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Author Archives: Van Hoang Nguyen
Weighted Moser-Trudinger inequalities
Recently, M. Ishiwata, M. Nakamura and H. Wadade (see this link) proved the following weighted Moser–Trudinger inequalities of the scaling invariant form in whole space (1) for any radial function with and for any where is the surface … Continue reading
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Hardy-Rellich inequality on functions which are orthogonal to radial functions.
Hardy inequality asserts that for any function . Rellich inequality asserts that for any function . The constants and are sharp. In this post, we will improve these inequalities when restricting to the functions which is orthogonal to all radial function, … Continue reading
Manifolds with nonnegative Ricci curvature and Sobolev inequalities
The main object of this post is the following nice result due to Michel Ledoux: Let be a complete dimensional Riemannian manifold with nonnegative Ricci curvature. If the following Sobolev inequality for some with and is the best constant in … Continue reading
An exercise on the affine functions
In this post, we solve the following exercise which gives a nice character of affine functions: Let be a smooth function on such that for any . Then is affine, i.e, there are and such that . Here is my … Continue reading
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Borell’s proof of Prékopa inequality
Prékopa inequality says that if are nonnegative, measurable functions on such that holds for any . Then (1) There are many proofs of (1). In this post, … Continue reading
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An improved Poincaré inequality for Gaussian measure.
Let denote standard Gaussian measure on , i. e The Poincaré inequality for states that for any Lipschitz function in , it holds This inequality is sharp and equality holds iff for some vector and . An improvement of Poincaré … Continue reading
L_p affine isoperimetric inequalities
In this post, we give an application of the shadow system (see here) to prove a affine isoperimetric inequality due to Lutwak, Yang and Zhang (see this paper). The proof we present here comes from the paper of Campi and … Continue reading
Đề thi Olympic sinh viên Toán 2015, bảng A
Câu A1: 1) Do nên là dãy giảm. 2) Do , và hàm với , do đó sử dụng qui nạp ta chứng minh được với mọi $n$. Từ đó suy ra dãy hội tụ. Dễ dàng suy ra giới … Continue reading
An integral inequality from a competition for Vietnamese student in Maths
Problem: Let be a continuous function such that . Prove that Following is my proof whose method I learnt from the proof of Kullback-Pinsker-Csiszar inequality in the information theory (in the book “measure theory” of Bogachev). Put and denote and … Continue reading
Nguyễn Văn Hoàng 