By Themistocles RASSIAS
Survey on Classical Inequalities offers a examine of a few of the well-known inequalities in classical mathematical research. topics handled comprise: Hardy-Littlewood-type inequalities, Hardy's and Carleman's inequalities, Lyapunov inequalities, Shannon's and similar inequalities, generalized Shannon practical inequality, operator inequalities linked to Jensen's inequality, weighted Lp -norm inequalities in convolutions, inequalities for polynomial zeros in addition to purposes in a couple of difficulties of natural and utilized arithmetic. it truly is my excitement to precise my appreciation to the prestigious mathematicians who contributed to this quantity. eventually, we want to recognize the wonderful assistance supplied by means of the employees of Kluwer educational Publishers. June 2000 Themistocles M. Rassias Vll LYAPUNOV INEQUALITIES AND THEIR purposes RICHARD C. BROWN division of arithmetic, collage of Alabama, Tuscaloosa, AL 35487-0350, united states. e-mail address:dicbrown@bama.ua.edu DON B. HINTON division of arithmetic, collage of Tennessee, Knoxville, TN 37996, united states. electronic mail handle: hinton@novell.math.utk.edu summary. for almost 50 years Lyapunov inequalities were a major device within the research of differential equations. during this survey, construction on a good 1991 ancient survey by means of Cheng, we cartoon a few new advancements within the thought of Lyapunov inequalities and current a few fresh disconjugacy effects with regards to moment and better order differential equations in addition to Hamiltonian structures. 1. advent Lyapunov's inequality has proved worthwhile within the research of spectral homes of standard differential equations. commonplace functions comprise bounds for eigenvalues, balance standards for periodic differential equations, and estimates for durations of disconjugacy.
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Now we are able to derive multivariable extensions of the one-dimensional results. The following theorem gives the n-dimensional analogue to Theorem 6. r Theorem 12. Let f be a non-negative measurable function on R. (~ - 0)0-' [(X - 0)-1 L. 37) Equality holds if and only if f is of the form f( ... , aj + Uj(Xj - aj)' ... ) = l1(u) . h(x), u E RO,l, X E Ra,b. , is stated by Theorem 13. Let h, ex, 'Y' ERn, h »0, and let f be a non-negative measurable function on R. 38) Note that in Theorem 12. and Theorem 13.
Primary 26DlO, 26D15, 26D20. Key words and phrases. Hardy's inequality, Carleman's inequality, Levin-Cochran-Lee inequality, power means, mixed power means, mixed-means inequalities. M. ), Survey on Classical Inequalities, 27-4j5. © 2000 Kluwer Academic Publishers. 28 Investigating limit values of p, another well-known inequality was obtained few years after Hardy's result. an)fo < e L:an , n=l n=l for sequences of positive real numbers, discovered by T. Carleman in [5]. During the same decade K.
5. G. Borg, On a Liapounoff criterion of stability, Amer. J. Math. 11 (1949), 67-70. 6. ___ , Ueber die Stabilitat gewisser Klassen von linearen Different ialgleichungen, Arkiv for Matematik Astronomi och Fysik 31 (1945), 1-31. 7. R. C. Brown, D. B. Hinton, and S. Schwabik, Applications of a one-dimensional Sobolev inequality to eigenvalue problems, J. Dilf. and Int. Equations 9 (1996), 481-98. 8. R. C. Brown and D. B. Hinton, Opial's inequality and oscillation of 2nd order equations, Proc. Amer.