002-9939-1986-0831388-8
M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894