Disconjugacy and integral inequalities

Author:
Achim Clausing

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

MSC:
Primary 26D15; Secondary 34B27

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

MathSciNet review:
570791

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**P. W. Bates and G. B. Gustaf son,*Green's function inequalities for two-point boundary value problems*, Pacific J. Math.**59**(1975), 327-343. MR**0437841 (55:10762)****[2]**W. Blaschke and G. Pick,*Distanzschätzungen im Funktionenraum*. II, Math. Ann.**77**(1916), 277-300. MR**1511857****[3]**A. Clausing,*Pólya's condition and polynomial expansions about two points*(in preparation).**[4]**W. A. Coppel,*Disconjugacy*, Lecture Notes in Math., vol. 220, Springer-Verlag, Berlin and New York, 1971. MR**0460785 (57:778)****[5]**Ph. Frank and G. Pick,*Distanzschätzungen im Funktionenraum*. I, Math. Ann.**76**(1915), 354-375. MR**1511829****[6]**Ph. Hartman,*Monotony properties and inequalities for Green's functions for multipoint boundary value problems*, SIAM J. Math. Anal.**9**(1978), 806-814. MR**506761 (80a:34015)****[7]**S. Karlin and Z. Ziegler,*Some inequalities for generalized concave functions*, J. Approximation Theory**13**(1975), 276-293. MR**0376981 (51:13156)****[8]**V. I. Levin and S. B. Stečkin,*Inequalities*, Amer. Math. Soc. Transl. (2)**14**(1960), 1-29. MR**0112925 (22:3771)****[9]**D. S. Mitrinovič,*Analytic inequalities*, Springer-Verlag, Berlin and New York, 1970. MR**0274686 (43:448)****[10]**R. R. Phelps,*Lectures on Choquet's theorem*. Van Nostrand-Reinhold, New York, 1966. MR**0193470 (33:1690)****[11]**G. Pólya,*On the mean-value theorem corresponding to a given linear homogeneous differential equation*, Trans. Amer. Math. Soc.**24**(1922), 312-324. MR**1501228**

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Additional Information

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

Keywords:
Disconjugate differential operator,
differential inequality,
monotonicity properties of Green's kernels,
integral representation,
integral inequality

Article copyright:
© Copyright 1980
American Mathematical Society