Disconjugacy and integral inequalities

Author:
Achim Clausing

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

MSC:
Primary 26D15; Secondary 34B27

MathSciNet review:
570791

Full-text PDF Free Access

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. Gustafson,*Green’s function inequalities for two-point boundary value problems*, Pacific J. Math.**59**(1975), no. 2, 327–343. MR**0437841****[2]**W. Blaschke and G. Pick,*Distanzschätzungen im Funktionenraum II*, Math. Ann.**77**(1916), no. 2, 277–300 (German). MR**1511857**, 10.1007/BF01456904**[3]**A. Clausing,*Pólya's condition and polynomial expansions about two points*(in preparation).**[4]**W. A. Coppel,*Disconjugacy*, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. MR**0460785****[5]**Ph. Frank and G. Pick,*Distanzschätzungen im Funktionenraum. I*, Math. Ann.**76**(1915), no. 2-3, 354–375 (German). MR**1511829**, 10.1007/BF01458149**[6]**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**, 10.1137/0509062**[7]**Samuel Karlin and Zvi Ziegler,*Some inequalities for generalized concave functions*, J. Approximation Theory**13**(1975), 276–293. Collection of articles dedicated to G. G. Lorentz on the occasion of his sixty-fifth birthday, III. MR**0376981****[8]**V. I. Levin and S. B. Stečkin,*Inequalities*, Amer. Math. Soc. Transl. (2)**14**(1960), 1–29. MR**0112925****[9]**D. S. Mitrinović,*Analytic inequalities*, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. MR**0274686****[10]**Robert R. Phelps,*Lectures on Choquet’s theorem*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR**0193470****[11]**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**, 10.1090/S0002-9947-1922-1501228-5

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
26D15,
34B27

Retrieve articles in all journals with MSC: 26D15, 34B27

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