A proof of the three geometric inequalities conjectured by Yu-Dong Wu and H.M. Srivastava

TL;DR

This paper addresses three geometric inequalities conjectured by Wu and Srivastava, disproves one, proves the other two, and introduces an optimal refinement for one of them.

Contribution

It provides proofs for two of the three conjectured inequalities and introduces an improved version for one, advancing understanding of these geometric problems.

Findings

Disproved the first inequality with a counterexample.

Proved the inequalities in Problems 2 and 3.

Introduced an optimal refinement of the inequality in Problem 3.

Abstract

In this short note the authors give answers to the three open problems formulated by Wu and Srivastava [{\it Appl. Math. Lett. 25 (2012), 1347--1353}]. We disprove the Problem 1, by showing that there exists a triangle which does not satisfies the proposed inequality. We prove the inequalities conjectured in Problems 2 and 3. Furthermore, we introduce an optimal refinement of the inequality conjectured on Problem 3.