Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Riemann's function has an exponential bound


Author: Paul R. Beesack
Journal: Proc. Amer. Math. Soc. 88 (1983), 313-316
MSC: Primary 26D15; Secondary 35L99
DOI: https://doi.org/10.1090/S0002-9939-1983-0695265-5
MathSciNet review: 695265
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Riemann function $ \upsilon (t;x)$ of a hyperbolic characteristic initial value problem has been much used in recent years to provide upper bounds for functions which satisfy Gronwall-type integral inequalities. This note gives a direct proof of the fact that $ \upsilon $ satisfies an inequality of the form $ \upsilon (t;x) \leqslant \exp (\int_t^x {b(s)ds}$.


References [Enhancements On Off] (What's this?)

  • [1] R. P. Agarwal, On an integral inequality in $ n$ independent variables, J. Math. Anal. Appl. 85 (1982), 192-196. MR 647567 (83g:26024)
  • [2] P. R. Beesack, On some Gronwall-type integral inequalities in $ n$ independent variables, J. Math. Anal. Appl. (to appear). MR 743329 (85k:26024)
  • [3] B. K. Bondge and B. G. Pachpatte, On some fundamental integral inequalities in $ n$ independent variables, Bull. Inst. Math. Acad. Sinica 8 (1980), 553-560. MR 615276 (82i:26010)
  • [4] -, On some partial integral inequalities in two independent variables, Funkcial. Ekvac. 23 (1980), 327-334. MR 621537 (82h:26021)
  • [5] J. Chandra and P. W. Davis, Linear generalizations of Gronwall's inequality, Proc. Amer. Math. Soc. 60 (1976), 157-160. MR 0422735 (54:10721)
  • [6] A. M. Fink, Wendroff's inequalities, Nonlinear Anal. 5 (1981), 873-874. MR 628107 (83a:26022)
  • [7] D. Y. Kasture and S. G. Deo, Inequalities of Gronwall type in two independent variables, J. Math. Anal. Appl. 58 (1977), 361-372. MR 0437895 (55:10816)
  • [8] B. G. Pachpatte, On some new integral and integrodifferential inequalities in two independent variables and their applications, J. Differential Equations 33 (1979), 249-272. MR 542673 (80f:26010)
  • [9] -, On some fundamental partial integral inequalities, J. Math. Anal. Appl. 73 (1980), 238-251. MR 560945 (81a:26012)
  • [10] -, On some new integral inequalities for nonselfadjoint hyperbolic partial integrodifferential equations, J. Math. Anal. Appl. 76 (1980), 58-71. MR 586643 (82a:45018)
  • [11] -, On some partial integral inequalities in $ n$ independent variables, J. Math. Anal. Appl. 79 (1981), 256-272. MR 603390 (82f:26018)
  • [12] -, On certain integral inequalities for partial differential and integral equations, An. Ştiinţ. Univ. "Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.) 27 (1981), 365-374. MR 640763 (83e:26016)
  • [13] D. R. Snow, A two independent variable Gronwall-type inequality, Inequalities III (Proc. Third Sympos. Univ. California, Los Angeles, Calif., 1969), Academic Press, New York, 1972, pp. 333-340. MR 0338537 (49:3301)
  • [14] -, Gronwall's inequality for systems of partial differential equations in two independent variables, Proc. Amer. Math. Soc. 33 (1972), 46-54. MR 0298188 (45:7240)
  • [15] E. Thandapani and R. P. Agarwal, On some new inequalities in $ n$ independent variables, J. Math. Anal. Appl. 86 (1982), 542-561. MR 652195 (83e:26025)
  • [16] W. Walter, Differential and integral inequalities, Ergebnisse Math. Grenzgeb., Band 55, Springer-Verlag, Berlin and New York, 1970. MR 0271508 (42:6391)
  • [17] C.-C. Yeh, On some integral inequalities in $ n$ independent variables and their applications, J. Math. Anal. Appl. 86 (1982), 387-410. MR 652184 (84f:26020)
  • [18] E. C. Young, Gronwall's inequality in $ n$ independent variables, Proc. Amer. Math. Soc. 41 (1973), 241-244. MR 0320493 (47:9030)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26D15, 35L99

Retrieve articles in all journals with MSC: 26D15, 35L99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0695265-5
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society