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Handbook of mathematics for engineers and scienteists part 48

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Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 48', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 7.2. Definite Integral 297 7. Holder s inequality at p 2 it translates into Bunyakovsky s inequality b b I b p 1 f x g x dx 1 f x p dx I I g x p-1 dx p 1. J a Ja Ja 8. Chebyshev s inequality I f x h x dx II g x h x dx I 1 h x dx II f x g x h x dx I J a Ja J a Ja where f x and g x are monotonically increasing functions and h x is a positive integrable function on a b . 9. Jensen s inequality f f fg g t x t dt 2 g t f x t dt v g t dt fa g t dt fa g t f x t dt fa g t dt f f fg g t x t dt g t f x t dt k g t dt fa g t dt if f x is convex f 0 if f x is concave f 0 fa g t f x t dt fa g t dt where x t is a continuous function a x b and g t 0. The equality is attained if and only if either x t const or f x is a linear function. Jensen s inequality serves as a general source for deriving various integral inequalities. 10. Steklov s inequality. Let f x be a continuous function on 0 n and let it have everywhere on 0 n except maybe at finitely many points a square integrable deriva- tive f x . If either of the conditions a f 0 f n 0 b 0 f x dx 0 is satisfied then the following inequality holds i- n n i- 0 fz x 2 dx f x 2 dx. 00 The equality is only attained for functions f x A sin x in case a and functions f x B cos x in case b . 11. A n-related inequality. If a 0 and f x 0 on 0 a then a 0 7.2.5-3. Arithmetic geometric harmonic and quadratic means of functions. Let f x be a positive function integrable on a b . Consider the values of f x on a discrete set of points fkn f a kdn Sn ------- k 1 . n . n 298 Integrals The arithmetic mean geometric mean harmonic mean and quadratic mean of a function f x on an interval a fi are introduced using the definitions of the respective mean values for finitely many numbers see Subsection 1.6.1 and going to the limit as n to to. 1. Arithmetic mean of a function f x on a fi lim - V fkn -T i f x dx. n x n fi - a Jn k i Ja This definition is in agreement with another definition of the mean value of a function f x on a fi given in Theorem 1 from .