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Young's Inequality

Let f be a real-valued, continuous, and strictly increasing function on [0,c] with c>0. If f(0)=0, a in [0,c], and b in [0,f(c)], then

int_0^af(x)dx+int_0^bf^(-1)(x)dx>=ab,
(1)

where f^(-1) is the inverse function of f. Equality holds iff b=f(a).

Taking the particular function f(x)=x^(p-1) gives the special case

(a^p)/p+((p-1)/p)b^(p/(p-1))>=ab,
(2)

which is often written in the symmetric form

(a^p)/p+(b^q)/q>=ab,
(3)

where a,b>=0, p>1, and

1/p+1/q=1.
(4)

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References

Cooper, R. "Notes on Certain Inequalities. I." J. London Math. Soc. 2, 17-21, 1927.Cooper, R. "Notes on Certain Inequalities. II." J. London Math. Soc. 2, 159-163, 1927.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. "A Theorem of W. H. Young." §8.3 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 198-200, 1988.Mitrinović, D. S. "Young's Inequality." §2.7 in Analytic Inequalities. New York: Springer-Verlag, pp. 48-50, 1970.Oppenheim, A. "Note on Mr. Cooper's Generalization of Young's Inequality." J. London Math. Soc. 2, 21-23, 1927.Riesz, F. "Su alcune disuguaglianze." Boll. Un. Mat. Ital. 7, 77-79, 1928.Takahashi, T. "Remarks on Some Inequalities." Tôhoku Math. J. 36, 99-106, 1932.Young, W. H. "On Classes of Summable Functions and Their Fourier Series." Proc. Roy. Soc. London Ser. A 87, 225-229, 1912.

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Young's Inequality

Cite this as:

Weisstein, Eric W. "Young's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/YoungsInequality.html

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