Download e-book for iPad: Convex Functions and their Applications: A Contemporary by Constantin Niculescu

By Constantin Niculescu

ISBN-10: 0387243003

ISBN-13: 9780387243009

Thorough advent to an immense quarter of arithmetic includes fresh effects contains many workouts

Show description

Read Online or Download Convex Functions and their Applications: A Contemporary Approach PDF

Similar functional analysis books

Get Regularization methods in Banach spaces PDF

Regularization tools aimed toward discovering reliable approximate suggestions are an important software to take on inverse and ill-posed difficulties. often the mathematical version of an inverse challenge involves an operator equation of the 1st sort and infrequently the linked ahead operator acts among Hilbert areas.

Download e-book for kindle: Bergman Spaces and Related Topics in Complex Analysis: by Hadenmalm & Zhu Borichev

This quantity grew out of a convention in honor of Boris Korenblum at the celebration of his eightieth birthday, held in Barcelona, Spain, November 20-22, 2003. The booklet is of curiosity to researchers and graduate scholars operating within the conception of areas of analytic functionality, and, particularly, within the concept of Bergman areas.

Read e-book online Functional and Shape Data Analysis PDF

This textbook for classes on functionality info research and form facts research describes how to find, evaluate, and mathematically signify shapes, with a spotlight on statistical modeling and inference. it truly is aimed toward graduate scholars in research in facts, engineering, utilized arithmetic, neuroscience, biology, bioinformatics, and different comparable components.

Extra info for Convex Functions and their Applications: A Contemporary Approach

Example text

10. 4 (The Hardy–Littlewood–P´ olya inequality) Suppose that f is a convex function on an interval I and consider two families x1 , . . , xn and y1 , . . , yn of points in I such that m m xk ≤ for m ∈ {1, . . , n} yk k=1 k=1 and n n xk = k=1 yk . k=1 If x1 ≥ · · · ≥ xn , then n n f (xk ) ≤ k=1 f (yk ), k=1 while if y1 ≤ · · · ≤ yn this inequality works in the reverse direction. Proof. We shall concentrate here on the first conclusion (concerning the decreasing families), which will be settled by mathematical induction.

N} and let p > 1. Prove that the maximum of n n ak xpk n p bk ykp k=1 k=1 ck xk yk k=1 for xk ∈ [mk , Mk ] and yk ∈ [mk , Mk ] (k ∈ {1, . . , n}) is attained at a 2n-tuple whose components are endpoints. 3. Assume that f : I → R is strictly convex and continuous and g : I → R is continuous. For a1 , . . , an > 0 and mk , Mk ∈ I, with mk < Mk for k ∈ {1, . . , n}, consider the function n h(x1 , . . , xn ) = n ak f (xk ) + g k=1 n ak xk k=1 ak k=1 n defined on k=1 [mk , Mk ]. Prove that a necessary condition for a point (y1 , .

9 Suppose that I is an open interval. A real-valued function f 2 is convex on I if and only if f is continuous and D f ≥ 0. Accordingly, if a function f : I → R is convex in the neighborhood of each point of I, then it is convex on the whole interval I. 2 Proof. If f is convex, then clearly D f ≥ D2 f ≥ 0. 3. 2 Now, suppose that D f > 0 on I. If f is not convex, then we can find a 2 point x0 such that D f (x0 ) ≤ 0, which will be a contradiction. In fact, in this case there exists a subinterval I0 = [a0 , b0 ] such that f ((a0 +b0 )/2) > (f (a0 )+ f (b0 ))/2.

Download PDF sample

Convex Functions and their Applications: A Contemporary Approach by Constantin Niculescu

by Anthony

Rated 4.65 of 5 – based on 42 votes