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  2. Jensen's inequality - Wikipedia

    en.wikipedia.org/wiki/Jensen's_inequality

    Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality In mathematics , Jensen's inequality , named after the Danish mathematician Johan Jensen , relates the value of a convex function of an integral to the integral of the convex function.

  3. QM-AM-GM-HM inequalities - Wikipedia

    en.wikipedia.org/wiki/QM-AM-GM-HM_Inequalities

    There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means.

  4. Convex function - Wikipedia

    en.wikipedia.org/wiki/Convex_function

    Indeed, convex functions are exactly those that satisfies the hypothesis of Jensen's inequality. A first-order homogeneous function of two positive variables x {\displaystyle x} and y , {\displaystyle y,} (that is, a function satisfying f ( a x , a y ) = a f ( x , y ) {\displaystyle f(ax,ay)=af(x,y)} for all positive real a , x , y > 0 ...

  5. Jensen's theorem - Wikipedia

    en.wikipedia.org/wiki/Jensen's_theorem

    Download as PDF; Printable version; ... In mathematics, Jensen's theorem may refer to: Johan Jensen's inequality for convex functions; Johan Jensen's formula in ...

  6. Trace inequality - Wikipedia

    en.wikipedia.org/wiki/Trace_inequality

    Download as PDF; Printable version; ... for example. Löwner–Heinz theorem ... satisfies Jensen's Operator Inequality if the following holds ...

  7. Bessel's correction - Wikipedia

    en.wikipedia.org/wiki/Bessel's_correction

    Firstly, while the sample variance (using Bessel's correction) is an unbiased estimator of the population variance, its square root, the sample standard deviation, is a biased estimate of the population standard deviation; because the square root is a concave function, the bias is downward, by Jensen's inequality.

  8. Unbiased estimation of standard deviation - Wikipedia

    en.wikipedia.org/wiki/Unbiased_estimation_of...

    Since the square root is a strictly concave function, it follows from Jensen's inequality that the square root of the sample variance is an underestimate. The use of n − 1 instead of n in the formula for the sample variance is known as Bessel's correction , which corrects the bias in the estimation of the population variance, and some, but ...

  9. Johan Jensen (mathematician) - Wikipedia

    en.wikipedia.org/wiki/Johan_Jensen_(mathematician)

    Instead, he was a successful engineer for the Copenhagen Telephone Company between 1881 and 1924, and became head of the technical department in 1890. All his mathematics research was carried out in his spare time. Jensen is mostly renowned for his famous inequality, Jensen's inequality. In 1915, Jensen also proved Jensen's formula in complex ...