On a class of new inequalities
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- by Daniel T. Shum PDF
- Trans. Amer. Math. Soc. 204 (1975), 299-341 Request permission
Abstract:
Inequalities of considerable interest are associated with the names of Beesack, Benson, Boyd, Calvert, Das, Hardy, Hua, Opial, Wong and Yang. In this note an elementary method used in a recent paper by Benson will be further investigated. The resultant new class of inequalities will bring a great number of inequalities—such as inequalities of Hardy’s and those of Opial’s—under one roof, so to speak.References
- Edwin F. Beckenbach and Richard Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 30, Springer-Verlag, Inc., New York, 1965. Second revised printing. MR 0192009, DOI 10.1007/978-3-662-35199-4
- Paul R. Beesack, Integral inequalities of the Wirtinger type, Duke Math. J. 25 (1958), 477–498. MR 97478 —, Extensions of Wirtinger’s inequality, Trans. Roy. Soc. Canada 53 (1959), 21-30.
- Paul R. Beesack, Elementary proofs of the extremal properties of the eigenvalues of the Sturm-Liouville equation, Canad. Math. Bull. 3 (1960), 59–77. MR 126014, DOI 10.4153/CMB-1960-010-3
- Paul R. Beesack, Hardy’s inequality and its extensions, Pacific J. Math. 11 (1961), 39–61. MR 121449, DOI 10.2140/pjm.1961.11.39
- Paul R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc. 104 (1962), 470–475. MR 139706, DOI 10.1090/S0002-9947-1962-0139706-1
- Paul R. Beesack, On certain discrete inequalities involving partial sums, Canadian J. Math. 21 (1969), 222–234. MR 240261, DOI 10.4153/CJM-1969-022-1
- P. R. Beesack, Integral inequalities involving a function and its derivative, Amer. Math. Monthly 78 (1971), 705–741. MR 325889, DOI 10.2307/2318009
- P. R. Beesack and K. M. Das, Extensions of Opial’s inequality, Pacific J. Math. 26 (1968), 215–232. MR 239026, DOI 10.2140/pjm.1968.26.215
- Donald C. Benson, Inequalities involving integrals of functions and their derivatives, J. Math. Anal. Appl. 17 (1967), 292–308. MR 202950, DOI 10.1016/0022-247X(67)90154-0
- D. W. Boyd and J. S. W. Wong, An extension of Opial’s inequality, J. Math. Anal. Appl. 19 (1967), 100–102. MR 212146, DOI 10.1016/0022-247X(67)90024-8
- David W. Boyd, Best constants in a class of integral inequalities, Pacific J. Math. 30 (1969), 367–383. MR 249556, DOI 10.2140/pjm.1969.30.367
- David W. Boyd, Best constants in inequalities related to Opial’s inequality, J. Math. Anal. Appl. 25 (1969), 378–387. MR 236337, DOI 10.1016/0022-247X(69)90241-8
- James Calvert, Some generalizations of Opial’s inequality, Proc. Amer. Math. Soc. 18 (1967), 72–75. MR 204594, DOI 10.1090/S0002-9939-1967-0204594-1
- K. M. Das, An inequality similar to Opial’s inequality, Proc. Amer. Math. Soc. 22 (1969), 258–261. MR 244448, DOI 10.1090/S0002-9939-1969-0244448-X G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, New York, 1952. MR 13, 727.
- Hua Luo-geng, On an inequality of Opial, Sci. Sinica 14 (1965), 789–790. MR 181725
- Cheng-ming Lee, On a discrete analogue of inequalities of Opial and Yang, Canad. Math. Bull. 11 (1968), 73–77. MR 239028, DOI 10.4153/CMB-1968-010-7
- N. Levinson, On an inequality of Opial and Beesack, Proc. Amer. Math. Soc. 15 (1964), 565–566. MR 166315, DOI 10.1090/S0002-9939-1964-0166315-8
- C. L. Mallows, An even simpler proof of Opial’s inequality, Proc. Amer. Math. Soc. 16 (1965), 173. MR 170989, DOI 10.1090/S0002-9939-1965-0170989-6 P. M. Maroni, Sur l’inégalité d’Opial-Beesack, C. R. Acad. Sci. Paris Sér. A 264 (1967), A62-A64. MR 34 #6007.
- 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, DOI 10.1007/978-3-642-99970-3
- Z. Olech, A simple proof of a certain result of Z. Opial, Ann. Polon. Math. 8 (1960), 61–63. MR 112927, DOI 10.4064/ap-8-1-61-63
- Z. Opial, Sur une inégalité, Ann. Polon. Math. 8 (1960), 29–32 (French). MR 112926, DOI 10.4064/ap-8-1-29-32
- R. N. Pederson, On an inequality of Opial, Beesack and Levinson, Proc. Amer. Math. Soc. 16 (1965), 174. MR 170990, DOI 10.1090/S0002-9939-1965-0170990-2
- Raymond Redheffer, Inequalities with three functions, J. Math. Anal. Appl. 16 (1966), 219–242. MR 209423, DOI 10.1016/0022-247X(66)90168-5
- D. T. Shum, On integral inequalities related to Hardy’s, Canad. Math. Bull. 14 (1971), 225–230. MR 313465, DOI 10.4153/CMB-1971-038-4 —, A general and sharpened form of Opial’s inequality, Canad. Math. Bull. (to appear).
- Giuseppe Tomaselli, A class of inequalities, Boll. Un. Mat. Ital. (4) 2 (1969), 622–631. MR 0255751
- James S. W. Wong, A discrete analogue of Opial’s inequality, Canad. Math. Bull. 10 (1967), 115–118. MR 212147, DOI 10.4153/CMB-1967-013-3
- Gou-sheng Yang, On a certain result of Z. Opial, Proc. Japan Acad. 42 (1966), 78–83. MR 197655
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 204 (1975), 299-341
- MSC: Primary 26A84
- DOI: https://doi.org/10.1090/S0002-9947-1975-0357715-3
- MathSciNet review: 0357715