Some Refinements of Jensen's Inequality on Product Spaces

Transcription

1 Journal of mathematics and comuter Science 5 (205) Article history: Received March 205 Acceted July 205 Available online July 205 Some Refinements of Jensen's Ineuality on Product Saces Peter O Olaniekun,*, Adesanmi A Mogbademu,+ Research Grou in Mathematics and Alications Deartment of Mathematics, University of Lagos, Nigeria * eterolaniekun@studentsunilagedung, + amogbademu@unilagedung Abstract In this aer, we give some refinements of the classical Jensen's ineuality which generalizes some results already obtained in literatures Keywords: Convex function, Jensen's ineuality, Fubini's theorem, L saces Introduction In [3] J Rooin refined the classical Jensen's ineuality as φ fdμ) φ f(x)ω(x, y)dμ(x)) dλ(y) (φ f)dμ, () where (, A, μ) and (, B, λ) are two robability measure saces, ω: [0, ) is a weight function on, I is an interval of the real line, f L (μ), f(x) I for all x and φ is a real-valued convex function on I Also, in [2], the authors roved a generalization of the classical Jensen's ineuality by Riemann-Stieltjes integration for two convex functions defined on an interval of R In this aer, we generalize the above aers to a very general case by considering a more general abstract sace ie the L saces and two fuctions in this sace 28

2 Peter O Olaniekun, Adesanmi A Mogbademu / J Math Comuter Sci 5 (205) Main Results We refine the classical Jensen's ineuality on the L saces and show how our result generalizes those in literature Theorem 2 Let be a measure sace, with measure μ Let f L (μ) and g L (μ) Suose φ is any convex function and + =, where < < and < < then the following ineuality holds Proof: Let φ fgdx) φ f dx) φ g dx) (2) A = φ f dx), B = φ g dx) The case when A = 0 is trivial Also A > 0 and B= is trivial So we consider the case 0 < A <, 0 < B < We set Now, F = φ f φ g, G = A B F φ f dx dx = φ f dx = φ g dx φ g dx = G dx = s t Let x 0 < F(x) < and 0 < G(x) < imlies that s, t R F(x) = e, G(x) = e This imlies s e +t e s + e t F(x)G(x) F (x) + G (x) x (22) Integrating both sides of (22), to obtain This imlies φ f φ g φ f dx) φ g dx) dx 282

3 Peter O Olaniekun, Adesanmi A Mogbademu / J Math Comuter Sci 5 (205) That is φ f φ gdx φ f dx) φ(fg)dx φ f φ gdx φ f dx) φ g dx) φ (fg)dx) φ(fg)dx φ f φ gdx φ f dx) φ g dx) φ g dx) Remark 22 If φ is an identity function then Theorem 2 gives Theorem 35 in [4] For simlicity, we state it as Corollary 23 Corollary 23 Let be a measurable sace, with measure μ Let f and g be measurable functions on with range [0, ] Suose φ is any identity function and + =, where <, < Then the following ineuality holds fgdx) f dx) g dx) Theorem 24 Let (, A, μ) and (, B, λ) be two measure saces and ω: [0, ) be a weight function on such that ω(x, y)dμ(x) = y, ω(x, y)dλ(y) = x If I is a measurable sace, f, g L (μ), f(x) I x and φ is a convex function in I, then φ fgdμ) φ f (x)ω(x, y)dμ(x)) dλ(y)] φ φ f dμ] g (x)ω(x, y)dμ(x)) dλ(y)] 283 φ g dμ]

4 Peter O Olaniekun, Adesanmi A Mogbademu / J Math Comuter Sci 5 (205) Proof: The functions ω and (x, y) f(x) and so (x, y) f (x)ω(x, y) is roduct-measurable on The same thing goes for g(x) We rove the first ineuality Clearly, f(x) ω(x, y)dλ(y)dμ(x)) = = f(x) ω(x, y)dλ(y)) dμ(x)) f(x) dμ(x)) Similarly forg(x), we have g(x) ω(x, y)dλ(y)dμ(x)) = f L (μ) = g(x) ω(x, y)dλ(y)) dμ(x)) = g(x) dμ(x)) = g L (μ) By Fubini's theorem we know that (x, y) f (x)ω(x, y) on belongs to L (μ λ) By the same argument, (x, y) g (x)ω(x, y) belongs to L (μ λ) Next, we define F: R and G: R by Now, φ F(y) = G(y) = f (x)ω(x, y)dμ(x)) dλ(y)] f (x)ω(x, y)dμ(x)) g (x)ω(x, y)dμ(x)) 284 φ, g (x)ω(x, y)dμ(x)) dλ(y)]

5 Peter O Olaniekun, Adesanmi A Mogbademu / J Math Comuter Sci 5 (205) = (φ F )(y)dλ(y)] (φ G )(y)dλ(y)] Using Theorem 2 we obtain (φ F )(y)dλ(y)] (φ G )(y)dλ(y)] φ F(y)dλ(y) G(y)dλ(y)) y = [φ = φ f (x)ω(x, y)dμ(x)) g (x)ω(x, y)dμ(x)) dλ(y)) [φ F (y)dλ(y))] [φ G (y)dλ(y))] f (x)ω(x, y)dμ(x)dλ(y)))] = [φ f (x) ω(x, y)dλ(y)) dμ(x))] [φ [φ = [φ f (x)dμ(x))] [φ g (x)dμ(x))] g (x)ω(x, y)dμ(x)dλ(y)))] g (x) ω(x, y)dλ(y)) dμ(x))] φ fgdμ(x)) Remark 25 Theorem 24 refines the result obtained by Hewitt and Stromberg on age 202 of [] and also generalizes [3] References [] E Hewitt, K Stromberg, Real and Abstract Analysis, Sringer-Verlag, New ork, (965) [2] P O Olaniekun, A A Mogbademu, A note on generalization of classical Jensen's ineuality, JMathComuter Sci 3(204),

6 Peter O Olaniekun, Adesanmi A Mogbademu / J Math Comuter Sci 5 (205) [3] J ROOIN, A refinement of Jensen's ineuality, J Ineual Pure and Al Math, 6(2) Art 38, (2005) [4] W RUDIN, Real and Comlex Analysis, 3 rd ed, McGraw-Hill, New ork, (974) 286