New high-dimensional generalizations of Nesbitt's inequality and relative applications
Junfeng Zhang, Jintao Wang
TL;DR
The paper advances Nesbitt's inequality by developing high-dimensional, multi-parameter generalizations for sums of the form $\sum_{i=1}^n \frac{a_i^m}{(ts-ra_i^p)^{\beta}}$, establishing rigorous lower/upper bounds across diverse parameter regimes using tools such as Jensen's inequality, generalized Radon inequalities, rearrangement, and the semiconcave-semiconvex theorem. It then applies these results to derive new inequalities, extend classical bounds in higher dimensions, and generate olympiad-style competition problems, while also connecting the framework to relations among Hurwitz-Lerch zeta functions. The work provides both a unifying theoretical approach for high-dimensional Nesbitt-type inequalities and practical applications in inequality problems and special functions. Overall, it broadens the scope of Nesbitt-type results and creates a versatile toolkit for analyzing parameterized, high-dimensional rational sums.
Abstract
Two kinds of novel generalizations of Nesbitt's inequality are explored in various cases regarding dimensions and parameters in this article. Some other cases are also discussed elaborately by using the semiconcave-semiconvex theorem. The general inequalities are then employed to deduce some alternate inequalities and mathematical competition questions. At last, a relation about Hurwitz-Lerch zeta functions is obtained.