Karamata's inequality

In mathematics, Karamata's inequality, named after Jovan Karamata, also known as the majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. It generalizes the discrete form of Jensen's inequality, and generalizes in turn to the concept of Schur-convex functions. f ( x 1 ) + ⋯ + f ( x n ) ≥ f ( y 1 ) + ⋯ + f ( y n ) . {displaystyle f(x_{1})+cdots +f(x_{n})geq f(y_{1})+cdots +f(y_{n}).}     (1) x 1 ≥ x 2 ≥ ⋯ ≥ x n {displaystyle x_{1}geq x_{2}geq cdots geq x_{n}}     and     y 1 ≥ y 2 ≥ ⋯ ≥ y n , {displaystyle y_{1}geq y_{2}geq cdots geq y_{n},}     (2) x 1 + ⋯ + x i ≥ y 1 + ⋯ + y i {displaystyle x_{1}+cdots +x_{i}geq y_{1}+cdots +y_{i}}      for all i ∈ {1, . . . , n − 1}.    (3) x 1 + ⋯ + x n = y 1 + ⋯ + y n {displaystyle x_{1}+cdots +x_{n}=y_{1}+cdots +y_{n}}     (4) x 1 + ⋯ + x n ≥ y 1 + ⋯ + y n . {displaystyle x_{1}+cdots +x_{n}geq y_{1}+cdots +y_{n}.}     (5) c i + 1 := f ( x i + 1 ) − f ( y i + 1 ) x i + 1 − y i + 1 ≤ f ( x i ) − f ( y i ) x i − y i =: c i {displaystyle c_{i+1}:={frac {f(x_{i+1})-f(y_{i+1})}{x_{i+1}-y_{i+1}}}leq {frac {f(x_{i})-f(y_{i})}{x_{i}-y_{i}}}=:c_{i}}     (6) ∑ i = 1 n ( f ( x i ) − f ( y i ) ) = ∑ i = 1 n c i ( x i − y i ) = ∑ i = 1 n c i ( A i − A i − 1 ⏟ = x i − ( B i − B i − 1 ⏟ = y i ) ) = ∑ i = 1 n c i ( A i − B i ) − ∑ i = 1 n c i ( A i − 1 − B i − 1 ) = c n ( A n − B n ⏟ = 0 ) + ∑ i = 1 n − 1 ( c i − c i + 1 ⏟ ≥ 0 ) ( A i − B i ⏟ ≥ 0 ) − c 1 ( A 0 − B 0 ⏟ = 0 ) ≥ 0 , {displaystyle {egin{aligned}sum _{i=1}^{n}{igl (}f(x_{i})-f(y_{i}){igr )}&=sum _{i=1}^{n}c_{i}(x_{i}-y_{i})\&=sum _{i=1}^{n}c_{i}{igl (}underbrace {A_{i}-A_{i-1}} _{=,x_{i}}{}-(underbrace {B_{i}-B_{i-1}} _{=,y_{i}}){igr )}\&=sum _{i=1}^{n}c_{i}(A_{i}-B_{i})-sum _{i=1}^{n}c_{i}(A_{i-1}-B_{i-1})\&=c_{n}(underbrace {A_{n}-B_{n}} _{=,0})+sum _{i=1}^{n-1}(underbrace {c_{i}-c_{i+1}} _{geq ,0})(underbrace {A_{i}-B_{i}} _{geq ,0})-c_{1}(underbrace {A_{0}-B_{0}} _{=,0})\&geq 0,end{aligned}}}     (7) In mathematics, Karamata's inequality, named after Jovan Karamata, also known as the majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. It generalizes the discrete form of Jensen's inequality, and generalizes in turn to the concept of Schur-convex functions. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I. If x1, . . . , xn and y1, . . . , yn are numbers in I such that (x1, . . . , xn) majorizes (y1, . . . , yn), then Here majorization means that x1, . . . , xn and y1, . . . , yn satisfies