The Vinogradov-Mordell-Tietäväinen inequalities

Erdős-Mordell-Type Inequalities in a Triangle

with equality if and only if the triangle is equilateral and P is its center. This inequality was conjectured by Erdős [1] and proved by Mordell and Barrow [2]. Oppenheim [3] established a number of additional inequalities relating the six distances p, q, r , x , y, and z. Such an inequality will be referred to as an Erdős-Mordell-type inequality. A survey of some of these inequalities can be f...

Reverse Inequalities of Erdös–mordell Type

This paper deals with the reverse inequalities of Erdös-Mordell type. Our result contains as special case the following reverse Erdös-Mordell inequality: R1 +R2 +R3 < √ 2 (ρ1 +ρ2 +ρ3) , where Ri and ρi (i=1, 2, 3) denote respectively the distances from an interior point Q of A1A2A3 to the vertexes A1, A2, A3 and to the circumcenters of A2QA3 , A3QA1 , A1QA2 . Some other closely related inequali...

On the Extension of the Erdös–mordell Type Inequalities

We discuss the extension of inequality RA c a rb + b a rc to the plane of triangle ABC . Based on the obtained extension, in regard to all three vertices of the triangle, we get the extension of Erdös-Mordell inequality, and some inequalities of Erdös-Mordell type. Mathematics subject classification (2010): 51M16, 51M04, 14H50.

The Bombieri-vinogradov Theorem

The Smoothed Pólya–Vinogradov Inequality

Let χ be a primitive Dirichlet character to the modulus q. Let Sχ(M,N) = ∑ M<n≤N χ(n). The Pólya-Vinogradov inequality states that |Sχ(M,N)| √ q log q. The smoothed Pólya–Vinogradov inequality, recently introduced by Levin, Pomerance and Soundararajan, is a numerically useful version of the Pólya–Vinogradov inequality that saves a log q factor. The smoothed Pólya–Vinogradov inequality has been ...