Disconjugacy and integral inequalities

Clausing, Achim

doi:10.1090/S0002-9947-1980-0570791-X

Disconjugacy and integral inequalities
HTML articles powered by AMS MathViewer

by Achim Clausing

Trans. Amer. Math. Soc. 260 (1980), 293-307

DOI: https://doi.org/10.1090/S0002-9947-1980-0570791-X

PDF | Request permission

Abstract:

The basic data in this paper are a disconjugate differential operator and an associated two-point boundary value problem. These define in a natural way a cone of functions satisfying a differential inequality with respect to the operator. By using a result of P. W. Bates and G. B. Gustafson on the monotonicity properties of Green’s kernels it is shown that such a cone has a compact convex base which is a Bauer simplex. This result is used to derive a variety of integral inequalities which include known inequalities of Frank and Pick, Levin and Steckin, Karlin and Ziegler, as well as several new ones.

References

P. W. Bates and G. B. Gustafson, Green’s function inequalities for two-point boundary value problems, Pacific J. Math. 59 (1975), no. 2, 327–343. MR 437841
W. Blaschke and G. Pick, Distanzschätzungen im Funktionenraum II, Math. Ann. 77 (1916), no. 2, 277–300 (German). MR 1511857, DOI 10.1007/BF01456904

Pólya’s condition and polynomial expansions about two points

W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. MR 0460785
Ph. Frank and G. Pick, Distanzschätzungen im Funktionenraum. I, Math. Ann. 76 (1915), no. 2-3, 354–375 (German). MR 1511829, DOI 10.1007/BF01458149
Philip Hartman, Monotony properties and inequalities for Green’s functions for multipoint boundary value problems, SIAM J. Math. Anal. 9 (1978), no. 5, 806–814. MR 506761, DOI 10.1137/0509062
Samuel Karlin and Zvi Ziegler, Some inequalities for generalized concave functions, J. Approximation Theory 13 (1975), 276–293. MR 376981, DOI 10.1016/0021-9045(75)90038-6
S. B. Stečkin and V. I. Levin, Inequalities, Amer. Math. Soc. Transl. (2) 14 (1960), 1–29. MR 0112925
D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686
Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
G. Pólya, On the mean-value theorem corresponding to a given linear homogeneous differential equation, Trans. Amer. Math. Soc. 24 (1922), no. 4, 312–324. MR 1501228, DOI 10.1090/S0002-9947-1922-1501228-5