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
where f-1 is the inverse function of f. Equality holds iff b=f(a). To prove this, draw the graph of f(x), and treat each integral as the area bounded by the x and y axes and the function. Clearly, all of the rectangle bounded by the axes, a, and b is included in the sum of these areas.
. . . . . . a picture will be added to this web page in due course.
Taking the particular function f(x)=xp-1 gives the special case
(ap)/p+((p-1)/p)bp/(p-1) ≥ ab,
which is often written in the symmetric form
(ap)/p+(bq)/q ≥ ab,
where a,b ≥ 0, p>1, and 1/p+1/q=1.
Mathworld: Young's Inequality
Wikipedia: Young's Inequality, which treats only the particular case (ap)/p+(bq)/q ≥ ab, and notes that Young's inequality is used in the proof of the H�lder inequality.
The Nondecreasing Sequence Two puzzle whose most elegant solution relies on Young's Inequality.
The AM-GM Inequality: the Arithmetic Mean of positive numbers is always greater than the Geometric Mean. This is proved using Jensen's Inequality.
The Cauchy-Schwarz inequality
The Triangle Inequality
The Quadratic Formula
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